Курс математики для технических высших учебных заведений. Часть 1. Аналитическая геометрия. Пределы и ряды. Функции и производные. Линейная и векторная алгебра: Учебное пособие [А. И. Мартыненко] (pdf) читать онлайн
- Курс математики для технических высших учебных заведений. Часть 1. Аналитическая геометрия. Пределы и ряды. Функции и производные. Линейная и векторная алгебра: Учебное пособие [2-е издание, исправленное] (и.с. Учебники для вузов. Специальная литература) 7.29 Мб, 544с. скачать: (pdf) - (pdf+fbd) читать: (полностью) - (постранично) - А. И. Мартыненко - В. Г. Зубков - В. Б. Миносцев - В. А. Ляховский
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Лауреат второго Всероссийского конкурса НМС по математике
Министерства образования и науки РФ «Лучшее учебное издание по математике
в номинации «Математика в технических вузах»
В. Г. ЗУБКОВ, В. А. ЛЯХОВСКИЙ,
А. И. МАРТЫНЕНКО, В. Б. МИНОСЦЕВ
КУРС МАТЕМАТИКИ
ДЛЯ ТЕХНИЧЕСКИХ
ВЫСШИХ УЧЕБНЫХ
ЗАВЕДЕНИЙ
Часть 1
Аналитическая геометрия. Пределы и ряды.
Функции и производные.
Линейная и векторная алгебра
Под редакцией
В. Б. Миносцева, Е. А. Пушкаря
Издание второе, исправленное
ДОПУЩЕНО
НМС по математике Министерства образования и науки РФ
в качестве учебного пособия для студентов вузов, обучающихся
по инженерно&техническим специальностям
•САНКТ4ПЕТЕРБУРГ•МОСКВА•КРАСНОДАР•
•2013•
ББК 22.1я73
К 93
Зубков В. Г., Ляховский В. А., Мартыненко А. И.,
Миносцев В. Б.
К 93
Курс математики для технических высших учебных
заведений. Часть 1. Аналитическая геометрия. Пределы и
ряды. Функции и производные. Линейная и векторная
алгебра: Учебное пособие / Под ред. В. Б. Миносцева,
Е. А. Пушкаря. — 2+е изд., испр. — СПб.: Издательство
«Лань», 2013. — 544 с.: ил. — (Учебники для вузов.
Специальная литература).
ISBN 978 5 8114 1558 8
Учебное пособие соответствует Государственному образовательному
стандарту, включает в себя лекции и практические занятия. Первая часть
пособия содержит 34 лекции и 34 практических занятия по следующим
разделам: «Множества», «Системы координат», «Функции одной
переменной», «Теория пределов и числовые ряды», «Дифференциальное
исчисление функций одной переменной», «Элементы линейной,
векторной и высшей алгебры, аналитической геометрии».
Пособие предназначено для студентов технических, физико+
математических и экономических направлений.
ББК 22.1я73
Рецензенты:
À. Â. ÑÅÒÓÕÀ äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðîôåññîð,
÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ;
À. À. ÏÓÍÒÓÑ ïðîôåññîð ôàêóëüòåòà ïðèêëàäíîé ìàòåìàòèêè è
ôèçèêè ÌÀÈ, ÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è
íàóêè ÐÔ; À. Â. ÍÀÓÌΠäîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê,
äîöåíò êàôåäðû òåîðèè âåðîÿòíîñòåé ÌÀÈ; À. Á. ÁÓÄÀÊ äîöåíò,
çàì. ïðåäñåäàòåëÿ îòäåëåíèÿ ó÷åáíèêîâ è ó÷åáíûõ ïîñîáèé ÍÌÑ ïî
ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ; Ó. Ã. ÏÈÐÓÌÎÂ
ïðîôåññîð, çàâ. êàôåäðîé âû÷èñëèòåëüíîé ìàòåìàòèêè è
ïðîãðàììèðîâàíèÿ ÌÀÈ (Òåõíè÷åñêèé óíèâåðñèòåò), ÷ëåíêîððåñïîíäåíò ÐÀÍ, çàñëóæåííûé äåÿòåëü íàóêè ÐÔ.
Обложка
Е. А. ВЛАСОВА
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Воспроизведение всей книги или любой ее части запрещается без письменного
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Любые попытки нарушения закона
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© Издательство «Лань», 2013
© Коллектив авторов, 2013
© Издательство «Лань»,
художественное оформление, 2013
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2/2 sin(−π/4) = − 2/2
√
√
√
2
2
2
2
+Y
, y = −X
+Y
,
2
2
2
2
√
x=X
√
√
2
2
(Y + X), y =
(Y − X).
2
2
√
x=
−3 < x < 3
! −3 < x < 3
#$ % & #
|x| < 3
' −3 < x < 3 ⇔ |x| < 3
"
|x − 1| < 5
! # (#) "
#$ % & #
# #
x − 1, x − 1 0,
|x − 1| =
−(x − 1), x − 1 < 0.
*+ # +
⎡
|x − 1| < 5 ⇐⇒ ⎣
x−10,
x−1 0} $ !
x2 − 3x + 2 = 0 =⇒ x1 = 1, x2 = 2 % !
x2 − 3x + 2 > 0 (−∞; 1) ∪ (2; +∞).
& " '#
R! $ E(f ) = {y|y ∈ R}
& D(f ) = (−∞; 1) ∪ (2; +∞) E(f ) = (−∞; +∞).
y=
(
1
.
(x + 1)(x − 2)
) " x
(x + 1)(x − 2) = 0
x1 = −1, x2 = 2!
D(f ) = (−∞; −1) ∪ (−1; 2) ∪ (2; +∞)
& D(f ) = (−∞; −1) ∪ (−1; 2) ∪ (2; +∞)
y = √1 1− x2 .
*
x
1 − x2 > 0 D(f ) =
= (−1; 1).
D(f ) = (−1; +1).
!" y = f (x) y = g(x) # $
D(f ) D(g) % &
!" y = f (x) + g(x) ! ' !& x #
!!
' f (x) g(x) #
!" y = f (x) + g(x) D(f ) ∩ D(g)
( !" y = f (x)g(x), y = f (x) − g(x)
f (x)
f (x)
y =
) # !" y =
g(x)
g(x)
& D(f ) ∩ D(g) ∩ {x|g(x) = 0}
√
y = x−1+
1
x−1
y = f (x) + g(x)
√
1
f (x) = x − 1, g(x) =
! "
x−1
D(f ) : x − 1 0 ⇐⇒ D(f ) = [1; +∞),
D(g) : x − 1 = 0 ⇐⇒ D(g) = (−∞; 1) ∪ (1; +∞).
# ! $ % & $ '
"
# (1; +∞)
u = f (x) ( % ! ! ! D(f )
) ! E(f ) y = g(u) ( % ! !
! D(g) E(f ) ⊂ D(g) ) ! E(g)
* " x ∈ D(f ) )%
y ∈ E(g) " x ∈ D(f ) ! u = f (x)
)% u ∈ E(f ) ! y = g(u)
)% y ∈ E(g) %! !
)+! " , ) $ )%! y = g(f (x)). . ! u = f (x) )+!
! y = g(u) (
u = x2 − 3x + 2 y = log2 u "
" y = log2 (x2 − 3x + 2)
u = 1 − x2 y = √1u
/
f (x) = 1 − x2 ⇒ D(f ) = R,
1
E(f ) = (−∞; 1], g(u) = √ ⇒ D(g) = (0; +∞), E(g) = (0; +∞) 0
u
E(f )⊂
1 D(g) #
" {x|1 − x2 > 0} = (−1; 1) & + '
1
# y = √
x ∈ (−1; 1)
1 − x2
X ⊂ R
−x ∈ X x ∈ X
O
!" y = f (x) #
#
f (−x) = f (x) $ x ∈ D(f )
(x; f (x)) (−x; f (−x))
Oy f (−x) = f (x)
D(f ) = (−∞; +∞)
y = x2 + 1
f (−x) = (−x)2 + 1 = x2 + 1 =
f (x)
y = x2 + 1 Oy
!"
y
2
y=x+1
3
2
1
-3 -2 -1
0
-1
1
2
3
x
-2
y = x2 + 1
# $ $ %
& ' $ (
%
y = ϕ(x)
ϕ(x) = f (x) + g(x),
ϕ(x) + ϕ(−x)
f (x) =
! g(x) =
D(f )
2
ϕ(x) − ϕ(−x)
" # y =
2
= x3 + 1 f (x) = 1
g(x) = x3 "
√
$"%& y = x
D(f ) = [0; +∞)
O
=
$"%$ y = x
2
+x
x
' ( ) * D(f ) = (−∞; 0)∪
O" +
,
f (−x) = f (x) f (−x) = −f (x)
x − x2
(−x)2 − x
=
= f (x)
f (−x) =
∪(0; +∞)
−x
x
x2 + x
= f (−x)"
−f (x) = −
x
x2 + x
) y =
x
"
$"%- y = x2 1− 1
' ( ) *
O"
+ ,
f (−x) = f (x) f (−x) = −f (x)"
1
1
= 2
= f (x)"
f (−x) =
2
D(f ) = (−∞; −1) ∪ (−1; 1) ∪ (1; +∞)
)
(−x) − 1
x −1
1
y= 2
x −1
"
y = f (x) T = 0
x − T x + T
f (x) = f (x ± T ) x ∈ D(f )
! " "
# $ % T
T0 : T = n · T0 n ∈ Z, n = 0 &
T0 > 0
' () y = sin x
T0 = 2π
x + 2π ∈ D(f ) x − 2π ∈ D(f ) sin(x ± 2π) = sin x
' (* y = {x}
T0 = 1
x + 1 ∈ D(f ) x − 1 ∈ D(f ) {x + 1} = {x}
+ "
T T
, y = {x} + 1
T = 1 y = {x} y = 1
- u = f (x)
T y = g(f (x)) .
/
, y = sin2 x
T0 = π
$ 0 (1 0 #
2
' ( y = f (x)
T y = Kf (kx + b) + a
T1 = T /|k| k ∈ R
' (3 y = 2 sin(3x + 2)
4 ! 2 y = sin x T = 2π k = 3 5%
T1 y = 2 sin(3x + 2) T1 = 2 3· π
2 T1 = 2·π3
' (6 y = √x
x = 0 T > 0 x−T
T < 0 x + T x = 0
!
" #$
y = x
D(f ) = (−∞; +∞) % x + T ∈ D(f )
x−T ∈ D(f ), x ∈ D(f ) & T0 f (x+T0 ) =
f (x) x + T0 = x '( T0 = 0
' y = x
" ) y = f (x)
M x ∈ D(f )
f (x) M y = f (x)
m
x ∈ D(f ) f (x) m
& y = x2 * m = −2
* + , y = −x4 * +
M = 1 * , y = sin x *
1
−1 sin x 1 , y = x, y = lg(x), y = tg(x), y =
*
x
!
" - y = f (x)
! " X ⊂ D(f ) # x1 ∈ X x2 ∈ X
x1 > x2 f (x1) > f (x2) $ %! #&' x !
#&' y() y = f (x) # ! "
X ⊂ D(f ) # x1 ∈ X x2 ∈ X x1 > x2
f (x1) < f (x2) $ %! #&' x ! ! &' y()
y = f (x) # ! " X ⊂ D(f )
# x1 ∈ X x2 ∈ X x1 > x2
f (x1 ) f (x2 )
y = f (x)
X ⊂ D(f )
x1 ∈ X x2 ∈ X
x1 > x2 f (x1) f (x2)
X
2
y = x (−∞; 0] [0; +∞)
y = √x
D(f ) = [0; +∞) x1 > x2 > 0 ! "
# $ !$ %! √
!
√ " f (x1) >
> f (x2 ) f (x1 ) − f (x2 ) = x1 − x2 & '
!$
√ √( √
√
( x1 − x2 ) · ( x1 + x2 )
x1 − x2
√
√
√
x1 − x2 =
=√
√
√
x + x
x + x
1
2
1
2
x2
√
√
) x1 > x2 ⇔ x1 − x2 > 0 ⇔ √xx1 −
√ > 0 ⇔ x1 − x2 > 0
x2
1+
⇔ f (x1 ) − f (x2 ) > 0 ⇔ f (x1 ) > f (x2 ) &'! !
* !"
(a; x1) (x2; x3) (x4; b) (x1; x2)
(x3 ; x4 )
y
y=f(x)
a
x1
x2
0
x3
x4 b x
f(x1)
y = f (x)
D(f )
E(f )
x ∈ D(f ) y ∈ E(f ).
! x1 ∈ D(f ) x2 ∈ D(f ) x1 = x2,
y ∈ E(f ). " y = x2
! x1 = 1
x2 = −1 y = 1
#$ y = f (x)
x1 = x2
y1 = y2
% y = f (x)
y ∈ E(f ) x ∈ D(f ). &
! ' ' y = f (x)
x = f −1(y). ( ' !
x y &
) * y = f −1(x).
% f −1
' ) f f −1
' f
' ) f −1. +
−1
f f ! ! , !' . *
D(f −1 ) = E(f ); f −1(f (x)) = x x ∈ D(f )/
E(f −1 ) = D(f ); f (f −1(x)) = x x ∈ D(f −1 ).
+
'
0
1!
!
2 !- ' !
' y = x 3 #45
! x y y
x
.
#4 y = f (x)
!
+ 1 ' 1
#44 " y = 2x − 1.
# $ %
y
-1
y=f (x)
y=f(x)
x
0
y=x
x y : y = 2x − 1 ⇐⇒ x =
y+1
. x y y x
2
x+1
. ! " ##
y =
2
y
y=2x-1
y=x
y= x+1
2
1
1
y = 2x − 1
x
y = x2
x ∈ [0; +∞), √
y = x
y
2
y=x (x>0)
y=x
y= x
x
0
y = x2 y =
√
x
! " # "$ % "$
! " &
'
(y = C) (y = xn, n ∈ R)
(y = ax) (y = loga x)
(y = sin x, y = cos x, y = tg x, y = ctg x)
y = arcsin x y = arccos x y = arctg x y = arcctg x
( !
" # # #
" $ $ % &
$ % $"# %
'$
) % "$ * & "
+ , $ & -
" + % -
. * % " /
⎧
⎨ 1 x > 0,
0 x = 0,
y=
⎩
−1 x < 0,
1 x ∈ Q,
y=
⎧ −1 x ∈ I,
⎨ x2 + 2x + 3, x < 0,
3 0 x < 5,
y=
⎩ √
x x 5.
y = kx + b.
D(f ) = (−∞; +∞); k = 0 E(f ) = (−∞; +∞),
b = 0 k > 0
k < 0 k = 0
! " # $%&'
y
y=kx+b, k>0
tg( ϕ )=k
b
y=(x-b)/k, k>0
ϕ
0
y = kx + b
x
k>0
b
− ;0
k
(0; b)
y = kx + b
Ox k = tg ϕ ! " k = 0 #$
b
y = x −
% & &
k
y
b
O
x
ϕ
y= x-b
k
, k 0
* D(f ) = (−∞; +∞),
E(f ) = (0; +∞)
+ y > 0!
a > 1 a < 1 ' , a = 1
y = ax -
& Oy+ (0; 1) #
') a .
y = loga x
y = loga x, a > 0, a = 1
, y = loga x y = ax
&/ +
D(f ) = (0; +∞) E(f ) = (−∞; +∞)
a > 1 a < 1
Ox (1; 0)
! " a
#$ % y = ax
y
y=a x, 01
1
0
1
x
y=log ax, 00
b
x
0
y = f (x) + b
! ! ! !"
" ! Ox b > 0 # b < 0 #
! ! Oy b
$
y = x2 − 1
Oy y = x2
2
y
y=x
2
y=x -1
0
-1
x
1
-1
y = x2 − 1
% y = f (x + a)
!&'
y = f (x) ( ) ) *!
! X = x + a Y = y +
!
Ox −a !" !
* *) Y = f (X)
a > 0 * * ' * !
Ox a x = X − a "
a > 0 y = f (x)
Ox
a a < 0
y = f (x) Ox
|a|
y=f(x+a)
y
y=f(x)
a>0
a
0
x
y = f (x + a)
Oy a > 0 a < 0
! " # Ox " a
$%
y = (x − 2)2
Ox y = x2
y
2
y=x
y=(x-2)
2
x
0
2
y = (x − 2)2
& y = kf (x) k ∈ R " '(
) * y = f (x) k !
Ox
k
y = f (x) k > 1
k Ox Oy 0 < k < 1
1
Ox Oy k −1
k
|k| ! "
Ox # $% −1 k < 0
1
Ox # &'$
|k|
( ) * y = −f (x) "
Ox * y = f (x)
( k > 0
+
Oy )
k k < 0 ,
"
)
% ,
y
π
- 2
- 2π
-1
3
2
1
0
y=-3sin x
π
2
π
y=sin x
2π x
-2
-3
y = −3 sin
x
- * y = f (kx)) k ∈ R) !
y = f (x) k Oy . +
k *
y = f (x) / ) ) ) f (1) = 0) )
X = kx) Y = y, ) * y = f (kx) !
1
kx = 1) x =
k
k > 1 * y = f (x) k Oy
Ox% 0 < k < 1 * y = f (x)
1
Oy Ox% k −1 " +
k
|k|
Oy #
−1 k < 0
1
|k|
Oy
y = f (−x)
y = f (x) Oy
! k > 0 " #
Ox
k $ k < 0 %
& '
y = cos 2x
Oy y = ln(−x)
y = ln x Oy
y = cos x
y
y=ln(-x)
y=ln x
0
-1
x
1
y = ln(−x)
$ "
y = f (kx + b)
!
y = f (x)(
•
y = f (x)
•
y = f (x + b)
& ))
•
y = f (kx+b) * "+
k Oy ,
& & √! "
y = 4 − 5x
-,
(
•
y=
√
x
#
√
•
y = x + 4
Ox√
•
y = −5x + 4
Oy
Oy
!
y
y
y
y= 4-5x
y= x+4
y= x
2
x
2
4/5 x
x
-4
y =
√
4 − 5x
y
y
y= f(x)
y=f(x)
x
x
y = |f (x)|
" # # y = f (x) $
# T # y = K ·f (kx+b)+a #
T1 =
T
%
|k|
& &' (
$
# Ox
# ) * + Ox * $
|k|'
, + |k| % T #
K ) a
# ,# T1 =
T
|k|
y=
√
1
−x + √
.
2+x
! ! "
! y = √−x −x 0 y = √21+ x #
2+x0
2+x=0
⎧
⎨ −x 0
2+x0 ⇔
⎩
2+x=0
(−2; 0].
x0
x > −2
y = arcsin(x + 1)3
! "
(x + 1)3 1
(x + 1)3 −1
! # $
x+11
⇔
x + 1 −1
D(f ) = [−2; 0]
x0
x −2
x
−x
y=
e +e
2
.
! D(f ) =
$ $ %
−x
x
& f (−x) = e 2+ e = f (x),
$'
$'
= (−∞; +∞)
y = x2 − 5x + 6
(−∞; +∞)
! !" !# !
f (−x) = (−x)2 − 5(−x) + 6 = x2 + 5x + 6 = ±f (x)
$ ! !! % %
& % ' ! ! ! " %
(y = x2 + 6) % (y = −5x)
! ( )# ! !! % %
*
T y = 5 sin 3x.
y = sin x ! 2π + !,
! * ' "
2π
y = 5 sin 3x !
3
2π
!
3
- !
y = 3 sin 5x + 4 cos 7x
+ !! . y = 3 sin 5x
2π
T1 =
y = 4 cos 7x / ,
5
2π
T2 =
0 1 1 !
7
2π 4π 6π
,
,
, . . . ! / 1
! 1
5
5
5
2π 4π
,
, . . . ! ) ' " ,
7
7
2 !2 ! 2π 3 ' ,
" 2 "
! T = 2π
4
√
5 y = 1 + x.
√
6 y = 3 1 + x.
1
.
4 − x2
√
= 4 9 − x2 .
√
= 2 + x − x2 .
√
2 − x2
.
=
x
2+x
).
= ln(
2−x
2x
).
= arccos(
1+x
y =
y
y
y
y
y = lg(
y
x2 − 3x + 2
).
x+1
x
y = arcsin(lg( ))
10
−x
e −2 e
√
y = 9 − x2
√
√
y = 1 + x + x2 − 1 − x + x2
y = (x + 1)2 + (x − 1)2
1+x
)
y = lg(
1−x
y = x2 − x + 1
y =
x
º
4
º
3
3
º
º
º
º
! "#$ % & " ' (
" # T )
" # !
y = 2 sin 3x + 7 cos 5x.
√
y = tg x
y = sin2 x
√
y = sin x
º
º
º
y = 2x + 3.
!
x ∈ R r
" x # y # x # y y # x
x 3
y 3
y = 2x + 3 ⇔ x = − ⇔ y = −
2 2
2 2
$ % R $
x 3
y = −
2 2
√
& f (x) = 1 + x2 2f (x) − f 2 (x)
f (x)
' (
− f 2 (x)
√ ( y = 2
2
f (x) √"
√ f (x) = 1 + x √ 2
2
' # y = 2 1+x − (√ 1 + x2 )2 y = 2 1+x − 1 − x2
2
$ 2f (x) − f 2 (x) = 2 1+x − 1 − x2
)
!" ! M(1; 1) #
135◦ Ox
* + ,- y − y0 = k(x − x0) .
. . /( Ox : k = tg(ϕ)
k = tg 135◦ = −1 ' + , " # M, # y − 1 = −(x − 1).
$ y = −x + 2
$%! #% % & #
% # % l1 : 18x + 6y − 17 = 0,
l2 : 14x − 7y + 15 = 0, l3 : 5x + 10y − 9 = 0
* 0 (1 # / A 2 #
# " l1 l2 B 2 # # " l2 l3 C 2
# # " l1 l3 3 . A ABC
k 1 − k2
tg ∠A =
. k1 , k2 2 . " " " l1 l2
1 + k1 k 2
17
*" y l1 l2 l1 : 18x+6y −17 = 0 ⇐⇒ y = −3x+ ;
6
15
l2 : 14x − 7y + 15 = 0 ⇐⇒ y = 2x + ; k1 = −3 k2 = 2
7
−3 − 2
= +1 =⇒ ∠A = 45◦
tg ∠A =
1−6
ABC
A
⎧
⎧
29
17
⎪
⎪
⎨ y = −3x + ,
⎨ x= ,
18x + 6y − 17 = 0,
42
6 ⇔
⇔
15
14x − 7y + 15 = 0
⎪
⎪
⎩ y = 2x +
⎩ y = 2 44 .
7
105
44
29
;2
)
A(
210 105
ΔABC A(1; 1) B(2; 3) C(3; 0)
ΔABC
! "
# $ !
x−1
y−1
=
⇔ y = 2x − 1.
AB :
3−1
2−1
x−2
y−3
=
⇔ y = −3x + 9.
BC :
0−3
3−2
x−1
x 3
y−1
=
⇔y=− + .
AC :
0−1
3−1
2 2
AB
Oxy
y 2x − 1
y 2x − 1 ΔABC
% ! # # "
C(3; 0)
C
! &# ! ΔABC %
$ y 2x − 1
'
ΔABC "
x 3
y −3x+9 y − + ! "
2 2
ΔABC
(#
⎧
⎪
⎨ y 2x − 1,
y −3x + 9,
⎪
⎩ y −x + 3.
2 2
) y = |x2 − 1|
*+ + , y = |f (x)|
+ , y = f (x) $ -! . )/
! +
•
y = f (x)
Ox,
•
$
! ! " #
|f (x)| =
f (x), f (x) 0,
−f (x), f (x) < 0.
% &
Ox 'f (x) 0( #
& Ox
!
" y = −f (x)
) ! * + ! &
,- . ! &
Ox |f (x)| 0
∀x ∈ D(f )
/
y = |x2 − 1| ! *0
y
y
y= x2 -1
y=x 2 -1
-1
1
1
x
1 0
-1
1
x
y = |x2 − 1|
(|x| − 2)2
2 $ 3
y = f (|x|) !
y = f (x) " ' *4($
•
y = f (x)
Oy
• " #
Oy
x, x 0,
|x| =
−x, x < 0.
x 0 x < 0 Oy! "
# y = f (−x)$ % $ &$' ( "
Oy$ )
Oy $$ # y = f (|x|) *
!$
y
y
y=f( x )
y=f(x)
x
x
y = f (|x|)
+ # y = (|x| − 2)2
y
y
y=(x-2)
$ &,$
y=( x -2) 2
2
4
4
2
x
-2
2
x
y = (|x| − 2)2
- (
#$
y=
√
sin x
sin x < 0 D(x) = {x| sin x
√ 0}
√
u > u 0 < u < 1
y = sin x
! !" # $ % $ y = sin x &
! !" ' % ' !'
u
u=sin x
1
−π
-1
y
π
2π
0
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2
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1
1 9
9
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4
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3
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1
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1
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3
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1
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4
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/ # . R M(a; b) 0 .
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2
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y
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1
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x2 − 2y 2 + 2x + 12y − 33 = 0
! x "
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8
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4
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1 3 14 144 1444
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x
1
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0
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1
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n
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·a
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1
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&
(
lim N (x) = −∞.
x→a
)
N (x) M(x)
*
+ !
, !-, # *
#
. ! N (x)
x → a ! N 1(x) " x → a
/ & 0 x → +∞ 1 2
ε > 0
0 ! ! # x
1
1
| < ε
|
3
N (x)
N (x)
- N (x) 4 , *
1
1
|
x > C 5 (! |
ε
N (x)
! # x
!
x→a
lim ϕ(x) = b lim ψ(x) = c
f (x) = ϕ(x) + ψ(x)
x→a
f (x) = ϕ(x) − ψ(x) x → a
lim [ϕ(x) ± ψ(x)] = lim ϕ(x) ± lim ψ(x).
x→a
x→a
x→a
x→a
lim ϕ(x) = b lim ψ(x) = c
x→a
x→a
lim [ϕ(x) + ψ(x)] = lim ϕ(x) + lim ψ(x)
x→a
x→a
lim ϕ(x) = b,
7.6
x→a
lim ψ(x) = c;
⇒
x→a
ϕ(x) = b + α(x),
ψ(x) = c + β(x).
! α(x) β(x) " # x → a
$!
f (x) = ϕ(x) + ψ(x) = [b + α(x)] + [c + β(x)] = (b + c) + [α(x) + β(x)],
lim f (x) = lim [ϕ(x) + ψ(x)] = lim {(b + c) + [α(x) + β(x)]} = b + c.
x→a
x→a
x→a
%
&
lim f (x) = lim [ϕ(x) + ψ(x)] = lim ϕ(x) + lim ψ(x).
x→a
x→a
x→a
' ! (
x→a
lim [ϕ(x) − ψ(x)] = lim ϕ(x) − lim ψ(x).
x→a
x→a
x→a
)
x→a
lim ϕ(x) = b lim ψ(x) = c
x→a
x → a *
( f (x) =
lim [ϕ(x) · ψ(x)] = lim ϕ(x) · lim ψ(x).
x→a
x→a
x→a
ϕ(x) · ψ(x)
+
x→a
lim ϕ(x) = b, lim ψ(x) = c
x→a
lim [ϕ(x) · ψ(x)] = lim ϕ(x) · lim ψ(x)
x→a
x→a
x→a
lim ϕ(x) = b,
x→a
lim ψ(x) = c;
x→a
7.6
⇒
ϕ(x) = b + α(x),
ψ(x) = c + β(x),
α(x) β(x)
x → a
f (x) = ϕ(x) · ψ(x) = [b + α(x)] · [c + β(x)] = (b · c) + [c · α(x) + b · β(x) +
+ α(x) · β(x)].
lim f (x) = lim [ϕ(x)·ψ(x)] = lim {(b·c)+[c·α(x)+b·β(x)+α(x)·β(x)]} =
x→a
x→a
x→a
= b · c + lim [c · α(x) + b · β(x) + α(x) · β(x)] = b · c = lim ϕ(x) · lim ψ(x).
x→a
x→a
x→a
lim [c · α(x) + b · β(x) + α(x) · β(x)] = 0
x→a
! " # # $ $ %
lim [k · ϕ(x)] = k · lim ϕ(x).
&'()
x→a
x→a
* $
'+
&',)
lim [f (x)]n = [lim f (x)]n.
x→a
"
'- '+ !.!
' /
0
x→a
.
%
lim ϕ(x) = b lim ψ(x) = c
x→a
c = 0
x→a
&'')
lim [ϕ(x)/ψ(x)] = lim ϕ(x)/ lim ψ(x).
x→a
'+
x→a
x→a
lim (x2 + 2x − 1)
x→2
1 2 3 $ !
•
4
lim (x2 + 2x − 1) = lim x2 + lim 2x − lim 1.
x→2
x→2
x→2
x→2
•
2
2
2
lim x = [lim x] = 2 = 4.
x→2
•
x→2
! " # # $ $
lim 2x = 2 lim x = 2 · 2 = 4.
x→2
x→2
y = f (x) 0 x
a
x→a
lim f (x) 0
sin x
= 1.
x
π
˘ x
0 < x < AC
2
AB
sin x !
˘
0 < AB < AC
0 < sin x < x
" #
x → 0 sin x $ %
lim sin x = 0 & lim cos x = 1
lim
x→0
x→0
x→0
x
x
lim cos x = lim (1 − 2 sin ) = 1 − 2 lim (sin2 ) = 1 − 2 · 0 = 1.
x→0
x→0
x→0
2
2
2
D
A
1
sin x
tg x
x
cos x B
O
'
sin x
x→0
lim
x
SOAB < S
!
C
SOAB =
S
( )
OAC
< SODC .
cos x sin x
OB · BA
=
,
2
2
OAC
1
1
x
= R 2 x = 12 x = ,
2
2
2
1 tg x
tg x
OC · CD
=
=
,
2
2
2
cos x sin x
x
tg x
< <
.
2
2
2
1
sin x
2
1
x
<
cos x <
sin x
cos x
1
sin x
>
> cos x.
cos x
x
x > 0 x < 0
x → 0 ⇒ lim cos x = 1
x→0
1
cos x
1
1
1
=
= = 1.
lim
x→0 cos x
lim cos x
1
SODC =
x→0
x → 0
%
sin x
'( ! "
x
! " #
$ "
! " &
x → 0
lim
x→0
+
sin x
= 1.
x
sin x
= 1
lim
x→0 x
tg x
),, lim
x→0 x
&)*(
%
- x → 0 %
$ .
+ ' $
sin x
tg x
sin x
1
= lim
= lim
·
· lim cos x = 1 · 1 = 1.
lim
x→0 x
x→0
x→0 x
x→0
x
cos x
5x
x→0
lim
arcsin 3x
x = 0
sin y
0
arcsin 3x = y sin y = 3x x =
0
3
x → 0 y → 0 !
sin y
5·
5x
3 = 5 lim sin y = 5 · 1 = 5 .
= lim
lim
x→0 arcsin 3x
y→0
y
3 y→0 y
3
3
x
1
"# lim 1 +
= e
x→+∞
x
$ % % &'
( )' ' *'% + +
+, %
&'- .
)
"# " - %+ # /
'* " +
"+ +
, %+ ' +,
"+ &'- #
0% % #
- % '* +
- % "+
+ ' ' % ,
1 12 3 4
• 3 4 5 "
yn = n
1 2 3 4
• , , , , · · · . 6 '
2 3 4 5
' -'
n
yn =
n+1
√ √
2, 2 2,
√
2 2 2.
1
3
7
15
2 2 , 2 4 , 2 8 , 2 16 .
!
"" 2n # 2 −1 " 2n −1"
$ yn = 2 2 "
%" yn
+1
y10, yn−1 , yn+1.
yn = 2n
n+3
& ' ()
*+ *)
() "
•
n
n
21
2n − 1
2 ∗ 10 + 1
2 (n − 1) + 1
= ,
yn−1 =
=
,
10 + 3
13
(n − 1) + 3
n+2
2n + 3
2 (n + 1) + 1
=
.
=
(n + 1) + 3
n+4
y10 =
yn+1
%",
0, 2; 0, 23; 0, 233; 0, 2333; . . . .
& $*)
-
yn = 0, 2 + [0, 03 + 0, 003 + 0, 0003 + . . .] = 0, 2 + S.
. - + *+ * * * (
)( ( ( q = 0, 1
b1 =n 0, 03" / + n 0 1n
2 Sn = b1(11 −− qq ) . / * * ()
Sn
b1
. *
S=
1−q
lim {yn } = lim [0, 2 + Sn ] =
n→+∞
n→+∞
0, 2 +
7
0, 03
= .
= 0, 2 +
1 − 0, 1
30
b1
1−q
=
lim
n→+∞
1
2
3
n−1
.
+
+
+
·
·
·
+
n2 n2 n2
n2
0 + 1 + 2 + 3 + . . . + n − 1. ! n
Sn =
0 + (n − 1)
n(n − 1)
a1 + an
·n=
·n=
.
2
2
2
"#
1+2+3+ ...+n −1
1
2
3
n−1
= lim
lim
+
+
+
.
.
.
+
=
n→+∞
n→+∞
n2 n2 n2
n2
n2
n(n − 1)
1
2
= .
= lim
n→+∞
n2
2
$
lim
n→+∞
1 1 1
1
+ + +...+ n
2 4 8
2
.
1 1 1
1
, , . . . n
2 4 8
2
% & n ' '
b1 (1 − q n )
. (
Sn =
1−q
1 1 1
b1 (1 − q n )
1
+ + + . . . + n = lim
=
lim
n→+∞
n→+∞
2 4 8
2
1−q
1
1
(1 − ( )n )
2
2
= 1.
= lim
1
n→+∞
1−
2
)
2n+1 + 3n+1
.
n→+∞
2n + 3n
lim
n → +∞ −→ 3n+1 :
2n+1 + 3n+1
lim
= lim
n→+∞
n→+∞
2n + 3n
2n+1 + 3n+1
3n+1
=
2n + 3n
3n+1
n+1
2
2n+1
+1
+1
n+1
0+1
3
3
=3
= lim
=
lim
= lim
n
n
1
3n
n→+∞ 2
n→+∞ 1
n→+∞ 1
2
1
·
0
+
+
·
+
3
3
3n+1 3n+1
3
3
3
2
n → +∞
3
1.
1
x
−
3;
lim 5
2.
x→3−0
1
x
−
3.
lim 5
x→3+0
1
→ −∞
! " x → 3 − 0# x − 3 → −0
x−3
$ %
#
1
lim 5 x − 3 = 5−∞ =
1
1
= 0.
=
5+∞
+∞
1
→ +∞#
x → 3 + 0# x − 3 → +0
x−3
1
lim 5 x − 3 = 5+∞ = +∞.
x→3−0
& "
#
x→3+0
' ( (
) * +
, %
sin x
=1
lim
x→0 x
lim
x→+∞
1
1+
x
x
1
y
= lim (1 + y) = e.
y→0
sin 5x
x→0
lim
x
! ! "# $
% & ! '
sin 5x
sin 5x · 5
sin 5x
= lim
= 5 lim
= 5 · 1 = 5.
lim
x→0
x→0
x→0 5x
x
x·5
y
( )* 5x = y + x = , x → 0 y → 0 :
5
sin 5x
sin y
sin y
= lim y = 5 lim
= 5.
lim
x→0
y→0
y→0 y
x
5
3πx
2
- lim
x→2
x
)
) ! * $
. ! x → 2
*! ! ! ! ,
! x = 2
3π · 2
3πx
sin
sin
2
2 = sin 3π = 0 = 0.
=
lim
x→2
x
2
2
2
%/
sin
sin x
x→+∞
lim
x
! ! # $
0 ! ! $
sin x
= 1 , x → +∞ $
lim
x→0 x
! *! * 1 lim x = +∞2 !
x→+∞
3 1| sin x| 12
*! * ! *! 4
sin x
= 0.
lim
x→+∞ x
5x
=y
2
lim
x→0
y→0:
5x
2 sin2 y
2 sin2 y
sin2 y
25
2 = lim
lim 2 =
=
2 = lim 4
2
y→0
y→0
x
2 y→0 y
2y
y2
25
5
2
sin y
25
25
25
lim
·1= .
=
=
2 y→0
y
2
2
2y
5
2 sin2
x→0
x=
x→π
lim
1 − sin
x
2
π−x
x = π
π
x
1 − sin
1 − sin
2 =
2 = 1 − 1 = 0,
lim
x→π
π−x
π−π
π−π
0
x−π = y
y → 0
x → π x−π → 0
"
x = y + π
!
y π
y+π
y
x
= sin
= sin
+
= cos ,
2
2
2 2
2
x
y
1 − sin
1 − cos
2 = lim
2.
lim
x→π
y→0
π−x
−y
0
#
$
%
0
y
2 y
= 2 sin
'( 1 − cos
2
4
y
y
1 − cos
2 sin2
2
4.
= lim
lim
y→0
y→0
−y
−y
y
= z y = 4z y → 0 z → 0 :
!
4
y
2 sin2
2
2
4 = lim 2 sin z = − 1 lim sin z =
lim
y→0
z→0 −4z
−y
2 z→0 z
1
1
sin z
· lim sin z = − · 1 · 0 = 0.
= − lim
z→0
2 z→0 z
2
sin
&
lim
x→π
1 − sin
π−x
x
2 = 0.
x→0
lim (cos(mx) − cos(nx))/x2
!
" 1 −0 1 = 00 ! ! #$ ! %&
m−n
m+n
x · sin
x
cos(mx) − cos(nx) = −2 sin
2
2
m−n
m+n
x · sin
x
−2 sin
cos(mx) − cos(nx)
2
2
lim
=
lim
=
x→0
x→0
x2
x2
m+n
m−n
x
x
sin
sin
2
2
· lim
=
= −2 lim
x→0
x
x
x→0
m+n
m−n
m+n
m−n
· sin
x
· sin
x
2
2
2
2
· lim
=
= −2 lim
m+n
m−n
x→0
x→0
·x
·x
2
2
m+n
m−n
x
x
sin
sin
m−n
m+n
2
2
·
= −2 ·
· lim m + n
· lim m − n
=
x→0
x→0
2
2
·x
·x
2
2
n2 − m2
2 2
2
.
= − (m − n ) · 1 · 1 =
4
2
x
3
' x→+∞
lim 1 +
x
! () ! ! &
) ! () *! ! + t = x3 + x = 3t
x → +∞ t = x3 → 0
lim
x→+∞
1+
3
x
x
⎛
⎞
3
1 3
= lim(1 + t) t = ⎝lim(1 + t) t ⎠ = e3 .
t→0
t→0
x→+∞
lim
x−1
x+1
x
!" # $ %
x
x
⎞
⎛
−1
1 x
x
1
+
lim
x · (1 − )
x→+∞
x−1
e−1
x
⎜
x ⎟
x =
= e−2 .
lim
= lim ⎝
=
⎠
1
x→+∞
x→+∞
x+1
e
1
x · (1 + )
lim 1 +
x
x→+∞
x
x
2x + 3
x→+∞
lim
2x − 1
y+1
&# % 2x−1 = y $ x =
$ 2x+3 = y+4
2
' x → +∞ y → +∞
x
y + 1
( y + 1 )
4 2 2
2x + 3
y+4
2
=
lim
= lim
= lim 1 +
x→+∞ 2x − 1
y→+∞
y→+∞
y
y
y
1
y 1 1
4 2
4 2
4
2 2
= lim 1 +
·1 =
· lim 1 +
=
lim 1 +
y→+∞
y→+∞
y→+∞
y
y
y
1
4 2
= (e ) · 1 = e2 .
(
x→+∞
lim (ln(2x + 1) − ln(x + 2))
' x → +∞
)
+∞ − ∞
* %
2x + 1
. + ",
ln(2x + 1) − ln(x + 2) = ln
x+2
* x - *
1
2+
2x + 1
x
= ln
ln
2
x+2
1+
x
⎞
1
⎜
x⎟
lim (ln(2x + 1) − ln(x + 2)) = lim ⎝ln
= ln 2,
2⎠
x→+∞
x→+∞
1+
x
1 2
x → +∞ x x
⎛
2+
! "
#$% 1, 12 , 212 , 213 , · · · .
#&' 2, 32 , 43 , 54 , 65 , · · · .
#&$ 1, 4, 9, 16, 25, · · · .
#&& 21 , 43 , 65 , 87 , · · · .
#&( 1, − 12 , 13 , − 14 , · · · .
#&) n
{yn} = 3n y3 , y5, yn+1
*
#&+
yn
1, 6; 1, 66; 1, 666; 1, 6666; . . .
#&,
lim
n→+∞
#&#
lim
n→+∞
#&-
1 + 3 + 5 + 7 + ... + (2n − 1)
.
n2
1−
1
(−1)n−1
1 1
+ −
+ ··· +
.
3 9 27
3n−1
lim
9n + 8n
.
+ 8n+1
n→+∞ 9n+1
√
2,
√
2,
√
2, . . . .
n sin n!
.
n2 + 1
lim
n→+∞
1
n→4−0
lim 2 n − 4 .
1
n
−
4.
lim 2
n→4+0
x→0 sinx4x .
lim
lim
lim
lim
5x
.
x→0 sin
sin 2x
x→2 sinx x .
x
.
x→0 sin
tg x
x
sin
3.
lim
x→0
x2
lim
cos x
.
lim
π π − 2x
x→
2
sin x − sin a
.
lim
x→a
x−a
x
x
lim
.
x→+∞ x + 1
x
2+x
lim
.
x→0
3−x
x2
2
x +2
lim
.
x→+∞ 2x2 + 1
lim
2
x→0 tg x −x3sin x .
x→+∞
lim x (ln(x + 1) − ln x) .
x→0 ln(1 x− 3x) .
α(x) β(x)
α(x)
=b=0
x → a x→a
lim
β(x)
= +∞
! α(x)
β(x) x → a
α(x)
= 0
lim
x→a β(x)
" α(x)
β(x) x → a
α(x)
= +∞
lim
x→a β(x)
# α(x) β(x)
α(x)
x → a x→a
lim
β(x)
+∞
y = x2 y = 3x x → 0.
$ % &
x2
x
0
= lim = = 0.
x→0 3x
x→0 3
3
y = x2
' y = 3x
lim
'
(
!
y=x
$ % &
2
x → 0
+ x − 6 y = 4 − x2 x → 2.
x2 + x − 6
(x − 2)(x + 3)
x+3
5
= − lim
= − = 0.
= lim
x→2
x→2 −(x − 2)(x + 2)
x→2 x + 2
4 − x2
4
lim
x → 2
cos x
y=
y = x1 x → +∞.
x
!
cos x
lim
x→+∞
x
1
x
= lim cos x.
x→+∞
" cos x x → +∞
x → +∞
# α(x) β(x),
α(x)
= 1!
x→a
lim
β(x)
x → a
" $ x % x = a &'
α(x)
≈ 1 α(x) ≈ β(x) (
x→a
lim
β(x)
& x a ) α(x) β(x)*+
x → a α(x) ∼ β(x)
, "
#
$ $ ! !
α ∼ α1 % β ∼ β1 x → a
lim
x→a
α1 (x)
α(x)
= lim
.
β1 (x) x→a β(x)
- ! α ∼ α1 . β ∼ β1 x → a
- !
lim
x→a
= lim
x→a
α(x)
α(x) α1 (x) β1 (x)
= lim
·
·
=
β(x) x→a β(x) α1 (x) β1 (x)
α(x) α1 (x) β1 (x)
α1 (x)
α1 (x)
·
·
=1·
· 1 = lim
.
x→a
α1 (x) β1 (x) β(x)
β1 (x)
β1 (x)
sin 5x
/ &$ x→
lim
!
0 sin 3x
sin 3x ∼ 3x x → 0
lim
lim
x→0
x→0
sin 5x
sin 3x
= 1 lim
=1
x→0 3x
5x
sin 5x ∼ 5x
sin 5x
5x
5
= lim
= .
sin 3x x→0 3x
3
α(x) β(x)
[α(x) − β(x)]
α(x) β(x)
lim α(x) = lim β(x) = 0 α(x) β(x)
x→a
x→a
x → a γ(x) = α(x) − β(x)
γ(x)
γ(x)
= lim
= 0
lim
α(x) ∼ β(x) x→a
α(x) x→a β(x)
β(x)
(ϕ(x) − ψ(x))
ψ(x)
= lim
= lim 1 −
=
x→a ϕ(x)
x→a
x→a
ϕ(x)
ϕ(x)
lim
= 1 − lim
x→a
ψ(x)
= 1 − 1 = 0.
ϕ(x)
! γ(x) " " #
"! α(x) $ # % !
γ(x)
= 0
lim
x→a
β(x)
&
!
x→a
lim γ(x) = lim α(x) = lim β(x) = 0
x→a
x→a
γ(x), α(x), β(x) x → a ' !" !()
γ(x) * " " # "!
α(x)
β(x)
= lim
= 0.
" # x→a
lim
x→a
+ ,
lim
x→a
γ(x)
γ(x)
γ(x) + α(x) + β(x)
= 1
lim
x→a
γ(x)
x → a - ! γ(x)
γ(x) + α(x) + β(x)
α(x)
β(x)
= lim 1 + lim
+ lim
= 1 + 0 + 0 = 1.
x→a
x→a γ(x)
x→a γ(x)
γ(x)
5x + 6x
x→0
lim
sin 2x
2
.
x → 0 5x + 6x2 ∼ 5x
5x + 6x2
5x
5
= lim
= .
sin 2x ∼ 2x lim
x→0 sin 2x
x→0 2x
2
! " # $ % $ & x → x0 %
' ! !% % " ( "
% ' &
)'*%'& ' ! + ' %" %$ "&" ,
$ "* % ( x → x0
"" % ' & # - + ' !
0
%" ./ %%'&*!
0
∞
∞
0 ' # # / % x → x0 ! (
/ % # # / " ( %
( ! ( / /(" 1 ./ +∞ − ∞
'2 # % $ x → x0 #
# /'& x → x0 %( ./ 0 · ∞
3 ! ( /! $ 2 %
1
lim (1 + α(x)) β(x) ! α(x) β(x) # $ % $
x→x0
x → x0 2 #$/ ( $ % ( % " α(x)
β(x) ' & x → x0 4 ./ 1+∞
$/ ./ 1 + ( / %
%$ 2" % "*! . !
/ % $" . (
!
3x + 5
5 lim
x→+∞ 2x + 7
- / ( / 1 #
# / % $ ./ ∞
∞
x
lim
x→+∞
3
3x + 5
3 + 5/x
= lim
= ,
x→+∞
2x + 7
2 + 7/x
2
x → +∞ x5 x7
!! " # $
!! !! % #
$
&' N (x) M(x)
x → a
N (x)
= 1
lim
x→a M(x)
(! Pn (x) = bnxn + bn−1xn−1 + · · · + b0 !
! x → +∞ !% bnxn
lim
x→+∞
Pn (x)
bn xn + bn−1 xn−1 + · · · + b0
= lim
=1
n
x→+∞
bn x
bn x n
)! % ! x → +∞
!
Qm (x)
am xm + am−1 xm−1 + ... + a0
= lim
=
x→+∞ Pn (x)
x→+∞ bn xn + bn−1 xn−1 + ... + b0
lim
*&+,
am xm
am
=
lim xm−n
x→+∞ bn xn
bn x→+∞
= lim
•
(m < n)
• (m > n)
+∞ −∞
• (m = n)
3
am
bm
8x + 3x − 5
&- x→+∞
lim
4x3 − 2x2 + 3
8x3 + 3x − 5
8x3
=
lim
= 2.
x→+∞ 4x3 − 2x2 + 3
x→+∞ 4x3
lim
5
2
5x − 2x + 3
x→−∞
lim
2x4 + 3x − 5
lim
x→−∞
5x5 − 2x2 + 3
5x5
5x
= lim
= −∞.
= lim
4
4
x→−∞
x→−∞
2x + 3x − 5
2x
2
!
√
3
√
3 x + 5x − 4 + x
" x→+∞
lim √
√
2x4 − 3x + 2 + 2 x
2
6
3
√
√
√ # $ $ $ √x → +∞√3 3 x2 +√5x − 4 ∼ 3 3 x2 = 3x2/3
√
6
x = x1/2 6 2x4 − 3x + 2 ∼ 2x4 = 6 2x√2/3 3 x = x1/3 %
& √x ' & (&
( 3 3 x2 + 5x − 4 ∼ 3x2/3 ) (
√
6
6
2/3
4
& 2x − 3x + 2 ∼ 2x &
√
√
3 3 x2 + 5x − 4 + x
3x2/3
3
= lim √
lim √
= √
.
√
6
6
6
3
4
x→+∞
2x − 3x + 2 + 2 x x→+∞ 2x2/3
2
' *
& $ ' & )+& +* (
• ! (! '
,& )( • & ) ) ( & ' &
& ,&( )+& .( & &
) ! ' $ !
√
√
√
√
√
√
( x + y) · ( x − y) = ( x)2 − ( y)2 = x − y (x 0, y 0),
√
√
√
√
√
√
3
( 3 x ± 3 y) · ( x2 ∓ 3 x · y + 3 y 2 ) = ( 3 x)3 ± ( 3 y)3 =
=x±y
(x 0, y 0).
√
lim ( x2 − 1 − x)
/ x→+∞
x2 + 3x − 1 − x2
3x − 1
√
= lim √
=
x→+∞ ( x2 + 3x − 1 + x)
x→+∞ ( x2 + 3x − 1 + x)
√
3
3x − 1
= .
x2 + 3x − 1 + x ∼ 2x| = lim
= |
x→+∞
2x
2
= lim
N (x) − M(x)
! x N (x) M(x)
"
!#
$ ! ! !
! x → x0 " !
0
" % ! & ! !
0
! (x − x0 ) " !# '
! " !
" x → x0 ! ! (x − x0 )
#!# ! ! #
( !
!
(x − x0 )#
2
x −9
)#*+ x→3
lim 2
x − 3x
, - ! !
! "# .
0
! # ,
0
! / (x − 3)x=3
x2 − 9
(x − 3)(x + 3)
x+3
= lim
= lim
= 2.
x→3 x2 − 3x
x→3
x→3
x(x − 3)
x
lim
!
#
2
5x + 2 sin 2x + tg
)#*0 x→0
lim
arctg 3x + 5x2
2
x
, - 1 x → 0 sin 2x ∼ 2x tg x ∼ x arctg 3x ∼ 3x#
. ! x
3 sin 2x 2 arctg 3x#3! )#+
3) = +∞
lim
x→+∞
⎛
⎞
1
1+ 2
1
x2 + 1
⎜
x ⎟
= lim ⎝
lim
=
3 ⎠ 4
x→+∞ 4x2 − 3
x→+∞
4− 2
x
x2 + 1
4x2 − 3
x+3
lim (x+
x→+∞
1
1
1
= 0.
= ( )+∞ = +∞ =
4
4
+∞
x→+∞
lim
x+8
x−2
x
8
1+
x+8
x
= lim
x→+∞
lim
= 1, lim x = +∞
2
x→+∞
x − 2 x→+∞
1−
x
1+∞
ϕ(x) = 1 + α(x)! "
x
x
x
x+8
x+8
10
−1
lim
= lim 1 +
= lim 1 +
=
x→+∞ x − 2
x→+∞
x→+∞
x−2
x−2
10
·x
x − 2⎫
x
−2
10x
⎪
⎬
lim
10
10
x→+∞
x
− 2 = e10 .
1+
= lim
=e
x→+∞ ⎪
⎪
x−2
⎭
⎩
⎧
⎪
⎨
# " $ % & ' (
%
⎡
⎤8
x
8 8⎥
⎢
lim ⎣ 1 +
⎦
x→+∞
x
lim
x→+∞
x+8
x−2
x
x
8
1+
x
x =
= lim
⎡
x→+∞
2
1−
⎢
x
lim ⎣ 1 +
x→+∞
# '
%
"
10
⎤−2 = e .
− x
−2
2⎥
⎦
x
lim
x→+∞
1+
k
x
x
= ek
x → +∞
∞
x
∞
5x − 4
4
5−
5x − 4
5
x
x
= lim
= ,
= lim
lim
2
x→+∞ 4x + 2
x→+∞ 4x + 2
x→+∞
4
4+
x
x
4 2
x → +∞ !
x x
3x − 4
5x
5
= lim
=
"#$% x→+∞
lim
4x + 2 x→+∞ 4x
4
2
3x + 2x + 1
#& x→+∞
lim
2x2 + 3
x → +∞ ' (
x2
∞
∞
1
2
3x2 + 2x + 1
3+ + 2
3x + 2x + 1
2
x
x x = 3,
= lim
lim
= lim
3
x→+∞
x→+∞
x→+∞
2x2 + 3
2x2 + 3
2
2+ 2
x
x2
2 1 3
x → +∞ ! , 2 , 2 → )
x x x
3x2 + 2x + 1
3x2
3
= lim
"#$% x→+∞
lim
= .
2
x→+∞ 2x2
2x + 3
2
2
2
x − 2x
#* x→−∞
lim
6x + 7
x → −∞
∞
( +
∞
! ' , -
'+ .
x2
x2 − 2x
2
1−
x − 2x
2
1
x
x
= lim
= −∞.
lim
= lim
=
7
x→−∞ 6x + 7
x→−∞ 6x + 7
x→−∞ 6
−0
+
x2
x x2
2
x2 − 2x
x2
6
= lim
= lim
= −∞
x→−∞ 6x + 7
x→−∞ 6x
x→−∞ x
lim
2
5x − 3x + 4
x→+∞
lim √
4x4 + 5
∞
∞
xn!
" n # $ " %&$!
x4 & ! x2
5x2 − 3x + 4
4
3
5− + 2
2
x
x
x
√
= lim
=
x→+∞
4x4 + 5
5
4+ 4
x2
x
5
5−0+0
= .
= √
2
4+0
5x2 − 3x + 4
= lim
lim √
x→+∞
x→+∞
4x4 + 5
'$ $ (" )
(
√
√ $
* *+ 4x4 + 5 ∼ 4x4 = 2x2
5x2 − 3x + 4
5x2 − 3x + 4
5x2
5
√
= lim
= lim
= .
2
4
x→+∞
x→+∞
x→+∞ 2x2
2x
2
4x + 5
P (x)
x→a
lim
! " P (x) Q(x) & "& +$ *&
Q(x)
+ a = 0 +$
, - " $ " x = a . )
0
P (x)
P (a) = Q(a) = 0 ! *
0
Q(x)
$ (x − a)
lim
3
x +1
/ x→−1
lim 2
x +1
x = −1 x3 + 1 =
= −13 + 1 = −1 + 1 = 0 x = −1 #
x2 + 1 = −12 + 1 = 1 + 1 = 2 0!
0
x3 + 1
= = 0.
lim 2
x→−1 x + 1
2
2
x −4
x→2
lim 2
x − 3x + 2
x = 2
0
0
x2 − 4
(x − 2)(x + 2)
x+2
= lim
= lim
= 4.
x→2 x2 − 3x + 2
x→2 (x − 2)(x − 1)
x→2 x − 1
lim
! " ! "
# $ % %
√
1+x−1
& lim √
3
x→0
1+x−1
x = 0
0
' 1 + x = y 6 # !
0
# # % " ( !
x → 0! y → 1!
√
y6 − 1
1+x−1
y3 − 1
.
lim √
=
lim
= lim 2
3
3
x→0
1 + x − 1 y→1 y 6 − 1 y→1 y − 1
#
# ! %
)
$
0
* y = 1+
0
!
y3 − 1
(y − 1)(y 2 + y + 1)
y2 + y + 1
=
lim
=
lim
=
y→1 y 2 − 1
y→1
y→1
(y − 1)(y + 1)
y+1
lim
=
3
1+1+1
= .
1+1
2
) # " #
"
!
!
,
- . % #
%
√
√
√ √
√
√
( x + y)( x − y) = ( x)2 − ( y)2 = x − y
(x 0, y 0)
√
√
√
√
√ √
3
( x ± y)( x2 ∓ 3 xy + 3 y 2 ) = ( 3 x)3 ± ( 3 y)3 = x ± y (x 0, y 0).
√
lim ( x2 − 5x + 6 − x)
x→+∞
x → +∞
+∞ − ∞
√ ! " # "$ %
& ( x2 − 5x + 6 + x) ' " ( )
* &
√
lim ( x2 − 5x + 6 − x) =
x→+∞
√
√
( x2 − 5x + 6 − x)( x2 − 5x + 6 + x)
√
=
= lim
x→+∞
x2 − 5x + 6 + x
x2 − 5x + 6 − x2
6 − 5x
= lim √
= lim √
.
x→+∞
x2 − 5x + 6 + x x→+∞ x2 − 5x + 6 + x
+& & x → +∞ * ! )
∞
&
∞
xn # n , #& % & x
6 − 5x
6 − 5x
x
=
lim √
= lim √
x→+∞
x2 − 5x + 6 + x x→+∞ x2 − 5x + 6 + x
x
6
−5
x
= lim
=
x→+∞
5
6
1− + 2 +1
x x
−5
−5
5
0−5
=
=√
=− .
=√
1+1
2
1−0+0+1
1+1
x→+∞
lim (x −
√
3
x3 + 8x2 )
%
x → +∞
)
+∞ − ∞
! * ! # " &(
# & & - ! ( " &
lim (x −
x→+∞
√
3
x3 + 8x2 ) =
= lim
(x −
√
3
x→+∞
√
x3 + 8x2 )(x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2 )
√
=
x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2
√
x3 − ( 3 x3 + 8x2 )3
√
=
x→+∞ x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2
= lim
x3 − (x3 + 8x2 )
√
=
3
x→+∞ x2 + x x3 + 8x2 + 3 (x3 + 8x2 )2
= lim
−8x2
√
=
3
x→+∞ x2 + x x3 + 8x2 + 3 (x3 + 8x2 )2
= lim
x→+∞
x→+∞
x2 (1 +
= −8 lim
1+
3
x2
=
8
8 2
3
1 + + (1 + ) )
x
x
= −8 lim
1+
1
8
+
x
3
3
8
(1 + )2
x
= −8 ·
8
1
=− .
1+1+1
3
√
3− 5+x
√
x→4
lim
1− 5−x
x = 4
0
0
!
"
!
#
$!
√
→ (3 + 5 + x)# √
% & " ! # $!
→ (1 + 5 − x)
√
√
3+ 5+x
√
√
(3 − 5 + x) ·
3− 5+x
3+ 5+x
√
√
= lim
lim
=
x→4 1 −
√
1+ 5−x
5 − x x→4
√
(1 − 5 − x) ·
1+ 5−x
√
√
√
(3 − 5 + x)(3 + 5 + x)(1 + 5 − x)
√
√
√
=
= lim
x→4 (1 −
5 − x)(1 + 5 − x)(3 + 5 + x)
√
√
(9 − ( 5 + x)2 )(1 + 5 − x)
√
√
=
= lim
x→4 (1 − ( 5 − x)2 )(3 +
5 + x)
√
(9 − (5 + x))(1 + 5 − x)
√
=
= lim
x→4 (1 − (5 − x))(3 +
5 + x)
√
√
(9 − 5 − x))(1 + 5 − x)
(4 − x))(1 + 5 − x)
√
√
= lim
= lim
=
x→4 (1 − 5 + x))(3 +
5 + x) x→4 (−4 + x))(3 + 5 + x)
√
1
1
1+1
1+ 5−x
√
= −1 · = − .
= −1 ·
= −1 · lim
x→4 3 +
3+3
3
3
5+x
+∞−∞
x→3
lim
1
6
−
x − 3 x2 − 9
! " x = 3 #
$ %
& ' ("
+∞ − ∞
lim
6
1
− 2
x−3 x −9
= lim
x−3
1
1
1
= lim
=
= .
(x − 3)(x + 3) x→3 x + 3
3+3
6
x→3
x→3
= lim
x→3
x+3−6
=
(x − 3)(x + 3)
# ' ' ' ( )
! &
( ' (
sin 5x
* x→0
lim
x
sin 5x ∼ 5x x → 0
5x
sin 5x
= lim
=5
lim
x→0
x→0 x
x
x
x→0
lim
tg 5x
tg 5x ∼ 5x x → 0
x
x
1
= lim
=
lim
x→0 tg 5x
x→0 5x
5
1 − cos 5x
x→0
lim
x2
5x
∼2
1 − cos 5x = 2 sin
2
x→0
2
5x
2
2
=2·
25x2
25x2
=
4
2
25 2
x
1 − cos 5x
25
2
lim
=
lim
= .
x→0
x→0 x2
x2
2
α = t2 tg t β = t2 sin2 t t → 0.
α = 5t2 + 2t5 β = 3t2 + 2t3 t → 0.
3
x +1
.
x→−1
lim 2
x +1
x
x→1
lim
2
+ 2x + 5
.
x2 + 1
3
4x − 2x
x→+∞
lim
3x3 − 5
2
2
.
x +x−1
.
! x→+∞
lim
2x + 5
lim
lim
lim
lim
3
x→1 xx −−11 .
2
x + 3x − 10
.
x→2 3x
2 − 5x − 2
2
3x − 2x − 1
.
x→+∞
lim
x3 + 4
x→3 x
2
− 5x + 6
.
x2 − 9
2
+ 10
.
x→2 xx2 −− 7x
8x + 12
√
3
x−1
.
lim √
x→1 4 x − 1
5
(1 + x)3 − 1
.
lim
x→0
x
√
1+x−1
.
lim
x→0
x
√
2x + 1 − 3
√ .
lim √
x→4
x−2− 2
x
lim √
.
x→0 3 1 + x − 1
√
√
1+x− 1−x
.
lim
x→0
x
√
√
lim 3 x + 1 − 3 x .
x→+∞
1
3
.
−
lim
x→1
1 − x 1 − x3
! !
! ! "#$ % $$" &
"' $ "
y = f (x)
x0
•
x0
•
x → x0
• x → x0
x0
! x0 x0
! " " y = ex
x = 1#
" # $ %& ' y = ex
x = 1, & ' ( )*
$
•
y = ex x = 1 ⇒ f (1) = e+
• * lim f (x) = lim ex = e+
x→1
x→1
• , )
x = 1 :
lim f (x) = f (x0 ).
x→x0
lim f (x) = f (1) = e.
x→1
- & y = ex
x = 1
$%#&' "
lim f (x) = f ( lim x),
.
x→x
x→x
x→x
lim x = x0 # ( )
) "
#
0
0
0
x0
x→x
lim f (x) = f (x0 )
−0
x0
Δx = x − x0 Δy = f (x) − f (x0) !"#$
%&
0
lim f (x) = f (x0 ) ⇒ lim [f (x) − f (x0)] = 0 ⇒ lim Δy = 0.
x−x0 →0
x→x0
Δx→0
"' y = f (x)
x0
!"($
lim Δy = 0.
Δx→0
"' y = x3
x
) * & +
Δy
Δy = (x + Δx)3 − x3 = x3 + 3x2 Δx + 3xΔx2 + Δx3 − x3 =
= 3x2 Δx + 3xΔx2 + Δx3 .
,
lim Δy = lim (3x2 Δx + 3xΔx2 + Δx3 ) = 0.
Δx→0
-
Δx→0
y = x3
−∞ < x < +∞
"# !"
# $ % ϕ(x) ψ(x)
x0 # &
x0
%
#
./
.& x0&
lim ϕ(x) = ϕ(x0 )
x→x0
x0
lim ψ(x) = ψ(x0 ).
x→x0
. f (x) = ϕ(x)·ψ(x)
x→x
lim f (x) = f (x0)
0
lim f (x) = lim [ϕ(x) · ψ(x)] = lim ϕ(x) · lim ψ(x) =
x→x0
x→x0
x→x0
x→x0
= ϕ(x0 ) · ψ(x0 ) = f (x0 ).
x
y = xn y = ax
y = sin x y = cos x ! !
(x ∈ R) y = loga x
x > 0 y = tg x
(− π2 +kπ; π2 +kπ) "" xk = (2k+1) π2
(k = 0; ±1; ±2; ...)
# $
u = ϕ(x)
u0 = ϕ(x0 )
x0
x0 y = f (u)
y = f [ϕ(x)]
%
& '
( '
) !*
loga (1 + x)
# + x→0
lim
x
, * - . x → 0
'
00 ( )
1
1
loga (1 + x)
x
= lim
loga (1 + x) = lim loga 1 + x .
x→0
x→0 x
x→0
x
lim
/
x = 0 !
( lim f (x) = f ( lim x))
x→xo
x→xo
lim loga 1 + x
1
x→0
1
x
lim 1 + x
=e
x→0
x
= loga
+
1
x
lim 1 + x
,
x→0
lim
x→0
loga (1 + x)
= loga e.
x
a=e
ln(1 + x)
= ln e = 1.
x
y = ln(1 + x) y = x
x → 0 $
lim
x→0
'
a
x→0
lim
x
(
)
,$ -. $
1=t
%
&
!" #
−1
x
0
0
ax −
*+
"
.
x = loga (t + 1) 0 x → 0 . t → 0
x
a −1
t
1
1
=
.
= lim
= lim
lim
x→0
t→0 loga (t + 1)
t→0 loga (t + 1)
loga (t + 1)
x
lim
t→0
t
t
/
" "
/ "1
loga (1 + t)
= loga e
lim
t→0
t
ax − 1
1
=
= ln a.
lim
x→0
x
loga e
a = e
ex − 1
= ln e = 1,
lim
x→0
x
x
y = e − 1 y = x
x → 0
%
2
!" #
x0
y = f (x)
y = (1 −1 x)2
x = 1
! x = 1, "
# $ !
y
1
0
1
x
y = (1 −1 x)2
% x0 y = f (x)
! x0
lim f (x) = lim f (x) = A
x→x −0
x→x +0
0
0
& " # #
y=
sin x
x
' ( ) * sinx x ! x = 0 !
x = 0 ! $ x → 0
" + )
lim
x→0+0
sin x
sin x
= 1, lim
= 1.
x→0−0
x
x
sin x
x = 0 f (0) = 1
x
f (x) =
sin x
,
x
x = 0;
x = 0
f (0) = 1.
1
0.8
0.6
0.4
0.2
-10
-5
5
10
-0.2
sin x
x
! x0
x0
" y = sin|x|x
# $ % x = 0 &'
( ) *
1
0.5
-10
-5
5
10
-0.5
-1
sin x
|x|
sin x
sin x
= lim
= −1
x→−0 −x
|x|
sin x
sin x
= lim
=1
lim
x→+0 |x|
x→+0 x
lim
x→−0
1
y =
2
(1 − x)
! ! ! x = 1 "
# $ % x = 1 $
&
1
y = sin x1
' ( ) y = sin $
x
x x = 0
*$ + x = 0 #
x → 0
1
% sin # , −1 1 ,
x
- .% / 0
1
− 7
−6
−5
− 4
−
3
− 2
−
1
1
x=±
k∈N
kπ
1
2
4
3
y = sin x1 y = 0
5
6
7
y = f (x)
a b
y = f (x)
[a, b] !
! "
[a, b]
y
y = f (x)
x = a = xΛ
x2 x = b
x
y = f (x)
[a, b] !
x = x1 = a" # ! x2
$ % y = f (x) [a, b]"
& " '
# ! M ! m "
m f (x) M (
) |f (x)| M " " y = f (x)
[a, b]
* y = f (x)
[a, b]
C
+
, -
y = f (x)" ,
[a, b]" . OX " ( #
OX '
x1, x2, x3
y
a
/
y = f (x)
x1
x2
x3
b
! " #
[a, b] f (a) = A f (b) = B
C,
A B
c f (c) = C
y
y = f (b)
y=C
f (b)
f (a)
a
c
x
b
º
y = f (x) [a, b]
x = f −1 (y)
OY
!
" # $
y=
$
3
1+x
.
1+x
x = −1
! !
0
" # $
0
# % (1 + x)! # 1 + x = 0 &"
!
x = 2
! y = 4x2 x = 2 ⇒ f (2) = 16"
#! $ x→2
lim f (x) = lim 4x2 = 16"
x→2
%!
& x = 2
lim f (x) = f (2) = 16.
x→2
'()
x
y
= sin x
*+ $ Δy ,
-
Δy = sin(x + Δx) − sin x = 2 sin
Δx
Δx
=
cos x +
2
2
Δx
sin
Δx
Δx
Δx
2 · cos x + Δx · Δx.
= 2 sin
·
cos x +
=
Δx
2
2
Δx
2
2
Δx
sin
Δx
2
= 1 cos x +
1
. Δx→0
lim
& x
Δx
2
2
⎞
⎛
Δx
sin
⎜
2 cos x + Δx · Δx⎟
lim Δy = lim ⎝
⎠=
Δx
Δx→0
Δx→0
2
2
Δx
sin
Δx
2
· lim cos x +
= lim
· lim Δx = 1 · cos x · 0 = 0.
Δx
Δx→0
Δx→0
Δx→0
2
2
/ y = sin x
−∞ < x < +∞(
'(0
x − 1,
f (x) =
3 − x,
0 x 3,
3 < x 4.
!"# $ % x = 3 ⇒ y = 2 &
%'!
x → 3 :
lim f (x) = 2,
lim f (x) = 0.
x→3−0
x→3+0
( )! x = 3 * %' + ,
' [0, 4] f (x) ' (x = 0) ' (x = 4)
y
2
0
4 x
1
3
-1
2
y = xx −− 25
5
. x = 5 ! ! '
! ) 0/0 / % )!
lim y = lim y = 10.
( )!
x→5−0
x=5
0
x→5+0
*
%'
y = x12
. x = 0 0 %'!
,1 +'
x = 0 ' +∞ ( )! x = 0 1
1 %'
u1 + u2 + u3 + · · · + un + . . . =
+∞
,
un .
n=1
u1, u2, u3, . . . , un, . . .
!
" un #$% &
n' un = f (n)
( ) '
1
11 + 12 + 13 + · · · + n1 + . . .
un = *
n
+ 2 + 6 + 18 + · · · + 2 · 3n−1 + . . .
un = 2 · 3n−1 *
n−1
, 1 − 1 + 1 − 1 + · · · + (−1) + . . .
un = (−1)n−1 *
π
π
π
π
- cos 1 + cos 2 + cos 3 + · · · + cos n + . . . un = cos πn
+ Sn n
n
Sn = u 1 + u 2 + u 3 + · · · + u n =
n
,
uk .
+
k=1
.& $ $ $$ / )
&
1
1
1
1
+
+
+ ··· +
+ ....
1·2 2·3 3·4
n(n + 1)
' 0 )) $ Sn
1& 2 1& / & "
/ ) $ " '
1
1
1
= −
.
n(n + 1)
n n+1
1
1
1
= =1− ;
1·2
2
2
1
1 1 1 1
1
1
+
= − + − =1− ;
S2 =
1·2 2·3
1 2 2 3
3
1
1
1 1 1 1 1 1
1
1
+
+
= − + − + − =1− .
S3 =
1·2 2·3 3·4
1 2 2 3 3 4
4
S1 =
Sn =
=
1
1
1
1
1
+
+
+ ··· +
+
=
1·2 2·3 3·4
(n − 1)n n(n + 1)
1
1
1
1
1
1 1 1 1 1 1
− + − + − + ··· +
− + −
=1−
.
1 2 2 3 3 4
n−1 n n n+1
n+1
lim Sn = lim
n→+∞
n→+∞
1−
!"#$
1
n+1
= 1 − lim
n→+∞
1
= 1.
n+1
2 + 6 + 18 + · · · + 2 · 3n−1 + . . . .
% &
'
S1 = 2, S2 = 2 + 6 = 8,
S3 = 2 + 6 + 18 = 26, . . . ,
(
Sn = 2 + 6 + 18 + · · · + 2 · 3n−1 .
)
S1 = 2 = 3 − 1, S2 = 8 = 32 − 1, S3 = 26 = 33 − 1, . . . , Sn = 3n − 1.
lim Sn = lim (3n − 1) = +∞.
n→+∞
!"#*
n→+∞
1 − 1 + 1 − 1 + · · · + (−1)n−1 + . . . .
|q| < 1
S = 1 b−1 q
|q| > 1 qn → +∞ n → +∞
b1 − b1 q n
= +∞.
n→+∞
1−q
lim Sn = lim
n→+∞
q = 1 !"#$
b1 + b1 + b1 + · · · + b1 + . . . .
% Sn = nb1 b1 = 0 n→+∞
lim Sn = +∞
& q = −1 !"#$
b1 − b 1 + b1 − b1 + . . . .
' Sn = 0 n ( Sn = b1 n ( )
b1 = 0 n→+∞
lim Sn *
|q| < 1
|q| 1
+ , - - )
,.
"#"
u 1 + u2 + u3 + · · · + un + . . .
!"#/
S
au1 + au2 + au3 + · · · + aun + . . . ,
σn
a
!"#0
aS
% 1 Sn n) !"#/
n) !"#0
σn = au1 + au2 + au3 + · · · + aun = a(u1 + u2 + u3 + · · · + un ) = aSn .
23
lim σn = lim aSn = a lim Sn = aS.
n→+∞
n→+∞
n→+∞
!"#0 aS
u 1 + u 2 + u3 + · · · + un + . . . ,
v 1 + v2 + v3 + · · · + vn + . . .
S S̄
(u1 + v1 ) + (u2 + v2 ) + (u3 + v3 ) + · · · + (un + vn ) + . . . ,
S + S̄
n
Sn S̄n σn !"
σn = (u1 + v1 ) + (u2 + v2 ) + (u3 + v3 ) + · · · + (un + vn ) = Sn + S̄n .
# $
% %
lim σn = lim (Sn + S̄n ) = lim Sn + lim S̄n = S + S̄.
n→+∞
n→+∞
n→+∞
n→+∞
!
$ & '
(u1 − v1 ) + (u2 − v2 ) + (u3 − v3 ) + · · · + (un − vn ) + . . .
!"#%$ !"#&$
S − S̄
!"#""$
&
u1 + u2 + u3 + · · · + uk−1 + uk + uk+1 + · · · + un−1 + un + . . .
uk+1 + · · · + un−1 + un + . . . .
(
( !"#"'$
!"#"($ !"#"'$ )
k *) !"#"($
!"#"'$
Sn n % $
Sk ) k *$ (k < n) σn−k
) n − k % $ ("
Sn = u1 + u2 + u3 + · · · + uk + uk+1 + · · · + un ,
Sk = u1 + u2 + u3 + · · · + uk , σn−k = uk+1 + uk+2 + · · · + un .
Sn = Sk + σn−k ,
Sk n
! S n→+∞
lim Sn = S
"# $
lim σn−k = lim (Sn − Sk ) = lim Sn − lim Sk = S − Sk .
n→+∞
n→+∞
n→+∞
n→+∞
% & & σn−k ' n → +∞ (
' !
lim σn−k = σ
' ! σ n→+∞
% $
lim Sn = lim (Sk + σn−k ) = Sk + lim σn−k = Sk + σ,
n→+∞
n→+∞
n→+∞
!
" ' ) * ) ( +
+! !
u1 +u2 +u3 +· · ·+un +. . .
n
un
, ! -
u 1 + u 2 + u3 + · · · + un + . . . ,
( - S . # & &
/(
Sn = u1 + u2 + u3 + · · · + un−1 + un
Sn−1 = u1 + u2 + u3 + · · · + un−1 .
un = Sn − Sn−1
lim un = lim (Sn − Sn−1 ) = lim Sn − lim Sn−1 .
n→+∞
n→+∞
n→+∞
n→+∞
lim Sn−1 = S n → +∞
lim Sn = S
n→+∞
n→+∞
n − 1 → +∞
lim un = S − S = 0
n→+∞
lim un = 0.
n→+∞
n
!
" ! # $ % # #
%
&
! !
1 2 3
n
+ + + ··· +
+ ...
2 3 4
n+1
' ( ) ' "
% n → +∞)
lim un = lim
n→+∞
n→+∞
! # un =
n
n+1
n
1
= lim
= 1.
n + 1 n→+∞ 1 + 1/n
* lim un = 0
n→+∞
# + $# # ! % !
lim un = 0
n→+∞
, ,
1
1
1
1
√ + √ + √ + ··· + √ + ....
-
n
1
2
3
1
lim un = lim √ = 0 . " $ #
n→+∞
n→+∞
n
"
# # %
1
1
1
1
Sn = √ + √ + √ + · · · + √ .
n
1
2
3
1
1
1
1
1
1
/ √ > √ √ > √ √ > √ , . . . # #
n
n
n
1
2
3
1
1
1
1
Sn > √ + √ + √ + · · · + √ ,
n
n
n
n
√
1
Sn > n · √ Sn > n
n
lim Sn = +∞
n→+∞
!
$
" #
un
'
%!
( )
)
!
n→+∞
%
%
lim un = 0
&
!
*+*
un = 2n!
n−1
&
,
1
-
23
1
, +3/*
%!
./0 -
*3/*
4
·
53/* 0 5
03/* 0
· ·
2
+
· · ·...·2
23/* 0 5
$
,
1
2
4
8
2n−1
+ + + + ··· +
+ ··· .
1! 2! 3! 4!
n!
*+0
1+
&
7
8
,
...
9
1 1 1
+ + + ··· .
3 5 7
6
%
%
%!
2 -
2
1
1
*
5
an = a1 + d(n − 1) a1 = 1, d = 2
an = 1 + 2(n − 1) = 2n − 1
1
un =
2n − 1
1+
1 · 4 1 · 4 · 9 1 · 4 · 9 · 16
+
+
+ ··· .
1 · 4 1 · 4 · 7 1 · 4 · 7 · 10
! "
# $ %
n2 " $
&#'
$
& &( '( &
) *a1 = 1, d = 3+ "
)
& ) * '
+ an = 3n − 2
un =
,
1 · 4 · 9 · 16 · · · · n2
1 · 4 · 7 · 10 · · · · (3n − 2)
1
1
1
1
+
+
+ ··· +
+ ....
1 · 12 12 · 23 23 · 34
(11n − 10) · (11n + 1)
! $ #
- '( &. !
(
A\11n+1
B \11n−10
1
=
+
=
(11n − 10) · (11n + 1)
11n − 10
11n + 1
=
A(11n + 1) + B(11n − 10)
.
(11n − 10) · (11n + 1)
/ ' ' ( $
!
1 = A(11n + 1) + B(11n − 10).
A − 10B = 1.
0 ) 12
12A + B = 1.
0 ) 12
%
$ '(
1
1
3 " A = , B = − 4
#
11
11
!
1
1
=
un =
(11n − 10) · (11n + 1)
11
1
1
−
.
11n − 10 11n + 1
u1 =
1
1
1
1−
, u2 =
11
12
11
1
1
1
1
1
−
, u3 =
−
,... .
12 23
11 23 34
1
1
1
1
1
1
1
1
1−
+
−
+
−
+ ···
Sn =
11
12
11 12 23
11 23 34
1
1
1
+
−
=
11 11n − 10 11n + 1
1
1
1
1
1
1
1
1
1−
+
−
+
−
+ ··· +
−
=
11
12 12 23 23 34
11n − 10 11n + 1
1
1
1−
.
=
11
11n + 1
1
1
1
1−
=
S = n→+∞
lim Sn = lim
n→+∞ 11
11n + 1
11
S = 111
2 1 1
1
1
+ + +
+
+ ··· .
3 3 6 12 24
! "! ! b1 = 32 , q = 21 , "# $ %
!
b1
=
S=
1−q
&
2
3
1
1−
2
+∞
,
n=1
5n + 1
.
4n − 1
4
= .
3
lim un = lim
n→+∞
n→+∞
5n + 1
5
= = 0.
4n − 1
4
1
1 1
+ ··· .
1− + −
4 9 16
2 3 4
2
3
4
5
+
+
+
+ ··· .
3
7
11
15
!
1
1
1
1
+
+
+ ··· +
+ ··· .
1·3 3·5 5·7
(2n − 1)(2n + 1)
1
1
1
1
+
+
+ ··· +
+ ··· .
1·2·3 2·3·4 3·4·5
n(n + 1)(n + 2)
1+
1 1 1
+ + + ··· .
2 4 8
"
+∞
,
n=1
n
.
3n − 1
S = n→+∞
lim Sn
!"
# ! $ $
Sn !" n %
! &
$ '(
#
) (
** *
+ (' )
S1 = u1
S2 = u1 +u2 S3 = u1 +u2 +u3 . . . Sn = u1 +u2 +u3 +· · ·+un (
n ) !
! ( ('( (
S 1 < S2 < S 3 < · · · < S n < . . . .
,
* + #
n→+∞
lim Sn = +∞
- + Sn < C
(! n #
) !
** * !
"
u 1 + u 2 + u3 + · · · + un + . . . ,
(U )
v 1 + v 2 + v3 + · · · + vn + . . . .
(V )
u1 v1 , u2 v2 , u3 v3 , . . . , un vn , . . . ,
!
U "
"
Sn
σn
n
Sn = u 1 + u 2 + u 3 + · · · + u n , σ n = v 1 + v 2 + v 3 + · · · + v n .
$
'
lim σn = σ
n→+∞
! !
!
U
Sn σ n
"
% & !
σn < σ !
# !
&
!
U
!
Sn < σ
!
!
V !
Sn < σ
!
u 1 + u 2 + u3 + · · · + un + . . . ,
(U )
v 1 + v 2 + v3 + · · · + vn + . . . .
(V )
$ !
u1 v1 , u2 v2 , u3 v3 , . . . , un vn , . . . ,
"
( " #
# !
#
V
Sn
(
!
σn
Sn = u 1 + u 2 + u 3 + · · · + u n , σ n = v 1 + v 2 + v 3 + · · · + v n .
(
!
Sn σn " V #
lim σn = +∞ *
) !
lim Sn = +∞
n→+∞
!
n→+∞
!
U
#
|q| < 1 |q| 1
1
1
1
1
+
+
+ ··· + p + ...
!!"#
1p 2p 3p
n
p > 1 p 1 p = 1
1 1 1
1
+ + + ··· + + ...,
1 2 3
n
!!$#
% !!"#
&
& '
!!!
1
1
1
1
+ 3 + 4 + ··· +
+ ....
2
2
3
4
(n + 1)n+1
!!(#
% ) * %
1
1
1
1
+
+
+ · · · + n+1 + . . . .
22 23 24
2
!!+#
% (11.6) & &
q = 1/2 < 1 ,
(11.5) & (11.6)
(11.5) -
!!.
√
1
1
1
1
+√
+√
+ ··· +
+ ....
ln 2
ln 3
ln 4
ln(n + 1)
!!/#
% ) * %
1
1
1
1
√ + √ + √ + ··· + √
+ ...,
n
+1
2
3
4
!!0#
(11.7)
(11.8)
√
√
1
1
>√ ,
ln n < n, ln n < n, √
n
ln n
(11.7)
! ! " !
#
$$ %
Un
lim
= k
n→+∞ Vn
n → +∞
$$ % "
+∞
,
#
un = 1 +
n=1
!
# $
1
1 1
+ + ··· +
+ ....
3 5
2n − 1
) )! ! !
1
vn = * ! *
&$$ '(
n! !
n
1
un
2n
− 1 = 1 = 0.
lim
= lim
1
n→+∞ vn
n→+∞
2
n
+ , - *
* ! * &$$ '(
. !
! !
./! ! *
! !
!
# '
¾
$$ 0
% ¾
u 1 + u2 + u3 + · · · + un + . . .
&
&$$ $1(
n
un+1
= ρ,
un
ρ > 1
lim
n→+∞
ρ < 1
#
ρ < 1
un+1
= ρ
n→+∞ un
lim
N
n N
"
#
!
ε
ε > 0
un+1
− ρ < ε $
un
un+1
un+1
− ρ < +ε, ρ − ε <
< ρ + ε.
un
un
un+1
! ρ + ε = q
< q % ρ
un
& '# ε ε #
# # q = ρ + ε < 1 % n N (
uN+1
uN+2
uN+3
< q,
< q,
< q, . . . ,
uN
uN+1
uN+2
−ε <
uN+1 < uN q, uN+2 < uN+1 q < uN q 2 , uN+3 < uN+2 q < uN q 3 , . . . .
) (
uN + uN+1 + uN+2 + uN+3 + . . . ,
2
*
3
uN + uN q + uN q + uN q + . . . .
) + !
|q| < 1
%
+
!
# *
"
+ !
*
, *
!
! -
u1 + u2 + u3 + · · · + uN−1
+ -
#
ρ > 1
un+1
lim
= ρ > 1 $
n→+∞ un
& n N #
/
.
un+1
> 1
un
un+1 > un
n
lim un = 0
n→+∞
lim
n→+∞
n→+∞
lim un = 0
n
un+1
= +∞
un
un+1
>1
un
!
! "
" ρ = 1 !
# $
$
#
$ %
%
"
1
3
5
7
2n − 1
+
+
+
+ ··· +
+ ....
3 32 33 34
3n
# & '
(
2(n + 1) − 1 2n − 1
un+1
3n (2n + 1)
=
lim
=
ρ = lim
= lim
:
n→+∞ un
n→+∞
n−to+∞ 3n+1 (2n − 1)
3n+1
3n
1
2n + 1
2 + 1/n
1
1
lim
=
lim
= .
3 n→+∞ 2n − 1
3 n→+∞ 2 − 1/n
3
ρ = 1/3 < 1
=
)
%
*
# & '
2
4
8
2n
+
+
+ ··· + 4 + ....
1 16 81
n
(
2n+1
un+1
2n+1 n4
2n
=
lim
= lim
:
=
n→+∞ un
n→+∞ (n + 1)4
n−to+∞ (n + 1)4 · 2n
n4
ρ = lim
1
n4
= 2 lim
= 2.
n→+∞ (1 + 1/n)4
n→+∞ (n + 1)4
ρ = 2 > 1
= 2 lim
n
a
un = k (a > 1, k > 1)
n
n+1
a
un+1
an+1 · nk
an
lim
= lim
:
= a.
=
lim
n→+∞ un
n→+∞ (n + 1)k
n→+∞ an · (n + 1)k
nk
n a > 1 ! " #
a
un = k " $ %
n
&
" ρ = 1 '
! # ( &
)
1
1
1
1
√ + √ + √ + ... + √ + ....
n
1
2
3
*
1
1
un+1
:√
=
= lim √
n→+∞ un
n→+∞
n
n+1
√
n
= lim √
= 1.
n→+∞
n+1
ρ = lim
* " #
! +
, & # ρ = 21 < 1 -./
0 # '
! " ," & 1 "&2
3 ¿
u 1 + u2 + u3 + · · · + un + . . .
¿
lim
√
n
n→+∞
un = q
q 1 q = 1
+∞
,
un =
n=1
n
+∞
,
1
1
·
1
+
.
2n
n
n=1
n
√
1
1
1
1
1
· 1+
= < 1.
lim n un = lim n n · 1 +
= lim
n→+∞
n→+∞
n→+∞ 2
2
n
n
2
!
" # #$ !
% &
' & (
"
&
+∞
,
n=2
un =
ln n
ln 2 ln 3
+
+ ··· +
+ ··· .
2
3
n
) & !
* & !
1
vn = " un + (#&
n
+∞
. 1
ln n
1
> -
vn ,
n → +∞
!
n
n
n=1 n
.
+∞
,
n=0
un =
+∞
,
n=0
1
.
4 · 2n − 3
+∞
.
n=0
+∞
.
1
n
2
n=0
vn =
1
q =
2
!"
1
n−3
un
2n
1
1
4
·
2
= lim
lim
= lim
= lim
= .
3
1
n→+∞ vn
n→+∞
n→+∞ 4 · 2n − 3
n→+∞
4
4− n
2n
2
#
%
+∞
.
n=0
un =
+∞
.
n=0
1
4 · 2n − 3
q =
+∞
.
n=0
1
2
1
2n
$
&
''(
+∞
,
un =
n=0
+∞
,
n=1
2n2
1
.
− 3n
+∞
.
+∞
.
1
vn =
2
n=1
n=1 n
)''(* !" !
1
2 − 3n
un
1
2n
= lim
= = 0.
lim
1
n→+∞ vn
n→+∞
2
n2
#
+∞
.
n=1
un =
'',
+∞
,
n=1
+∞
.
n=1
1
2n2 − 3n
um =
+∞
,
1
sin .
n
n=1
+∞
.
n=1
vn =
+∞
.
n=1
1
n2
+
=
+∞
.
1
n
n=1
+∞
.
n=1
vn =
1
sin
un
sin m
n
= 1 = 0.
lim
= lim
= lim
1
n→+∞ vn
n→+∞
m→ 0 m
n
+∞
+∞
.
.
1
! un = sin
n
n=1
n=1
""#
+∞
,
n=1
2 · 5 · 8 . . . (3n − 1)
.
1 · 5 · 9 . . . (4n − 3)
$ n + 1 %
un+1 =
2 · 5 · 8 . . . (3n − 1)(3(n + 1) − 1)
=
1 · 5 · 9 . . . (4n − 3)(4(n + 1) − 3)
3n + 2
2 · 5 · 8 . . . (3n − 1) · (3n + 2)
= un
.
1 · 5 · 9 . . . (4n − 3) · (4n + 1)
4n + 1
un+1
3n
3
= < 1
lim
= lim
n→+∞ un
n→+∞ 4n + 1
4
&%
""'
+∞
,
n!
n=1
(
en
!
.
un+1
(n + 1)!/en+1
(n + 1)! ·en
= lim
=
lim
=
n→+∞ un
n→+∞
n→+∞
n!/en
n! · en+1
lim
= lim
n→+∞
n+1
= +∞ > 1.
e
!
)
+∞
,
n=1
n+1
2n − 1
n
.
-
n
√
n+1
n+1
1
n
n
= < 1.
lim
un = lim
= lim
n→+∞
n→+∞
n→+∞ 2n − 1
2n − 1
2
+∞
.
1
.
n=2 ln n
+∞
. 2n
.
n
n=0 5 + 1
+∞
.
1
.
2
n=2 n ln n
+∞
.
1
tg .
n
n=1
10n
.
n=1 n!
+∞
.
n
.
n/2
3
n=1
+∞
.
+∞
.
n=1
2n
3n + 1
n
.
! "# $ " !
% &
' $ " "% ! " "
&
(% # ! ) *
" ) "% "% " (
! #
+ !
& !
1
1
1
1
1
1
1
1
1
1
− − + + − − + + − · · · + (−1)n(n−1)/2 2 + . . . .
12 22 32 42 52 62 72 82 92
n
!
!
"# u1, u2, . . . , un, . . . #
! $
! # %
u1 − u2 + u3 − u4 + · · · + (−1)n−1 un + . . . .
&'( ')
* !
+!#
'( '
u1 > u2 > u3 > · · · > un > . . .
n→+∞
lim un = 0
! " # ! $ % ! ! #
* ,
S2m = u1 − u2 + u3 − u4 + · · · + u2m−1 − u2m .
-
%
S2m = (u1 − u2 ) + (u3 − u4 ) + · · · + (u2m−1 − u2m ).
.
# #
# S2m
m
/ $ S2m # %
u1 − [(u2 − u3 ) + (u4 − u5 ) + · · · + (u2m−2 − u2m−1 ) + u2m ].
- ! # ! 0 1
S2m < u1 # m . #
S2m m
1 ! - S2m !
|u1 | + |u2 | + |u3 | + · · · + |un | + . . . ,
u1 + |u1 | u2 + |u2 |
un + |un |
+
+ ··· +
+ ... .
2
2
2
!
un > 0 : |un | = un
un < 0 : |un | = −un
"
un + |un |
|un | + |un |
=
= |un |;
2
2
un + (−un )
un + |un |
=
= 0.
2
2
#
% &
" '
" $
% (# ) %
*+ % &
) % &
1
2
$
|u1 | |u2 |
|un |
+
+ ··· +
+ ... .
2
2
2
,&
, # %
u2 + |u2 | |u2 |
un + |un | |un |
u1 + |u1 | |u1 |
−
+
−
+· · ·+
−
+. . . .
2
2
2
2
2
2
-
+ %
. )
2·
/
)+ %
un
un + |un | |un |
−
=2·
= un .
2
2
2
% %
+ %
1
1
1
1
1
1
−
−
+
+
−
−... .
12 22 32 42 52 62
0
1
1
1
1
+
+
+
+ ... .
12 22 32 42
p = 2 > 1 !
"
$ !
# #
1 1 1
1
+ − + · · · + (−1)n−1 · + . . . ,
2 3 4
n
%&' ()
# *+ ,-#
1−
1+
1
1 1 1
+ + + ··· + + ...,
2 3 4
n
%&' ()
!
.
%&' /) %&' ()
%&' /)
%&' ()
0 1 #
&' & u1 +u2+u3 +· · ·+un+. . .
|u1 | + |u2 | +
+ |u3 | + · · · + |un | + . . .
2
&' ' u1 +u2+u3 +· · ·+un+. . .
|u1 | + |u2 | +
+ |u3 | + · · · + |un | + . . .
! " # $
!
% ! ! &
' ( &
) ! $ * )
!&
' ) + $ &
$ $
, ! &
- !
$ ! $
$ !)
. $ &
$
/ $ !) "
1 1 1 1 1 1 1 1
+ − + − + − + − ...,
2 3 4 5 6 7 8 9
S
1−
$ $ 0
1 2 0 0 &
0 3 ! 1 $
1 1 1 1 1 1
1
1
1
−
+ − ... .
1− − + − − + −
4
2 4 3 6 8 5 10 12 7
* $ " Sn 4 5
σn 60
1
7
1
1 1 1
= , S4 = 1 − + − = ,
2
2
2 3 4
12
37
1 1 1 1 1
S6 = 1 − + − + − = , . . . ;
2 3 4 5 6
60
1
7
1 1
1 1 1 1 1
σ3 = 1 − − = , σ6 = 1 − − + − − = ,
2 4
4
2 4 3 6 8
24
S2 = 1 −
1
1
1
37
7
+ −
−
=
,....
24 5 10 12
120
σ3 = 0, 5S2 , σ6 = 0, 5S4 , σ9 = 0, 5S6 , . . .
σ3m = 0, 5S2m lim S2m = S lim σ3m =
σ9 =
m→+∞
= 0, 5 lim S2m = 0, 5S
m→+∞
m→+∞
!"
0, 5S
#
lim σ3m+1 = lim
m→+∞
m→+∞
lim σ3m+2 = lim
m→+∞
m→+∞
σ3m +
1
2m + 1
= 0, 5S
1
1
−
= 0, 5S.
σ3m +
2m + 1 4m + 2
lim σn %
$ n→+∞
n & !"
' ( ) *"
) +
, +
-"
. ) S
Sn = u1 +u2 +u3 +· · ·+un n → +∞ S = n→+∞
lim Sn
'( / n 0
S ≈ Sn ,
1"
) n # 2
0 ) 1" )
3
S n
u1 + u2 + u3 + · · · + un + un+1 + un+2 + . . . .
Sn n
n n
! " # "
|u1 | + |u2 | + |u3 | + · · · + |un | + . . .
$
% n $&
rn = un+1 + un+2 + un+3 + . . . ; rn = |un+1 | + |un+2 | + |un+3 | . . . .
p &
|un+1 + un+2 + · · · + un+p | |un+1 | + |un+2 | + · · · + |un+p |.
' p → +∞# "
lim |un+1 + un+2 + · · · + un+p | lim |un+1 | + |un+2 | + · · · + |un+p |,
p→+∞
p→+∞
|rn | # "
( r3
rn |
sin 1 sin 2 sin 3
sin n
+ 2 + 3 + ··· + n + ....
2
2
2
2
% ) & * # # #
sin 1 > 0, sin 2 > 0, sin 3 > 0, sin 4 < 0, sin 5 < 0, sin 6 < 0, sin 7 > 0, . . . .
%
sin 1 sin 2 sin 3
+
+ · · · + sin n + . . . .
+
2 22 23
2n
sin n
1
2n 2n # , "
+
- " , " ,
1
1
1
1
+
+
+ ··· + n + ....
2 22 23
2
'
. " , # ,
" # , " , " r3, r3 , r3#
|r3 | < r3 < r3
|r3 | < r3 =
1
1
1
1
1
1/24
= .
+ 5 + 6 + ··· + n + ··· =
4
2
2
2
2
1 − 1/2
8
!"# $
"
!"
" %
& ' ' ( (
1
1
1
1
− + − · · · + (−1)n−1
+ ....
1! 3! 5!
(2n − 1)!
!
"
ΔS = |S − Sn | = |rn | un+1 .
#$ % &# & ' ((
& & % ## %% |rn | un+1 0, 01 #
1
0, 01.
(2n + 1)!
)
%%
%
# & *+
1
1
S ≈ S2 = − ≈ 1 − 0, 17 = 0, 83.
1! 3!
) ' '
1−
1
1
1
1
1
− 2 + 3 − 4 − 5 + ··· .
2 2
2
2
2
, %
1+
- #&
#' %#& &# %
1
1
1
1
1
+
+
+
+
+ ··· .
2 22 23 24 25
.
&
%'/
&
1
q = < 1 # %# 2
%
% #
0
+∞
,
(−1)n+1
5n + 6
n=1
.
! " #
$ % &
1
= 0,
• ' lim |un | = 0
n→+∞
lim
n→+∞
5n + 6
1
1
1
>
>
> ··· .
11
16
21
• ' |un | |un+1 |
( $ % &
(
1
" un =
#
"
(−1)n+1
un =
5n + 6
5n + 6
)
+∞
,
(−1)n−1 · n
n=1
10n + 9
.
*
n
=
% & n→+∞
lim |un | = 0 * $ lim
n→+∞
=
10n + 9
1
= 0
10
( $
+
+∞
,
n=1
(−1)n+1
3 · 5 · 7...(2n + 1)
.
2 · 5 · 8...(3n − 1)
, *
"
3 · 5 · 7...(2n + 1) · (2n + 3)
|un+1 |
2n + 3
2
2 · 5 · 8...(3n − 1) · (3n + 2)
= lim
= lim
= < 1.
lim
n→+∞ |un |
n→+∞
n→+∞ 3n + 2
3 · 5 · 7...(2n + 1)
3
2 · 5 · 8...(3n − 1)
+∞
,
sin nα
.
(ln
10)n
n=1
! " # $ % sin nα & $ %
' | sin nα| 1 ( ' )*
! ++ + vn = (ln 110)n
,!"
lim
√
n
n→+∞
-
vn = lim
n→+∞
n
1
1
1
=
< 1.
= lim
(ln 10)n n→+∞ ln 10
ln 10
- + ) + *
. . (−1)n
+∞
n−1
! " /% *
( % '
' " S = |S − Sn| = |rn| |un+1| 0, 001
n=1
|un+1 | =
0+" #1222
1
1
0, 001 =
⇒ n + 1 1000 ⇒ n 999.
n+1
1000
3 1 − 214 − 314 + 414 − 514 − 614 + · · ·
+∞
.
4
(−1)n+1
.
n=1 ln(n + 1)
2
+∞
.
(−1)n+1 · (n + 1)
.
(n3 + 1)
n=1
+∞
.
(−1)n
n=1
+∞
.
3n + 1
2n + 1
n
.
1 · 4 · 7...(3n − 2)
.
7 · 9 · 11...(2n + 5)
+∞
.
1
(−1)n+1 1 + n .
10
n=1
(−1)n
n=1
+∞
.
(−1)n n2−n .
n=1
+∞
.
n
n
n=1 (2n + 1) · 5
+∞
.
cos 5n
.
n+1
5
n=1
! "
#$%
$ & #$% y = f (x)
M0 (x0 , y0) !
'($ M0 (x0, y0 ) &$ )# $ #$*
% +
& $ $ &*
$ , " "
& $ " $ " (" &"," )#
#$% - ./0 , $ $ M(x, y) " M0
M $$ 1 " " " $ M M0 & + "
x−x0 = Δx → 0. ' M0 + $ ($
& $) M0 " + 23
454
M0 (x0 , y0 )
M0 M
M(x, y) M0 (x0, y0 )
MP
=
M0M k = M
0P
= tg ϕ
Δy
.
kT = tg α = lim
Δx→0 Δx
!
M0(x0, y0) "
# $ y = f (x)
M0 (x0, y0) %
& !
" ! ! ' s = s(t). ( t
y
T
N
M(x,y)
M 0(x0 ;y 0)
ϕ
O
P
α
x
x0
x
s(t)
t+ Δt s(t+ Δt).
Δt Δs = s(t + Δt) − s(t).
v
Δt
v =
Δs
Δt
!
" # t $ # %
Δt # & %
' Δt.
v
v Δt → 0
v = lim
Δt→0
Δs
.
Δt
t
!
(# ) $ %
)*
' $ +
'*
* , )* )
#) &) )%
-. . $
'
(a; b) y = f (x).
x ∈ (a; b).
y = f (x). x
Δx Δx y
Δy. ! "
#
Δy
Δy → 0
Δx → 0. $ % & Δx Δx → 0
'" 00 .
Δy
Δx→0
lim
Δx
() ) y = f (x) x
Δy
Δx
Δx → 0.
f (x).
*%
y
f (x) = lim
= f (x)
x
Δy
f (x + Δx) − f (x)
= lim
.
Δx Δx→0
Δx
f (x) "
Δx→0
, $
dy df (x)
,
.
dx
dx
# x0
() +
$ -
y , y (x),
! %
-
$
dy
df (x0 )
|x=x0 =
.
dx
dx
() .
* $
"%
# $ , %
y = y(x) y (x) $ yx % x = x(y)
x(y) $ xy
/ % 0
# # #( 1 # 2 #
$# x0 " # 2# #
x0 () (
y |x=x0 = y (x0 ) = f (x0 ) =
t s
t
y = x2.
! " #$
% &' Δy "
Δy = (x + Δx)2 − x2 = 2xΔx + Δx2 .
% Δx $ Δx → 0,
Δy
2xΔx + Δx2
= lim
= 2x.
f (x) = lim
Δx→0 Δx
Δx→0
Δx
( )% &'*
+* &' y = x2
x = 0, 5 '
Δy
( * lim
%
Δx→0 Δx
) Δx , * * &
x0
Δy
y − y0
= lim
f (x0 ) = lim
.
Δx→0 Δx
Δx→0 x − x0
- $ x0 +
). x0 ), x → x0 − 0
x → x0 + 0 ' /
0
y = f (x) x0
f (x0 + 0) = lim
x→x0 +0
Δy
.
Δx
y = f (x)
x0
Δy
.
x→x0 −0 Δx
f (x0 − 0) = lim
( f (x) %
x0 *
. %
, .* "
Δy
Δy
= lim
.
lim
x→x0 −0 Δx
x→x0 +0 Δx
f (x) x0
+∞ −∞
x0
+∞ −∞
!"# y = f (x)
!"$ y = f (x)
(a; b)
f (a + 0) f (b − 0)
[a; b]
% &
!"! y = f (x)
' ( ) x Δx = 0. *
+ &
Δy =
)
lim Δy = lim
Δx→0
Δx→0
Δy
Δx.
Δx
Δy
lim Δx = f (x) · 0 = 0,
Δx Δx→0
, (
*
-
! . (−∞; +∞) y = |x|
+ x = 0, +
(
f (−0) = −1, f (+0) = 1.
√
3
x
1
y = √
x = 0,
3
3 x2
! ! ! "
! Δx " # ! !
! $ !
(−∞; +∞)
y =
y = C %
y = C !
! ! !$ "
" #
Δy
0
= lim
= 0.
(C) = lim
'()
Δx→0 Δx
Δx→0 Δx
y = xn * !
$ '+
&
!
(xn ) = n · xn−1 .
'(,
- ! '(, !
! " &! ! & ! .
√
1
1
n = : ( x) = (x1/2 ) = 1/2 · x−1/2 = √ ,
'(/
2
2 x
1
1
= (x−1 ) = −x−2 = − 2 .
n = −1 :
'(0
x
x
y = ax
Δy = ax+Δx − ax = ax (aΔx − 1).
%1
!
!! 1 0
ax − 1
= ln a,
lim
x→0
x
Δy
ax (aΔx − 1)
= lim
= ax ln a.
Δx→0 Δx
Δx→0
Δx
3 ! !$ a = e
(ax ) = lim
(ex ) = ex .
'('2
'(''
y = log
Δy = loga (x + Δx) − loga x = loga
a
x
x + Δx
Δx
= loga 1 +
.
x
x
Δx
loga 1 +
x
.
(loga x) = lim
Δx→0
Δx
loga 1 +
Δx
x
1
=
x
Δx
loga 1 +
Δx
x
Δx
x
Δx → 0,
lim
x→0
' (
x
Δx Δx
1
.
= loga 1 +
x
x
loga (1 + x)
1
= loga e =
,
x
ln a
1
.
x ln a
!"#$"%&
1
.
x
y = sin x y = cos x
!"#$"#&
(loga x) =
a=e
(
(ln x) =
Δy = sin(x + Δx) − sin x = 2 sin
2 sin
(sin x) = lim
Δx→0
) *
lim
Δx→0
Δx
Δx
· cos x +
.
2
2
Δx
Δx
· cos x +
2
2
.
Δx
Δx
2 lim cos x + Δx ,
Δx Δx→0
2
2
sin
(sin x) = cos x,
(cos x) = − sin x.
!
" # $
%
2x sin 2x
.
lim
x→0 1 − cos2 x
√
x+9−3
.
& lim
x→0
x x
x
lim
.
x→+∞ 1 + x
1
1
1
+ ... + n−1 .
lim 1 + +
n→+∞
4 16
4
%
!
%
%
%
%
%
+∞
.
n
n(n + 1)
(−1)n
.
'
n=1 (n + 3)!
.
n=1
+∞
.
( ) *
" + ,
lim
x→0
2x sin 2x
lim
x→0 1 − cos2 x
2x sin 2x
4x sin x cos x
x
= lim
lim cos x = 4 · 1 · 1 = 4.
= 4 lim
2
2
x→0
x→0
1 − cos x
sin x x→0
sin x
√
lim
x→0
x+9−3
.
x
√
√
√
x+9−3
x+9−3
x+9+3
= lim
·√
lim
=
x→0
x→0
x
x
x+9+3
√
( x + 9)2 − 32
x+9−9
1
1
= lim √
= lim √
= lim √
= .
x→0 x( x + 9 + 3)
x→0 x( x + 9 + 3)
x→0
6
x+9+3
lim
x→+∞
x
1+x
x
.
x
−x
x
1+x
= lim
=
x→+∞ 1 + x
x→+∞
x
−x
x −1
1
1
1
1+
= lim 1 +
= lim
= e−1 = .
x→+∞
x→+∞
x
x
e
lim
lim
n→+∞
1+
1
1
1
+
+ ... + n−1
4 16
4
.
1 1
1
4 16
1
! " q = n → +∞ "
4
b1
S =
# !
1−q
4
1
1
1
1
b1
= .
+ ... + n−1 =
=
lim 1 + +
1
n→+∞
4 16
4
1−q
3
1−
4
+∞
.
n
.
n(n + 1)
n=1
n
1
= lim
= 1 = 0,
1
n(n + 1) x→+∞
1+
n
lim Un = lim
x→+∞
x→+∞
" % &
lim |Un | = lim
n→+∞
1
=0
(n + 3)!
' (
lim
n→+∞
(−1)n
.
n=1 (n + 3)!
|Un | > |Un+1 |.
+∞
.
1
(
n=1 (n + 3)!
Un+1
1/(n + 4)!
1
= lim
= 0 < 1,
= lim
n→+∞ 1/(n + 3)!
n→+∞ n + 4
Un
" ! "#$
+∞
.
! "#$
n→+∞
"
)
#
*"& y = f (x) !
x0
+ ,"& y = f (x) ! x0
- ! .
/+ ,"& x → x0 .
+ lim f (x) = f (x0 )
x→x0
x→0
0 lim ctg2
x
5x
· tg2 .
8
4
2
2
z→a za4 −− az4 .
lim
x+a
1
lim 1 +
.
x→+∞
x−1
(−1)n−1
1 1 1
lim 1 − + − + · · · +
.
n→+∞
2 4 8
2n−1
+∞
.
√
n=1
n
.
5n − 1
. √(−1)
.
2n + 9
n=1
+∞
n
3
#
#
! " # $
% &$ ' ! (
%
)' *
+ '
% '
% , % ' '
u(x) v(x)
(u + v) = u + v .
- .
/ % * / x % y(x) = u(x) + v(x)
+ Δx &
Δy = (u(x + Δx) − u(x)) + (v(x + Δx) − v(x)) = Δu + Δv.
0 *
Δy
Δu + Δv
Δu
Δv
= lim
= lim
+ lim
= u + v .
y = lim
Δx→0 Δx
Δx→0
Δx→0 Δx
Δx→0 Δx
Δx
u(x) v(x)
(u · v) = u · v + v · u.
! " " x #" $
% Δx.
y(x) = u(x) · v(x)
Δy = (u + Δu) · (v + Δv) − u · v.
&! %! % ' %"
Δy = v · Δu + u · Δv + Δu · Δv.
(! '
(u · v) = lim
Δx→0
v · Δu + u · Δv + Δu · Δv
.
Δx
) ! ' %! % * %
! ! " +
lim
Δx→0
Δu
·v
Δx
= u · v,
Δv
· u = v · u,
Δx→0
Δx
Δu
· Δv = u · 0 = 0.
lim
Δx→0 Δx
lim
,
lim Δv = 0,
Δx→0
## $" #" $ v " %!
(-! ' %" " #""
(c · u(x)) = c u(x) + c · u (x) = c · u (x).
(u − v) = u − v ,
!
!
" #
(−v) = −v .
(u · v · w) = u · v · w + u · v · w + u · v · w .
u(x) v(x)
x v(x) = 0,
u
v
$
Δx.
u v − v u
.
v2
$ % %
Δy =
)!
=
x
&% '
y(x) =
u(x)
v(x)
(
u(x + Δx) u(x)
−
.
v(x + Δx) v(x)
#
u(x + Δx) = u + Δu;
* %
v(x + Δx) = v + Δv;
vΔu − uΔv
u + Δu u
− =
.
v + Δv
v
v(v + Δv)
Δx,
%!
Δy =
+
!
Δx → 0
lim
Δx→0
lim Δv = 0
Δx→0
,
# %!
#
lim
Δx→0
&& ' %
/! , # !
Δu
= u ,
Δx
, # !
Δv
= v
Δx
&% '
v
% -
(
%. & %%
,
" "
! ,0 %!
Cv
C
=− 2 ,
v
v
u
C
=
u
.
C
,
y = tg x
y = ctg x
!
(tg x) =
sin x
cos x
=
1
(sin x) cos x − (cos x) sin x
=
.
2
cos x
cos2 x
" ! #
(ctg x) = −
1
.
sin2 x
$
%! & & ' ( ( & & '
& & & &)*
&&
y = y(x)
+&&& & ( , '
- &. *& -
& ' / ' - x = x(y) & & &
(a; b)
&& & , & y 0 ! & ) x (y),
& ) ) / #& & )*&, & x ( ' - y = y(x) && ) y (x) .
y (x) =
1
x (y)
yx =
1
.
xy
1
2 ' - x = x(y)
&&' & &
& && ' - y = y(x) *& &
&& 3 !& x *& & Δx = 0. 2!
' - y = y(x) *& & Δy = 0. & & #& &&
lim Δy = 0.
Δx→0
2!
Δy
=
Δx
1
1
= .
Δx
x (y)
lim
Δy→0 Δy
arcsin x arctg x
4 ' - x = sin y (y ∈ [−π/2; π/2]) ( ' y = arcsin x.
+
& & (−π/2; π/2). 5. y (x) = lim
Δx→0
x = cos y = 0 0 & &.
yx =
1
1
1
.
=
=
xy
cos y
1 − sin2 y
sin y = x,
(arcsin x) = √
1
.
1 − x2
! " arctg x
yx =
1
1
1
1
= cos2 y =
=
=1:
.
xy
cos2 y
1 + tg2 y
1 + x2
#
arccos x arcctg x
$ % &!% %'( )(* )
) % & ! "
arcsin x + arccos x =
+
&
π
π
, arctg x + arcctg x = .
2
2
" ! , !-& (π/2) = 0 *.
1
(arccos x) = − √
,
1 − x2
(arcctg x) = −
1
.
1 + x2
/
0! y = y(u) u = u(x). 1 y 2 ! " & .
* x, & u 3 &2! %* !& y = y(u) = y(u(x)).
4 * - !'5 6 , - !'(
&
ux
u = u(x)
x y = y(u)
yu
u y = y(u(x)) x
yx
yx = yu · ux .
7
x Δx u y
Δu Δy. Δx → 0
Δu
"
Δy
Δy Δu
=
·
.
Δx
Δu Δx
! u = u(x) !
Δx → 0 Δu → 0.
lim
Δx→0
Δy
Δy
Δu
= lim
· lim
,
Δx Δu→0 Δu Δx→0 Δx
#$% $&'
( #$% $&'
Δu Δx → 0
$% $ y = sin3 x.
) * +
y = u3 , u = sin x, y = (u3 )u · (sin x)x = 3u2 · cos x = 3 sin2 x · cos x.
$% &
) * +
y = sin x3.
y = sin u, u = x3 , y = (sin u)u · (x3 )x = cos u · 3x2 = 3x2 cos x3 .
! , ! #-. '
! / - ! - 0
! # ' - *
! y = sin5 √x 1 ! 0
y x - * +
• √
x2
• sin x2 √
• sin x
/ - 0
u /
, " "
√
u = sin x y = u5 . 3 #$% $&'+
√
√
√
yx = (u5 )u · (sin x)x = 5 sin4 x · (sin x)x .
sin √x
√ x
y = sin u, u = x!
√
√
1
1
(sin x)x = cos u · √ = cos x · √ .
2 x
2 x
"# #
(sin5
√
x)x = 5 sin4
√
1
x · cos x · √ .
2 x
u #
√
# $ %
& '$ ! $
$(
(sin3 x) = 3 sin2 x(sin x) = 3 sin2 x cos x ,
(sin x3 ) = cos x3 · (x3 ) = 3x2 cos x3 .
& ( )* % ' $ % +
' ,
(arcsin 5x) =
1
· 5,
1 − (5x)2
−1
1
·
( arcctg 3x) = √
· 3,
2 arctg 3x 1 + (3x)2
1
1 −1
cos
= − sin · 2 ,
x
x x
tg ln x
1
1
tg ln x
2
,
=2
· ln 2 ·
2 ln x x
cos
√
(ctg3 2x − x2 ) =
√
−1
1
√
· √
· (2 − 2x).
3ctg2 2x − x2 ·
2
2
sin 2x − x 2 2x − x2
# )* %( $ +
#$ $ -
u = u(x).
1. (C) = 0,
√
1
2. (un ) = nun−1 u , ( u) = √ u ,
2 u
3. (au ) = au ln a · u , (eu ) = eu · u ,
u
u
4. (loga u) =
, (ln u) = ,
u ln a
u
5. (sin u) = cos u · u ,
6. (cos u) = − sin u · u ,
u
,
7. (tg u) =
cos2 u
u
8. (ctg u) = − 2 ,
sin u
u
,
9. (arcsin u) = √
1 − u2
u
,
10. (arccos u) = − √
1 − u2
u
,
11. (arctg u) =
1 + u2
u
12.(arcctg u) = −
1 + u2
1
u
= − 2,
u
u
! " # y = f (x)!
$ # " !
# $ " %
" & ! " ! '
# & '!
% " " #
y=2
sin2 x3
· arctg
√
√
tg 3 x
.
x+ 2
ln (x + 3)
√
2 3
2 3
y = 2sin x ln 2 · 2 sin x3 · cos x3 · 3x2 · arctg x + 2sin x
1
1
√ +
1+x2 x
ln(x + 3) √
1
√
−2
tg 3 x
3
2
x3 x
x+3
=
+
ln4 (x + 3)
√
1
2 3
+
= 2sin x 3x2 ln 2 · sin 2x3 · arctg x + √
2 x(1 + x)
√
1
ln(x + 3)
2 tg 3 x
√
+ 3
.
−
√
x+3
ln (x + 3) 3 3 x2 · cos2 3 x
ln2 (x + 3)
1
cos2
√
3
!
y = x3
• f (x + Δx) = (x + Δx)3 ,
• Δy = f (x+Δx)−f (x) = (x+Δx)3 −x3 = 3x2 Δx+3xΔx2 +Δx3 ,
Δy
= 3x2 + 3xΔx + Δx2 ,
•
Δx
Δy
= lim (3x2 + 3xΔx + Δx2 ) = 3x2 .
• y = lim
Δx→0 Δx
Δx→0
√
" y = x.
√
• f (x + Δx) = x + Δx,
√
√
• Δy = f√
(x + Δx) − f (x) =√ x + Δx − x, √
√
√
√
x + Δx − x
x + Δx − x
x + Δx + x
Δy
=
=
·√
•
√ =
Δx
Δx
Δx
x + Δx + x
1
=√
√ ,
x + Δx + x
• y = lim
Δx→0
Δy
1
1
= lim √
√ = √ .
Δx Δx→0 x + Δx + x
2 x
!
"#
$%&'(
%)& y = x5.
" # * y = (x5) = 5x5−1 = 5x4.
√
%)) y = x3.
√
" # * y = ( x3) = (x3/7) = 3/7x3/7−1 = √3 4 .
7
7
7
7 x
%)+ y = x12 .
" # *
y =
1
x2
%), y = x −x22
" # *
=−
y =
1
3
+ 2√ .
2
x
x x
= (x−2 ) = −2x−3 = −
√
x
2
.
x3
.
√
x−2 x
= (x−1 − 2x−3/2 ) = −x−2 + 3x−5/2 =
x2
%)' y = ax−5.
" # * y = (ax−5) = a(x−5) = −5ax−6 = 5a
.
x6
%)- y = √x.
" # * y = ( √x) = (x1/n ) = 1/nx1/n−1 = √1 n−1 .
n
n
n
%). y =
"#
√
3
n x
x.
√
* y = ( 3 x) = (x1/6) = 1/6x−5/6 = √61 5 .
6 x
√
3 5
%)%/ y = x x.
√
16x2 5 x
√
5
.
y = (x3 x) = (x16/5) = 16/5x11/5 =
5
y = sin x + cos x.
y = (sin x + cos x) = (sin x) + (cos x) = cos x − sin x.
y = tgxx .
y =
tg x
x
x
− tg x
2x
(tg x) x − x tg x
x − sin x · cos x
cos
.
=
=
=
2
2
x
x
x cos2 x
! y = ctg x · arccos x.
y = (ctg x · arccos x) = (ctg x) arccos x + (arccos x) ctg x =
arccos x
ctg x
.
=−
−√
sin2 x
1 − x2
"# $% # &
y = log2 x · 2x.
y = (log2 x) 2x + (2x ) log2 x =
$' '
2x
+ 2x · ln 2 · log2 x.
x ln 2
( y = lne x .
x
(ex ) ln x − (ln x) ex
ex (x ln x − 1)
=
.
y = (
ln2 x
x ln2 x
) % %' %* ' #
+ ,
- % & %& &
$
%* ' #
. y = cos3 x.
2
y = 3/ cos
·
(− sin x)
=
01 x2
/ 01 2
$% *
$% *
2
= −3 sin x cos x.
y =
y =
y =
√
tg x.
√
3
1
√
·
2 tg x
/ 01 2
1
1
√
.
=
2x
2 x tg x
cos
2
cos
/ 01 2
arctg x − (arcsin x)3 .
1
1
1
y =
− 3(arcsin x)2 · √
.
3
2 1 + x2
1 − x2
3 arctg x
y = lg sin x.
1
· cos x.
ln 10 · sin x
y = arcctg(ln x) + ln(arctg x).
y =
y = −
1
1
1
1
+
.
1 + ln2 x x arctg x 1 + x2
y = (e5x − ctg 4x)5.
y = 5(e5x − ctg 4x)4 · 5e5x +
4
.
sin2 4x
y = cos e3x .
y = − sin e3x · e3x · 3 = −3e3x sin e3x .
√
y = arctg −x.
y =
1
1
1
√ .
· (−1) = −
· √
1 − x 2 −x
2(1 − x) −x
y = ln
(x − 2)5
.
(x + 1)3
(x − 2)5
3
5
−
.
) = (5 ln(x − 2) − 3 ln(x + 1)) =
y = (ln
3
(x + 1)
x−2 x+1
y = 2arcsin 3x + (1 − arccos 3x)2.
y = (2arcsin 3x + (1 − arccos 3x)2 ) =
1
1
· 3 + 2(1 − arccos 3x) √
· 3.
= 2arcsin 3x ln 2 √
2
1 − 9x
1 − 9x2
y = x1 .
y = cos x.
y = √x.
y = x(1 − x2).
y = sin x + 3 cos x.
y = x arctg x.
sin x
.
y = log
x
3
3
y = arcctg x + x ln x − tgxx .
y = x5 − 4x3 + 2x − 3.
√
y = x2 x2.
+3
.
y = x2 2x
− 5x + 5
x
y=
.
2 + ex
y = arcsin x + arccos x.
3
6
y = √axa2 ++ bb2 .
y = x ctg x.
2
x−x
.
y = (1 + x ) arctg
2
3
y = x3 ln x − x3 .
y = (x2 − 2x + 2)ex.
y = ln(ex − 5 sin x − 4 arcsin x).
1
1
cos(5x2 ) − cos x2 .
20
4
y = arctg ln x.
1
y = ln arcsin x + ln2 x + arcsin ln x.
2
y = −
!" # !$
!
# ! " %
!"
& ' !
% " % !
( !"
!
$
) ' % " *
y = f (x)
ln y = ln f (x).
x, y = f (x)*
1
(ln y)x = (ln f (x))x =⇒ · y = (ln f (x))x,
y
y = y(ln f (x)) = f (x)(ln f (x)).
+ $
, -
y = f (x)
!
" "
. /$( !
% !
" %
! , 0- )%! ) % *
1
n
y = xn =⇒ ln y = n ln x =⇒ y = =⇒ y = (xn ) = nxn−1 .
y
x
u = u(x) v = v(x)
y = u(x)v(x)
!" #
$
y = u(x)v(x) =⇒ ln y = v ln u =⇒
v
v
1
=⇒ y = v ln u + u =⇒ y = (uv ) = uv v ln u + u .
y
u
u
%&% y = (sin x)cos x.
' (
$
y = (sin xcos x ) = sin xcos x
cos2 x
− sin x ln sin x .
sin x
) #
*
# +
+
cos x
%&, y = sintg xx√
' (
$
2
4
√
3
ln x2
arcsin x3
.
√
3
sin xcos x ln x2
√
ln y = ln
= ln sin xcos x +
4
2tg x arcsin x3
√
√
4
3
+ ln 2 ln x − ln 2tg x − ln arcsin x3 =
1
1
1
= cos x ln sin x + ln 2 + ln ln x − tg x ln 2 − ln arcsin x3 =⇒
3
3
4
1 cos2 x
1
ln 2
3x2
√
y =
− sin x ln sin x +
−
−
=⇒
2
y
sin x
3x ln x cos x 4 arcsin x3 1 − x6
√
3
sin xcos x ln x2
√
·
y =
4
2tg x arcsin x3
2
1
ln 2
3x2
cos x
√
.
− sin x ln sin x +
−
−
·
sin x
3x ln x cos2 x 4 arcsin x3 1 − x6
-# " *
# * #
t
y = y(t),
x = x(t).
x = x(t) y = y(t)
t ! "
Δx → 0, Δt → 0.
#
yx
Δy
Δy/Δt
Δy
= lim
= lim
= lim
Δx→0 Δx
Δt→0 Δx/Δt
Δx→0 Δt
3
Δx
y
= t .
Δt→0 Δt
xt
lim
$ % #
#
yx =
yt
y (t)
= .
xt
x (t)
&'()*
'(
x = a(t − sin t),
y = a(1 − cos t),
+ , &'()*
yx
t
t
2 sin cos
a(1 − cos t)
sin t
2
2 = ctg t .
=
=
=
a(t − sin t)
1 − cos t
2
2 t
2 sin
2
! # %
# #
- F (x, y) = 0.
x y
x
y = yx ,
y
x sin y − y 2 ln x = 0.
!
"
(x sin y − y 2 ln x)x = sin y + x cos y · y − 2yy ln x −
# $
y =
y2
= 0.
x
"
2
y − x sin y
.
x(x cos y − 2y ln x)
%
&
'
(
) * T
y = f (x) M0 (x0 ; y0 ) +
,-.
/ 01 L $ 2 0
M0 (x0 ; y0 )
)
k
'"
y − y0 = k(x − x0 ). 3 *
y = f (x)
M0 (x0 ; y0 ),
' )
kT = y (x0 ) = f (x0 ). 4 y −f (x0 ) = f (x0 )(x−x0 )
* T 1 "
y = f (x0 ) + f (x0 )(x − x0 ).
+ '.
5 L1 L2
$
)
! k1 · k2 = −1. 4
)
N M0 (x0 ; y0 )
y = f (x) 1 * kN
1
1
=−
kT
f (x0 )
N M0 (x0 ; y0 )
1
kN = −
*
y = f (x) "
y = f (x0 ) −
1
(x − x0 ).
f (x0 )
+ .
θ λ1 λ2
T1 T2 λ1 λ2 ,
M0 (x0; y0).
y
T2
22
T1
Mo( x o ; y o )
α1
O
θ
α2
θ = α2 − α1 .
tg θ = tg(α2 − α1 ) =
x
tg α2 − tg α1
y2 − y1
=
.
1 + tg α2 · tg α1
1 + y1 · y2 M0
θ = arctg
21
y2 − y1
.
1 + y1 · y2 x=x0
!!
!!
M0 (2; 4).
" # $ x0 = 2, y0 = f (x0) = 4. % !& !'
( ( T $
y = x2
N $
y = f (x0 ) + f (x0 )(x − x0 ) =⇒ y = 4x − 4
y = f (x0 ) −
1
f (x
x 9
(x − x0 ) =⇒ y = − + .
4 2
0)
!) θ
2
λ1 : y = (x − 2)2
λ2 : y = −4 + 6x − x .
√
ds
1
s = t,
v= = √
dt
t
2
Q = a(1 + be−kt ).
dQ
= −abke−kt . !
dt
" Q #$ % " Q = a(1 +
dQ
−kt
+be ) abe−kt = Q − a.
= k(a − Q).
dt
& ' % %% (
' & ) & &
* & "* & + &"* % $
% $ ,* ) $
y = (sin x)arctg x.
ln y = ln(sin x)arctg x =⇒ ln y = arctg x · ln sin x.
#%% ) & '
1 ln sin x
·y =
+ arctg x · ctg x.
y
1 + x2
- . ) & ' y = (sin x)arctg x ,
$
ln sin x
+ arctg x · ctg x .
1 + x2
√
x2 (x − 1)3 2x + 3
√
y=
.
(3x − 4)2 4 3x + 2
y = (sin x)arctg x
/
1
1
ln y = 2 ln x + 3 ln(x − 1) + ln(2x + 3) − 2 ln(3x − 4) − ln(3x + 2) ⇒
2
4
3
2
2·3
3
1 2
+
−
−
⇒
(ln y)x = y = +
y
√x x −1 2(2x + 3) 3x − 4 4(3x + 2)
2
3
3
1
6
3
x (x − 1) 2x + 3 2
√
y =
+
+
−
−
.
(3x − 4)2 4 3x + 2 x x − 1 2x + 3 3x − 4 4(3x + 2)
y2 cos x = a2 sin 3x.
! " # # y $ x
2y · y cos x − y 2 sin x = 3a2 cos 3x ⇒ y =
3a2 cos 3x + y 2 sin x
.
2y cos x
% x3 + y3 − 3axy = 0.
!
" # # y $ x
3x2 + 3y 2 y − 3a(y + xy ) = 0 ⇒ y =
&$
x2 − ay
.
ax − y 2
$ #
x = cos2 t,
y = sin2 t.
' ( ( $ )
yx =
(sin2 t)t
= −1.
(cos2 t)t
( # * # #
( + $
,
f (x) = y = x3 − 2x2 + 3 x0 = 1.
- # x0 = 1 . #
/
$
f (x0 ) = y0 = f (1) = 2,
f (x) = 3x2 − 4x, y (x0 ) = y0 = f (1) = −1.
s
s1 = 100 + 5t s2 = t2/2.
s1 = s2 ⇒ 100 + 5t = t2 /2 ⇒ t2 − 10t − 200 = 0.
t = 20.
ds1
ds2
= 5 !, v2 (t) =
= t ⇒ v2 (20) = 20 !.
v1 (t) =
dt
dt
" # $ v2 − v1 = 15 !.
% ! " # $%&
' & Q = 2t2 +3t+1. (
) *
!' (
N
" J =
dQ
= 4t + 3. & t = 3 J = 15
dt
( + ), y = xsin x.
x3 · sin x
.
) ( + ), y =
ln x · arctg x
* ( yx x3 + ln y − x2 ey = 0.
+ ( yx x2 + y 2 = 1.
( yx + )
⎧
3at
⎪
⎨x =
,
1 + t3
2
⎪
⎩y = 3at .
1 + t3
, T
x2 + 2xy2 + 3y4 = 6 M(1; −1).
− 4x
-
3
2
- y = x3 − x2 −
"++)& .
y = x2 x + y = 2
s t s = 14 t4 − 4t3 + 16t2,
t = t0
t = tk !"
#
s $ t $
! % ! mv2/2 &
s = 1+t+t2 ,
!
" #
$ #%
!
& "
" & y = f (x)
(a; b)' # ( ) *
Δy
.
f (x) = lim
Δx→0 Δx
+
&) # ' %
$
Δy
= f (x) + α(Δx),
Δx
α(Δx) , $ & Δx → 0. "(
!
Δy = f (x)Δx + α(Δx)Δx
. / 0
# $ 1 1' )
1
* Δx (
!
' f (x) ) &- ' '
x' && & $ $ & '
Δx.
Δx = dx,
!
y = f (x) $
dy = y · dx = f (x)dx.
" #
x0
dy = f (x0 )dx.
% y = f (x) x
√
& y = sin x x = π4
' (
√
√
cos x
· dx.
dy = d sin x = ( sin x) dx = √
2 sin x
) x = π/4 dy = √1 dx.
2 2
* !
4
dy
,
dx
y = f (x)
y =
+
x (
,# $
# # ) - +
. #
)- x / . M # y = f (x)
0 1 x 2 Δx. . # x + Δx $
N ) M - MP. 3 $
Δx = dx = MK Δy = N K, y = tg ∠P MK.
- MP K
MK · tg ∠P MK = y dx = P K = dy.
x
x + Δx.
y
MK= Δ x
PK=dy
NK= Δy
N
M
x
O
P
K
x+Δx x
1.dC = 0;
2.d(u + v) = du + dv;
3.d(uv) = v · du + u · dv;
4.d(cu) = cdu;
u
v · du − u · dv
.
v2
u v !
" #! !
5.d
v
=
x
d(uv) = (uv)dx = (u v + v u)dx = v · u dx + u · v dx = v · du + u · dv.
$
dy = ydx
x
!
x ! u"
x
dy = y dx.
y = y(x), x = x(u).
! "
#
$ %& (16.8) (16.10) "'
( & " ' ) *
)
dy = yu · du = yx · (xu · du) = yx · dx.
+ & ' , " *
, '- " & y = f (u(v(x))) "
" ).& " "& &/
dy = f (u)du = f (u)u(v)dv = f (u)u(v)v (x)dx.
0
sin x
.
x
$ % /
x2
e
arcsin2 x3
dex = ex dx2 = ex 2xdx = 2xex dx
dx
2 3
d arcsin x = 2 arcsin x3 d arcsin x3 = 2 arcsin x3 √
dx3 =
1 − x6
2 arcsin x3 2
6x2 arcsin x3
= √
3x dx = √
dx
6
6
1−x
1 − x
sin x
x cos x − sin x
sin x
=
dx =
dx
d
x
x
x2
2
2
2
2
1 " )2& 3& 3*
"2' 0 2 '- *
! "
0 - "' '
2
3
6x arcsin x
√
1 − x6
x cos x − sin x
x2
2xex
2
2xex dx = ex 2x dx = ex dex2 = dex
6x arcsin x
dx3
2 arcsin x3 2
√
dx = √
3x dx = 2 arcsin x3 √
=
1 − x6
1 − x6
1 − x6
x cos x − sin x
dx =
= 2 arcsin x3 d arcsin x3 = d arcsin2 x3 =
x2
sin x
sin x
=
dx = d
x
x
2
2
2
2
2
3
! ! " !
# $ ! % &
! &
$ ! $ %$
! ! & "!
'((
e sinx x
x2
) ex * & x ∈ (−∞; +∞)
sin x
& x ∈ (−∞; 0) ∪ (0; +∞)
x
%$# # & F (x)
sin x
dx $ %
Φ(x) &# dF (x) = ex dx dΦ(x) =
x
% "! &! # #
sin x
ex
+ ,
x
! " ! +, ! & "!
&# #
2
2
2
- $
% %$%
! & ! Δx.
." ! ! &# Δx ! ! / !
Δy ≈ dy.
0'('12
3 !& 0'(42 !
f (x + Δx) ≈ dy + f (x) ≈ f (x) + y dx.
0'('52
! "# " # "
$ sin 29◦.
% & ' () sin 29◦
! y = f (x) = sin x. * +#
! ! #
x = 30◦ = π/6. * x + Δx 29◦ , )
" ,)#
x + Δx =
π
29π
⇒ Δx = −
≈ −0, 01745.
180
180
& )'
sin(x + Δx) ≈ sin x + (sin x) Δx, sin(x + Δx) ≈ sin x + Δx cos x.
-) .) x = π/6 = 0, 52359#
Δx = −0, 01745#
sin 29◦ ≈ sin 30◦ − cos 30◦ · 0, 01745 ≈ 0, 4849.
√
3
/ 8, 05.
%√ & ' () √8, 05 !
y = x x + Δx = 8, 05. 0+) x = 8, Δx = 0, 05.
1 )'
√
√
√
√
√
1
x + Δx ≈ x + ( x) · Δx,
x + Δx ≈ x + √ · Δx.
2
3
3
3
3
3
3
3
3
3 x
-) .) x Δx, #
√
3
8 = 2,
0, 05
3
≈ 2, 0041.
8, 05 ≈ 2 +
3·4
- ) ) ! ! y = f (x)
2 ) ! !
y
f (x)
f (x) = (f (x)).
y = x
5
y = (x5 ) = 5x4 , y = (5x4 ) = 20x3 .
N
n
n − 1.
y (n) = f (n) (x) = (f (n−1) (x)).
n y = sin x
πn
.
y = cos x, y = − sin x, y (3) = − cos x, . . . , y (n) = sin x +
2
! " #
! " # ! " ! " $%
& ' % ##
! ! !
d2 y = d2 f (x) = d(dy) = d(y dx) = (y dx) dx = y dxdx = y dx2 .
% #
( (y dx) )' % dx x
dx dx $ % ) dx2 .
* n! ! ! n
n − 1.
y = sin 3x
dn y = dn f (x) = d(dn−1 y) = y (n) dxn .
"
dy = y · dx = 3 cos 3x · dx,
d2 y = y · dx2 = −9 sin 3x · dx2 ,
d3 y = y (3) · dx3 = −27 cos 3x · dx3 .
tg 46◦
y = f (x) = tg x.
x = 45◦ = π/4. x + Δx 46◦ .
x + Δx =
π
46π
⇒ Δx =
≈ 0, 01745.
180
180
!"#$"%&
Δx
tg(x + Δx) ≈ tg x + 2 .
cos x
' ( ) x = π/4 = 0, 78538 Δx =
= 0, 01745
tg(x + Δx) ≈ tg x + (tg x) Δx,
tg 46◦ ≈ tg 45◦ +
0, 01745
≈ 1 + 2 · 0, 01745 ≈ 1, 0349.
cos2 45◦
tg 46◦ ( ) *+ ",%--$
√
"#$"" 70.
√
√ 70
y = x x + Δx = 70. . x = 64, Δx = 6.
!"#$"%& /
3
3
3
√
√
√
3
x + Δx ≈ 3 x + ( 3 x) · Δx,
' (
0
)
√
3
√
3
70
70 ≈ 4 +
√
√
1
3
x + Δx ≈ 3 x + √
· Δx.
3
3 x2
√
x Δx, 3 64 = 4,
6
= 4, 125.
3 · 16
( 10−3
1"2"$
3 4 ) 4
"#$"2 y = x2√x.
"#$"% y = x arctg x.
"#$"1 y = sinx x .
"#$"- ctg x ln sin x
y = (1 + x2 ) arctg x.
n
y = 1 +1 x .
2 x
y = x 2e
y − 2y + y = ex ?
d3y.
x4
, d4 y.
y =
2−x
y = x2 e−x ,
ln 1, 02.
√
35.
" arctg 1, 05.
!
5
#$ % & %$ & ' (
) *+ ,- * . $ /
0 12 3 *
4 $ 3 50 $
1+
3 2 %$ 0 2 + $3
3 2 %/
$
/
y = f (x)
(a; b)
ξ ∈ (a; b) ! "
# $
% & ξ ' &
$
•
•
f (ξ) = lim
f (ξ + Δx) − f (ξ)
.
Δx
f (ξ) = M
Δx→0
$%& (a; b)
'% ! Δx
! " #
f (ξ) f (ξ + Δx)
( )
− f (ξ)
0 ⇒ f (ξ) 0;
Δx > 0, f (ξ + Δx)
Δx
*
− f (ξ)
0 ⇒ f (ξ) 0.
+
Δx < 0, f (ξ + Δx)
Δx
, ξ %- %
Δx. !% % % (17.2) (17.3)
% f (ξ) = 0. ,
*
y = f (x) :
• [a; b]
• (a; b)
•
f (a) = f (b) = 0.
(a; b) !
"
#
ξ ! f (ξ) = 0.
, $%& ( .
. ! ". ". M m
/ M = m, $%&
% !
% M = m. ,. '0 M
% ,. ' # %
ξ, &0 $%& % 1
( 2 f (ξ) = 0. ,
ξ OX
f (a) = f (b) = 0.
y = f (x) :
• [a; b]
• (a; b)
! "
#
f (b) − f (a) = f (ξ)(b − a).
! " # AB $%
ξ$
yx − f (a)
f (b) − f (a)
f (b) − f (a)
=
, ! : yx = f (a) +
(x − a).
x−a
b−a
b−a
& '! y = f (x) yx ( F (x)
F (x) = y − yx = f (x) − f (a) −
f (b) − f (a)
(x − a).
b−a
,! - F (x) !
*+ !
[a; b] ! # &
. F (x) :
• # [a; b]/
• ,,-! (a; b) 01
f (b) − f (a)
b−a
(a; b), 1 !3
F (x) = f (x) −
!3 !
• F (a) = F (b) = 0
)
2
! f (x)/
4 5 ! # & ! [a; b] .6
" (# ξ . F (ξ) = 0. 7
2 "
f (b) − f (a)
= 0,
F (ξ) = f (ξ) −
b−a
! ! 8
y
B
A
D
0
x
a
c
b
f (b) − f (a)
b−a
! "
y = f (x) c # $ %!
f (ξ) =
AB
# ! & '(%!
)*+ (a; b) !' , - ' . !
/ ,'. '(%! '
! "
# " $ % &
' ( ( [a; b]
) * ! # )
( ! ) +
! " #
, ( f (x) ϕ(x) - [a; b]
* #! ϕ(x) = 0*
(a; b) ! ξ *
f (ξ)
f (b) − f (a)
=
.
0
ϕ (ξ)
ϕ(b) − ϕ(a)
F (x) = f (x) − f (a) −
!"
f (b) − f (a)
· (ϕ(x) − ϕ(a)).
ϕ(b) − ϕ(a)
#
.
$ 00 ∞
∞
%&"'
f (x) ϕ(x)
x0
! "
x → x0
# # " $% &
'# x → x0 '# & &
( " )
f (x)
f (x)
= lim .
x→x0 ϕ(x)
x→x0 ϕ (x)
lim
(%&"%)*
+," - , "
0
0
%&"% *
lim
. , /
x→0
sin 3x
.
x
sin 3x
(sin 3x)
3 cos 3x
= lim
= 3.
= lim
x→0
x→0
x→0
x
x
1
lim
lim
x→0
1 − cos x
.
x2
1 − cos x
(1 − cos x)
sin x
.
=
lim
= lim
x→0
x→0
x→0 2x
x2
(x2 )
lim
!
" #
sin x
(sin x)
cos x
1
1
= lim
= lim cos x = .
= lim
x→0 2x
x→0 (2x)
x→0
2
2 x→0
2
lim
∞
∞
lim
x→+∞
x3
.
ex
x3
3x2
6x
6
= lim x = lim x = lim x = 0.
x
x→+∞ e
x→+∞ e
x→+∞ e
x→+∞ e
lim
! ! " #
$ !
x → +∞
% & ' ( ! ) * +
,
lim
x→+∞
lim
x→+∞
x + sin x
.
x
x + sin x
(x + sin x)
1 + cos x
= lim
.
= lim
x→+∞
x→+∞
x
x
1
lim cos x.
x→+∞
! "
x ! #$
+∞ − ∞
% ! & "
.
' 00 ∞
∞
()$*
lim
π
−0
2
x→
1
− tg x .
cos x
+ , & x → π2 !
- . +∞ − ∞.
/ - & +
1
sin x
1 − sin x
1
− tg x =
−
=
.
cos x
cos x cos x
cos x
x → π2 ! &
-$ 0 ' +∞ − ∞
' 00 . +
lim
π
x→
2
1 − sin x
− cos x
= lim
= 0.
π − sin x
cos x
x→
2
0 · +∞
% ! &
.
' 00 ∞
∞
()$1
lim x ln x.
x→+0
1
ln x
x
= lim
lim x ln x = lim
= − lim x = 0.
1
x→+0
x→+0
x→+0 1
x→+0
− 2
x
x
! !
" #
1
$%#%
+∞
n
lim (1 + axm )b/x , m > 0, n > 0.
x→0
n
y = (1 + axm )b/x ⇒ ln y =
b ln(1 + axm )
.
xn
& ' "
0
.
x → 0# '
0
(
abm
(b ln(1 + axm ))
abmxm−1
lim xm−n =
lim ln y = lim
= lim
=
n
m
n−1
x→0
x→0
x→0
(x
)
(1
+
ax
)nx
n
⎧
⎪
⎪0, m > n,
⎪
⎨ab, m = n,
=
) *# +#,⎪
+∞, m < n, ab > 0,
⎪
⎪
⎩
−∞, m < n, ab < 0.
⎧
e0 = 1, m > n,
⎪
⎪
⎪ ab
⎨
e , m = n,
n
lim (1 + axm )b/x =
*$%#$$x→0
⎪
e+∞ = +∞, m < n, ab > 0,
⎪
⎪
⎩ −∞
= 0, m < n, ab < 0.
e
x→0
& ' ! a = b = 1 m = n ' " '
lim (1 + x)1/x = e.
*$%#$.x→0
0
0
lim xx .
x→+0
y = xx .
x
ln y = ln x = x ln x. !"# ln y $
% " &# ! ' #
lim y = lim xx = 1.
x→+0
x→+0
+∞
0
(
lim (tg x)cos x .
x→π/2−0
y = (tg x)cos x .
ln sin x − ln cos x
ln y = cos x ln tg x =
. ' ln sin x − ln cos x#
1
cos x
1
'
% % x → π/2 − 0.
cos x
)
(ln sin x − ln cos x)
= lim
x→π/2−0
x→π/2−0
1
cos x
*% %% ln y % " &#
lim
lim
+
cos x sin x
+
sin x cos x
·
cos2 x
= 0.
sin x
lim (tg x)cos x = 1.
y=
x→π/2−0
x→π/2−0
ϕ(x) → 0, ψ(x) → +∞ x → x0
lim ϕ(x)ψ(x) = 0
x→x0
0+∞
,- ' # "
ln y = ψ(x) ln ϕ(x) → −∞, y → 0.
y = f (x) = x3 + 4x2 − 7x − 10
[−1; 2]
•
[−1; 2]
• !""
• #$! % " $ % f (−1) = 0
f (2) = 0
& ' ! !
!( "
)
x3 −3x+c = 0
[0; 1].
y = x3 − 3x + c
% [0; 1] ! * +!
! a, b ∈ (0; 1) ⇒ f (a) − f (b) = 0. ,!
(a; b) ! - $ ! ( %!
f (b) − f (a)
=
# f (a) = f (0) = c, f (b) = f (1) = c−2,
b−a
= −2 = 0. .!- % * ! $!
- ! $ %$ ( % [0; 1]
/ 01 * 0 23)1
+ ! ! -%- %
% %- !
4
lim
x→0
ln x
ctg x
∞
∞
.
ln x
(ln x)
sin2 x
= lim
.
= − lim
x→0 ctg x
x→0 (ctg x)
x→0
x
lim
0
+ !- ! -
0
* !
sin2 x
sin x
= lim
· sin x = 1 · 0 = 0.
x→0
x→0 x
x
lim
lim
x→0
lim
x→0
1
1
−
sin2 x x2
ln x
= 0.
ctg x
(
+∞ − ∞) .
lim
x→0
1
1
−
sin2 x x2
= lim
x→0
x2 − sin2 x
x2 sin2 x
!
0
.
0
" # $
! % & % ! $% x4
x2 sin2 x ∼ x4 .
lim
x→0
1
1
−
sin2 x x2
= lim
x→0
#
lim
x→0
x2 − sin2 x
x4
1
1
−
sin2 x x2
= lim
x→0
!
0
.
0
2x − sin 2x
=
4x3
1 − cos 2x
2 sin2 x
1
= lim
= .
2
x→0
x→0
6x
6x2
3
= lim
'
2
lim (cos 2x)3/x (
x→0
1+∞) .
#() #
2
lim ln(cos 2x)3/x = lim
x→0
*
x→0
3 ln cos 2x
tg 2x
= −6.
= −6 lim
x→0 2x
x2
2
lim (cos 2x)3/x = e−6 .
x→0
+
lim (tg x)sin x (
x→+0
00) .
x→1
lim x1/x .
x→+∞
lim ln x ln(x − 1).
x→1
lim (1 − x) tg
πx
.
2
! " #
'
$ #%
& ' (
$ #
u1 (x) + u2 (x) + ... + un (x) + ... =
+∞
,
un (x).
) * +
n=1
*
! " x
" #
,
! x # %
' x0 %
# !& # ' '
!
*- $ x = x0
%& " %
#
*. ' ( % %
# " ! # %
/ '
& # n
'
n
,
Sn (x) =
uk (x) = u1 (x) + u2 (x) + ... + un (x)
) *-+
k=1
! # ! x& # 0!
x
Sn(x) n → +∞ ! " "
Sn(x)
S(x)
S(x)−Sn(x) n
+∞
.
rn(x) =
uk (x)
# " $ $
$
lim un (x) = 0
%&' ()
n→+∞
"
lim rn (x) = 0.
%&' *)
n→+∞
$
% && *) $
$ %&' &)+
k=n+1
u (x)
n+1
lim
< 1.
n→+∞
un (x)
%&' ,)
! x %&' ,)
-
$ "
. $ "
%&' ,) " "
u (x)
n+1
lim
%&' /)
= 1.
n→+∞
un (x)
! %&' /)
0 %&' &)
&' &
+∞
,
n=1
1
.
n(x + 3)n
un =
1
1
, un+1 =
.
n(x + 3)n
(n + 1)(x + 3)n+1
u
n(x + 3)n
1
n+1
< 1.
lim
= lim
= lim
n→+∞
n→+∞ (n + 1)(x + 3)n+1
n→+∞ |x + 3|
un
|x + 3| > 1
• x = −2 :
⇒ x > −2 x < −4
x = −2 x = −4 :
+∞
+∞
.
. 1
1
=
n
n(x
+
3)
n=1
n=1 n
+∞
.
+∞
.
1
1
=
n
n
n=1 n(x + 3)
n=1 n(−1)
!
" " #$%
& $ &
• x = −4 :
+∞
,
1
,
n
n=1
!
+∞
.
1
' $ $
n
n=1 n(x + 3)
x ∈ (−∞, −4] ∪ (−2, +∞)
()*
+∞
.
√
n 3 cosn x.
n=1
√
√
3
un = n 3 cosn x un+1 = (n + 1) cosn+1 x
3
(n + 1)√
un+1
√
cosn+1 x
√
= lim
lim
= lim | 3 cos x| .
3
n
n→+∞ un
n→+∞
n→+∞
n cos x
+ " %& " x xk = kπ
k = 0, ±1, .. . %& k " &
+∞
. √
• " , k
n 3 cosn xk = 1 + 2 + 3 + ...
•
k=1
+∞
.
" , k
k=1
√
n 3 cosn xk = −1 + 2 − 3 + ... + (−1)n n + ... .
+∞ √
. n cosn x
n=1
xk = kπ, k = 0, ±1, ...
n
+∞
.
5−x
1
3n + 2 8x − 3
3
n=0
! " #$ % &
' ()
un+1 (x)
< 1.
lim
n→+∞ un (x)
" $
' ()
n
n+1
5−x
5−x
1
1
un (x) =
, un+1 =
3n + 2 8x − 3
3n + 5 8x − 3
un+1 (x)
= lim 3n + 2 5 − x < 1.
lim
n→+∞
n→+∞
un (x)
3n + 5 8x − 3
*
lim
n→+∞
3n + 2
= 1,
3n + 5
5−x
8x − 3 < 1
$!
⎧
5−x
⎪
⎪
> −1,
⎪
⎨ 8x − 3
⎪
⎪
⎪
⎩ 5 − x < 1.
8x − 3
⇒
−1<
⎧
5−x
⎪
⎪
+ 1 > 0,
⎪
⎨ 8x − 3
⎪
⎪
⎪
⎩ 5 − x − 1 < 0.
8x − 3
5−x
< 1.
8x − 3
⇒
⎧
7x + 2
⎪
⎪
> 0,
⎪
⎨ 8x − 3
⎪
⎪
⎪
⎩ 9x − 8 > 0.
8x − 3
+
, $
2 4 8
, +∞ -$
x ∈ −∞, − 7
9
& $.&
$.$ %!
8
⇒
9
+∞
.
1
.
3n
+2
n=0
n
+∞
. (−1)
2
⇒
• x = − ⇒
7
n=0 3n + 2
• x =
x∈
2 5 8
, +∞
−∞, −
7
9
!! "
" ! ! "#
! " $ ! %
! % ! !
!
&' (
D ! "
# $ % $
# $ # $ "
! "
) ! * ! #
% ! +* ! #
% +* ) ! #
, !
- !
* " ! +*
!
&' & & '
D # # $ %
&' ( ( $ . sinn2nx #
n=1
# x ∈ (−∞, +∞)
+∞
sin nx
2
n
1
x
n
+∞
. 1
!
2
+∞
.
sin nx
n2
n=1
n=1
2
n
x!
!"
D
!# ! "
#$
D
S(x) %
S (x)
% & % %
' % D
$% & '(
) * + ' !
!, (
+∞
,
an (x − x0 )n = a0 + a1 (x − x0 ) + ... + an (x − x0 )n + ...,
n=0
% #
(x − x0) #$
a1 an , . . . #
- x0 = 0, ) ! *
+∞
,
n=0
) ! *
a0 a1 an
n )
an xn = a0 + a1 x + ... + an xn + ...,
) !.*
' / !
0
1 ! 1
%' / %
) !.* ' %'
1
2
-R
0
3
R
n+1
un+1
= lim an+1 x = |x| lim an+1 < 1.
lim
n
n→+∞
n→+∞
n→+∞
un
an x
an
a
n
|x| < lim
= R.
n→+∞ an+1
an
R = n→+∞
lim
an+1
|x| < R
|x| > R (−R; R)
! " x = ±R #$%
$ &$ ' "$( $ & " ##( & & %
#$ $ ) x = −R x = R * & ##
) $ # $ (−R; R) $ +$
# #( " $ ( $ #( # #, %
+ [−R; R] ! - ' " # $ (−R; R) )
*#
" $ # *' & ##' $ ) & x = 0
. " # R = 0 .
- & *( x ∈ (−∞; +∞) " "
R = +∞
n ! ! "
+∞
.
√x n .
n=1
xn
xn+1
1
1
un = √ un+1 = √
an = √ an+1 = √
n
n
n+1
n+1
√
an
n+1
= lim
√
R = lim
=1
n→+∞ an+1
n→+∞
n
(−1; 1)
+∞
+∞
. xn
. 1
√ =
√
• x = 1
+∞
.
1
α
n=1 n
n
n=1
n=1
n
α 1
• x = −1
+∞
.
+∞
. (−1)n
xn
√ =
√
n n=1
n
n=1
!" ! #
$ % % [−1; 1)
&' (
an =
(n + 1)!
=
n!
= lim (n + 1) = +∞.
+∞
.
xn
.
n=1 n!
1
1
an+1 =
R =
n!
(n + 1)!
lim
n→+∞
an
=
an+1
lim
n→+∞
n→+∞
$ % % x ∈ (−∞; +∞)
! ) %
(n + 1)! = n!(n + 1)
* + %
&' ,
(−R; R) r < R !
" # $ [−r; r]
# %
• & ' ' $
( (
$ ( $$ " (
) " "
$ $( $ (
( ( (
• & " ( *$
+ $ $ "
•
an
,
R = lim
n→+∞ an+1
(x0 − R; x0 + R),
un+1
an+1 (x − x0 )n+1
an+1
0 > 0 "
4 x3
0 < x < +∞
• #
√
$%& y = 2x − 3 x2.
' ( ) *
√
2 x−2
2
√
.
• + , y = f (x) = 2 − √ =
x
x
• )
•
4
4
3
3
3
f (x) = 0,
√
23x−2
√
= 0,
3
x
√
3
x = 1,
3
x = 1.
x = 1
- x = 0
x = 0 #
)
−∞ < x < 0;
•
0 < x < 1;
1 < x < +∞.
. / " ,
√ )
2 x−2
0 −∞ < x < 0 y = √x > 0
3
3
2(−1 − 1)
= 4 > 0
y (−1) =
−1
* "
√
0 0 < x < 1 y = 2 √x x− 2 < 0
" < 0
> 0 * " 1
√
0 1 < x < +∞ y = 2 √x x− 2 > 0
> 0
* "
3
3
3
3
x = 0
x −∞; −3 −3; 0
0; +∞
y
0
>0
27
y
− 3
e
!"
# $ y = x3 − 3x2.
# % y = x(1 + √x)
# & y = 1 + √x.
√
# ' y = x2 − x.
3
1
y = e x − x.
# (
3
# (( y = x x+2 4
º
) * +
, - ., .,
+ /0 . 1 .
0 . 2 0
# 3 * *
4 . *5 6
. 7
8 . .
(x − x0 )2k+1
x > x0 :
x > x0 x < x0
f (2k+1) (ξ)
(x − x0 )2k+1 > 0,
(2k + 1)!
x < x0 :
f (2k+1) (ξ)
(x − x0 )2k+1 < 0.
(2k + 1)!
x0
! f (x) < f (x0) x < x0
f (x) > f (x0 ) x > x0 " # $
% f (x)
&
f (2k+1) (x0 ) < 0
x0 $ % f (x)
f (x) = x4
' ( ) * + + f (x) = 4x3 . ,
x = 0 x = 0 %
- x = 0 + . f (0) = 12x2 |x=0 = 0,
f (0) = 24x|x=0 = 0 f (4) (0) = 24 > 0
$ % f (x) = x4 x = 0 #
f (x) = x5
' ( ) -
f (0) = f (0) = f (0) = f (4) (0) = 0 f (5) (0) = 120 > 0
# $ % f (x) = x5
+ / x
, +
0 y = x2k
x = 0 +
0 y = x2k+1
(−∞; +∞) + $ %
1 +
%
! # . $
)
x0 f (x),
x0 f (x) = 0
f (x0) < 0 x0
f (x0) > 0 x0
f (x)
y = f (x) = x3 − 4x + 2
x1 = − √2
2
x2 = √
3
f (x) = 6x
3
! " # $
# % f
2
x1 = − √
3
12
2
√
= √ > 0
f
3
3
2
−√
3
12
= − √ < 0.
3
& $ '(
2
x2 = √
3
(
#
)
f (x) =
√
3
!
x2 − x3 .
"
*# +% % '
$ x = 23 '
2
,- f 23 = − 2
= − - < 0 $ (
4
5
5
9 3 x4 (1 − x)
'
x=
2
3
$
2
1
3
3
√
3
f (x) = x2 − x3
93
. / 0 ' ,1)- $$ 2 0 '(
# [a; b] / $ 0 ' %
% (a; b) 0 % x = a x = b 3 (
# ' ' #
# [a; b] % $
•
xk (a; b)
f (x) f (xk ),
f (x) f (a) f (b),
•
• M m
f (xk ) f (a) f (b)
xk
!"# M m f (x) = x3 − 4x + 2
[0; 1] , [−1; 1] , [−2; 2]
$ % & ' !(" ' ) '
(0; 1)
'
f (x) * f (0) = 2, f (1) = −1
+ M = f (0) = 2, m = f (1) = −1 * *
[0; 1]
, (−1; 1)
f (−1) = 5, f (1) = −1 M = f (−1) = 5, m = f (1) = −1 $
' [−2; 2] , (−2; 2)
2
2
x1 = − √ x2 = √
3
3
2
2
−√
≈ 5, 0792 √
≈ −1, 0792 +
3
3
*
f(−2) = 2 f (2) = 2
2
M = f − √
3
2
m=f √
* [−2; 2]
3
, ' ' *
!""
y = f (x) (a; b)
! "
# x0 $ % &&'(
!"!
y = f (x) (a; b)
y
yT
f(xo)
y
T
Mo
y=f(x)
x
a
0
x
x
0
b
x0
y
y=f(x)
y
T
yT
f(xo)
Mo
x
0
a
xo
x b
x0 !
"# "
$ (x0 − δ; x0 + δ),
!
x0
" " %
"
y = f (x)
(a; x0 ) (x0 ; b) x0
&" '"( y = f (x) ) " !
" f (x) * * (a; b) + * *
f (x) < 0 ' '"(
)
" f (x) > 0 % "
x ∈ (−∞; +∞)
x ∈ (−∞; +∞)
y = 2k(2k − 1)x(2k−2) > 0
!
y = x(2k−1)
y = 2(k − 1)(2k − 2)x(2k−3) < 0
x < 0 y > 0 x > 0
" x < 0 x > 0
#
y = x3 − 4x + 2
y = 6x < 0 x < 0
y = 6x > 0 x > 0 $" % $ "&
"' x < 0 ' x > 0 ( )*
+
√
y =
3
x2 − x3
2
f (x) = − 4
'
9 x (1 − x)5
" " x = 1 ' f (x) < 0 x < 1
f (x) > 0 x > 1 $" % $ "&
"' x < 1 ' x > 1 ( *
,
3
x0
y = f (x) f (x0) f (x0) = 0.
- " % ' ' .
/ " ' 0 "0 "
"
1
2 " " f (x) = 0 1 3 " 4
"' " '
5 "
1 3 " % "
$ ' ' 1 "
6 ! x0 "
y = f (x)
#
$ (x0 − δ; x0 + δ)
x0 " f (x) > 0 f (x) < 0 $
x ∈ (x0 − δ; x0)
x ∈ (x0; x0 + δ) f (x) < 0
x0 y = f (x)
!
x0
f (x) = 0 ! "
"
# x0 $
"# #
$ ## # ! % & ' # !
( y = x2k #
# ) % ( y = x(2k+1) #
* x = 0 + x = 0
y = x3 − 4x + 2 # √,
- # & y = x2 − x3 + #
*
. x = 1 .( .
f (x) . x = 0 '
f (x) ( #*
/
√
01 y = x3 − 4x + 2 y = x2 − x3 1
" 1 01 y = x3 − 4x + 2
f (x) > 0
3
3
x
y
0
−∞; 0
0
√
" 1 01 y = x2 − x3
y = f (x)
3
x
y
0
y = f (x)
−∞; 0
0
f (0) = 2 > 0. " #
- 13 < x < +∞ f (x) < 0 !
f ( 23 ) < 0. #
1
• + x =
3
x = 13 "! #
/ !
•
−∞;
x
y
1
3
>0
y = f (x)
y = x(x + √x).
1
3
1
; +∞
3
0
+∞
>0
$ ! $
y = f (x)
)*+ y = x3ex.
, - "
• .
y = f (x) = x2 ex (3 + x), y = (x3 + 6x2 + 6x)ex .
•
/ $ x = −3
x = 0 0 ' '
f (−3) = (−27 + 54 − 18)e−3 =
9
> 0.
e3
/
f (0) = 0 .!'
"
f (0) = (x3 + 9x2 + 18x + 6)ex |x=0 = 6 > 0.
1 x = 0
2
f (0) > 0
•
f (x) = 0, (x3 + 6x2 + 6x)ex = 0, ex = 0, x(x2 + 6x + 6) = 0, x1 = 0,
√
√
x2 + 6x + 6 = 0, x2,3 = −3 ± 9 − 6 = −3 ± 3.
√
√
√
−∞ < x < −3 − 3; −3 − 3 < x < −3 + 3;
−3 +
√
3 < x < 0; 0 < x < +∞.
√
" "
• !
# −∞ < x < −3 − 3
f (x) = (x3 + 6x2 + 6x)ex < 0 ex > 0 $
(x3 +6x2 +6x) < 0 x < 0 % $
√
√
# −3 − 3 < x < −3 + 3
9
3
2
x
f (x) = (x + 6x + 6x)e > 0 f (−3) = 3 > 0
e
$
$
√
# −3 + 3 < x < 0
f (x) =
3
2
x
= (x + 6x + 6x)e < 0 f (−1) = (−1 + 6 −
1
− 6)e−1 = − < 0 $
e
# 0 < x < +∞
f (x) =
3
2
x
= (x + 6x + 6x)e > 0 f (1) = (1 + 6 + 6)e =
$ √
= 13e > 0 $
• & √ " x = −3 − 3 ≈ −4, 732;
x = −3 + 3 ≈ −1, 268 x = 0
$
%
√
x
−∞; −3 − 3
y
4
x !! ! "# $ !
! %! 2x + x2 = 0,
&!" "# $ y = 2x y = −x2 '# ! (
!
ξ2 = 2 ξ3 = 4
) ! ! # # ! &!" &
* #
(a, b)
f (x)
x = a x = b f (x)
+ ! , ! !
! ! # #! f (x) = 0, &!
&!" - ! ! "# $
! ! ! .! #!
x3 − 4x + 2 = 0.
/
x3 − 4x + 2 = 0
y
y=ψ(x)= x
2
y=ϕ( x)=2
y=ϕ( x)= 2
x
x
x
ξ1 0 ξ2 ξ3
y=-x 2
x2 = 2x
y = x3 − 4x + 2
y → −∞ x → −∞ y → +∞ x → +∞,
x3 − 4x + 2 = 0 ξ1 ∈ (−∞; − √2 )
2 2
ξ2 ∈ (− √ ; √ )
3 3
2
ξ3 ∈ ( √ ; +∞).
3
3
!"
# !$
" ! # % & %
& % ' ( ) f (−3) = −13 < 0
f (−2) = 2 > 0 ) f (0) = 2 > 0 f (1) = −1 < 0 !)
f (1) = −1 < 0 f (2) = 2 > 0' * ! ξ1 ∈ (−3; −2) ξ2 ∈ (0; 1)
ξ3 ∈ (1; 2)'
+ ! ! ,-'./ &
(a; b). + # ) #0 % x̄ ∈ (a; b).
+ ! % ) ξ ∈ (a; b), )
1)# ,.2'3/ % [ξ; x̄],
|f (ξ) − f (x̄)| = |f (ζ)||ξ − x̄|,
ζ ∈ (ξ, x̄). ξ
f (ξ) = 0
|f (x̄)| = |f (ζ)||ξ − x̄|.
m1
f (x)
[a; b], |f (x̄)| m1 |ξ − x̄|
f (x̄)
|ξ − x̄|
.
m1
! "
x̄ ξ.
#$!
x̄ = 0, 5
%
x3 − 4x + 2 = 0.
& ' x̄ ∈ (0; 1),
( ξ2 . )
f (x̄) = f (0,5) = 0, 125,
f (x) = 3x − 4 f (0) = −4 f (1) =
= −1 m1 = |f (x)| [0; 1] = 1.
|ξ − 0,5| 0,125. * #+
$! x̄ = 0, 5 ξ #
0,125, , 0, 375 ξ 0,625
2
- .( , . +
/ #$ / . "
- " , / / "+
0 / $ .
1 .# . , / +
.
1 2
/ /' , +
3 . /
(a; b). ! f (a) < 0" f (b) > 0. #$
$ [a; b] f (a + b −2 a ). % & $
" ξ = a + b −2 a , f (a + b −2 a ) > 0, ξ ∈ (a; a + b −2 a );
f (a + b −2 a ) < 0, ξ ∈ (a + b −2 a ; b). '
() $ " * " + ,
" * $) )
+ $ "
() " +
-
|b − a|
|x − ξ| <
,
.
2
* n/ +" x 0 $ n) +
,$ - . 1 2 $ !*
2 3 $ $ +"
2 = 8 2 2 = 16 > 10.
4
n
n
n
3
4
x3 − 4x + 2
(0; 1).
= 0,
# 2 1 5 $
"
! $
6 ( !
$
7" )
$
$ 8
7
72 * . 2
* "
" 2 |x − ξ| 2
ξ2 , f (0) = 2 > 0
f (1) = −1 < 0,
x1 = 0, 5,
f (0, 5) = 0, 125 > 0 ⇒ ξ2 ∈ (0, 5; 1, 0).
x2 = 0, 75, f (0, 75) = 0, 753 − 3 + 2 < 0 ⇒
ξ2 ∈ (0, 5; 0, 75).
f (xn ),
x3 = 0, 625; f (0, 625) = 0, 6253 − 2, 5 + 2 < 0 ⇒ ξ3 ∈ (0, 5; 0, 625).
x4 = 0, 5625
1
=
ξ
x4
16
= 0, 0625.
n
!"# (a; b)
$ f (x) f (x) % & ' ( ) %
y = f (x) ξ * ( +, "-.# *
h = ξ − xn , xn − n *
f (ξ) = f (xn ) + f (xn )h + O(h2 ) = 0,
!/#
$ % ξ , !"#
!/# O(h2 ), & h hn = xn+1−xn, xn+1
% * % ξ :
f (xn ) + f (xn )hn = f (xn ) + f (xn )(xn+1 − xn ) = 0.
xn+1 = xn −
f (xn )
, n = 0, 1, 2, . . .
f (xn )
!.#
( 0 %
y
T
y=f(x)
yn
xn
ξ
xn
0
x n+1
1 "!" ' &
, y = f (x) :
x n+1
x
(xn ; yn )
yT = f (xn ) + f (xn )(x − xn ).
% T
xn+1, yT (xn+1) = 0.
! " # " ξ !
y = f (x) $ $ ! (xn ; yn )
%&'
f (x) [a; b]
xn f (xn),
xn+1, !"#$%
ξ.
( ! $ ) xn
* f (xn) > 0, ! $ * + '&'+
f (x) > 0 + ! + xn+1 ξ. !
x̄n, ! $ f (x̄n) < 0,
! $+ ! $ y = f (x) ! x̄n
! x̄n+1, $ , ξ, xn,
! $+ x̄n+1
, + ! + $
- , , ! f (x) = 0
x̄ = xn+1 ! * .%&&/
|xn+1 − ξ|
|f (xn+1 )|
.
m1
0 , * 1$ .'2%/
xn+1
n=2
f (ζ)
f (xn+1 ) = f (xn) + f (xn )(xn+1 − xn ) +
(xn+1 − xn )2 ,
2
ζ ∈ (xn , xn+1 ) ! ! +
.%&3/ 1
|f (xn+1 )| =
M2 !
|f (ζ)|
M2
(xn+1 − xn )2
(xn+1 − xn )2 ,
2
2
, |f (x)| [a; b],
|xn+1 − ξ|
M2
(xn+1 − xn )2 .
2m1
! +
.%&4/
5 .%&4/ ! +
! , !
!" ! " $
x1 = −3 −
x0 = −3,
13
= −2, 4348,
23
x2 = −2, 4348 −
(−2, 4348)3 + 4 · 2, 4348 + 2
= −2, 2415,
3 · (−2, 4348)2 − 4
x3 = −2, 2415 −
(−2, 2415)3 + 4 · 2, 2415 + 2
= −2, 2151.
3 · (−2, 2415)2 − 4
!" # $
18
(2, 2415 − 2, 2151)2 < 10−3 .
2·8
% & 10−3 ξ1 = −2, 215.
ξ3 ∈ (1; 2) :
|x3 − ξ1 |
f (1) = −1, f (1) = −1, f (1) = 6,
f (2) = 2, f (2) = 8, f (2) = 12.
x0 = 2.
x1 = 2 −
2
= 1, 75,
8
x2 = 1, 75 −
1, 753 − 4 · 1, 75 + 2
= 1, 6811,
3 · 1, 752 − 4
x3 = 1, 6811 −
1.68113 − 4 · 1, 6811 + 2
= 1, 675.
3 · 1, 68112 − 4
(1; 2)
!"#$ [1, 5; 2]. % & '
M2 = f (2) = 12, m1 = f (1, 5) = 2, 75.
(
12
(1, 681 − 1, 675)2 < 10−3 .
2 · 2, 75
) * 10−3 ξ3 = 1, 675.
|x3 − ξ| 2
f (x) =
x = √a.
! "
xm − a
1
a
xn+1 = xn − n m−1 = ((m − 1)xn + m−1 ), n = 0, 1, 2, . . . #
mxn
m
xn
$ % x = √a, m = 2
1
a
xn+1 = (xn + ), n = 0, 1, 2, . . .
&'
2
x
= xm − a = 0
10−4 .
m
(
√
n
5
) " * ! a = 5 &' + x0 = 2
,
1
5
1
a
1
x1 = (x0 + ) = (2 + ) = 2, 25,
2
x0
2
2
5
1
a
1
) = 2, 23611,
x2 = (x1 + ) = (2, 25 +
2
x1
2
2, 25
5
1
a
1
x3 = (x2 + ) = (2, 23611 +
) = 2, 23607.
2
x2
2
2, 2361
x3 = 2, 2361 - .
2 - 10−4.
/0
) . 1 3
40 54 . !
√
& y = 1 − x3
3
y=
y=
1 − x3
x2
3
3
(x − 1)2
6
M m
y = (x + 1)2 − (x − 1)2 [−1; 1] [−2; 0] [−2; 2]
3
3
(x + 1)2 −
R
y =
√
3
1 − x3
•
D(y) = (−∞; +∞)
•
!"
!"
• #! ! $ Oy : x = 0 y = 1 $ Ox : y = 0
x = 1
• #! %& &' (
) ( & (& % ( (
*
√
3
1 − x3
1
1
3
= lim (
k = lim
− 1) = − lim (1 − 3 )1/3 = −1,
3
x→+∞
x→+∞
x→+∞
x
x
x
√
1
3
b = lim ( 1 − x3 + x) = lim (−x(1 − 3 )1/3 + x) =
x→+∞
x→+∞
x
1x
1
1
1
= lim −x +
+
x
=
lim
= 0.
+
x
·
O
+
O
3
6
2
x→+∞
x→+∞
3x
x
3x
x5
+
• ,' ( % $
y =
( y = −x
1 (−3x2)
x2
= −
.
3
3
3
2
3 (1 − x )
(1 − x3 )2
- ! ! x = 0
3
y = −
x = 1 . %
(1 − x3 )2 2x −
3
x2 2(−3x2)
√
3 3 1 − x3
(1 − x3 )4
=
2x
(1 − x3 )2x + 2x4
= −
3
3
3
5
(1 − x )
(1 − x3 )5
( / (& ! !
=−
•
x −∞; 0
y
0
y
∪
•
0
0
0
0; 1
0
±+∞
0
+∞
max
√
3
4
>0
∪
∪
• '
( %
) *+,.
" %
" %
/ /
[a; b]
0
$+ ,
y =
3
(x + 1)2 −
3
(x − 1)2
M
[−1; 1] [−2; 0] [−2; 2]
y
1
-1
0
-1
1
y =
x
3
(x + 1)2 −
3
(x − 1)2
•
(−1; 1)
√
√ y(−1) = − 4 = m
3
y(+1) =
3
4=M
(−2; 0)
! y(−2) =
y(0) = 0 " [−2; 0]
√
m = y(−1) = − 4 M = y(0) = 0
•
(−2; √
2) √
ymax = y(1) = 3 ymin = y(−1)
√ = − 4 #
√
$ y(−2)
=
1
−
9
%
y(2)
=
9
−
1
&
√
√
m = ymin = y(−1) = − 4 M = ymax = y(1) = 4
'( ) R
*
r
+ h " , - .
h2
/() 0 12 V = πr2 h * r2 + = R2
•
x = −1
√
1− 39
3
4
3
3
3
3
h2
r =R −
4
2
2
3
4
*
2
r
h
h 3
V = π R2 −
4
3h2
dV
2
=π R −
= 0
dh
4
2
V ,
- 12
2R
h= √
3
r=
R2 −
h2
=
4
R
h
O
2r
= R
2
3
V =0
h = 0
r = R h = 2R r = 0
2R
2
h = √ ;r = R
3
3
y
=
y
= x2 e−x
x + arctg x
!" $
!" %
x2 − 4x
− 4x + 8
!" #
x2
M m
y = x2e−x [0; 1]! [0; 3]! [−1; 2]
" # a #
$ ! $ $
!" &
!" '(
!
⎛
a11
⎜ a21
A = (aij ) = ⎜
⎝
m × n
a12
a22
...
...
⎞
a1n
a2n ⎟
⎟
⎠,
am1 am2 . . . amn
m × n aij
" !#
$ aij % & ' ( )
'* + ' *
, -
. i , "i = 1, 2, ..., m)
j , - "j = 1, 2, ..., n) & -'+*
' -& A, B
. &( (
&( -& *
'* *
•
+ + & -
"m = n# ' &*
•
+ + & - "m = n#
' . +/ + / -
'
0
A=
a11 a12
a21 a22
i = j
(a11 , a22, . . . , ann )
•
aij = 0 i = j
⎛
⎞
•
0 ...
a11 0
⎜ 0 a22 0 . . .
⎜
⎜ 0
0 a33 . . .
⎜
⎝
0
•
0
0
0
0 ⎟
⎟
0 ⎟
⎟
⎠
. . . ann
!
E
⎛
⎞
1 0 0 ... 0
⎜ 0 1 0 ... 0 ⎟
⎜
⎟
0 0 1 ... 0 ⎟
E=⎜
⎜
⎟
⎝ ⎠
0 0 0 ... 1
"
# "
• $
• $ %
•
A = (a11 , a12 , . . . , a1n )
•
$ % &
&
⎛
⎞
a11
⎜ a21 ⎟
⎟
A=⎜
⎝ ⎠
am1
•
$ B #
A ! A
AT
a11 a12 a13
a
a
a23
⎛ 21 22 ⎞
a11 a21
AT = ⎝ a12 a22 ⎠
a13 a23
A=
(A = B)
aij = bij
A
B
!
!
"
(m × n)
C # !
"
A B$
A
B #
B=
%
$
2 4 1
3 0 5
A + B = C, cij = aij + bij .
1 2 3
! %
A=
2 4 5
.
&
" #
1 2 3
2 4 5
'(
&
+
2 4 1
3 0 5
=
A + B = B + A,
3 6 4
5 4 10
.
) )*
(A + B) + C = A + (B + C).
)* (
( &
) +
A + 0 = A.
3 −1
=
−1 2
1 · 3 + 1 · (−1) 1 · (−1) + 1 · 2
2 1
=
=
3 · 3 + 1 · (−1) 3 · (−1) + 1 · 2
8 −1
BA =
1 1
3 1
AB = BA.
AB
= BA
A(BC) = (AC)B
(A + B)C = AC + BC.
!
!
1 1
"! A = 1 1
B=
1
1
−1 −1
AB =
1 1
1 1
1
1
0 0
·
=
−1 −1
0 0
a11 a12
a21 a22
A=
.
a11 a22 − a21 a12
a
a
|A| = 11 12
a21 a22
.
! "#
" $ %
a
a
|A| = 11 12
a21 a22
= a11 a22 − a21 a12 .
&
' a11 , a12 , a21 , a22 " (
& !
) %
2 5
3 −4
2 5
3 −4
.
= 2 · (−4) − 5 · 3 = −23.
* +
• ,
-
a11 a12
a21 a22
•
.
* .
! "+ .
#
a11 a12
a21 a22
•
a11 a21
=
a12 a22
,
= − a21 a22
a11 a12
.
"
•
a11 a12
a11 ka12
a21 ka22 = k a21 a22 .
!"
!#
•
! $
" % "
%" # #
•
a11 + λa12 a12
a21 + λa22 a22
a11 a12
=
a21 a22
.
& % !
#
' ⎛! ($
⎞
a11
⎜ a21
⎜
A=⎝
...
an1
a12
a22
...
an2
...
...
...
...
|A| =
a12
a22
···
an2
···
···
···
···
a1n
a2n ⎟
⎟.
... ⎠
ann
)*#+
.
)*#.
)*#, n
! "
- % %
a11
a21
···
an1
a1n
a2n
···
ann
/ " % 0
1 $ " $ % $ #
)*#2 #
n $ (n − 1)
%
& "
M
M
a
ij
12
!"#$ M
12
a21
a
= 31
···
an1
n
a23
a33
···
an3
···
···
···
···
12
a2n
a3n
···
ann
%
&
!"'(
) a A
* i j &
+ +
, -
- .
A = (−1) M .
!"/$
0 . A = (−1) M = M 1 A = (−1) M = −M
2 -
.
,
a A
|A| =
+ i
!"3$
,
|A| =
a A
+ j.
!"'($
4 - 5
ij
i+j
ij
11
1+1
11
11
12
ij
ij
1+2
12
12
n
ij
ij
ij
ij
j=1
n
i=1
! n
" #
!"3 !"'(
n 6
n − 1 5 !"3$ !"'($
! n n
n − 1
!
a11 a12 a13
|A| = a21 a22 a23
a31 a32 a33
" # |A| = a11 A11 +a12 A12 +a13 A13, "
! $% |A| = a12 A12 + a22 A22 + a32 A32 .
aa1121 aa1222
A11 = a22 A12 = −a21
a
a
A21 = −a12 A22 = a11 11 12
a21 a22
! "!! #
& $ % % %
−1 −2 −10
9
10 .
|A| = 1
1
2
0
' ( ) #
*
* " # (i = 3) :
+
|A| = a31 A31 + a32 A32 + a33 A33 .
, !$ * - .
+
−2 −10
= (−2)10 − 9(−10) = −20 + 90 = 70;
A31 = (−1) M31 = 1
9
10
−1 −10
= −[(−1)10 − 1(−10)] = 0;
A32 = (−1)3+2 M32 = −1
1
10
−1 −2
= (−1)9 − 1(−2) = −9 + 2 = −7.
A33 = (−1)3+3 M33 = 1
1
9
3+1
,
#
!$ -
+
|A| = 1 · 70 + 2 · 0 + 0 · 7 = 70.
!"
!
"
# $
! " #
# $
$
a11 A21 + a12 A22 + a13 A23 = 0%
a12 A11 + a22 A21 + a32 A31 = 0.
& ' $ $
(
)
$*
a11 A21 + a12 A22 + a13 A23 = a11 (−M21 ) + a12 M22 + a13 (−M23 ) =
a12 a13
a11 a13
a11 a12
=
= −a11 ·
+ a12 ·
− a13 ·
a32 a33
a31 a33
a31 a32
= −a11 (a12 a33 − a13 a32 ) + a12 (a11 a33 − a13 a31 ) − a13 (a11 a32 − a12 a31 ) =
= −a11 a12 a33 + a11 a13 a32 + a12 a11 a33 −
−a12 a13 a31 − a13 a11 a32 + a13 a12 a31 = 0.
+
$ $ (
, -. )
$ /
& )
(
, 0 .*
−1 −2 −10
9
10 1
|A| = 1
1
2
0
2 3 * & ) )
$ ( ) ) $
0 0 −10
10
|A| = 0 7
1 2
0
!
|A| = a11 A11 + a21 A21 + a31 A31
0 −10
= 70.
= 1
7
10
" # $
n
n − 1 %
" n
&
'⎛
⎞ ⎛
⎞
1
5 3
3 2 4
A = ⎝ 2 −1 0 ⎠ B = ⎝ 2 3 −2 ⎠
−1 2 2
4 0 2
( ) *
+
" + ) cij = aij + bij
⎞
⎞ ⎛
4 7 7
1+3
5+2 3+4
C = A + B = ⎝ 2 + 2 −1 + 3 0 − 2 ⎠ = ⎝ 4 2 −2 ⎠ .
3 2 4
−1 + 4 2 + 0 2 + 2
⎛
!"#$
A=
( ) %
0 5
4 1
,B =
1 3
2 −2
.
C = α · A ⇒ cij = α · aij
0 5
0 10
1 3
5 15
2A = 2 ·
=
, 5B = 5 ·
=
4 1
8 2
2 −2
10 −10
0 + 5 10 + 15
5 25
D = 2A + 5B =
=
.
8 + 10 2 − 10
18 −8
⎛
⎞
A · B B · A
⎛
⎞
2 4 0
2 1 0
A = ⎝ 2 0 4 ⎠ , B = ⎝ 1 −1 2 ⎠ .
1 2 3
3 2 1
i j !
" i
# ⎛ j $ #
⎞
2 · 2 + 4 · 1 + 0 · 3 2 · 1 + 4 · (−1) + 0 · 2 2 · 0 + 4 · 2 + 0 · 1
%& A · B = ⎝ 2 · 2 + 0 · 1 + 4 · 3 2 · 1 + 0 · (−1) + 4 · 2 2 · 0 + 0 · 2 + 4 · 1 ⎠ =
1 · 2 + 2 · 1 + 3 · 3 1 · 1 + 2 · (−1) + 3 · 2 1 · 0 + 2 · 2 + 3 · 1
⎛
⎞
8 −2 8
= ⎝ 16 10 4 ⎠ .
13 5 7
⎛
2·2+1·2+0·1
¾µ B·A = ⎝ 1 · 2 + (−1) · 2 + 2 · 1
3·2+2·2+1·1
⎞
2·0+1·4+0·3
1 · 0 + (−1) · 4 + 2 · 3 ⎠ =
3·0+2·4+1·3
2·4+1·0+0·2
1 · 4 + (−1) · 0 + 2 · 2
3·4+2·0+1·2
⎛
⎞
6 8 4
= ⎝ 2 8 2 ⎠.
11 14 11
' !( ) ! ( *
* (! ! A · B = B · A
A=
2 −1 3 5
4 0 1 2
⎛
−2
⎜ −1
⎜
,B = ⎝
4
3
⎞
1 0
5 −2 ⎟
⎟.
0 1 ⎠
1 −1
+ * !" !$(
" # $ ( ( !* $ *
* !
' ! *
, * ' # '
!#
2 · (−2) − 1 · (−1) + 3 · 4 + 5 · 3 2 · 1 + (−1) · 5 + 3 · 0+
A·B =
4 · (−2) + 0 · (−1) + 1 · 4 + 2 · 3 4 · 1 + 0 · 5 + 1 · 0+
5 · 1 + 2 · 0 + (−1) · (−2) + 3 · 1 + 5 · (−1)
2 · 1 + 4 · 0 + 0 · (−2) + 1 · 1 + 2 · (−1)
=
24 2 0
2 6 −1
.
B · A
! M13 M32
A13 A32
⎛
⎞
2 −1 0
A = ⎝ −1 2 3 ⎠ .
4
1 3
" # $ %
&' ij
&' ( (
i j ) Aij =
= (−1)i+j Mij
M13
−1 2
=
4 1
= −1 · 1 − 4 · 2 = −1 − 8 = −9.
A13 = (−1)1+3 M13 = (−1)4 · (−9) = −9.
2 0
= 2 · 3 − (−1) · 0 = 6.
M32 =
−1 3
A32 = (−1)3+2 M32 = (−1)5 · 6 = −6.
*
5 6 3
|A| = 0 1 0
7 4 5
.
" # $ +
,
|A| = a21 A21 + a22 A22 + a23 A23 = −a21 M21 + a22 M22 − a23 M23 =
6 3
+1· 5 3 −0· 5 6 =
= −0 ·
4 5
7 5
7 4
5 3
= 5 · 5 − 3 · 7 = 25 − 21 = 4.
=
7 5
|A| =
4
2 1 4
1 −2 0 3
−2 −3 2 1
3
2 0 1
.
|A| = a13 A13 +a23 A23 +a33 A33 +a43 A43 = 1·A13 +0·A23 +2·A33 +0·A43 =
= A13 + 2A33 = M13 + 2M33 .
M13 M33 ! " #
!
1 −2 3
−3 1
− (−2) · −2 1 + 3 · −2 −3 =
M13 = −2 −3 1 = 1 ·
3
3 1
2
2
1
3
2 1
= 1(−3 · 1 − 2 · 1) + 2(−2 · 1 − 3 · 1) + 3(−2 · 2 − 3 · (−3)) = −5 − 10 + 15 = 0.
4 2 4
−2 −3
− 2 · 1 3 + 4 · 1 −2 =
M33 = 1 −2 3 = 4 ·
3 2
2
1
3 1
3 2 1
= 4(−2 · 1 − 2 · 3) − 2(1 · 1 − 3 · 3) + 4(1 · 2 − 3 · (−2)) = −32 + 16 + 32 = 16.
|A| = M13 + 2M33 = 0 + 2 · 16 = 32
! $ % $ #
! $ !
• & ! !!$'( #
! $ ) −2 :
4
2 1 4
1 −2 0 3
.
|A| =
−10 −7 0 −7
3
2 0 1
•
1
|A| = 1 · (−1)1+3 −10
3
−2 3 1 −2 3
−7 −7 = −10 −7 −7 .
2
1
2
1 3
•
−3 :
1
0
0
|A| = −10 −27 23
3
8 −8
•
.
−27 23
= 8 −27 23 = 8(27 − 23) = 32.
|A| = 1 · (−1)1+1
1 −1
8 −8
!
0 −1 5
3 4 7 = 12.
x x 8
" #
$ % "$
#
%
4 7
− (−1) · 3 7 + 5 · 3 4 = 12 ⇒ 24 − 7x + 5(3x − 4x) = 12
0 ·
x x
x 8
x 8
⇒ 24 − 12x = 12 ⇒ x = 1.
&
A=
2 4
−1 3
1 0
B=
.
2 3
'(
⎛
⎞
5 8 4
A = ⎝ 3 2 5 ⎠ .
7 6 0
A
B
! "
''
⎛ D = 3A + 5E
⎞ E !
! A = ⎝
1 1 2
0 1 4 ⎠.
−1 2 3
⎛
1 1 2
3 1 ⎠.
4 1 1
A2 A = ⎝ 1
⎞
⎛ ⎞
1 3 2
2
A = ⎝ 2 0 4 ⎠ B = ⎝ 1 ⎠ .
1 2 0
3
⎛
⎞
A · B
M12 M22
⎛A12 A22 ⎞
1
3 1
A = ⎝ −2 −1 2 ⎠ .
0
1 3
1 0 4
|A| = −2 2 3 .
1 0 5
|A| =
3
2 1
0
2 3
−2 −3 2
4
1 2
2
0
1
4
1 1 0
4 x 6 = 0.
3x 1 4
.
! A
AA−1 = E.
'
$% & A
A
−1
A−1
" !#
A−1 A = E.
( )
!
A
A ! "" # $
"
* ( % &
) A−1 (% & &
A % % + |A| = 0
* |A| = 0
A
|AA−1 | = |A||A−1 | = 0.
, % & |AA−1| = |E| = 1
*
&
(
⎛
⎞
a11 a12 a13
A = ⎝ a21 a22 a23 ⎠
a31 a32 a33
- % +
a11 a12 a13
|A| = a21 a22 a23
a31 a32 a33
= 0.
Aij aij
A−1 A
• ! B A "# $ %
aij Aij $"
|A| " A&
⎛
⎞
A11 /|A| A12 /|A| A13 /|A|
B = ⎝ A21 /|A| A22 /|A| A23 /|A| ⎠ .
A31 /|A| A32 /|A| A33 /|A|
•
' B T (
B )
⎛
⎞
A11 /|A| A21 /|A| A31 /|A|
B = ⎝ A12 /|A| A22 /|A| A32 /|A| ⎠ .
A13 /|A| A23 /|A| A33 /|A|
T
B T # A
*
⎞
⎞⎛
a11 a12 a13
A11 /|A| A21 /|A| A31 /|A|
AB = ⎝ a21 a22 a23 ⎠ ⎝ A12 /|A| A22 /|A| A32 /|A| ⎠ =
a31 a32 a33
A13 /|A| A23 /|A| A33 /|A|
⎛
T
⎛
⎜
=⎝
a11 A11 +a12 A12 +a13 A13
|A|
a21 A11 +a22 A12 +a23 A13
|A|
a31 A11 +a32 A12 +a33 A13
|A|
a11 A21 +a12 A22 +a13 A23
|A|
a21 A21 +a22 A22 +a23 A23
|A|
a31 A21 +a32 A22 +a33 A23
|A|
a11 A31 +a12 A32 +a13 A33
|A|
a21 A31 +a22 A32 +a23 A33
|A|
a31 A31 +a32 A32 +a33 A33
|A|
⎞
⎟
⎠=
⎞
1 0 0
= ⎝ 0 1 0 ⎠ = E,
0 0 1
⎛
# " |A| + %
" , "- " + # %
# + ,
# + , ,
AB T = E
A
−1
B T = A−1
⎛ A
11
⎜ |A|
⎜ A
⎜ 12
=⎜
⎜ |A|
⎝ A13
|A|
A21
|A|
A22
|A|
A23
|A|
A31
|A|
A32
|A|
A33
|A|
⎞
⎟
⎟
⎟
⎟.
⎟
⎠
|A| = 0
⎛
A−1
A11 A21
1 ⎜
A12 A22
⎜
=
|A| ⎝ . . . . . .
A1n A2n
⎞
. . . An1
. . . An2 ⎟
⎟
... ... ⎠
. . . Ann
⎛
"#%
⎞
3 2 2
A = ⎝ 1 3 1 ⎠
5 3 4
!"#"$
& ' ( ) A
3 2 2
|A| = 1 3 1
5 3 4
= 27 + 2 − 24 = 5.
*+ , ,
Aij = (−1)i+j · Mij (
3 1
A11 =
3 4
1
A12 = −
5
1 3
A13 =
5 3
= 9, A21 = − 2
3
3
1
= 1, A22 = 5
4
= −12, A23 = −
2
4
2
4
3
5
= −2, A31 = 2
3
= 2, A32 = − 3
1
3
2
= 1, A33 =
3
1
2
= −4,
1
2
= −1,
1
2
= 7.
3
⎛
A−1
9
5
⎜
⎜
⎜
⎜ 1
=⎜
⎜ 5
⎜
⎜
⎝ −12
5
−2
5
2
5
1
5
⎞
−4
5 ⎟
⎟
⎟
−1 ⎟
⎟.
5 ⎟
⎟
⎟
7 ⎠
5
⎛
a11
⎜ a21
⎜
⎜
A=⎜
⎜ ai1
⎝
a12
a22
. . . a1k
. . . a2k
ai2
. . . aik
⎞
. . . a1n
. . . a2n ⎟
⎟
⎟
⎟,
. . . ain ⎟
⎠
am1 am2 . . . amk . . . amn
m n ! "# $ %
! k k & ! ' !%
!( $ k
) ' k% ! A $! %
# ! ' # $ # !
! $ !( k k
) !
⎛
⎞
2 3
4 5
A = ⎝ 0 −2 3 1 ⎠ ,
0 2
2 4
# '!
$ %
2
3 4
0 −2 3
0
2 2
' !# !%
# # #
!
A *
3 4
−2 3 " ! ! + %
! ) ! $ !
! ! , ' !
A r
A
r
r ! "
A
r(A)
"
⎛
⎞
1
2 5 3
⎜ 0
1 7 4 ⎟
⎟.
A=⎜
⎝ 0
0 0 0 ⎠
0 −1 0 0
#
1
2 5 3
0
1 7 4
=0
0
0 0 0
0 −1 0 0
$
! %
1
2 5
1 7 = 7 = 0. &
M34 = 0
0 −1 0
' r(A) = 3
(
$
)
! (*
+
! !, • * $ . /
$
* 0
•
$ . /
$
!, . /
*
* 0
•
. /
0
•
!
"
#
$
" $ $
%&%
⎛
⎞
2 3
5 −3 −2
3 −1 −3 ⎠ .
A=⎝ 3 4
5 6 −1
3 −5
' (
) * % ! + $
+
⎛
⎞
1 1 −2
2 −1
3 −1 −3 ⎠ .
A1 = ⎝ 3 4
5 6 −1
3 −5
* ! $ ! A1
#
, &
⎛
⎞
1 1 −2
2 −1
9 −7
0 ⎠.
A2 = ⎝ 0 1
0 1
9 −7
0
* $ !
A2
⎛
⎞
1 1 −2
2 −1
9 −7
0 ⎠.
A3 = ⎝ 0 1
0 0
0
0
0
- A3 !
1 1 −2
2 −1
A4 =
0 1
9 −7
0
! . $
!
A # r(A) = 2
/ $ $ 0
! !
⎛
⎞
1 2 3 4
A = ⎝ 2 4 6 8 ⎠.
3 6 9 12
!
"
# $ %
&
' ( r(A) = 1
⎛
⎞
3 5 7
A = ⎝ 1 2 3 ⎠.
1 3 5
(
)*
+ )* , -"
- ). ' / * "
! )* 0 ' % )& "
- * 1 !
% )& $ '
• ( $
- " 2"
*
⎛
⎞
⎛
⎞
⎛
⎞
3 5 7
3+1 5+3 7+5
4 8 12
⎝ 1 2 3 ⎠ ∼ ⎝ 1
2
3 ⎠ = ⎝ 1 2 3 ⎠.
1 3 5
1
3
5
1 3 5
• *
3 % $
⎞
3 5 7
⎝ 1 2 3 ⎠
1 3 5
⎛
1
& "
4
⎞
⎛
1 2 3
∼ ⎝ 1 2 3 ⎠.
1 3 5
• 4* 0 '
0
*
-
⎞
3 5 7
⎝ 1 2 3 ⎠
1 3 5
⎛
⎞
0 0 0
⎝ 1 2 3 ⎠.
1 3 5
⎛
•
∼
⎛
⎞
3 5 7
⎝ 1 2 3 ⎠
1 3 5
∼
1 2 3
1 3 5
.
1 2
1 3 = 3 − 2 = 1 = 0
!
" #
⎛
2 −4 3
1
⎜ 1 −2 1 −4
A=⎜
⎝ 0 1 −1 3
4 −7 4 −4
⎞
0
2 ⎟
⎟.
1 ⎠
5
$ % $ &' (
• )
%
⎛
2 −4 3
1
⎜ 1 −2 1 −4
⎜
A=⎝
0 1 −1 3
4 −7 4 −4
•
⎞
0
2 ⎟
⎟
1 ⎠
5
⎛
∼
1 −2 1 −4
⎜ 2 −4 3
1
⎜
⎝ 0 1 −1 3
4 −7 4 −4
⎞
2
0 ⎟
⎟.
1 ⎠
5
! *
& &
+( &
, (−2)
(−4) '
⎛
⎞
1 −2 1 4 −2
⎜ 0 0
1 9 −4 ⎟
⎟
A∼⎜
⎝ 0 1 −1 3 1 ⎠ .
0 1
0 12 −3
•
⎛
⎞
1 −2 1 4 −2
⎜ 0 1 −1 3 1 ⎟
⎟.
A∼⎜
⎝ 0 0
1 9 −4 ⎠
0 1
0 12 −3
•
⎛
⎞
1 −2 1 4 −2
⎜ 0 1 −1 3 1 ⎟
⎟.
A∼⎜
⎝ 0 0
1 9 −4 ⎠
0 0
1 9 −4
•
!
⎛
⎞
⎛
⎞
1 −2 1 4 −2
1 −2 1 4 −2
⎜ 0 1 −1 3 1 ⎟
⎟ ∼ ⎝ 0 1 −1 3 1 ⎠ .
A∼⎜
⎝ 0 0
1 9 −4 ⎠
0 0
1 9 −4
0 0
0 0 0
•
" # $ %&
# '
%& & (
1 −2 1
1 −1
= 1 = 0.
M = 0 1 −1 = 1
0 1
0 0
1
) & #
*+ ,
⎛
⎞
1 2 0
A = ⎝ 3 2 1 ⎠.
0 1 2
" $ ( -
1 2 0
|A| = 3 2 1
0 1 2
= −9.
|A| = 0
A11 = (−1)
2 1
1 2
1+1
= 3,
A12 = (−1)
3 1
0 2
1+2
= −6,
3 2
= 3.
A13 = (−1)1+3
0 1
2 0
= −4,
A21 = (−1)2+1
1 2
A23 = (−1)2+3
1 0
= 2,
A22 = (−1)2+2
0 2
1 2
= −1.
0 1
2 0
= 2,
A31 = (−1)3+1
2 1
A33
!
1 0
= −1,
A32 = (−1)3+2
3 1
3+3 1 2
= −4.
= (−1)
3 2
⎛
⎞
−3/9 6/9 −3/9
B = ⎝ 4/9 −2/9 1/9 ⎠ .
−2/9 1/9
4/9
" # $
⎛
⎞
−1/3 4/9 −2/9
A−1 = ⎝ 2/3 −2/9 1/9 ⎠ .
−1/3 1/9
4/9
%&&
⎞
⎛
1 0 0 0 5
A = ⎝ 0 0 0 0 0 ⎠.
2 0 0 0 11
⎛
⎛
⎛
⎞
4 3 2 2
A = ⎝ 0 2 1 1 ⎠.
0 0 3 3
⎞
1 2 3 6
A = ⎝ 2 3 1 6 ⎠.
3 1 2 6
⎞
0 2 0 0
A = ⎝ 1 0 0 4 ⎠.
0 0 3 0
⎞
10 20 −30
A = ⎝ 0 10 20 ⎠ .
0 0 10
⎛
⎞
1 2
2
A = ⎝ 2 1 −2 ⎠ .
2 −2 1
⎛
! " #
$ %
&
m '" !
( x1 , x2 , . . . , xn :
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
a11 x1 + a12 x2 + · · · + a1k xk + · · · + a1n xn = c1 ,
a21 x1 + a22 x2 + · · · + a2k xk + · · · + a2n xn = c2 ,
ai1 x1 + ai2 x2 + · · · + aik xk + · · · + ain xn = ci ,
am1 x1 + am2 x2 + · · · + amk xk + · · · + amn xn = cm .
n
) *
aij
x !
" # $
# %
& # &
" # $
#! %
# &
#! #
# &
'
#
# & # &
& # &
( # %
# )
$ # $ *
$ #
#%
# # #*
• # $ $ %
*
• #
$ $
# # #
•
•
+ # # %
#
, # # $ $
#
# - # # # %
$ & , #
# # %
&
.# # # $
%
#
#
# # #
#
$
## # - %
# #
.# A # %
$ /26.10)
⎛
⎞
a11 a12 . . . a1n
⎜ a21 a22 . . . a2n ⎟
⎟
A=⎜
⎝
⎠
am1 am2 . . . amn
X
⎛
⎞
x1
⎜ x2 ⎟
⎜
⎟
X=⎝
⎠
xn
⎛
⎞
c1
⎜ c2 ⎟
⎟
C=⎜
⎝ ... ⎠
cn
!
A·X
"
⎞
a11 x1 + a12 x2 + · · · + a1n xn
a21 x1 + a22 x2 + · · · + a2n xn ⎟
⎟
⎟
⎟
ai1 x1 + ai2 x2 + · · · + ain xn ⎟
⎠
⎛
⎜
⎜
⎜
A·X =⎜
⎜
⎝
#
am1 x1 + am2 x2 + · · · + amn xn
$%
& &
'
"
(
A · X = C.
)
&
. ,
(
% * &
( ,
*
+,
-
"
/
⎛
⎞
a11 a12 . . . a1n
⎜ a21 a22 . . . a2n ⎟
⎟
A=⎜
⎝ ºººººººººººººº ⎠
am1 am2 . . . amn
⎛
⎞
a11 a12 . . . a1n c1
⎜ a21 a22 . . . a2n c2 ⎟
⎟
B=⎜
⎝ ⎠,
am1 am2 . . . amn cm
r(A) = r(B)
"
r(B) = n
A
B
r(A) = r(B) < n
$
"
! &
' $
# (
•
! %#
# !
r(A) =
! #
" $ $
!
* $
A:
|A| = 0
!
A·X = C
)
A−1
(
A−1 · (A · X) = A−1 · C.
•
+
%
% ! &
$ $
%)
(A−1 · A) · X = A−1 · C.
•
, & & &
A−1 · A = E
-
E · X = X
#
)
X = A−1 · C.
.
⎧
⎨ 3x1 + 2x2 + 2x3 = 5,
x1 + 3x2 + x3 = 0,
⎩
5x1 + 3x2 + 4x3 = 10.
⎞
⎛ ⎞
⎛ ⎞ ⎛
x1
5
3 2 2
X = ⎝ x2 ⎠
C = ⎝ 0 ⎠.
AX = C A = ⎝ 1 3 1 ⎠
10
5 3 4
x3
A−1
⎛
⎞
9/5 − 2/5 − 4/5
−1
2/5 − 1/5 ⎠ .
A = ⎝ 1/5
−12/5 1/5
7/5
⎞
⎞ ⎛
⎞ ⎛
⎛
1
5
9/5 − 2/5 − 4/5
2/5 − 1/5 ⎠ · ⎝ 0 ⎠ = ⎝ −1 ⎠ .
X = ⎝ 1/5
2
10
−12/5 1/5
7/5
!" #
#$%
x1 = 1% x2 = −1% x3 = 2 & ' $() %
* $#"
+, $ n $ n
A·
- ./
X = C % # ' |A| = 0
#$"+
⎛ A
An1
11 A21
...
⎜ |A| |A|
|A|
⎜ A A
An2
12 22
⎜
...
⎜
−1
X = A ·C = ⎜ |A| |A|
|A|
⎜ ºººººººººººººººº
⎜
⎝ A1n A2n
Ann
...
|A| |A|
|A|
0#
⎞
x1
⎜ x2 ⎟
1
⎟
⎜
⎝ ⎠ = |A|
xn
⎛
⎛
⎞
⎛
⎞
⎟⎛
c1
A11 A21 . . . An1
⎟
⎟⎜
⎟
1 ⎜
⎟ ⎜ c2 ⎟
⎜ A12 A22 . . . An2
·
=
⎟·⎝
ºº ⎠
⎟
|A| ⎝ º º º º º º º º º º º º º º º
⎟
cn
A1n A2n . . . Ann
⎠
⎞
A11 c1 + A21 c2 + · · · + An1 cn
⎜ A12 c1 + A22 c2 + · · · + An2 cn ⎟
⎟
·⎜
⎝ ⎠.
A1n c1 + A2n c2 + · · · + Ann cn
⎞⎛
⎞
c1
⎟ ⎜ c2 ⎟
⎟·⎜
⎟
⎠ ⎝ ºº ⎠.
cn
⎧
1
⎪
⎪
x1 =
(A11 c1 + A21 c2 + · · · + An1 cn )
⎪
⎪
|A|
⎪
⎪
⎪
1
⎨
x2 =
(A12 c1 + A22 c2 + · · · + An2 cn )
|A|
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
⎩ xn =
(A1n c1 + A2n c2 + · · · + Ann cn )
|A|
! " # $% $!$
# !& '
# |A| ' & % ' %
! ! ! & &
!' !'(
c1 a12 . . . a1n
c2 a22 . . . a2n
,
Δx1 = A11 c1 + A21 c2 + · · · + An1 cn =
cn an2 . . . ann
a11 c1 . . . a1n
a c . . . a2n
Δx2 = A12 c1 + A22 c2 + · · · + An2 cn = 21 2
an1 cn . . . ann
) * !' & + " ,
(
x1 =
Δ x1
Δ x2
Δ xn
, x2 =
, . . . , xn =
.
|A|
|A|
|A|
- ' # + . ½½ # # #
+ /
⎧
⎨ 3x1 + 2x2 + 2x3 = 5,
x1 + 3x2 + x3 = 0,
⎩
5x1 + 3x2 + 4x3 = 10.
½½
x1
! a11 = 0"
# $
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
a1k
a1n
c1
a12
x2 + · · · +
xk + · · · +
xn =
,
a11
a11
a11
a11
a21 x1 + a22 x2 + · · · + a2k xk + · · · + a2n xn = c2 ,
x1 +
ai1 x1 + ai2 x2 + · · · + aik xk + · · · + ain xn = ci ,
%&'(&)
am1 x1 + am2 x2 + · · · + amk xk + · · · + amn xn = cm .
* #
%26.12) a21
+
a31
#
, #
# $
⎧
x1 + a12 x2 + · · · + a1k xk + · · · + a1n xn = c1 ,
⎪
⎪
⎪
a22 x2 + · · · + a2k xk + · · · + a2n xn = c2 ,
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
+ #
ai2 x2 + · · · + aik xk + · · · + ain xn = ci ,
%&'(-)
am2 x2 + · · · + amk xk + · · · + amn xn = cm .
$
a1k =
a1k
,
a11
aik = aik −
c1 =
c1
,
a11
ci = ci −
a1k
ai1 ;
a11
c1
ai1 ;
a11
i = 2, 3, . . . , n;
i = 2, 3, . . . , m.
. #
%26.13) !
a22 "
# a32 , . . . , ai2, . . . , am2
/
0 ! /
!
.
! A %&'(&)
1 ! B
r(A) = r(B).
xr+1 , . . . , xn
αr+1 , αr+2, . . . , αn
x1 , x2 , . . . , xr αr+1 , αr+2
. . . , αn
! "#
$ %
! "
! &
⎧
⎨ 3x1 + 2x2 + 2x3 = 5,
x1 + 3x2 + x3 = 0,
⎩
5x1 + 3x2 + 4x3 = 10.
' % ( ) % *
$
•
% (
⎧
⎨
x1 + 3x2 + x3 = 0,
3x1 + 2x2 + 2x3 = 5,
⎩ 5x + 3x + 4x = 10.
1
2
3
•
+$ &
$ , (
⎧
⎨ x1 + 3x2 + x3 = 0,
−7x2 − x3 = 5,
⎩
−12x2 − x3 = 10.
•
$-
x2
x3
⎧
⎨ x1 + x3 + 3x2 = 0,
x3 + 7x2 = −5,
⎩ −x − 12x = 10.
3
2
•
)$
⎧
⎨ x1 + x3 + 3x2 = 0,
x3 + 7x2 = −5,
⎩
−5x2 = −5.
x2 = −1
x3 = −5 − 7x2 = 2
x1 = −3x2 − x3 = (−3)(−1) − 2 = 3 − 2 = 1.
!" #
$%& !
"
⎧
⎨ x1 + 5x2 + 4x3 + 3x4 = 1,
2x1 − x2 + 2x3 − x4 = 0,
⎩
5x1 + 3x2 + 8x3 + x4 = 1.
' ! ( ! " )
*
#
⎛
1
5 4
⎝ 2 −1 2
5
3 8
⎛
1
∼⎝ 0
0
⎞ ⎛
1
5
4
3 1
−1 0 ⎠ ∼ ⎝ 0 −11 −6
1 1
0 −22 −12
⎞ ⎛
5 4 3 1
1 5 4
11 6 7 2 ⎠ ∼ ⎝ 0 11 6
11 6 7 2
0 0 0
1 5 4 3 1
∼
.
0 11 6 7 2
⎞
3
1
−7 −2 ⎠ ∼
−14 −4
⎞
3 1
7 2 ⎠∼
0 0
+ ! *) *
*) ,,)
*)
r(A) = r(B) = 2,
!"" # " "
x1 + 5x2 + 4x3 + 3x4 = 1,
11x2 + 6x3 + 7x4 = 2.
$ # x1
x3 x4 # %
#
x2
⎧
14
1
2
⎪
⎪
⎨ x1 = 11 − 11 x3 + 11 x4 ,
⎪
⎪
6
2
7
⎩
− x3 − x4 .
x2 =
11 11
11
x3 = 1, x4 = 1 # x1 = −1, x2 = −1
! " #
&'(
⎧
2x1 + x2 + x3 − x4 = 8,
3x1 + 2x3 + 5x4 = 12,
x1 − x2 + x3 = 4,
⎪
⎪
⎩ 8x + x + 5x + 3x = 10.
1
2
3
4
⎪
⎪
⎨
) * " #"
+ % # " ! # #
, "
-
⎞ ⎛
⎞
⎛
x1 x2 x3 x4
⎜ 2
1 1 −1
⎜
⎜ 3
0 2
5
⎜
⎝ 1 −1 1
0
8
1 5
3
⎛
x2 x1 x3 x4
⎜ 1 2 1 −1
⎜
5
∼⎜
⎜ 0 3 2
⎝ 0 3 2 −1
0 6 4
4
x2 x1 x3
⎟ ⎜ 1 2 1
⎟ ⎜
⎟∼⎜ 0 3 2
⎟ ⎜
⎠ ⎝ −1 1 1
1 8 5
⎞ ⎛
x2 x1 x3
⎜
8 ⎟
⎟ ⎜ 1 2 1
⎜
12 ⎟
⎟∼⎜ 0 3 2
12 ⎠ ⎝ 0 0 0
0 0 0
2
8
12
4
10
x4
−1 8 ⎟
⎟
5 12 ⎟
⎟∼
0 4 ⎠
3 10
⎞
x4
−1
8 ⎟
⎟
5
12 ⎟
⎟∼
0 ⎠
6
−6 −22
⎛
x2 x1 x3 x4
⎜ 1 2 1 −1
8
⎜
12
0
3
2
5
∼⎜
⎜
⎝ 0 0 0
0
1
0 0 0
0 −22
⎧
⎪
⎪
⎨
⎞
⎟
⎟
⎟.
⎟
⎠
x2 + 2x1 + x3 − x4 = 8,
3x1 + 2x3 + 5x4 = 12,
x4 = 0,
⎪
⎪
⎩ 0 · x + 0 · x + 0 · x + 0 · x = −22.
2
1
3
4
⎧
⎨ x + 2y + z = 0,
2x + y + z = 1,
⎩
x + 3y + z = 0.
! " #
⎛
⎞
1 2 1
A = ⎝ 2 1 1 ⎠,
1 3 1
⎛
⎞
0
C = ⎝ 1 ⎠,
0
⎛
⎞
x
X = ⎝ y ⎠.
z
$%
&' (
' )"
⎛
A−1
*
X=A
⎞
−2
1
1
0
1 ⎠.
= ⎝ −1
5 −1 −3
⎛
−1
⎞ ⎛ ⎞ ⎛
⎞
−2
1
1
0
1
0
1 ⎠· ⎝ 1 ⎠ = ⎝ 0 ⎠.
· C = ⎝ −1
5 −1 −3
0
−1
$ " x = 1 y = 0 z = −1
⎧
⎨
x + 2y − z = 2,
2x − 3y + 2z = 2,
⎩
3x + y + z = 8.
1
|A| = 2
3
1
Δy = 2
3
2 −1
−3
2 = −8,
1
1
2 −1
2
2 = −16,
8
1
2
2 −1
2 −3
2 = −8.
8
1
1
1
2 2
2 −3 2 = −24.
3
1 8
x=
Δx =
Δz =
−16
−24
−8
= 1, y =
= 2, z =
= 3.
−8
−8
−8
⎧
2x1 + x2 − x3 = 1,
3x1 + 2x2 − 2x3 = 1,
⎩
x1 − x2 + 2x3 = 5.
⎨
! "#$ %
⎞ ⎛
⎞
⎛
1 −1
2 5
2
1 −1 1
⎝ 3
2 −2 1 ⎠ ∼ ⎝ 2
1 −1 1 ⎠ ∼
1 −1
2 5
3
2 −2 1
⎛
⎞ ⎛
⎞
1 −2
2
1 −1
2
5
5
3 −5 −9 ⎠ ∼ ⎝ 0
6 −10 −18 ⎠ ∼
∼⎝ 0
0
5 −8 −14
0
5 −8 −14
⎞ ⎛
⎞
⎛
1 −1
2
1 −1
2
5
5
1 −2 −4 ⎠ ∼ ⎝ 0
1 −2 −4 ⎠ ∼
∼⎝ 0
0
5 −8 −14
0
0
2
6
⎛
⎞ ⎛
⎞
1 −1
2
5
&$
⎠
1 −2 −4 ⎠ ⎝
∼⎝ 0
0
0
1
3
⎧
⎨ x1 − x2 + 2x3 = 5,
x2 − 2x3 = −4,
⎩
x3 = 3,
x3 = 3, x2 = −4 + 6 = 2, x1 = 5 + x2 − 2x3 = 1.
x1 = 1 x2 = 2 x3 = 3
!" #
⎧
⎨ x1 + 2x2 + 4x3 − x4 − 3x5 = 7,
2x1 + x3 + x5 = 4,
⎩
x2 + 2x4 − x5 = 6.
$
% & '
( )
*
⎞ ⎛
⎞
1
2
4 −1 −3
1 2 4 −1 −3 7
7
⎝ 2 0 1
0
1 4 ⎠ ∼ ⎝ 0 −4 −7
2
7 −10 ⎠ ∼
6
0 1 0
2 −1 6
0
1
0
2 −1
⎛
⎞ ⎛
⎞
1 2
4 −1 −3 7
1
2
4 −1 −3
7
6 ⎠∼⎝ 0 1
1
0
2 −1
0
2 −1 6 ⎠ .
∼⎝ 0
0 −4 −7
2
7 −10
0 0 −7 10
3 14
⎛
+
&, )
• & !
•
& &
• )
& - & #
.
'
⎧
⎨ x1 + 2x2 + 4x3 − x4 − 3x5 = 7,
x2 + 2x4 − x5 = 6,
⎩
−7x3 + 10x4 + 3x5 = 14,
. &
)
⎧
⎨ x1 + 2x2 + 4x3 = 7 + x4 + 3x5 ,
x2 = 6 − 2x4 + x5 ,
⎩
−7x3 = 14 − 10x4 − 3x5 .
&,
x
x x x
1
2
3
x4
⎧
3
10
⎪
⎪
⎨ x3 = −2 + 7 x4 + 7 x5 ,
x2 = 6 − 2x4 + x5 ,
⎪
⎪
⎩ x = 3 − 5x − 5x .
1
4
5
7
7
5
x x
x x x x x
x x x
!" #
$# % x = 0 x = 0 x = −2 x = 6
x = 3 x = 7 x = 7 x = 11 x = −1 x = −7
&' (
⎧
4
1
5
2
3
4
1
2
5
3
5
1
5
4
3
2
4
3
2
1
x − y − z = 5,
2x + y + 3z = 3,
⎩
x − 4y − 6z = 7.
⎨
) % * % +
" ⎞ ⎛ ,
⎞ ⎛
⎞
⎛
1 −1
1
1 −1 1
1 −1
1 5
5
5
⎝ 2
1
3 3 ⎠∼⎝ 0
3
5 −7 ⎠ ∼ ⎝ 0
3 5 −7 ⎠ .
1 −4 −6 7
0 −3 −5
0
0 0 −5
2
+
• - . " −2 & /
& "
• 0 1 / " 1
"
• *
& 1 " /
/ +⎧
x − y − z = 5,
3y + 5z = −7,
⎩
0x + 0y + 0z = −5.
⎨
!" "" /
% /
!
"
" # $%&
B
_
a
A
a
' "
# (& ! # )&
* AB. + , * - " "
* . a, , " / 0 .
a
|a|
$
0
|0| = 0 1
2 , - -
-
* , "
3 a b
! " "
# $ a b
4 a b
#
$ a b
% a b
#
$ a ↑↓ b
a b
#
$ a = b
BC = AD AB ↑↓ CD
B
C
A
D
! ! " !
# a + b !
" # $% b !
" " & !
a ' ()
_
b
_
a
_ _
a+b
_
b
$ %!& ' a + b (
a " ) "
b
* % (
'+ b ! a
, , - .#/ a + b
a
a (−b) a b !"#
_
b
_
-b
_ _
a-b
_
a
$ O % OACB OC
"# # %
a + b BA
"# # % a − b
A
_
a
O
C
_ _
a-b
__
a+b
_
b
B
&' a
λ c a
|c| = |λ| · |a| a λ > 0,
a λ < 0
&'&2a a
a, − 12 a
a a
( # −a % %
a $ λ = −1) −a = (−1)a
λa = aλ
λ1 (λ2 a) = (λ1 λ2 )a
λ(a + b) = λa + λb
$ !
% a
b = λa
a = λb
b
!
"#
& b = λa a = λb b a
' !
& b a a = 0 b = λa ( ' ) '
b = 0 0 = 0 · a & a = 0 '
a = 0 = 0 · b
%
" a * ea
#
a + ! ' !
a = |a|ea
a
.
ea =
|a|
l
el
O
_
a
O
_
el
ϕ
1
l
a b
ϕ
!"
a
_
a
b
ϕ
_
b
ϕ
0 ϕ π
!" a l
a el
!# ! " l
# # α$ # %&! ' # #
l$ l ! AB l
$ # ! (
l AB = x2 − x1.
$ %
__
AB
_
el
α A
A1
x1
B
β
B1
x2
l
& ϕ
AB '( l
!"" x2 > x1 ) l AB ' ϕ
) l AB * ' AB ⊥ l (ϕ = 90◦) ) l AB = 0
AB l
AB l
el
AB l
l AB = !l AB · el = A1 B1 .
"
! a l #$ a $#%
#$ $ $& #%$ #
!l a = |a| cos ϕ.
' (
! a l (# )# &
# * #$ # # $ & a
% O +
B
a
X
O
B1
l
, -#$ !l a = x−0 = x .( #$&
x
& OBB1 : cos ϕ =
$ x = |a| cos ϕ !l a = |a| cos ϕ
|a|
! $##/ $##
!l (a + b) = !l a + !l b.‘
+
' (
!$ AC = AB +BC 0& !l AB = x2 −x1 !l BC =
= x3 − x2 !l AC = x3 − x1 1 2#
!l AC = x3 − x1 = (x2 − x1 ) + (x3 − x2 ) = !l AB + !l BC
0#
)& & #/3
B
C
A
X 1 X2
O A1
B2
X3
C3
e
l (λa) = λ l a.
!"
#
$ λ > 0
a % l & ϕ
λa % l & ϕ λ < 0
λa % l & π − ϕ
# & (λ > 0) l (λa) = |λa| cos ϕ =
= |λ||a| cos ϕ = λ l a
#
& (λ < 0)
l (λa) = |λa| cos(π − ϕ) = |λ||a| cos(π − ϕ) = −λ|a|(− cos ϕ) =
= λ|a| cos ϕ = λ l a
'
# l (a − b) = l (a + (−1)b) = l a + l (−1)b =
= l a + (−1) l b = l a − l b
(
⎧
⎨ 2x1 + 3x2 + 2x3 = 9,
x1 + 2x2 − 3x3 = 14,
⎩
3x1 + 4x2 + x3 = 16.
⎧
2x − x2 + x3 + 2x4 + 3x5 = 2,
⎪
⎪
⎨ 1
6x1 − 3x2 + 2x3 + 4x4 + 5x5 = 3,
6x1 − 3x2 + 4x3 + 8x4 + 13x5 = 9,
⎪
⎪
⎩ 4x − 2x + x + x + 2x = 1.
1
2
3
4
5
⎛
⎧
2x + x2 − x3 = 4,
⎪
⎪
⎨ 1
−x1 − 3x2 + 2x3 = 3,
3x1 + 4x2 − 3x3 = 1,
⎪
⎪
⎩ 4x + 7x − 5x = −1.
1
2
3
⎧
⎨ 2x1 + 3x2 + 2x3 = 9,
x1 + 2x2 − 3x3 = 14,
⎩
3x1 + 4x2 + x3 = 16.
⎛
2 3
A = ⎝ 1 2
3 4
⎛
⎞
x1
X = ⎝ x2 ⎠
x3
⎞
2
3 ⎠
1
⎞
9
B = ⎝ 14 ⎠ ! "
16
# ! X = A−1 B $%! & A−1
&' A
!" "( A
2 3 2
2 −3
− 3 1 −3 + 2 1 2 =
|A| = 1 2 −3 = 2
3 4
3 1
4
1
3 4 1
= 2 · 14 − 3 · 10 + 2 · (−2) = 28 − 30 − 4 = −6.
2 −3
3 2
= 5,
A11 =
= 14, A21 = −
4 1
4 1
3 2
1 −3
= −10,
= −13, A12 = −
A31 =
3 1
2 −3
2 2
= −4, A32 = − 2 2 = 8,
A22 =
1 −3
3 1
1 2
= −2, A23 = − 2 2 = 1,
A13 =
3 4
3 4
2 3
= 1,
A33 =
1 2
⎛
⎞
14
5 −13
1
A−1 = − 6 ⎝ −10 −4 8 ⎠ .
−2 1
1
⎛
⎞
⎛
⎞⎛
⎞
x1
14
5 −13
9
1
X = ⎝ x2 ⎠ = A−1B = − 6 ⎝ −10 −4 8 ⎠ ⎝ 14 ⎠ =
x3
−2 1
1
16
⎛
⎞
⎛
⎞ ⎛
⎞
126 + 70 − 208
−12
2
1
1
= − ⎝ −90 − 56 + 128 ⎠ = − ⎝ −18 ⎠ = ⎝ 3 ⎠ .
6
6
18 + 14 + 16
12
−2
! " # $
" % & %"
⎛
2 3 2 9
⎝ 1 2 −3 14
3 4 1 16
⎛
6
∼⎝ 0
6
⎞
⎛
⎞ ⎛
2 3 2 9
2
⎠ ∼ ⎝ 2 4 −6 28 ⎠ ∼ ⎝ 0
3 4 1 16
3
⎞ ⎛
2 3
2
9 6 27
1 −8 19 ⎠ ∼ ⎝ 0 1 −8
0 −1 −4
8 2 32
⎛
⎞
2 3 2
9
∼ ⎝ 0 1 −8 19 ⎠ .
0 0 −12 24
' ( " " ) %
• * +&
⎞
3 2 9
1 −8 19 ⎠ ∼
4 1 16
⎞
9
19 ⎠ ∼
5
•
•
•
•
⎧
⎨ 2x1 +3x2 +2x3 = 9,
−8x3 = 19,
x2
⎩
−12x3 = 24,
!
9 − 3x2 − 2x3
=
x3 = −2 x2 = 19 + 8x3 = 19 − 16 = 3 x1 =
2
9−9+4
= 2.
2
"
⎧
2x − x2 + x3 + 2x4 + 3x5 = 2,
⎪
⎪
⎨ 1
6x1 − 3x2 + 2x3 + 4x4 + 5x5 = 3,
6x1 − 3x2 + 4x3 + 8x4 + 13x5 = 9,
⎪
⎪
⎩ 4x − 2x + x + x + 2x = 1.
1
2
3
4
5
# $
⎛
⎜
⎜
⎜
⎜
⎝
⎛
x1
2
6
6
4
1
⎜ 0
∼⎜
⎝ 0
0
⎛
x3
⎜ 1
⎜
∼⎝
0
0
!
⎞
⎛
x3
⎜
2 ⎟
⎟ ⎜ 1
⎜
3 ⎟
⎟∼⎜ 2
9 ⎠ ⎝ 4
1
1
⎞ ⎛
2 −1 2
3
2
1
⎜ 0
2 −1 0 −1 −1 ⎟
⎟∼⎜
−2 1
0
1
1 ⎠ ⎝ 0
2 −1 −1 −1 −1
0
⎞ ⎛
x1 x2 x4 x5
x3
⎜
2 −1 2
3
2 ⎟
⎟∼⎜ 1
2 −1 0 −1 −1 ⎠ ⎝ 0
0
2 −1 −1 −1 −1
x2 x3 x4 x5
−1 1 2 3
−3 2 4 5
−3 4 8 13
−2 1 1 2
x1
2
6
6
4
x2 x4 x5
−1 2 3
−3 4 5
−3 8 13
−2 1 2
⎞
2
3
9
1
⎟
⎟
⎟∼
⎟
⎠
⎞
−1 2
3
2
−1 0 −1 −1 ⎟
⎟∼
−1 0 −1 −1 ⎠
−1 −1 −1 −1
⎞
x2 x1 x4 x5
−1 2
2
3
2 ⎟
⎟∼
1 −2 0
1
1 ⎠
−1 2 −1 −1 −1
2
2
2
2
⎛
⎞
x3 x2 x1 x4 x5
⎜ 1 −1 2
2 3 2 ⎟
⎟
∼⎜
⎝ 0 1 −2 0 1 1 ⎠ ⇒ x4 = 0.
0 0
0 −1 0 0
•
•
!"
# ! $ !
• # (−1)
•
•
• %
& ! $ $ x2 x3 x4 x1
x5
x2 = 1 + 2x1 − x5 x3 = 2 + x2 − 2x1 − 3x5 = 3 − 4x5 %
' xn = x5 = 0 ! (
⎛
⎜
⎜
X=⎜
⎜
⎝
0
1
3
0
0
⎞
⎟
⎟
⎟.
⎟
⎠
)%#
⎧
2x + x2 − x3 = 4,
⎪
⎪
⎨ 1
−x1 − 3x2 + 2x3 = 3,
3x1 + 4x2 − 3x3 = 1,
⎪
⎪
⎩ 4x + 7x − 5x = −1.
1
2
3
* (
⎛
⎜
⎜
∼⎜
⎜
⎝
x1 x2
2
1
−1 −3
3
4
4
7
x3
−1 4
2
3
−3 1
−5 −1
⎞
⎛
⎞
1 3 −2 −3
⎟
⎟ ⎜ 2 1 −1 4 ⎟
⎟∼⎜
⎟
⎟ ⎝ 3 4 −3 1 ⎠ ∼
⎠
4 7 −5 −1
⎛
⎞
⎛
⎞
1 3 −2 −3
1 3 −2 −3
⎜ 0 −5 3 10 ⎟
⎟ ⎝ 0 −5 3 10 ⎠ ∼
∼⎜
⎝ 0 −5 3 10 ⎠ ∼
0 −5 3 11
0 −5 3 11
⎞
1 3 −2 −3
∼ ⎝ 0 −5 3 10 ⎠ .
0 0
0
1
⎛
•
•
•
• !
" 0 · x1 + 0 · x2 + 0 · x3 = 1
!!
# !
$!
⎧
⎨ 2x1 − x2 + x3 = 2,
3x1 + 2x2 + 2x3 = −2,
⎩
x1 − 2x2 + x3 = 1.
$!%
⎧
⎪
x1 + 5x2 − 2x3 − 3x4 = 1,
⎪
⎪
⎪
⎨ 7x1 + 2x2 − 3x3 − 4x4 = 3,
x1 + x2 + x3 + x4 = 5,
⎪
⎪
2x1 + 3x2 + 2x3 − 3x4 = 4,
⎪
⎪
⎩ x − x − x − x = −2.
1
2
3
4
a
! "#$% & '
(
a
) "#$
a = OM = OP + P M = OP + OM3 = (OM1 + OM2 ) + OM3 .
* OM1 OM2 OM3 + a
* , !-$ .% / OM1 =
= (Ox a · i OM2 = (Oy a · j OM3 = (Oz a · k (Ox a = ax
(Oy a = ay (Oz a = az /
!-0 "%
a = ax · i + ay · j + az · k.
ax ay az
a ! "
# a = (ax; ay ; az )
-0 " $ rM % M(a1 ; a2; a3)
OM % % &
' %
* , -0 "
rM = OM = (a1; a2; a3 ).
AB
A(x1 ; y1 ; z1 ) ' 1 B(x2 ; y2 ; z2 ) ) '
(Ox AB = x2 − x1 (Oy AB = y2 − y1 (Oz AB =
AB / ax = x2 − x1
= z2 − z1 . ( 2
ay = y2 − y1 az = z2 − z1 +
/
!-0 -%
AB = (x2 − x1 ) · i + (y2 − y1 ) · j + (z2 − z1 ) · k.
3 AB
' , /
AB = rB − rA .
AB = OB − OA = rB − rA .
|M1 M |
|M1 M |
= λ M1 M = λMM2
|MM2 |
|MM2 |
(x − x1 )i + (y − y1 )j + (z − z1 )k = λ((x2 − x)i + (y2 − y)j + (z2 − z)k)
=
M1
r
r1
M
r2
O
M2
= λ(x2 − x), y − y1 = λ(y2 − y), z − z1 = λ(z2 − z)
z
x=
x1 + λx2
,
1+λ
y=
y1 + λy2
,
1+λ
!
Oxy
x − x1 =
x, y
z1 + λz2
.
1+λ
M1 M2 !" #
z=
$ !%
!& '
M1(1; 2; 3) M2(2; 1; 1)
M1M2
( )
*
M1 M2 λ =
!+
x=
-
1+2
2+1
3+1
= 1, 5, y =
= 1, 5, z =
= 2
2
2
2
M(1, 5; 1, 5; 2)
. #
0
!
% '
*
1 2 !
#
α β
γ
#
#
345 . + #
!
M1 M
= 1
MM2
%
!
/
! /
"
/
cos α cos β
cos γ
6
a = ax · i + ay · j + az · k 7#
ax = 6Ox a =
= |a| · cos α, ay = 6Oy a = |a| · cos β , az = 6Oz a = |a| · cos γ
ay
az
ax
; cos β =
; cos γ =
.
cos α =
|a|
|a|
|a|
z
γ
β
O
y
α
x
|a|
ax
ay
cos α = 2
; cos β = 2
;
ax + a2y + a2z
ax + a2y + a2z
az
cos γ = 2
.
ax + a2y + a2z
! "
cos2 α + cos2 β + cos2 γ = 1.
#$
% & '( ) α β γ *
+ ! , #$
-) + *&) ) "
ea ) *. )
ea = cos α · i + cos β · j + cos γ · k.
##
% *. a !
, /#
'( '
0 ) &+ ) +
1) 2 "
"
)
3 AB
A(1; 2; −2) B(2; −1; 0)
AB AB = (2 − 1)i + (−1 − 2)j + (0 − (−2))k = i − 3j + 2k
√
3
2
AB
1
|AB| = 12 + (−3)2 + 22 = 14 eAB =
= √ i − √ j + √ k
|AB|
14
1
3
2
cos α = √ cos β = − √ cos γ = √
14
14
14
14
14
a = ax i + ay j + az k b = bxi + by j +
!" b = λa a =
bz k
λb # $
%&'
bx = λax by = λay bz = λaz ax = λbx ay = λby az = λbz
# λ # (' $
bx
by
bz
=
=
ax
ay
az
ax
ay
az
=
= .
bx
by
bz
a b
) *" +
, -# $ -
-
-' - ' -
.
*/ a = (1; 3; 5)
b = (2; 6; 0)
1
3
5
= =
2
6
0
5 = λ · 0
1
$$ λ =
2
a b
*0 a = (1; 3; 0)
b = (2; 6; 0)
%- λ
a b
1
3
0
= =
2
6
0
0 = λ·0
- # 1
. 2$
a b
a · b = |a| · |b| · cos ϕ.
a
b
b
a
a b
a · b
(a, b)!
!a b = |b| cos ϕ " #
" $% & ' a · b = |a| · !a b
'
a·b
.
!a b =
|a|
a · b = b · a $
( ) ' a · b = |a| · |b| · cos ϕ = |b| · |a| · cos ϕ = b · a
λ(a · b) = (λa) · b = a · (λb) * $
( ) λ > 0' λ(a · b) = λ|a| · |b| · cos ϕ =
= |λa| · |b| · cos ϕ = (λa) · b + )$ & #
, &
a(b + c) = a · b + a · c
- $
( ) )
*
& ' a(b + c) = |a| · !a (b + c) = |a|(!a b + !a c) = |a| · !a b +
+ |a| · !a c = a · b + a · c
( , & -$ $. -$ $
-.
& & -$ . #
) '
a ⊥ b,
|a| = |0,
|b| = 0
⇐⇒
a · b = 0.
/
( )
a ⊥ b , ϕ = (a;
b) = 90◦ =⇒ cos ϕ = 0
a · b = |a| · |b| · cos ϕ = 0 0 -
a · b = 0 |a| · |b| · cos ϕ = 0 =⇒
% 1 |a| = |0
|b| = 0
cos ϕ = 0 =⇒ ϕ = 90◦ a ⊥ b
a · b > 0 ⇐⇒ (a;
b)
a · b < 0 ⇐⇒ (a; b)
! " " "! " #
a2 = |a|2 .
$
2
◦
2
%"&! a = a·a = |a|·|a|·cos 0 = |a| '
$
√
a2 = |a|.
) "" "&!*
√
x2
= |x|
(
$ (a − 2b)2 |a| = 1 |b| = 2 (a;
b) = 60◦
2
+ , ! (a−2b)2 = a2 −4a·b+4b = |a|2 −4|a|·|b|·cos 60◦ +4|b|2 =
1
= 1 − 4 · 1 · 2 · + 4 · 4 = 13
2
-" (a − 2b)2 = 13
% ! " .! ! # "! " * " /
2
" * ! i2 = j 2 = k = 1 i·j = i·k =
= j · k = 0
0 a = ax · i + ay · j + az · k b = bx · i + by · j + bz · k a · b =
2
2
= (ax · i + ay · j + az · k)(bx · i + by · j + bz · k) = ax · bx · i + ay · by · j +
2
+ az · bz · k + ax · by · i · j + ... + ay · bz · j · k = ax · bx + ay · by + az · bz
- !&!
a · b = ax · bx + ay · by + az · bz .
(
a = (1; 0; 2) b = (−2; 1; 3)
+ , ! a · b = 1 · (−2) + 0 · 1 + 2 · 3 = 4 > 0 =⇒ # (a;
b)
1"
" " 2 ! # "!
! ! " * " " ! 3*
! 3 " ! & "!"
ax · bx + ay · by + az · bz = 0.
4
a = (1; −3; m)
b = (2; 1; 4)!
1
a ⊥ b ⇐⇒ a·b = 0 ⇒ 1·2+(−3)·1+4m = 0 ⇒ m =
4
1
a⊥b m=
4
! "
# $
cos ϕ =
ax bx + ay by + az bz
a·b
.
= 2
ax + a2y + a2z · b2x + b2y + b2z
|a| · |b|
%
& ' ( cos ϕ ( ) ( *
% ' ' )' ) '
+ + , - (
(
.
a = (1; 2; 3)
b = (−2; −1; −1)
/ " + %
cos ϕ = √
7
=− √
2 21
12
1 · (−2) + 2 · (−1) + 3 · (−1)
−7
√ =
=√
2
2
2
2
2
14 · 6
+ 2 + 3 · (−2) + (−1) + (−1)
$ 0 ) ( '
( ( cos ϕ(0 ≤ ϕ ≤ π) /
+' cos ϕ < 0
$ *1 + *2 ( ϕ 3)4
) ' ( *2
7
cos ϕ = − √ ≈ −0, 764 =⇒ ϕ ≈ 2, 44
2 21
cos ϕ = −
7
√ ,
2 21
ϕ ≈ 2, 44
ABCD O
AB = a AD = b.
!" # CD CB CO BD
CD = −a |CD| = |AB| = |a| CD ↑↓ a CB = −b
1
1
1
CO = CA = (−AC) = − (a + b) AC = a + b
2
2
2
O CA = −AC BD = BC + CD =
= AD + CD = b − a
B
C
_
a
A
_
b
O
D
AC = b
BC AN KO
ABC O
AB = a
a b
B
A
M
N
a O
A
C
b K
BC = BA + AC = −a + b = b − a AN = AB + BN =
1
1
1
1
1
1
1
= a + BC = a + (b − a) = a + b − a = a + b = (a + b)
2
2
2
2
2
2
2
ABC
1
1
ABA C AN = AA = (a + b)
2
2
KO !
1
1
2:1,
! " # KO = |KB| KO = KB =
3
3
1
1 1
1 1
1
1
1
1
= (KA + AB) = (− AC + a) = (− b + a) = (a − b) = a − b
3
3 2
3 2
3
2
3
6
$ % ! CN , BO, CO, OM
a−b
b − 2a
a − 2b
a − 2b
BO =
CO =
OM =
& CN =
2
3
3
6
O
AB = a AF = b. !
" " # ! $% & DE OB OC AD BC CF
' # !
B
C
_
a
A
D
_
b
O
F
E
( & DE = −a OB = −b OC = a AD = 2(a + b) BC = a + b
CF = −2a
ABC CA = a CB = b M N
AB ! ) CM *
B
N
M
A
b
a
C
1
1
AB = b−a AM = AB = (b−a) CM = CA+AM =
3
3
1
2a + b
= a + (b − a) =
3
3
AC = c
ABC AM
) AM
+ , AB
= b
|BM|
|BM |
|AB|
|b|
|b|
⇒
,
=
=
=
|c|
|MC|
|AC|
|BC|
|b| + |c|
|BC| = |BM | + |MC|
BC = c − b
|b|
(c − b) ⇒,
|b| + |c|
BM =
⇒ AM = AB + BM = b +
|b|
|b| · c + |c| · b
(c − b) =
.
|b| + |c|
|b| + |c|
B
b
A
M
C
c
! " ABC r1 r2 r3
r M
O
B
_
r2
_
r
_
O
_
r1
M
D
r3
A
C
1
1
BC = r3 − r2 BD = BC = (r3 − r2 )
2
2
r3 − r2
1
+r 2 −r1 = (r3 +r2 −2r1 )
BC AB = r2 −r 1 AD = BD +AB =
2
2
2
1
1
AM = AD = (r3 + r2 − 2r1 ) ⇒ r = OM = r1 + AM = r1 + (r3 + r2 −
3
3
3
1
− 2r1 ) = (r1 + r2 + r3 )
3
a = 3i + 4j + 5k
√
√
32 + 42 + 52 = 50 = 5 2,
√
√
√
3 2
4 2
5 2
3
; cos β =
; cos γ =
.
cos α = √ =
10
10
10
5 2
|a| =
√
ABCD
A(3; 2; −2) B(4; 4; 1) C(−1; 2; 0) D(−3; −2; −6)
! " # $%
% ! # &'( (
) %#!
a = i + 3j − k
√
|a| = 12 + 32 + (−1)2 = 11
! '! * & $ +
'! a '
ea =
a
1
3
1
= √ i + √ j − √ k.
|a|
11
11
11
) '! '! '! a,
−ea
1
3
1
−ea = − √ i − √ j + √ k.
11
11
11
, a · b
a = 2i − 3j + k b = −i + j + 3k
+
a · b = 2 · (−1) + (−3) · 1 + 1 · 3 = −2.
a · b < 0 a b
a − b l
|a| = 2 |b| = 1 a b
l
π/3
π/4
√
l a = |a| · cos(π/3) = 2 · (1/2)√= 1 l b = |b| ·
2/2 l (a − b) = l a − l b = 1 − 2/2
m a(2; 3; 5)
· cos(π/4) =
b(−2; 1; m)
! " ! # $ !"
%
1
a · b = 0 ⇒ 2 · (−2) + 3 · 1 + 5 · m = 0 ⇒ m = .
5
& (3a − 2b) · (2a + 3b) |a| = 1 |b| = 2
a ⊥ b
a ⊥ b ⇒ a · b = 0 a2 = |a|2 = 1 b = |b|2 = 4 '
2
2
2
(3a − 2b)(2a + 3b) = 6a − 6b + 9ab − 4ab = 6 · 1 − 6 · 4 = −18.
( a = (−1; 0; 3)
b = (2; 1; 0)
) ! ! !
*
√
ab
2
2
−1 · 2 + 0 + 0
√
=−
.
cos ϕ =
√
=−
=
5
|a||b|
50
(−1)2 + 32 22 + 12
6 √ 7
√
2
2
ϕ = arccos −
= π − arccos
.
5
5
+ a b
|a| = 2 |b| = 3
(2a − b) ⊥ (a + 2b)
ABC M BC
AB = a AC = c AM
ABC r1 = i +
+2j + 3k r2 = 3i + 2j + k r3 = i + 4j + k
ABC ! " #
A(1; −5; 3)
B AB = i − j + 5k
" $
%&' " %(
2a − 3b a = −i + 2k b = 2i + j − 2k
" %&'
) " # ) AB A(2; −3; 1)
B(1; −1; −1)
O ! " % (
ABC AO + BO + CO = 0
" $ & 2a − 3b # l
a b !
* +&' #& l , 2π/3 3π/4
" $ & - b(3a−2b) a = 2i+j−3k
b = j − k . " #+ " $ / ( 0
1" ! " , %
& AB CD A(1; 2; 1) B(−2; 0; 2) C(−3; 1; 2) D(2; 5; 0)
- a+b(2a − 3b) a = (1; −2; −3)
b = (−3; 2; 1)
- ( + " ( s (2a−b) ⊥
a = (0,5; 2; 5) b = (s; 2; −1)
⊥ (a + b)
, ( ( AC BD %
) " ( A B C D
, B ABC % )
" ( 2 A B C
a, b, c
a, b, c c
b ! a
" !
# ! $ a ! b
% a, b, c ! ! !
& a, b, c
! '
! "
!
( !
a ! b c
)
*% c ! )
c ⊥ a c ⊥ b+
,% ! c - !)
|c| = |a| · |b| · sin (a;
b);
.% a, b, c ! !
!
a b
a × b
[a, b]
a
!
"#
a × b $
b
_
b
_ _
a xb
_
a
a × b = −b × a % &
' $
# a×b b×a
( |a| · |b| · sin (a;
b) = |b| · |a| · sin (b;
a)
)# $
λ(a × b) = (λa) × b %" &
' ) * $ λ > 0(
λ(a × b)
)
a × b (λa) × b λ > 0
) '# *
) ( |λ(a × b)| =
= |λ| · |a × b| = λ · |a| · |b| · sin (a;
b), |(λa) × b| = |λa| · |b| · sin (a;
b) =
= λ|a| · |b| · sin (a; b)
+, $ $ $ λ < 0
- a × (b + c) = a × b + a × c % &
.
/ 0
a b,
|a| = |0,
|b| = 0
⇐⇒ a × b = 0.
! a b ϕ = (a;
b) = 0◦ =⇒ sin ϕ = 0
a×b = 0 |a×b| = 0 " a×b = 0 |a|·|b|·sin (a;
b) = 0 =⇒
#! $ % |a| = |0 |b| = 0
8b) = 0 (a;
sin(a;
b) = 0 (a;
b) = π a b
& ' ! a × a = 0
( ! ! )
* +!
, -
# .&/
_
b
_
a
! |a × b| = |a| · |b| · sin (a;
b)
0!
- ! ! 1
. (2i+ j − k) × j − i × (j − 2k)
2 1 % !
(2i+j −k)×j−i×(j−2k) = 2i×j +j ×j −k×j −i×j +2i×k =
= i × j + 2i × k + j × k
# i × j |i × j| = |i| · |j| · sin 90◦ = 1
k .&3 i × j = k 4
i × k = −j j × k = i 5
i × j + 2i × k + j × k = k − 2j + i = i − 2j + k
6 i − 2j + k
_
k
_
j
_
i
i j j
i×i=j×j =k×k = 0
i × j = k,
j × i = −k,
j × k = i,
k × i = j,
k × j = −i,
i × k = −j.
!
"#$%
a × b = (ax · i + ay · j + az · k) × (bx · i + by · j + bz · k) =
= ax by · i × j + ax bz · i × k + ay bx · j × i + ay bz · j × k+
+ az bx · k × i + az by · k × j = ax by · k − ax bz · j − ay bx · k+
+ ay bz · i + az bx · j − az by · i = (ay bz − az by )i−
− (ax bz − az bx )j + (ax by − ay bx )k.
&
ay az
ax az
ax ay
· k.
·i−
·j+
a×b=
by bz
bx bz
bx by
'"#$()
!
" # ! $%&
' $ ()*
L
c
d=a b
ϕ
_
b
_
a
+ # ' d = a × b ,
,
a b
- |a × b| = S .
! (a b c) = (a × b) · c = |a × b| · |c| · cos ϕ
π
ϕ / d c 0 ()* $/ ϕ <
2
+ ' h = |c| cos ϕ
1 #- (a b c) = |a × b| · |c| · cos ϕ = S · h = V $%& ,
π
$/& ()* 2 ϕ >
2
h = −|c| · cos ϕ cos ϕ < 0 a · b · c = −V 1 # V = |(a b c)|
* 3
A(2; −1; −1) B(5; −1; 2) C(3; 0; 3)
D(6; 0; −1)
4 ! - 4 DA = (−4; −1; 0) DB =
(−1; −1; 3) DC = (−3; 0; −2) . ! # ,
$%& $ DA DB DC !#
#! $%& ' $-
a b c
(a × b) × c (a × b) × c =
= a × (b × c) = a × b × c a × b × c = c × a × b = b × c × a a × b × c =
= −b× a × c = −c× b × a = −a× c × b = ! " #
$ i × j × k % i × j = k ! &
'( )# i × j × k = 0 *
&
a × b × c × d
$ + ,
'( )
a = 2i − 3j + k b = −i + j + 3k
-
a×b
$ . !'( /#
i
j
a × b = 2 −3
−1 1
k
1
3
= i −3 1
1
3
2
1
−3
−j 2
+
k
−1 3
−1 1 =
= i(−9 − 1) − j(6 + 1) + k(2 − 3) = −10i − 7j − k.
j × i + 3j × k − 5k × i + (3i + 5j − k) × (i − 6j + 5k).
i×j = k i×k = −j j ×k = i
! "
−i × j + 3j × k + 5i × k + 3i × i − 18i × j + 15i × k + 5j × i − 30j × j+
+25j × k − k × i + 6k · j − 5k · k = −k + 3i − 5j + 3 · 0 − 18k − 15j − 5k−
−30 · 0 + 25i − j − 6i − 5 · 0 = 22i − 21j − 24k.
A(1; 1; 1) B(1; 2; 3) C(−1; 2; 1)
ABC
ABC
AB AC
!
1
SABC = AB × AC .
2
"# $ AB AC
AB(0; 1; 2), AC(−2; 1; 0),
i
j k
1 2
0
0
2
1
AB × AC = 0
−j
+k
=
1 2 = i
1 0
−2 0
−2 1
−2 1 0
= −2i − 4j + 2k.
%$& ! ' ( )*+
√
√
AB × AC = (−2)2 + (−4)2 + 22 = 24 = 2 6.
SABC =
, &
√
1 √
· 2 6 = 6
2
-
p − q 2p + q p q
60◦
"# &
(p − q) × (2p + q) = 2p × p + p × q − 2q × p − q × q =
= 2 · 0 + p × q + 2p × q − 0 = 3p × q.
. & # $
&
S = |3p × q| = 3|p| · |q| · sin 60◦ = 3 · 1 · 1 ·
√
√
3 3
3
=
.
2
2
CD ΔABC !"
/
CD = h
AB
1
SΔABC = |AB| · h
2
√
!" SΔABC = 6!
# AB = (0; 1; 2)
|AB| =
√
1√
6=
5h
2
√
12 + 22 =
√
5,
$
√
√
2 30
2 6
.
h= √ =
5
5
!%
a = (2; −1; 5) b = (−2; 3; 0)
a
b, & $
i
j
k
−1 5
a × b = 2
−1 5 = i
3
0
−2 3
0
2
5
−1
−j 2
+
k
−2 0
−2 3 =
= −15i − 10j + 4k !
' (
a×b
−15i − 10j + 4k
15
10
4
= √
= −√
i− √
j+√
k.
|a × b|
152 + 102 + 42
341
341
341
e ) &
−e!
e=
#
)
!*
a = (1; −1; 2) b = (0; 1; 2) c = (2; 0; 1)
+ !,- $
1 −1 2
1 2
2 = 1 ·
(abc) = 0 1
0 1
2 0
1
+1· 0 2
2 1
= 1 − 4 − 4 = −7.
+2· 0 1
2 0
=
A(2; 2; 2), B(4; 3; 3), C(4; 5; 4), D(5; 5; 6)
!" #
V $ " % & !!!
V $ ' AB AC AD ("$! V =
1
= S H V = S H S S H ) $$ !
3
* $+ # !!! ' *+ #$ ,$!
$ S = 21 S -!* ! !
!* !!! . ! V = 16 V /#$!
# $!#' AB = (2; 1; 1) AC = (2; 3; 2)
AD = (3; 3, 4) " $
2 1 1
(AB AC AD) = 2 3 2
3 3 4
=2· 3 2
3 4
−1· 2 2
3 4
+1· 2 3
3 3
=
= 2 · 6 − 1 · 2 − 1 · 3 = 7.
! V = 61 V = 16 |AB AC CD| = 76
A
0 #$ # #$ !
!! " H 1 !# V = S H (!+ '2+
S !* $+ !!! #$!
BC ×BD /#+ 3 # !
! BC = (0; 2; 1), BD = (1; 2; 3)
i j k
BC × BD = 0 2 1
1 2 3
=i· 2 1
2 3
−j· 0 1
1 3
= 4i + j − 2k.
√
S = |BC × BD| = 42 + 12 + (−2)2 = 21.
√
7
7 = 21H ! H = √ .
21
+k· 0 2
1 2
4 +
=
N
.
Mo ( x o ; yo ; z o)
M(x;y;z)
−y0 )j+(z −z0 )k
r = OM r0 = OM0
M
M0 M = r − r0
!
M0
(r − r0 ) · N = 0.
$ %
"#
&
'( &
)
*
"
%
A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0.
& M
+(
"
M0 M ⊥ N
"#
M0 (1; 2; 3) N = (2; −1; 1)
,
-
"
$ )
A = 2 B = −1 C = 1 .
2(x − 1) − 1(y − 2) + 1(z − 3) = 0
2x − y + z − 3 = 0
/
2x − y + z − 3 = 0
"# )
r · N + D = 0,
0
"
D = −r0 · N
$
'
%
(
()
#
Ax + By + Cz + D = 0.
"
A = 0
A B C
%
# $
!
&'
D
+ B(y − 0) + C(z − 0) = 0.
A x+
A
(
N = (A; B; C)
)
!
*
M0 −
D
; 0; 0
A
+
! " # " $
#
! %
,)
% *
'
*
M1 (x1 ; y1 ; z1 ) M2 (x2 ; y2 ; z2 ) M3 (x3 ; y3 ; z3 )
+-+
./ r1 , r2 , r3 , r 0 ! )
M(x; y; z)
M1
M
r
- r1 r M2
2
* !
M3
r3
O
,
= M1 M2 × M1 M3
+
)/
)
'
(r − r1 )((r2 − r1 ) × (r3 − r1 )) = 0.
,
N =
!
((r − r1 )(r3 − r1 )(r2 − r1 )) = 0.
!"
# $% & ' " (
) &
x − x1 y − y1 z − z1
x2 − x1 y2 − y1 z2 − z1
x3 − x1 y3 − y1 z3 − z1
= 0.
!
!*"
+ !" && , ' A B C
%-' )
. && , D = 0 (
' O(0; 0; 0) % %
Ax + By + Cz = 0.
. && , A = 0 ')
Ox
N = (0; B; C; ) i = (1; 0; 0)"
By + Cz + D = 0 Ox
. && , ' A = 0 D = 0 By + Cz = 0
( Ox Ox (
. && , ' A = 0 B = 0 ')
Oz
N = (0; 0; C) k = (0; 0; 1)
Cz + D = 0 Oxy
Oz
. && , ' A = B = D = 0
z = 0 ) % Oxy
( "
Oxy
ϕ
α1
α2
ϕ
_
N2
_
N1
α1 : A1 x + B1 y + C1 z + D1 = 0,
α2 : A2 x + B2 y + C2 z + D2 = 0,
N 1 = (A1 ; B1 ; C1 )
N 2 = (A2 ; B2 ; C2 )
A 1 A 2 + B1 B2 + C 1 C 2
N1 · N2
cos ϕ = = 2
.
N 1 · N 2
A1 + B12 + C12 · A22 + B22 + C22
+1=0
x − y + 3z = 0
! "
2x + y − z +
#
2 · 1 + 1 · (−1) − 1 · 3
2
⇒
cos ϕ =
= −√
2
2
2
2
2
2
66
2 + 1 + (−1) · 1 + (−1) + 3
2
⇒ ϕ = π − arccos( √ ) ≈ 1, 82.
66
2
$ cos ϕ = − √ .
66
! "" " #!$
#!
%
" " !
& "
"
"
! "" " ' !$
A1 · A2 + B1 · B2 + C1 · C2 = 0.
' !
2x−3y+5 = 0
mx + 7y − 6 = 0
A1
B1
C1
=
=
,
A2
B2
C2
0
= .
( ) $ % " #!$ m2 = −3
7
0
14
=⇒ m = − % *
() m2 = −3
7
3
2x − 3y + 5 = 0 − 143 x + 7y − 6 = 0 +
2x − 3y + 5 = − 143 x + 7y − 6 "" " ,
- $ m = − 143
% M0(x0; y0; z0) α : Ax + By + Cz +
, " d M0 α
" M0 M1 α '. ! " "
/ $
|Axo + Byo + Czo + D|
√
d=
.
''!
2
2
2
+D = 0
A +B +C
M0
_
N
M1
α
N
α : N = (A; B; C) M1 (x1 ; y1 ; z1 ) d = M1 M0
M1M0 M1M0 = (x0 − x1)ı + (y0 − y1)j + (z0 −
−z1 )k N M1 M0 ϕ = 0 ϕ = π !
" "
N · M1 M0 = N · M1 M0 · cos ϕ = ± N · M1 M0 .
' ( &)%)
N · M1 M0 = A(x0 − x1 ) + B(y0 − y1 ) + C(z0 − z1 ).
#$%&
#$%#
* M1 ∈ α + Ax1 + By1 + Cz1 + D = 0 =⇒ −Ax1 −
− By1 − Cz1 = D N · M1 M0 = Ax0 + By
0+
1 − Cz1 =
√ Cz0 − Ax1 − By
= Ax0 +By0 +Cz0 +D , " N = A2 + B 2 + C 2 , M1 M0 = d
( #$%& #$%#
√
± A2 + B 2 + C 2 · d = Ax0 + By0 + Cz0 + D,
d=±
Ax0 + By0 + Cz0 + D
√
A2 + B 2 + C 2
d ≥ 0 ( #$%% - ( (
"" " . #$#
d=
r 0 · N + D
|N |
.
#$%/
0 1 ! #$/
/2
Ax + By + C = 0,
A2 + B 2 = 0
N = (A; B)
!
A1 x+B1 y+C1 = 0 A2 x+B2 y+C2 = 0
!
"
#
N1 · N2
A 1 A 2 + B1 B2
cos ϕ = = 2
.
N 1 · N 2
A1 + B12 · A22 + B22
$%&'()
*"
" +
" " N1 N2 #
B1
C1
A1
=
=
,
A2
B2
C2
"
$%&',)
N1 ⊥N2 #
A1 · A2 + B1 · B2 = 0.
$%&'-)
. M0 (x0 ; y0 ) Ax + By + C = 0
Oxy / $%&'0)#
d=
|Ax0 + By0 + C|
√
.
A2 + B 2
$%&'0)
1 " /
"
%&'
2x − 3y + z − 1 = 0
M1 (1, 0, −1)
2 +
M2 (1, 1, 1)
α
:
.2
# 3 M1 +
# 2 − 1 − 1 = 0 M1 ∈ α 1 M2
! # 2 − 3 + 1 − 1 = 0 M2 ∈
/ α
%&4
.2
# 3 /
$%&'') #
|2 − 3 + 1 − 1|
1
d=
=√ .
2
2
2
14
2 + (−3) + 1
M2
α
M0(2; 1; 1) 2x − 2y + 2z + 1 = 0
! " ! M0 # ! $ %
&
N = (2, −2, 2) '
& ( !
2(x − 2) − 2(y − 2) + 2(z − 1) = 0.
x − y + z − 2 = 0.
)
M0(2; 0; 3) N = (2; 2; −2)
*
!
2(x − 2) + 2(y − 0) − 2(z − 3) = 0 + x + y − z + 1 = 0
,
O(0; 0; 0) M1(−4; 2; −1) M2(−2; −4; 3)
* - !
x
x−0
y
z
y
−
0
z − 0
−4 − 0 2 − 0
−1 = 0 ⇔
−1 − 0 = 0 ⇔ −4 2
−2 −4 3
−2 − 0 −4 − 0 3 − 0
2
−4 −1
−1
+ z −4 2 = 0 ⇔
⇔ x ·
−
y
·
−4 3
−2 3
−2 −4
⇔ x(6 − 4) − y(−12 − 2) + z(16 + 4) = 0 ⇔ x + 7y + 10z = 0.
M1 (2; 0; −1) M2 (1; −1; 3)
3x + 2y − z − 5 = 0
*
! " ! # ! $ '
% N & M1 M2
N1 = (3; 2; −1)
M1 M2 × N1 . &
M1 M2 = (−1, −1, 4) &
i
N = M1 M2 ×N1 = −1
3
= −7i + 11j + k
j
−1
2
k
4
−1
= i· −1
2
−1
4
−j·
−1
3
−1 −1
4
+k·
−1
3
2
=
M1
N
−7(x − 2) + 11(y − 0) + 1(z + 1) = 0 − 7x + 11y + z + 15 = 0.
α1
α2 : 2x + y = 0
!"
# N1 = (1, −1, 2) N2 = (2, 1, 0)
$ % &'
x − y + 2z − 3 = 0
cos ϕ =
N1 N2
1
1 · 2 + (−1) · 1 + 2 · 0
=√ .
√
=
|N1 ||N2 |
30
12 + (−1)2 + 22 · 22 + 12 + 02
&
M0(0, 2, 1) a = i + j + k b = i + j − k
! " ( )
i j k
N = a × b = 1 1 1
1 1 −1
= −2i + 2j
(
=i· 1 1
1 −1
−j· 1 1
1 −1
+k· 1 1
1 1
=
$ % *'
x − y + 2 = 0
+
AB
A(−7; 2; −2), B(3; 4; 10)
!"
, Mo - AB
. $ %*& ' xo = −72+ 3 = −2 yo = 2 +2 4 = 3
−2 + 10
= 4 ( ) /
zo =
2
AB = (10; 2; 12)
10(x+2)+2(y−3)+12(z−4)
5x + y + 6z − 17 = 0
0 !
α1 : 2x + y − z + 3 = 0 α2 : −2x − y + z − 5 = 0
−2(x − 0) + 2(y − 2) = 0
M α1
α2
M xo = 0 yo = 0 zo = 3 M(0, 0, 3)
! "#$%%&
|3 − 5|
2
d=
=√ .
2
2
2
6
(−2) + (−1) + 1
#$%% M
AB 1 : 2 x − 3y + z − 6 = 0
A(2, 0, 1) B(−1, 3, 1)
#$%' A AB
A(−1, 2, −3) B(0, 2, 4)
#$%#
M1 (2, −5, 0) M2 (6, 0, 2) x + 5y +
+2z − 10 = 0
( )
*+ , -
-
- . /
" %01& "#%%& ! "#%'&
N1 · r + D1 = 0,
N2 · r + D2 = 0.
A1 x + B1 y + C1 z + D1 = 0,
A2 x + B2 y + C2 z + D2 = 0.
"#%%&
"#%'&
3
2 3
N 1 = (A1; B1; C1), N 2 = (A2; B2; C2), N 1
"#%%& "#%'& . . .
∦ N 2,
α1
l
α2
N1 N 2
__
N2
__
N1
M0 (x0 ; y0 ; z0 )
s = (m; n; p),
s
!
M(x; y; z) l " #$% &
OM = OM0 + M0 M . ' M0 M s =⇒ M0 M = s · t, (
t ∈ (−∞; +∞). ) M0 M
r0 = OM0 r = OM , &
r = r0 + t · s.
" %
z
_
s
M
l
M0
ro
r
0
y
x
M ∈ l.
t
r
*
r = OM = xi + yj + zk, r0 = OM0 = x0 i + y0 j + z0 k, ts =
= tmi + tnj + tpk.
⎧
⎨ x = x0 + tm,
y = y0 + tn,
⎩
z = z0 + tp.
M0 ! s. "
t x; y; z M(x; y; z)
! t ! "# $
t=
! &' &
y − y0
z − z0
x − x0
, t=
, t=
.
m
n
p
x − x0
y − y0
z − z0
=
=
,
m
n
p
%
#
( ) % * $
⎧
x − x1
y − y1
⎪
⎨
=
,
m
n
y
−
y
z
−
z
1
1
⎪
=
.
⎩
n
p
z − z1
x − x1
=
m
p
+
, *
-
, . ! ) % ! & & $
) ! +)
. & . ! * Oz, /
Ox 0 1 0
x − x1
y − y1
z − z1
=
=
,
0
n
p
s = (0; n; p).
s
x = x1 .
y = y1 ).
x − x1
y − y1
z − z1
=
=
0
0
p
Oz, s ⊥ Ox
s ⊥ Ox
s ⊥ Oy (x = x1,
4x − y − z + 12 = 0,
y − z − 2 = 0.
! " # $ %#
$ M0 &'$ s $
#( M0 $ ) *%
) + + z = 0, $ ) *
4x − y + 12 = 0,
y − 2 = 0,
⇔
5
x0 = − ,
2
y0 = 2.
,
- % ) M0(− 52 ; 2; 0) ∈ l $
#( &' s ) +
) # N 1 = (4; −1; −1) N 2 = (0; 1; −1)
# # $ ( &'
$ # + s l ./
s = N 1 × N 2 , s ⊥ N 1 , s ⊥ N 2 .
i j
k
0 s = 4 −1 −1
0 1 −1
= i(1 + 1) − j(−4) + k(4) = 2i + 4j + 4k.
1 + $ &
5
2 = y − 2 = z.
2
4
4
x+
5
2 = y−2 = z
2
4
4
x+
5
x+
2 = y − 2 = z = t,
t :
2
4
4
⎧
5
⎪
x
+
⎪
⎧
⎪
5
⎪
2 =t,
⎪
⎪
⎨
⎨ x = − + 2t,
2
2
⇔
y−2
y = 2 + 4t,
⎪
⎪
= t,
⎪
⎩
⎪
4
⎪
z = 4t.
z
⎪
⎩
= t.
4
⎧
5
⎪
⎨ x = − + 2t,
2
y = 2 + 4t,
⎪
⎩
z = 4t.
! " t
⎧
5
⎪
x+
⎧
⎪
⎪
5
⎪
5
2,
⎪
⎪
⎨ x = − + 2t,
⎨ t=
x+
y−2
z
2
2
2
y = 2 + 4t, ⇐⇒ ⎪ t = y − 2 , ⇐⇒ 2 = 4 = 4 .
⎪
⎩
⎪
⎪
4z
⎪
z = 4t.
⎪
⎩
t= .
4
⎧
5
⎪
x
+
⎪
⎨
2 = y − 2,
2x + 5 = y − 2,
2x − y + 7 = 0,
⇐⇒
⇐⇒
2
4
y
−
2
=
z.
y − z − 2 = 0.
⎪
z
y−2
⎪
⎩
= .
4
4
2x − y + 7 = 0,
y − z − 2 = 0.
!"#
$ %
!"# &
!'# (
! !
M0 (− 52 ; 2; 0) M1 (−1; 5; 3)
!"# !$# %
!"# !$# M0M1 .
% & ' ' &( '
M0 (x0; y0; z0) M1 (x1; y1; z1),
(
s = M0 M1 = (x1 − x0 )i + (y1 − y0 )j + (z1 − z0 )k. )
' &( '
x − x0
y − y0
z − z0
=
=
.
!!*#
x1 − x0
y1 − y0
z1 − z0
!+
M0 (− 52 ; 2; 0) M1(−1; 5; 3).
, - . / !!*#
.
5
5
x+
2 = y − 2 = z ⇐⇒
2 = y − 2 = z.
3
5
5−2
3
3
3
−1 +
2
2
x+
% (& ' & 0&
⎧ '.
5
⎪
x+
⎪
⎪
⎪
2 = y − 2,
⎨
3
3
⇐⇒
⎪
2
⎪
⎪
y−2
z
⎪
⎩
= ,
3
3
2x + 5 = y − 2,
⇐⇒
y − 2 = z,
2x − y + 7 = 0,
y − z − 2 = 0.
1 !$#
2 .
x − x1
y − y1
z − z1
=
=
,
m1
n1
p1
x − x2
y − y2
z − z2
l2 :
=
=
.
m2
n2
p2
l1 :
!!!#
s1 = (m1; n1; p1) s2 = (m2 ; n2; p2) :
cos ϕ =
s1 · s2
m1 m2 + n1 n2 + p1 p2
= 2
.
|s1 | · |s2 |
m1 + n21 + p21 m22 + n22 + p22
! "
m1
n1
p1
=
=
m
n
p
2
2
2
# $ %
& & ' $
&
"
m1 · m2 + n1 · n2 + p1 · p2 = 0.
(
%
l α"
y − y0
z − z0
x − x0
=
=
,
m
n
p
α : Ax + By + Cz + D = 0.
)
*
+& P (x1; y1; z1 ) & l
α# ' , & & -
( # ' x, y, z * #
& ' t1 #
) * "
l:
A(x0 + mt) + B(y0 + nt) + C(z0 + pt) + D = 0 =⇒
Ax0 + By0 + Cz0 + D
.
Am + Bn + Cp
.
% ' $ ' - ( #
"
x1 = x0 + t1 m, y1 = y0 + t1 n, z = z0 + t1 p.
/
=⇒ t1 = −
!" "
#
r = r0 + t · s,
r · N + D = 0.
$ r %
#
t1
t1 = −
r0 · N + D
.
N ·s
&% ϕ '
( ) %
% ' ! s
( N
#
cos(N ; s) = cos(90◦ − ϕ) = sin ϕ.
N
l
S
α
ϕ
p
* %
+,+-
#
Am + Bn + Cp
cos(N ; s) = √
.
A2 + B 2 + C 2 m2 + n2 + p2
sin ϕ = √
A2
Am + Bn + Cp
.
+ B 2 + C 2 m2 + n2 + p2
!"#
s # N
Am + Bn + Cm = 0
$%
& '
# (
!"# s #
N
m
n
p
=
= .
$
A
B
C
)
l,
l
!"
' * +"
A1 x + B1 y + C1 z + D1 = 0,
A2 x + B2 y + C2 z + D2 = 0.
$$
,& ( "( *
! $$
(A1 x + B1 y + C1 z + D1 ) + λ(A2 x + B2 y + C2 z + D2 ) = 0,
$
# λ !+- - * −∞
+∞.
#
$ !+ *
λ !
%) # A = A1 + λA2 , B = B1 + λB2, C = C1 + λC2, D = D1 + λD2
. ( * ! $$ * x; y; z
!" $$ +"! &
$ -& +( - !
! M0 (x0; y0; z0) ∈/ α2, A2x0 + B2 y0 +
+C2 z0 + D2 = 0 "
λ0
M0 !
α2 A2 x + B2 y + C2 z + D2 = 0)
A1 x0 + B1 y0 + C1 z0 + D1 + λ0 (A2 x0 + B2 y0 + C2 z0 + D2 ) = 0 =⇒
=⇒ λ0 =
A1 x0 + B1 y0 + C1 z0 + D1
.
A2 x0 + B2 y0 + C2 z0 + D2
# λ = λ0
$ %& " !
! M0 λ0 '
$( ) M0 , '
& $" " M0 ∈/ α2 ), !
! % "* ( '
&!& '
% %
# ! M1 (x1; y1; z1) M2(x2; y2; z2 ) '
!& S 1(m1 ; n1; p1) S 2(m2; n2; p2) S1 ∦ S2
+ "* h
'
( % M1 M2 S 1 S 2
! '
$ & % ,-
S
M2 1
S2
M1
S1
S2
z = 0 Oxy
z = 0
l
M0 (0; −1; 2), M1 (1; 1; 1)
l:
x + y − z + 3 = 0,
2x − y − 1 = 0.
! M0
−1 − 2 + 3 = 0,
M0 ∈ l.
1 − 1 = 0,
" M1
−1 − 1 + 3 = 0,
2 − 1 − 1 = 0,
M1 ∈/ l.
" #
$% " & x0 = 0
y0 = −1 z0 = 2 ' M0 ( ) #
$%* +
, & s = N1 × N2 * N1 = (1, 1, −1) N2 = (2, −1, 0)
i j
s = N1 × N2 = 1 1
2 −1
1
+k
2
-
k
1
1 −1
−1
−
j
−1 = i
2 0
−1
0
0
1
= −i − 2j − 3k.
−1
x
y+1
z−2
=
=
−1
−2
−3
+
$
x
y+1
z−2
=
=
.
1
2
3
.
- %
t
x
y+1
z−2
=
=
= t.
1
2
3
⎧ x
⎪
⎧
= t,
⎪
⎪
⎨ y1 + 1
⎨ x = t,
= t, ⇔
y = −1 + 2t,
2
⎪
⎩ z = 2 + 3t.
⎪
⎪
⎩ z − 2 = t;
3
l1 :
y−1
z+2
x
=
=
,
2
3
2
l1 l2
l2 :
y+1
z−1
x
=
=
.
−3
1
0
s1 = (2, 3, 2) s2 = (−3, 1, 0) !" #
cos ϕ =
s1 · s2
2 · (−3) + 3 · 1 + 2 · 0
3
.
=√
= −√
2
2
2
2
2
|s1 ||s2 |
170
2 + 3 + 2 · (−3) + 1
$
l:
y+1
z−1
x
=
=
;
2
3
1
l α
α:
3x − y + 2z − 1 = 0.
% &
' s = (2, 3, 1) (% N = (3, −1, 2)
' )"
s·N
2 · 3 + 3 · (−1) + 1 · 2
5
=√
= ,
14
|s||N |
22 + 32 + 12 · 32 + (−1)2 + 22
5
.
ϕ = arcsin
14
sin ϕ =
* l
+ ' l , $
#
α
⎧ x
⎧
⎪
= t,
⎪
⎪
⎨ x = 2t,
⎨ y2 + 1
x
y+1
z−1
= t, ⇔
y = −1 + 3t,
=
=
=t⇔
⎩
⎪
3
2
3
1
⎪
z = 1 + t.
⎪
⎩ z − 1 = t;
1
t
2
3 · 2t − (−1 + 3t) + 2(1 + t) − 1 = 0 ⇒ t = − .
5
⎧
2
4
⎪
⎪
x
=
2
−
=− ,
⎪
⎪
⎪
5 5
⎪
⎨
11
2
=− ,
y = −1 + 3 −
⎪
5
5
⎪
⎪
⎪
3
2
⎪
⎪
= .
⎩ z =1+ −
5
5
4 11 3
M − 5 ; − 5 ; 5
!" #
x2 = y −3 1 = z1 M(1, 1, 2).
$ % & %' ( M0
M1
M, M0 , M1 ) M0 ( '
M0 (0, 1, 0) *
& M1 %
⎧ x
⎪
⎧
= t,
⎪
⎪
⎨ 2
⎨ x = 2t,
x
y−1
z
y−1
y = 1 + 3t,
=
= =t⇔
= t, ⇔
⎪
⎩ z = t.
2
3
1
⎪
⎪ z 3
⎩
= t;
1
(' t ∈/ 0, t = 1
M1 x1 = 2 y1 = 4 z1 = 1 M1(2, 4, 1)
+, M M1 M2
-!. /0
x − 1 y − 1 z − 2,
0 − 1 1 − 1 0 − 2,
2 − 1 4 − 1 1 − 2,
x − 1 y − 1 z − 2,
= 0 ⇔ −1
0
−2,
1
3
−1.
=0⇔
0 −2
⇔ (x − 1)
3 −1
− (y − 1) −1 −2 + (z − 2) −1 0
1
1
−1
3
⇔ 2x − y − z + 1 = 0.
=0⇔
! " "
#$
⎧
⎪
⎨ x = y − 1,
2
3
y−1
z ⇔
⎪
⎩
=
3
1
3x − 2y + 2 = 0,
y − 3z − 1 = 0,
% " $
3x − 2y + 2 + λ(y − 3z − 1) = 0.
& λ " #
'" M :
M(1; 1; 2)
1
3 − 2 + 2 + λ(1 − 6 − 1) = 0 ⇔ λ = .
2
( " # " # )
λ = 21 $
1
3x − 2y + 2 + (y − 3z − 1) = 0 ⇔ 2x − y − z − 1 = 0.
2
*
ABC,
C(3; 4; 0)
A
A(1; 2; −1) B(1; −2; −2)
+ $ % ,,
A BC $
1 + 3 −2 + 4 −2 + 0
;
;
, M(2; 1; 1).
M
2
2
2
( " AM # " #
" - ./$
"
x−1
y−2
z+1
x−1
y−2
z+1
=
=
⇔
=
=
.
2−1
1−2
−1 + 1
1
−1
0
0 ## 1 -234/$
d=
(2 − 1)2 + (1 − 2)2 + (−1 + 1)2 =
√
2.
x + y + z + 1 = 0,
x − y − z − 1 = 0.
M(2, 1, 3)
x + 2y − z + 5 = 0
l : x −5 2 = y −1 3 = z +2 1
α : x + 4y − 3z + 1 = 0
n
n
! " ! # $%& ' ' !
! ( ) a = (a1 , a2 , a3 ) * ! !
'! & ' ! ' !) a = (a1 , a2 ) +! !"
, " x1 , x2 , x3 - a = (x1 ; x2 )
& a = (x1 ; x2 ; x3 ) ( ./
n !
$ n x1, ..., xn
n!
" n! #
a = (x1 ; ...; xn) $ x1 , ..., xn
- & !0& !"
& ! ' !.) 0 = (0; 0; ...; 0)
- (−x1 ; −x2 ; ...; −xn) 0 ' '!
a = (x1 ; x2 ; ...; xn) 0 −a
$ %
⎧
a x + a12 x2 + ... + a1n xn = b1 ,
⎪
⎪
⎨ 11 1
a21 x1 + a22 x2 + ... + a2n xn = b2 ,
.................................................
⎪
⎪
⎩ a x + a x + ... + a x = b .
m1 1
m2 2
mn n
m
x1 = α1; x2 = α2 ; ...; xn = αn
n x = (α1 ; α2 ; ...; αn)
b1; b2; ...; bm m
b = (b1 ; b2 ; ...; bm)
a = (x1; x2; ...; xm)
b = (y1 ; y2 ; ...; ym) a = b
xi = yi(i = 1, 2, ..., n)
! " #
n
b = (y1 ; y2 ; ...; yn)
a = (x1 ; x2 ; ...; xn)
c = a + b = (x1 + y1 ; x2 + y2 ; ...; xn + yn ),
$ %&
d = a − b = (x1 − y1 ; x2 − y2 ; ...; xn − yn ).
"
a = (x1 ; x2 ; ...; xn)
$ &
λ
λa = {λx1 ; λx2 ; ...; λxn}.
$ &
' n (
!
) n
! "
# " $ "
Rn
(1)
(1)
(1)
(2)
(2)
(2)
* a1 = (x1 ; x2 ; . . . ; xn ), a2 = (x1 ; x2 ; . . . ; xn ), ...,
(k)
(k)
(k)
ak = (x1 ; x2 ; ...; xn )
λ1 ; λ2 ; ...; λk
!
λ1 a1 + λ2 a2 + ... + λk ak = 0.
$ )&
λ1, λ2, ..., λn
a a1 , a2, ..., an
a = λ1 a1 + λ2 a2 + ... + λn an = 0.
λ1 , λ2 , ..., λn e1 =
(1; 0; 0; ...; 0) e2 = (0; 1; 0; ...; 0), ..., en = (0; 0; ...; 0; 1)
a = λ1 a1 +λ2 a2 +...+λnan
!
⎧
x = λ1 · 1 + λ2 · 0 + ... + λ2 · 0,
⎪
⎪
⎨ 1
x2 = λ1 · 0 + λ2 · 1 + ... + λ2 · 0,
.................................................
⎪
⎪
⎩ x = λ · 0 + λ · 0 + ... + λ · 1.
n
1
2
2
"
# λ1 = x1 , λ2 = x2 , ..., λn = xn $
%
Rn
& n% ' a = (x1 ; x2 ; ...; xn) x1 ; x2 ; ...; xn
& &(& ' e1 , e2 , ..., en
"
# n%
& & &
' ) *' ' %
+ # * ' %
, -&! . /
0 & n% ! ,
-( & ' %
& & ' , '
- /1 a = (x1 ; x2 ; ...; xn)
b = (y1 ; y2 ; ...; yn) a · b = (x1 + y1 ; x2 + y2 ; ...; xn + yn ) 2 & 3
# * ' ,
- / & 4! !
5 #
Q
& ' #&
0 & 6 &
*
# x y x1 x2 y1 y2
i j 7 e1 e2
%
8 & &(6
x1 x2
y1 y2
y1 = a11 x1 + a12 x2 ,
y2 = a21 x1 + a22 x2 .
' a11 , a12 , a21 , a22 % &
M Q x1 x2
N
y1 y2
! N
M
M
Q L
λ
!"#$%
Q
" #
$ $ x1 x2 % #
" # Ox1 x2 %
& # '
A=
a11 a12
a21 a22
,
# ((' )
%
y1
x1
* ' %' X = x1 , Y = y1 ,
# $ + ( ,
Y = AX.
!
- & ' A
'
( |A| = aa11 aa12 ) (
21 22
*
( ! ++
% ' ( |A| = 0
|A| = 0
(
. (( % # #
$ x1 x2 . ( ,
y1 a12
y2 a22
a
a
= 22 y1 − 12 y2 .
x1 =
|A|
|A|
a
a
11
12
a21 a22
a11 y1
a21 y2
a
a
= − 21 y1 + 11 y2 .
x2 =
|A|
|A|
a11 a12
a21 a22
⎧
a22
a12
⎪
y1 −
y2 ,
⎨ x1 =
|A|
|A|
a
a
21
11
⎪
y2 .
⎩ x 2 = − y1 +
|A|
|A|
N (y1 ; y2)
M(x1; x2) M,
N ! "" #
Q
"" $ % & % % A'
A−1 =
a22 / |A| − a12 / |A|
−a21 / |A| + a11 / |A|
.
( " &
$ ) % A−1'
! A−1A =
A−1 Y = A−1 AX.
10
= E EX = X
01
A−1Y
= X.
*+
X = A−1 Y.
,
y1 = x1 ,
y2 = x2
% E =
**
10
01
.
y1 = 2x1 + 3x2 ,
y2 = 3x1 + 5x2
*
23
|A| =
35
= 1
A =
23
35
!"!# x1
%
x1 = 5y1 − 3y2 ,
x2 = −3y1 + 2y2 .
A−1 =
x2 $
5 −3
−3
2
y1 = 2 · 1 +
M(1; 2) N
+3 · 2 = 8 y2 = 3 · 1 + 5 · 2 = 13 L : x1 + 2x2 − 2 = 0
λ
x1 + 2x2 − 2 = 0
x1
x2 y1
y2
!" #$
(5y1 − 3y2 ) + 2(−3y1 + 2y2 ) − 2 = 0,
− y1 + y2 − 2 = 0.
& '
(
$ )
A=
23
46
y1 = 2x1 + 3x2 ,
y2 = 4x1 + 6x2
23
,
|A| =
46
* %
)
)
+
!,# ) *
M 2x1 +3x2 = 0
y1 = 2x1 + 3x2 y2 = 4x1 + 6x2 = 2(2x1 + 3x2 ) = 2 · 0 = 0
(
y1 = x1 ,
y2 = −x2
10
0 −1
M(x1 ; x2 ) N
Ox1
M(1; 2) N (1; −2)
A =
OM M(x1; x2) : OM = x1e1 + x2e2
ON
N ! M : ON = y1 e1 + y2 e2 "
x1 , x2 y1 , y2 # $
!% &
' ! ( Ox1x2x3
"% "% " )
⎛
⎛
⎞
⎞
⎞
y1
a11 a12 a13
x1
X = ⎝ x2 ⎠ , Y = ⎝ y2 ⎠ , A = ⎝ a21 a22 a23 ⎠ ,
x3
y3
a31 a32 a33
⎛
% * (|A| = 0) (
* % + ,-)
Y = AX, X = A−1 Y.
&& ( ( !
! ( ( ! ( . (
/ ( Q ( *
e1 e2 0 Ox1 x2
!% Ox1 x2
e1 e2 0 (
,1+
'!2 ( Q ( ! M x1, x2 . 2
% x1, x2 . 0 !
% % 3 #* !
OM
M ( * % ! e1 , e2
e1 , e2 )
Ox1 x2
OM = x1 e1 + x2 e2 , OM = x1 e1 + x2 e2 .
4 !
x1 e1 + x2 e2 = x1 e1 + x2 e2 .
,5
x2
x,1
x’2
.M
e2
,
e2
,
e1
α
0
x1
e1
α11 = e1 e1 ; α12 = e1 e2 ; α21 = e2 e1 ; α22 = e2 e2 .
! " #
x1 = d11 x1 + d12 x2 ,
x2 = d21 x1 + d22 x2 .
$!
% " $! & ' " #
(
L=
d11 d12
d21 d22
)!
* (+
, (# ( X = xx12 X = xx1 . -
2
$! + ' .
X = LX .
L
! "
d d
# L = d d ,
$ L %
L L &$ ' $
L $
T
11
21
12
22
T
d11 d21
d12 d22
LT L =
d11 d12
=
d21 d22
10
=
= E.
01
() *+
, '
- $
! α ! " $ . () */
!$ .
)
0" " Ox x
1 () + Q Y = AX $
M(x ; x ) $ N (y ; y )
- $" X Y $
(X = LX ) 1 " $
.
Y = L ALX .
() *2
3! $ M(x ; x )
$ N (y ; y ) 1
a a
1 A = a a :
Y = A X ! 1
' A = L AL
4
! $ 5
!
d211 + d221 d11 d12 + d21 d22
d12 d11 + d22 d21 d212 + d222
LT
L
LT = L−1 .
=
1 2
1
2
1
2
1
−1
1
2
−1
11
21
12
22
2
x1 x2
! "! # $#
" "! $% " %
&' () $) " $ " x1, x2,
$ "
"
*+ ! ' " %
* a12 = a21 , '
F (x1 , x2 ) = a11 x21 + 2a12 x1 x2 + a22 x22 .
F (x1 , x2 ) = (a11 x1 + a12 x2 )x1 + (a21 x1 + a22 x2 )x2 .
,( A = aa1121 aa1222 ( "
- ( $( X = xx12 %
( " XT = (x1; x2)! !
" %
+ )
F (x1 , x2 ) = XT AX.
.
* ) / ) ! $ (
(
0 " ) x1 x2 "" " "
) " Ox1 x2 1 %
" Ox1 x2 *) " "
+ $
$ .2
x1 = a11 x1 + a12 x2 ,
x2 = a21 x1 + a22 x2 .
)
( $ L =
$ .2 + ) %#
) X =
x1
x2
X = LX .
x1
.
, X =
x2
a11 a21
a12 a22
3
.
x1 x2
! x1 x2"
x1 x2# F (x1, x2)
$ % & ! # %
Ox1 x2 " %
& F (x1, x2)
!
" " % '
(& #
2
2
F (x1 , x2 ) = λ1 x1 + λ2 x2 ,
& !
)
( ! % ! % '
& * + ,' X = (x1 x2)
-+ % " ( #
XT = X T L−1 .
.
$ XT X
! . #
F (x1 , x2 ) = X T (L−1 AL)X .
/ %
, A ( #
A =
Ox1x2 " % '
λ1 0
0 λ2
.
/ 0 + " ,
+ '
$ 0
F (x1, x2) ! 1
2 "
% % ! " '
% , L % ! 1
λ1 0
0 λ2
3 % 0 +
L
2 " ,
L
λ1 0
0 λ2
= L−1 AL.
, L#
= LL−1 AL = EAL = AL.
% !
L
λ1 0
0 λ2
= AL.
λ1 0
0 λ2
=
α11 α12
α21 α22
AL =
=
a11 a12
a21 a22
λ1 0
0 λ2
=
α11 α12
α21 α22
α11 λ1 α12 λ2
α21 λ1 α22 λ2
=
a11 α11 + a12 α21 a11 α12 + a12 α22
a12 α11 + a22 α21 a21 α12 + a22 α22
α11 λ1 α12 λ2
α21 λ1 α22 λ2
=
,
a11 α11 + a12 α21 a11 α12 + a12 α22
a12 α11 + a22 α21 a21 α12 + a22 α22
.
α11 λ1 = a11 α11 + a12 α21 ,
α21 λ1 = a12 α11 + a22 α21 .
α12 λ2 = a11 α12 + a12 α22 ,
α22 λ2 = a21 α12 + a22 α22 .
α11 (a11 − λ1 ) + α21 a12 = 0,
α11 a21 + α21 (a22 − λ1 ) = 0;
α12 (a11 − λ1 ) + α22 a12 = 0,
α12 a21 + α22 (a22 − λ1 ) = 0.
!"" α11
α12 α21 α22 # $ %&'
!# $ ( )
* # '
+ & $ !# ,
a11 − λ1 a12
a21 a22 − λ1
= 0,
a11 − λ2 a12
a21 a22 − λ2
= 0.
λ1 λ2 ) '
a11 − λ a12
-
a21 a22 − λ = 0,
λ2 − (a11 + a22 )λ + (a11 a22 − a12 a21 ) = 0.
.
λ1 , λ2 λ3
a11 − λ a12 a13
a21 a22 − λ a23
a31 a32 a33 − λ
= 0.
2
2
F (x1 , x2 ) = λ1 x1 + λ2 x2
λ1 0
,
0 λ2
!"#$%& Ox1 x2
A
' (
y1 = λ1 x1 + 0 · x2 ,
y2 = 0 · x1 + λ2 x2 .
y1 = λ1 x1 ,
y2 = λ2 x2 .
!"#$")&
*
Ox1 x2 M1 (1; 0)
M2 (0; 1)$ + OM1 = e1 OM2 = e2 $ , !"#$")&
M1 M2 - Q1 (λ1 ; 0)
Q2 (0; λ2 )$ , OQ1 = λ1 e1 OM1 = e1
OQ2 = λ2 e2 OM2 = e2 $
. - !"#$")& e1 e2
λ1 e1 λ2 e2 $
"#$))
r
λ1r r
λ
/ e1 e2
- !"#$")&$
, $
"#$% !
y1 , y2 x1, x2 " "
# $
% N (y1; y2) "
" M(x1; x2)&
α11 (1 − (−2)) + α21 · 3 = 0,
α11 · 3 + α21 (1 − (−2)) = 0;
α12 (1 − 4) + α22 · 3 = 0,
α12 · 3 + α22 (1 − 4) = 0.
α11 = −α21 , α22 = α12 α21 = 1
α11 = −1 α12 = 1 = α22
L=
−1 1
1 1
,
!
x = −x + y ,
y = x + y .
"# $
% &
(y − x )2 + 6(y 2 − x 2 ) + (x + y )2 + 6y − x + 2(x + y ) − 1 = 0 ⇔
⇔ −4x 2 + 8y 2 − 4x + 8y − 1 = 0
' & %
# ( # xy # ) # #
# * ! + #(
& & $ #) ,
&
1
1
2
2
2
2
−4x +8y −4x +8y −1 = 0 ⇔ −4(x +x + )+8(y +y + ) = 2 ⇔
4
4
1
1
(y + )2 (x + )2
1 2
1 2
2
2 = 1.
⇔ −2(x + ) + 4(y + ) = 1 ⇔
−
1
1
2
2
4
2
1
1
- X = x + , Y = y + , . ,
2
2
Y 2 X2
. X, Y # ( 1 − 1 = 1
4
2
' # # ) ) & ,
# # # /* #
,
# & ( # % %
# #
Y2
X2
4
2
0 ( 1 − 1 = 1
xy − 2x − 3y + 6 = 0
1
a11 = 0 a12 = a21 =
2
−α 1
2 = 0 ⇔
1
−α
2
a22 = 0
1
1
1
= 0 ⇒ α1 = α2 = − .
4
2
2
! "# ! $# %
⇔ α2 −
⎧ 1
⎪
⎪
⎨ − 2 α11 +
⎪
⎪ 1
⎩
α11 −
2
1
α21 = 0,
2
1
α21 = 0;
2
⎧ 1
⎪
⎪
⎨ 2 α12 +
⎪
⎪ 1
⎩
α11 −
2
1
α22 = 0,
2
1
α22 = 0,
2
,
α11 = α21 , α22 = −α12 .
&% α11 = α21
= α12= 1 α22 = −1 ' ( %) *
1 1
L =
+ ! $# %)
1 −1
x = x + y ,
y = x − y .
, -
%
(x +y )(x −y )−2(x +y )−3(x −y )+6 = 0 ⇔ x 2 −y 2 −5x +y +6 = 0
%) - %*
% *
xy . %
25
2
% (x − 5x + ) −
4
1
5
1
−(y 2 −y + ) = 0 ⇔ (x − )2 −(y − )2 = 0. /
4
2
2
5
1
X = x − Y = y − X Y % *
2
2
% % X 2 − Y 2 = 0 ⇔ (X − Y )(X + Y ) = 0
) % + Y = X Y = −X
. % . %
0 %12 % Y = X Y = −X.
AB
2a − 3b A(1; 2; 0) B(−2; 2; 1) a = (2; 1; 3) b = (0; 2; 1)
ΔABC A(1; 2; 3) B(2; −1; 0)
C(1; 1; −1)
l
l:
4x − y − z + 12 = 0,
y − z − 2 = 0.
M(1; 0; 0) α :
x + y + z + 1 = 0.
A(1; 2; 0) B(−2; 2; 1) a = (2; 1; 3) b = (0; 2; 1)
2a−3b AB
=
AB(2a − 3b)
AB = (−3; 0; 1)
|2a − 3b|
2a − 3b = (4; −4; 3)
9
−12 + 0 + 3
= − √ ≈ −1,4
2a−3b AB = √
16 + 16 + 9
41
A(1; 2; 3) B(2; −1; 0) C(1; 1; −1)
1
SΔABC = |AB × AC|. AB = (1; −3; −3), AC = (0; −1; −4).
2
i j
k
AB × AC = 1 −3 −3 = 9i + 4j − k.
0 −1 −4
√
98
1√
SΔABC =
≈ 4,9.
81 + 16 + 1 =
2
2
¿¾º¿
l:
4x − y − z + 12 = 0,
cos α =?, cos β =?, cos γ =?
y − z − 2 = 0.
S = 2i + 4j +√4k
! "! |S| = 4 + 16 + 16 = 6
eS =
S
1
2
2
= i + j + k.
3
3
3
|S|
# ! "
$ l
1
cos α = ,
3
2
cos β = ,
3
% & M(1; 0; 0) α :
2
cos γ = .
3
x + y + z + 1 = 0.
"
l$ ' " ( " M
α )* + !
N = (1; 1; 1)$ " l
⎧
⎨ x = 1 + t,
y = t,
l=
⎩
z = t.
" " l α
1+t+t+t+1=0⇒t=−
2
3
,
1
2
2
x= , y=− , z=− .
3
3
3
1 2 2
! M ; − ; −
3 3 3
⇒
% - a = (4; −2; −4) b = (6; −3; 2)
(2a − 3b) · (a + 2b)
% . A B C D
A(3; 1; 1) B(−2; 1; −2) C(−3; −1; 0) D(2; 0; 17)
% /
y−1
z−2
y+2
z−3
x+1
x
=
=
=
=
−1
2
−1
3
2
−3
% 0 ! "
M(0; 1; 2) x −2 1 = y1 = z +0 1
!"
"
#
$%
" %
!
& & !
& &
''( z
(x; y) x
x = Re z y
y = Im z
)''(*
# 1 = (1; 0) i = (0; 1)
& ! , & x = (x; 0) %
iy = (0; y) & . ! % &
, ! # i & ! & & &/
i2 = −1.
)''0*
z = (x; y)
+ , & !
- %
&
+
''0 z = x + iy
! "
)'''*
1 &, & ! ,
& ! & ,
+ !&" z1 = x1 + iy1 z2 = x2 + iy2 & ! &/
z = x + iy.
•
z1 = x1 + iy1 z2 = x2 + iy2
"
z1 = z2 ⇐⇒ x1 = x2 , y1 = y2 .
)''2*
•
z1 ± z2 = (x1 ± x2 ) + i(y1 ± y2 ).
•
!
z1 · z2 = (x1 + iy1 )(x2 + iy2 ) = x1 x2 − y1 y2 + i(x1 y2 + x2 y1 ).
•
" z = x−iy #
z = x + iy # $%
zz = (x + iy)(x − iy) = x2 + y 2 .
& '#%
#
z1
• ( z =
)
z2
) * # ' zz1
2
z2% # % #
z1
x1 x2 + y1 y2 x2 y1 − x1 y2
z1 · z2
(x1 + iy1 )(x2 − iy2 )
=
=
=
+i
.
z2
z2 · z2
(x2 + iy2 )(x2 − iy2 )
x22 + y22
x22 + y22
+% %
z1 = 2 + 3i z2 = 3 − 2i
z1 + z2 = 5 + i, z1 − z2 = −1 + 5i,
z1 z2 = 6 + 6 + i(9 − 4) = 12 + 5i,
z1
(2 + 3i)(3 + 2i)
6 − 6 + i(9 + 4)
2 + 3i
=
=
= i.
=
z2
3 − 2i
(3 − 2i)(3 + 2i)
9+4
2 x2 + px + q = 0
D = p4 − q < 0
z 2 + pz + q = 0
2
p4 − q < 0
z1,2 = α ± βi,
!
"
p
α=− β=
2
q−
p2
4
# $ $ %&
'
z 2 + pz + q = (z − z1 )(z − z2 ) = ((z − α) − βi)((z − α) + βi).
( z2 + 8z + 25 = 0
) * √ + , - ! +
z1,2 = −4± 16 − 25 = −4±3i z 2 +8z +25 = ((z +4)−3i)((z +4)+3i).
z 2 + pz + q
. / 0 & Re z = x "
& z Ox 0 Im z = y Oy
Oxy 1
y=Im z
z
y
z
r
ϕ
x
x=Re z
z = x + iy
Oxy
Oxy
z
z z
z
!
z = ∞
z
!"# $
z O z
% % z# % z
&'%& &' % z
z % N %& O
z %
N
N
z
z
y
0
z
x
(% ) |z|
% )
*
" !#$
|z| = r =
%
r z.
x2 + y 2 .
z
r z Ox
⎧
y
⎪
arctg
x > 0,
⎪
⎪
⎪
x
⎪
y
⎪
⎨ π + arctg
x < 0,
x
ϕo = arg z =
π
⎪
y > 0, x = 0,
⎪
⎪
⎪ 2π
⎪
⎪
⎩ −
y < 0, x = 0.
2
√ √
√
!
√
z1 = 1+i 3" z2 = −1+i 3" z3 = −1−i 3 z4 = 1−i 3 #
! $
y
x √
3
π
= z3
z1 ϕo1 = arctg
1√ 3
4
− 3
= π.
ϕo3 = π + arctg
−1
3
z2 z4
√ !
√
3
2π o
π
− 3
o
=
= − $#%
"# ϕ2 = π + arctg
ϕ4 = arctg
−1
3
1
3
√
! &! ! r = x2 + y 2 = 1 + 3 = 2. '
% ( # # r = 2
) * "# +
z1
z3
y
2
z2
-2
-1
z1
3
0
1
2
- 3
z3
-2
z4
x
z
R
R
!
" # # $
|z| < R,
%
&
|z| = R,
'
( ! $ ) z0
*+ &
|z − z0 | = R
*
|z| > R.
y
1111111
0000000
0000000
1111111
R
0000000
1111111
z
0000000
1111111
0000000
1111111
1111111
0000000
0000000
1111111
0000000
1111111
0000000
1111111
R
0000000
1111111
0000000
1111111
0
0000000
1111111
0000000
1111111
0000000
1111111
0
|z| < R
x
|z − z0 | < R.
" # ,# $
|z − z0 | < R,
-
&
|z − z0 | > R.
.
( % - /
# *+
z
!"#
$ % & z
2πk k ∈ Z ' $ $ $(
ϕ = Arg z
ϕ = Arg z = arg z + 2πk = ϕo + 2πk, k ∈ Z,
)*
k = 0
+ ,-
x = r cos ϕ, y = r sin ϕ
)
. $ r
ϕ
z = r(cos ϕ + i sin ϕ),
))
/( 0 )
1
2. .
3
4
eiϕ = cos ϕ + i sin ϕ.
)
,
z = reiϕ ,
)5
!
"
6 )) )5 /
/ z
.
5 # " "
$ z% &
' √( z1 = 3) z2 = −3) z3 = 3i) * z4 = −3i
+ z5 = 3 − i 3
7 8 9 : /$
. ϕ◦ = 0
)) )5 0 ) )* z1 = 3 = 3(cos 2πk +
+i sin 2πk) = 3ei2πk .
ϕo = π
z2 = −3 = 3(cos(2k + 1)π + i sin(2k + 1)π) = 3ei(2k+1)π .
π
x = 0 y > 0 → ϕo =
2
1
1
π + i sin 2k +
π) = 3i(2k+1/2)π .
z3 = 3i = 3(cos 2k +
2
2
y < 0 → ϕo = −
π
2
1
1
1
π + i sin 2k −
π = 3i(2k− 2 )π .
z4 = −3i = 3 cos 2k −
2
2
√
z5 = 3 − i 3
√
− 3
o
= − π6
ϕ = arctg
3
√ 2k− 1 π
√
1
1
6
π + i sin 2k −
π = 2 3e
z5 = 2 3 cos 2k −
.
6
6
!
! # !!$% &
! '
z1 = r1 eiϕ1 z2 = r2 eiϕ2
"
z = reiϕ = z1 z2 = r1 r2 ei(ϕ1 +ϕ2 ) = r1 r2 (cos(ϕ1 + ϕ2 ) + i sin(ϕ1 + ϕ2 ),
('
z1
r1
r1
z = reiϕ =
= ei(ϕ1 −ϕ2 ) = (cos(ϕ1 − ϕ2 ) + i sin(ϕ1 − ϕ2 )) (' )
z2
r2
r2
r = r1 r2
ϕ = ϕ1 + ϕ2
r1
r =
r2
ϕ = ϕ1 − ϕ2 .
* !! z = reiϕ
z n = (reiϕ )n = rn einϕ .
sin ϕ cos ϕ eiϕ
! " # " 2π $
%&' " # n $ ϕ
( ) ! ϕo = arg z.
*) " ) " ( n "
+ +$ $ ($ ) ,
" - ",
. ϕ = Arg z = arg z + 2πk. "
n, " " z = reiϕ "
(r(cos ϕ + i sin ϕ))n = rn (cos nϕ + i sin nϕ).
√
n
z=r
1/n
ϕo + 2πk
√
ϕo + 2πk
ϕo + 2πk
n
+ i sin
.
e
= n r cos
n
n
i
k
"
√
0, 1, 2, ..., n − 1
/
" n
z
z
√
r n
0 k " ,
n
n
( 1 n − 1 + " !
2 n, "
z n − a = 0,
√
z = n a
-
" ,
a = reiϕ & " n
)$ " /
3 ! "#
"
√
z = 3 − i 3.
4 1√ 5 6 √
z = 3 − i 3 ) 1 " 7 ) 2 3
− π6 8 "
√
√
π
(3 − i 3)6 = (2 3e−i 6 )6 =
= 26 33 ei11π = 1728(cos π − i sin π) = 1728(−1 − i0) = −1728,
1/6 √ √ (−1+12k)π
√
√
π
6
12
6
36
3 − i 3 = 2 3e−i( 6 +2πk)
= 2 3ei
.
√
6
k 3 − i 3
√
√
√ −i π
√
11
6
6
12
12
36
z1 = ( 3 − i 3)1 = 12e
, z2 = ( 3 − i 3)2 = 12ei 36 π ,
√
√
√
√
23
35
6
6
12
12
z3 = ( 3 − i 3)3 = 12ei 36 π , z4 = ( 3 − i 3)4 = 12ei 36 π ,
√
√
√
√
47
59
6
6
12
12
z5 = ( 3 − i 3)5 = 12ei 36 π , z6 = ( 3 − i 3)6 = 12ei 36 π .
!" #$ % %" "%"$ z "& '
( √
)* ( "$*)% " * %&"$) '
" 12 12 " +
y
z3
z2
11π
36
z4
12
12
z1
12
z5
x
z6
12
z4 + 1 = 0.
√
, (
z 4 + 1 = 0 → z 4 = −1 → z = 4 −1.
"%)% -.* $/ $ )* "$$ )* "
arg z = π,
−1 = 1ei(π+2πk)
" $ )
z = ei
π+2πk
4
.
k
π
π
π
+ i sin =
4
4
3π
3π
+ i sin
= cos
4
4
5π
5π
+ i sin
= cos
4
4
7π
7π
+ i sin
= cos
4
4
z1 = ei 4 = cos
z2 = ei
3π
4
z3 = ei
5π
4
z4 = ei
7π
4
√
2
(1 + i),
2 √
2
=
(−1 + i),
2
√
2
=
(−1 − i),
√2
2
=
(1 − i).
2
z
! "#$!
y
1
z1
z2
π
4
1
-1
z3
-1
x
z4
!"
z .
% & z5 = 32 = 25 ' ( )
z1 = 2! *
)
! "+$!
, ' - . !/$ )
- 01 sin nx cos nx
' sin x cos x.
!/ sin 2x cos 2x sin x cos x.
z 5 − 32 = 0
y
2 z2
z3
2π
5
-2
2
z1
x
z4
-2 z 5
2
n = 2 : (cos ϕ + i sin ϕ) = cos2 ϕ + 2i sin ϕ cos ϕ − sin2 ϕ =
= cos 2ϕ + i sin 2ϕ.
! "#
"
cos 2ϕ = cos2 ϕ − sin2 ϕ sin 2ϕ = 2 sin ϕ cos ϕ.
$ % &' ' ! '
& %( !
)
z1 = −3 + 4i z2 = 4 − 2i z1
z1 /z2
z2 z1 + z2 z1 − z2 z1 · z2
*% + ,
z1 + z2 = 1 + 2i, z1 − z2 = −7 + 6i,
z1 z2 = (−3 + 4i)(4 − 2i) = −12 + 8 + i(6 + 16) = −4 + 22i,
z1
−3 + 4i 2 + i
−6 − 4 + i(8 − 3)
1
−3 + 4i
=
·
=
= −1 + i.
=
z2
4 − 2i
2(2 − i) 2 + i
2(4 + 1)
2
1
= (4 − 2i) −1 + i =
2
1
z1 = z2 −1 + i
2
= −(2 − i)2 = −(4 − 4i − 1) = −3 + 4i.
z1 z2 z1 +z2 z1 −z2 14 z1 z2 zz1
2
y
6
z1 z2
z1
-7
-5
z 1z 2
4
4
z1 z2
z1 2
z2
-3 -1
-2
1
4
z2
x
z2 + 4z + 13 = 0.
! " # $ z1,2 = −2 ±
= 2 ± 3i.
√
4 − 13 =
% %$ &
' & "$
z = 1 + i z = 1 − i
! "
π
1
= ,
1
4
π
arg z = arg(1 − i) = arctg(−1) = − ,
4
arg z = arg(1 + i) = arctg
"
|z| =
√
1+1=
√
2,
|z| =
√
2.
z=
z=
√
√
π
√
π
π
,
2 cos + i sin
4
4
√
π
π
.
= 2 cos − i sin
4
4
2e 4 i =
π
2e− 4 i
π
2 < |z| < 4 2 < arg z < π
2 < |z| < 4
! " # ! $ $
! $ " % $ $ r = 2
r = 4 &''
y
4
2
-4
-2
0
2
4
x
-2
-4
π2 < arg z < π ! $(
) *# + * *
# ! $ $ r = 2 r = 4
$ #" &''
# " !
#! # " !
, (1 + i)4
! "
!"#
$% m = 4, a = 1 b = i"
4·3 2 4·3·2 3 4·3·2·1 4
i +
i +
i =
1·2
1·2·3
1·2·3·4
= 1 + 4i − 6 − 4i + 1 = −4.
(1 + i)4 = 1 + 4i +
& ' 1 + i $ % ( $%
) & ** *# $
k =0: 1+i =
√ πi √
π
2e 4 = 2(cos + i sin π4 ).
4
+( ** ,-" ** ,."
√
(1 + i)4 = ( 2)4 eπi = 4(cos π + i sin π) = −4,
# # & &
** / √1 + i
4
0 $' $$( # $$ &
$ $ **# ** , "
√
4
1$ #
'
1+i =
k
4
√ i ( π +2πk)
2e 4 4
.
& 2# #,#*# ) & &)
√
√
√
π
π
π
8
8
4
+ i sin
,
1 + i = 2ei 16 = 2 cos
16
16
1
√
√
√
9π
9π
9π
8
8
+ i sin
,
z2 = 4 1 + i = 2ei 16 = 2 cos
2
16
16
√
√
√
17π
17π
17π
8
8
4
i
16
,
1 + i = 2e
= 2 cos
z3 =
+ i sin
3
16
16
√
√
√
25π
25π
25π
8
8
z4 = 4 1 + i = 2ei 16 = 2 cos
+ i sin
.
16
16
4
√
3 $ & $ 8 2
z1 =
) $ -."
** -
z 3 + 27 = 0
z
y
8
z2
2
z1
x
π
16
0
z3
8
2
z4
√
→z=
3
−27.
z3 = −27
→
z2 = −3 z2 = −3
arg(−3) = −π
!
r = 3 " #$%& ' (
π
5π
z1 z3 arg z1 = arg z3 =
|z| = 3
3
3
y
3
z2
-3
x
3
0
-3
z1
z3
) * ' sin 3ϕ cos 3ϕ sin ϕ
cos ϕ
n = 3 :
3
(cos ϕ + i sin ϕ) = cos 3ϕ + i sin 3ϕ → cos3 ϕ + 3i cos2 ϕ sin ϕ+
+ 3i2 sin2 ϕ cos ϕ + i3 sin3 ϕ = cos3 ϕ − 3 sin2 ϕ cos ϕ+
+ i(3cos2 ϕ sin ϕ − sin3 ϕ) = cos 3ϕ + i sin 3ϕ.
cos 3ϕ = cos3 ϕ − 3 sin2 ϕ cos ϕ sin3ϕ = 3 cos2 ϕ sin ϕ − sin3 ϕ.
z1 = 4 + 3i z2 = 2 − i z1 z2
z1
z1 + z2 z1 − z2 z1 z2
z2
! z2 + 2z + 10 = 0
2
3z + 2z + 4 = 0
"# √
z1 = i z2 = −2 z3 = 1 + i 3
√
"" (1 + i 3)3 !
" z6 + 1 = 0 z3 − i = 0
"
$% & ' ( ) &
) ( ) * ' +
* %* * + ,- (
%*- ' &
R(x) =
Qm (x)
,
Pn (x)
."
Pn (x) = an xn + an−1 xn−1 + an−2 xn−2 + · · · + a1 x + a0 =
Qm (x) = bm xm +bm−1 xm−1 +bm−2 xm−2 +· · ·+b1 x+b0 =
n
,
k=0
m
,
ak xk ,
bk xk .
k=0
!
!
R(x) =
x4 + 5x3 − 6x + 5
.
x3 + 2x2 − 1
" # $ "% % &
x + 3 −6x2 − 5x + 8&
R(x) = x + 3 +
−6x2 − 5x + 8
.
x3 + 2x2 − 1
' Pn (x) % x − a
&
(
! "# $ % &'
x − a Pn(x) x = a.
( '
P (x) = 3x9 − 2x5 + 3x2 + 4x − 8 x + 1
Pn (x)
" # $
) a = −1. *+
$
9
P (−1) = 3(−1) − 2(−1)5 + 3(−1)2 + 4(−1) − 8 =
= −3 + 2 + 3 − 4 − 8 = −10.
x2 −5x − 6 = 0,
−3 2
x1 = 1, x2 = −3, x3 = 2.
= 5(x − 1)(x − 2)(x − 2)(x − 3)º
5x4 −4x3 +115x2 −140x+60 =
x3 + x = x(x − i)(x + i)
! " #
$
Pn (x) = an (x − a)k1 (x − b)k2 ...(x − p)ks ,
%
" a, b...p
# " ! "! n
& a, b...p '" # (
a ) " k1 b ) " k2 p ) (
" ks
! "
α ± βi * " $ " α + βi " (
# " k "! + # " α − βi "
, # "
% -, "
% " (x−(α+βi))k
,
! "" (x − (α − βi))k
- ""'. ! " "(
!
! # " /$
(x − (α + βi))k (x − (α − βi))k = ((x − α) − βi)k ((x − α) + βi)k =
= ((x − α)2 + β 2 )2 = (x2 − 2αx + α2 + β 2 )k = (x2 + px + q)k ,
p = −2α, q = α2 + β 2
"
# +# x2 + px + q
0
"
*" 1" ! "2 $ " (
# " " ,22 0 !"(
" '. $
Pn (x) = an (x − a)k1 (x − b)k2 ...(x2 + p1 x + q1 )s1 (x2 + p2 x + q2 )s2 ....
* 2 n = k1 + k2 + · · · + ks + 2(s1 + s2 + · · · + sm ).
! "
A
,
x−a
A
II.
(n = 2, 3, ...),
(x − a)n
Mx + N
III. 2
(D = p2 − 4q < 0),
x + px + q
Mx + N
IV. 2
(D = p2 − 4q < 0, n = 2, 3...)
(x + px + q)n
I.
# $ %
$% % &
(x)
QPm(x)
n
Pn (x) = (x − a)k · · · (x2 + px + q)l · · ·
! "
!
Qm (x)
A1
A2
Ak
=
+
+ ··· +
+ ···
2
Pn (x)
x − a (x − a)
(x − a)k
M1 x + N1
M2 x + N2
Ml x + Nl
+
+ ··· + 2
,··· ,
x2 + px + q (x2 + px + q)2
(x + px + q)l
Ai,
Bi , Mi , Ni (i = 1, 2, ...)
#
' ! ()
* ' ! () %
a
+% * ' ! () %
%
$ , ,,
# $ ,,, ,-
Pn (x)
1
1
A = 1, B = − , C = .
5
5
! ""# ! $%&'(
1
1
x2 + 2x − 6
1
5
= −
+ 5 .
x3 + x2 − 6x
x x+3 x−2
$%&)(
%&'
*
3x2 + 5
.
(x − 1)2 (x2 + 2x + 5)
A
B
Mx + N
3x2 + 5
=
+
.
+
(x − 1)2 (x2 + 2x + 5)
x − 1 (x − 1)2 x2 + 2x + 5
-
$%&+,(
3x2 + 5
=
(x − 1)2 (x2 + 2x + 5)
A(x − 1)(x2 + 2x + 5) + Bx(x2 + 2x + 5) + (Mx + N )(x − 1)2
=
.
(x − 1)2 (x2 + 2x + 5)
.
x2 + 5 = A(x3 + x2 + 3x − 5) + B(x2 + 2x + 5)+
+M(x3 − 2x2 + x) + N (x2 − 2x + 1)
x2 + 5 = (A + M)x3 + (A + B − 2M + N )x2 +
+(3A + 2B + M − 2N )x + (−5A + 5B + N ).
. ""# ! !/ ./ x
⎧
x3 : A + M = 0,
⎪
⎪
⎪
⎨x2 : A + B − 2M + N = 3,
⎪
x : 3A + 2B + M − 2N = 0,
⎪
⎪
⎩
− 5A + 5B + N = 5.
-! !
A = 1 , B = 1, M = − 1 , N = 5
4
4
4
!
"#$
5
1
1
− x+
3x2 + 5
1
4
4 .
= 4 +
+
(x − 1)2 (x2 + 2x + 5)
x − 1 (x − 1)2 x2 + 2x + 5
% & $ #!
$ "' $ ( # '
P (x) ≡ Q(x) )* $ +# +$ !
$ x = a : P (a) = Q(a) ( a ! + #
, $" $ - .
-. + + ' / $ ' - "
$ x + . #
& $ * ( $ ( # Pn (x) -0
+$ $ + # #
+ . +# *"- # $ !
-$ +$ 1
!"
2 2 + $ .
* "#
x2 + 2x − 6 = A(x + 3)(x − 2) + Bx(x − 2) + Cx(x + 3).
, $ ' +#- x
x = 0 : −6 = −6A =⇒ A = 1,
1
x = 2 : 2 = 10C =⇒ C = ,
5
1
x = −3 : −3 = 15B =⇒ B = − .
5
3- "#$
2$ $ $ $ - + '- +$!
- $ ' $ ' + ) + #- - +
$ ( #
x4 1+ 1 .
x4 +1
1
x1 = √ (1+i),
1
x2 = √ (−1 + i),
2
1
x3 = − √ (1 + i),
2
1
x4 = √ (1 − i).
2
2
1
1
1
1
√
√
√
√
−i
−i
x +1= x−
x+
·
2
2
2
2
1
1
1
1
x − √ + i√ .
· x + √ + i√
2
2
2
2
4
! "# " ! "# "
1
1
1
1
− i√
x− √
+ i√ ·
x− √
2
2
2
2
1
1
1
1
·
x+ √
− i√
x+ √
+ i√
=
2
2
2
2
7 6
7
6
2
2
1
i2
i2
1
x+ √
=
−
−
=
x− √
2
2
2
2
x4 + 1 =
=
1 1
1 1
2
2
x2 + √ x + +
=
x2 − √ x + +
2 2
2 2
2
2
√
√
= (x2 − 2x + 1)(x2 + 2x + 1).
$ %" & '()* ! ! +, .%% ,!
1
Ax + B
Cx + D
√
√
=
+
=
x4 + 1
x2 − 2x + 1 x2 + 2x + 1
√
√
(Ax + B)(x2 + 2x + 1) + (Cx + D)(x2 − 2x + 1)
=
x4 + 1
!
√
√
Ax3 + A 2x2 + Ax + Bx2 + B 2x + B + Cx3 −
√
√
−C 2x2 + Cx + Dx2 − D 2x + D = 1.
& '(//*
x
A B C D
⎧
A + C = 0,
⎪
⎪
⎪
⎨A√2 + B − C √2 + D = 0,
√
√
⎪
A + B 2 + C − D 2 = 0,
⎪
⎪
⎩
B + D = 1.
C A D
C = −A D = 1 − B
B
√
√
√
A 2 + B + A 2 + 1 − B = 0 ⇒ 2A 2 = −1 ⇒
√
√
1
2
2
⇒ A=− √ =−
⇒ C = −A =
,
4
4
2 2
√
√
√
A+B 2−A− 2+B 2=0 ⇒
√
√
1
1
2B 2 = 2 ⇒ B =
⇒ D= .
2
2
A, B,
& ' x4 1+ 1 (
C, D
!"#$$%
√
√
2 1
2 1
+
+
−
1
4√ 2 +
4√ 2 .
=
4
x +1
x2 − 2x + 1 x2 + 2x + 1
!"#$)%
!"#$
P4(x) = x4 + 1
! "
# "$% & " $ " '
" " " x2 +
+px + q D < 0% ()
x4 + 1 = (x2 + p1 x + q1 )(x2 + p2 x + q2 ) =
= x 4 + p 1 x 3 + q1 x 2 + p 2 x 3 + p 1 p 2 x 2 + p 2 q 1 x + q 2 x 2 + p 1 q 2 x + q 1 q 2 .
* " "$ ! " &$ x &"
&" " &"% " $ $ $
p1 p2 q1 q2
⎧
√
⎪
p + p2 = 0 ⇒ p2 = −p1 ⇒ p2 = − 2,
⎪
⎪ 1
⎪
⎪
p1 p2 = −q1 − q2 = −2 ⇒
⎪
⎪q1 + p1 p2 + q2 = 0 ⇒
⎪
⎨⇒ p2 = 2 ⇒ p = √2,
1
1
⎪
p2 q1 + p1 q2 = 0 ⇒ −p1 q1 + p1 q2 = p1 (q2 − q1 ) = 0 ⇒ q1 = q2 ,
⎪
⎪
⎪
⎪
p1 = 0,
⎪
⎪
⎪
⎩q q = 1 ⇒ q 2 = q 2 = 1 ⇒ q = q = 1.
1 2
1
2
1
2
q1 = q2 = −1,
2
p = −2,
!"# $%&
'
*
2x
2
x4 + 1
()
+ ()
√
x4 + 1 = x4 + 2x2 + 1 − 2x2 = (x2 + 1)2 − ( 2x)2 =
√
√
= (x2 + 1 − 2x)(x2 + 1 + 2x).
!# $
" !#
%
*
-
!
,
& !
! !
3
2
2
x + 2x − x + 1 | x + x − 1
'
−−−−−−−
| x+ 1
x3 + x2 − x
− − − − − − −−
(x2 + 1
x2 + x − 1
− − − − −−
−x + 2
x3 + 2x2 − x + 1
x2 + x − 1
!
x3 + 2x2 − x + 1
−x + 2
=x+1+ 2
.
x2 + x − 1
x +x−1
!"
x3 (x
& ' (
x3 (x
#$"#
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(x2
x2 + 1
.
− 1)2
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Ex + F
.
+ 2x + 5)2
x2 + 1
E
A B
C
D
+
= + 2+ 3+
.
2
− 1)
x x
x
(x − 1) (x − 1)2
x+5
.
x(x + 3)(x2 + 2x + 5)2
x+5
B
Cx + D
A
+
+
= +
x(x + 3)(x2 + 2x + 5)2
x x + 3 x2 + 2x + 5
) !
' ! )
* !"
+3
.
#$"$ x(x +x1)(x
− 2)
A
B
C
+3
= +
+
.
& ' ( x(x +x1)(x
− 2)
x x+1 x−2
! * )
* (
x + 3 = A(x + 1)(x − 2) + Bx(x − 2) + Cx(x + 1).
+ ⎧ * ) * x(
3
⎪
⎪
x = 0 ⇒ 3 = −2A ⇒ A = − ;
⎪
⎪
2
⎨
2
x = −1 ⇒ 2 = 3B ⇒ B = ;
⎪
3
⎪
⎪
⎪
⎩ x = 2 ⇒ 5 = 6C ⇒ C = 5 .
6
3
2
5
x+3
=− +
+
.
x(x + 1)(x − 2)
2x 3(x + 1) 6(x − 2)
x+5
.
x2 (x − 1)
A B
C
x+5
= + 2+
.
x2 (x − 1)
x x
x−1
x + 5 = Ax(x − 1) + B(x − 1) + Cx2 .
x = 0 ⇒ 5 = −B ⇒ B = −5;
x = 1 ⇒ 6 = C ⇒ C = 6;
x = −1 ⇒ 4 = 2A − 2B + C ⇒ A = −6.
x+5
6
5
6
=− − 2+
.
− 1)
x x
x−1
x2 (x
!"
! #$%% & ' (' #"
$%% &
)
2x + 1
.
+ 2x + 5)
x(x2
2x + 1
A
Bx + C
= + 2
.
x(x2 + 2x + 5)
x x + 2x + 5
* ! + #
+
2x + 1 = A(x2 + 2x + 5) + x(Bx + C).
# # #
,
' + '
2x + 1 = (A + B)x2 + (2A + C)x + 5A.
* , #$%% & ' #' , x
⎧
⎪
⎨A + B = 0,
2A + C = 2,
⎪
⎩5A = 1.
$ !
1
8
1
A= , B=− , C= .
5
5
5
2x + 1
1
−x + 8
=
+
.
x(x2 + 2x + 5)
5x 5(x2 + 2x + 5)
x−3
.
x2 (x2 + 1)
x−3
A B
Cx + D
= + 2+ 2
.
x2 (x2 + 1)
x x
x +1
x − 3 = Ax(x2 + 1) + B(x2 + 1) + x2 (Cx + D).
x − 3 = (A + C)x3 + (B + D)x2 + Ax + (A + B),
⎧
A + C = 0,
⎪
⎪
⎪
⎨B + D = 0,
⎪
A = 1,
⎪
⎪
⎩
A + B = −3,
A = 1 B = −4 C = −1 D = 4
x−3
1
4
−x + 4
= − 2+ 2
.
x2 (x2 + 1)
x x
x +1
x5 − x4 + 2x + 3
x2 − 1
! ""! " #$% & ' (() *
+ " "$ & " &! "$,
-
x+5
.
x4 (x + 1)
x
.
(x2 + 5)2 (x + 3)
./
! ""! "%0 " "$ ",*
, (() & "% &! % ,
x
.
(x − 1)(x + 2)
x2 (x
x+3
.
+ 1)
!!" # $#
!!"
x+5
.
− 5)
x2 (x
x(x2
x+4
.
+ 4x + 8)
un =
S=2
(−1)n+1
n2
x ∈ R
n
n+1
un = 4n−1
1
2
1
4
|r2 | 0, 01
− 2a12 e
2
.
3
− x12 . − sin x. 3 √31x2 . 1 − 3x2. cos x −
x −xex
x log3 x−sin x
x
−3 sin x. 1+x
x ln 3 cos
. 2+e
.
2 + arctg x.
(2+ex )2
x ln 3 log23 x
1
x−sin x cos x
4
2
0. − 1+x
+
ln
x
+
1
+
.
5x
−
12x
+
2.
2
(x cos x)2
2 −6x+25
5
√6ax .
83 x5/3 . −2x
.
ctg x − sinx2 x .
2
2
(x −5x+5)
a2 +b2
√
x
2
(e −5 cos x) 1−x −4
√
x arctg x. 3x2 ln x. x2 ex . (ex −5
.
sin x−4 arcsin x) 1−x2
x
1
1√
2
2
2 (sin(5x ) + sin(x )). x(1+ln2 x) . arcsin x 1−x2 .
sin x−1
y = xsin x cos
.
x ln x + sin xx
x3 ·sin x
3
1
1
.
ln x·arctg x x + ctg x − x ln x − arctg x(1+x
2)
3
(2xey −3x2 )y
.
1−x2 yey
t(2−t )
.
− xy .
T : x − 4y − 5 = 0, N : 4x + y − 3 = 0.
1−2t3
15
10
( 3; − 2 , −2; 3 .
θ1 = − arctg 0, 6; θ2 = arctg 3.
t0 = 8, t1 = 0, t2 = 4, t3 = 8. 181, 5
√
x
x
dx. x cos x−sin
52 x xdx. arctg x + 1+x
dx.
2
x2
n n!
(−1)
2x
2 arctg x + 1+x2 . (1+x)n+1 .
−e−x (x2 − 6x +
284
3
4
≈ 0, 81
+6)dx . (2−x)5 dx .
y < 0 x ∈ (−∞; 0) ∪ (0; +∞),
y = f (x) x = − 12 −
y = f (x) x < − 12
4
x > − 21 , y (0) y > 0 x ∈ (−∞; 0) (0; ∞)
m = f (−1) = −4, M = f (0) = 0.
m = f (−2) = −20, M = f (4) = 16.
!
"
#
x = 0 x+y =41. $
x = − 12 %
D(y) = (−∞; 0) (0; ∞) 4&
' x = 0 y = x () (−∞; 0)4 (2; ∞)
(0; 2) ymin = y(2) = 3 * (−∞; 0) (0; ∞)
$ + , -.
) arccos k1 arctg hd .
y = 1 ymin = y(2) = −1 &
y = 0 x → +∞ ymin = y(0) = 0 ymax = y(2) = e44
y = x + π2 x → +∞
y = x − π2 x → −∞ & /
0 [0; 1]
M = y(1) = 1e 1 m = y(0) = 0 [0; 3] M = y(2) = e42 m = y(0) = 0
[−1; 2] M = y(−1) = e m = y(0) = 0
*2' a2
⎛
⎞
⎛
⎞
−4 −8 −4
8 3 6
3 4
1 6
⎝ −3 −1 −5 ⎠ . ⎝ 0 8 12 ⎠ .
−7 −6 1
−3 6 14
⎛
⎞
⎛
⎞
M12 = −6
10 6 5
11
M22 = 3
⎝ 8 11 6 ⎠ . ⎝ 16 ⎠ .
A12 = 6
9 8 10
4
A22 = 3.
/
⎛
⎞
1/10 −1/5 7/10
1/10 −1/5 ⎠
2 3 3 3 ⎝ 0
0
0
1/10
⎛
⎞
1 2
2
19 · ⎝ 2 1 −2 ⎠
2 −2 1
x = 2 y = 3 z = −2 x = −3, y = 2, z = −1
x1 = 1, x2 = 5, x3 = 2
x1 = 2 x2 = −1 x3 = −3
x1 = 25 + 35 x3 x2 = 14 + 34 x3
x4 =
7
20
+
13
x
20 3
AM = 5a+c
. B(2; −6; 8). 2a − 3b = −8i − 3j + 10k.
6
e1,2 = ±(− 13 i + 23 j − 23 k), cos α = − 13 , cos β = 23 , cos γ = − 23
√
√
√
3 22−2 . 4 2. √− 2. 7, − 152
cos(AC; BD) = − 9√2310 . cos B = 2 1010 .
√
20i + 14j + 2k. 25 3. {ha =
{±(−2i + j + 3k)} {ha = 1; hb =
12
√7 .
11
x−1
1
◦
x + 7z + 22 = 0.
√
42
,
3
√
3 5
,
5
hb =
hc =
√
70
}.
5
√
3 2
}.
2
2y − 5z + 10 = 0.
x
= y+1
= z−1
.
= y−1
= z−2
. α = 90◦ ,
3
1
−1
2
1
β = 45 , γ = 135◦ M (1; −1; 4). 11x − 17y − 19z + 10 = 0.
200
5x = − 52 (y − 1) = −(z − 2)
6 + 2i, 2 + 4i, 11 + 2i, 1 + 2i
x+1
−1
=
y−1
2
=
z−2
−3
−1 ± 3i,
√
−1±i 11
3
i( π3 +2πk)
π
ei( 2 +2πk) = cos π2 + i sin π2 2ei(π+2πk) = 2(cos
π + i sin π)
2e
√
√
± 3±i
i± 3
π
π
, −i, 2 .
= 2 cos 3 + i sin 3 ±i,
2
=
E
+ C. Ax + xB2 + xC3 + xD4 + x+1
.
x3 − x2 + x − 1 + 3x+2
x2 −1
Ax+B
Cx+D
E
1
2
2
3
2
+
+
.
+
.
−
+
+ x+1
.
x2 +5
(x2 +5)2
x+3
3(x−1)
3(x+2)
x
x2
2
1
2
1
x+4
− 5x − x2 + 5(x−5) . 2x − 2(x2 +4x+8) .
y
2
1
x
1
y = (x − 1)3 + 2
y
1
x
-2
0
y = x−2
x+2
y
1
x
-1
0
y =
√
−x
y
x
0
y = ln(1 − x)
1
− π
3
1
π
− 12
y
0 π
6
π
12
2π
3
x
−1
y = 5 sin(2x − π/3)
y
0
x
y = x21+1
y
0
x
y = x2 + x1
y
1
−2π
−π
0
−1
π
2π
y = sin1 x
3π x
y
−5π/2 −3π/2 −π/2
π/2
1
3π/2
5π/2
0
x
y = lg(cos x)
x
0
r = ϕ
y
π
x
0
r = ϕπ
y
r
ϕ
0
1
2
x
r = 2 cos ϕ
y
1
-1
1
0
x
-1
r = 1
0
x
3
r = 3 cos 4ϕ
y
x
0
-1
1
y = | lg |x||
y
1
0
x
y = 2|x|
y
1
-1
x
0
1
-1
y = x · |x|
y
0
x
3
y = x3 − 3x2
y
O
x
y = x(1 +
√
x)
y
x
O
y = 1 +
√
x
y
4/27
x
0
1
8/27 0,5
y =
√
3
x2 − x
y
1
0
x
1
y = e x − x
1
y
3
-2
0
2
x
y = (x3 + 4)/x2
y
1
4
2
x
−1
2 −4x
y = x2x−4x+8
y
4/e 2
2
x
y = x2 e−x
y
π/2
x
−π/2
y = x + arctg x
Aa −
Gg −
Mm −
Ss −
Y y −
Aα −
Zζ −
Λλ −
π −
φ −
Bb −
Hh −
N n −
T t −
Zz −
Cc −
Ii −
Oo −
Uu −
Dd −
Jj −
P p −
V v −
Bβ −
Hη −
Mμ −
Rρ −
Ξχ −
Γγ −
Θθ −
N ν −
Σσ −
Ψψ −
Ee −
Kk −
Qq −
W w −
F f −
Ll −
Rr −
Xx −
Δδ −
Iι −
Ξξ −
Tτ −
Ωω −
E −
Kκ −
Oo −
Υυ −
!
"##
"
$ "%
& '( $ !%"
& $ !%
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0 $ $ %" %
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' + ! # ,
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' " %
' " %
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0 +
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0 + " %
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0 !
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0 1 /
0 #
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0
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2
2! !
2
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2 /
2
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!(
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x → +∞ !
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Âèêòîð Ãåîðãèåâè÷ ÇÓÁÊÎÂ,
Âëàäèìèð Àíàòîëüåâè÷ ËßÕÎÂÑÊÈÉ,
Àíàòîëèé Èâàíîâè÷ ÌÀÐÒÛÍÅÍÊÎ,
Âåíèàìèí Áîðèñîâè÷ ÌÈÍÎÑÖÅÂ
ÊÓÐÑ ÌÀÒÅÌÀÒÈÊÈ
ÄËß ÒÅÕÍÈ×ÅÑÊÈÕ ÂÛÑØÈÕ
Ó×ÅÁÍÛÕ ÇÀÂÅÄÅÍÈÉ
×ÀÑÒÜ 1
Àíàëèòè÷åñêàÿ ãåîìåòðèÿ.
Ïðåäåëû è ðÿäû.
Ôóíêöèè è ïðîèçâîäíûå.
Ëèíåéíàÿ è âåêòîðíàÿ àëãåáðà
Ïîä ðåä. Â. Á. Ìèíîñöåâà, Å. À. Ïóøêàðÿ
Ó÷åáíîå ïîñîáèå
Èçäàíèå âòîðîå, èñïðàâëåííîå
ËÐ ¹ 065466 îò 21.10.97
Ãèãèåíè÷åñêèé ñåðòèôèêàò 78.01.07.953.Ï.007216.04.10
îò 21.04.2010 ã., âûäàí ÖÃÑÝÍ â ÑÏá
Èçäàòåëüñòâî «ËÀÍÜ»
lan@lanbook.ru; www.lanbook.com
192029, Ñàíêò-Ïåòåðáóðã, Îáùåñòâåííûé ïåð., 5.
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Министерства образования и науки РФ «Лучшее учебное издание по математике
в номинации «Математика в технических вузах»
В. Г. ЗУБКОВ, В. А. ЛЯХОВСКИЙ,
А. И. МАРТЫНЕНКО, В. Б. МИНОСЦЕВ
КУРС МАТЕМАТИКИ
ДЛЯ ТЕХНИЧЕСКИХ
ВЫСШИХ УЧЕБНЫХ
ЗАВЕДЕНИЙ
Часть 1
Аналитическая геометрия. Пределы и ряды.
Функции и производные.
Линейная и векторная алгебра
Под редакцией
В. Б. Миносцева, Е. А. Пушкаря
Издание второе, исправленное
ДОПУЩЕНО
НМС по математике Министерства образования и науки РФ
в качестве учебного пособия для студентов вузов, обучающихся
по инженерно&техническим специальностям
•САНКТ4ПЕТЕРБУРГ•МОСКВА•КРАСНОДАР•
•2013•
ББК 22.1я73
К 93
Зубков В. Г., Ляховский В. А., Мартыненко А. И.,
Миносцев В. Б.
К 93
Курс математики для технических высших учебных
заведений. Часть 1. Аналитическая геометрия. Пределы и
ряды. Функции и производные. Линейная и векторная
алгебра: Учебное пособие / Под ред. В. Б. Миносцева,
Е. А. Пушкаря. — 2+е изд., испр. — СПб.: Издательство
«Лань», 2013. — 544 с.: ил. — (Учебники для вузов.
Специальная литература).
ISBN 978 5 8114 1558 8
Учебное пособие соответствует Государственному образовательному
стандарту, включает в себя лекции и практические занятия. Первая часть
пособия содержит 34 лекции и 34 практических занятия по следующим
разделам: «Множества», «Системы координат», «Функции одной
переменной», «Теория пределов и числовые ряды», «Дифференциальное
исчисление функций одной переменной», «Элементы линейной,
векторной и высшей алгебры, аналитической геометрии».
Пособие предназначено для студентов технических, физико+
математических и экономических направлений.
ББК 22.1я73
Рецензенты:
À. Â. ÑÅÒÓÕÀ äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðîôåññîð,
÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ;
À. À. ÏÓÍÒÓÑ ïðîôåññîð ôàêóëüòåòà ïðèêëàäíîé ìàòåìàòèêè è
ôèçèêè ÌÀÈ, ÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è
íàóêè ÐÔ; À. Â. ÍÀÓÌΠäîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê,
äîöåíò êàôåäðû òåîðèè âåðîÿòíîñòåé ÌÀÈ; À. Á. ÁÓÄÀÊ äîöåíò,
çàì. ïðåäñåäàòåëÿ îòäåëåíèÿ ó÷åáíèêîâ è ó÷åáíûõ ïîñîáèé ÍÌÑ ïî
ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ; Ó. Ã. ÏÈÐÓÌÎÂ
ïðîôåññîð, çàâ. êàôåäðîé âû÷èñëèòåëüíîé ìàòåìàòèêè è
ïðîãðàììèðîâàíèÿ ÌÀÈ (Òåõíè÷åñêèé óíèâåðñèòåò), ÷ëåíêîððåñïîíäåíò ÐÀÍ, çàñëóæåííûé äåÿòåëü íàóêè ÐÔ.
Обложка
Е. А. ВЛАСОВА
Охраняется законом РФ об авторском праве.
Воспроизведение всей книги или любой ее части запрещается без письменного
разрешения издателя.
Любые попытки нарушения закона
будут преследоваться в судебном порядке.
© Издательство «Лань», 2013
© Коллектив авторов, 2013
© Издательство «Лань»,
художественное оформление, 2013
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Ox, Oy, Oz, + +
O # + ' $
% Ox Oy Oz
, Oxyz.
- M Oxyz %
x . Ox #
+' M
Ox /' $
y . Oy z .
Oz # !0$ 1 ' #x; y; z $
M
Oxyz
' + ,
x, y, z
, '
Oxy Oyz Ozx +
2
!0 , 0 %
& x > 0 y > 0 z > 0 && x < 0 y > 0 z > 0 &&& x < 0 y < 0
z > 0 &( x > 0 y < 0 z > 0 ( x > 0 y > 0 z < 0 (&
x < 0 y > 0 z < 0 (&& x < 0 y < 0 z < 0 (&&& x > 0 y < 0
z 0,
⎪
⎪
⎪
x
⎪
⎪
⎨ π + arctg y x < 0,
x
ϕ=
π
⎪
y
> 0, x = 0,
⎪
⎪
2π
⎪
⎪
⎪
⎩ − y < 0, x = 0.
2
*
M
x = 2, y = −2
+√ , - . $ - r = 22 + (−2)2 =
π
= 2 2, tg ϕ = −1 . ϕ = arctg(−1) = −
4
√
/ - M 2 2; − π4
. # ) %
) O- Oxy OXY
( α 0 #
(Ox;
OX) = α 1 (Oy;
OY ) = α 2 (
x y
X Y
M
y
Y
M
X
r
ϕ
P
α
O
x
N
Ox OX ! M
" ϕ
r " M α + ϕ
!#
$%&'
x = r cos(α + ϕ), y = r sin(α + ϕ).
( "
) "
x = r(cos α cos ϕ − sin α sin ϕ) = (r cos ϕ) cos α − (r sin ϕ) sin α;
y = r(sin α cos ϕ + cos α sin ϕ) = (r cos ϕ) sin α + (r sin ϕ) cos α.
* r cos ϕ = X r sin ϕ = Y #
$%+%'
x = X cos α − Y sin α, y = X sin α + Y cos α.
, $%+%'
X Y
X = x cos α + y sin α, Y = −x sin α + y cos α.
%.
$%+-'
x y
X Y
α = π/4
/ 0 1 cos(π/4) =
$%+%'
√
2/2 sin(−π/4) = − 2/2
√
√
√
2
2
2
2
+Y
, y = −X
+Y
,
2
2
2
2
√
x=X
√
√
2
2
(Y + X), y =
(Y − X).
2
2
√
x=
−3 < x < 3
! −3 < x < 3
#$ % & #
|x| < 3
' −3 < x < 3 ⇔ |x| < 3
"
|x − 1| < 5
! # (#) "
#$ % & #
# #
x − 1, x − 1 0,
|x − 1| =
−(x − 1), x − 1 < 0.
*+ # +
⎡
|x − 1| < 5 ⇐⇒ ⎣
x−10,
x−1 0} $ !
x2 − 3x + 2 = 0 =⇒ x1 = 1, x2 = 2 % !
x2 − 3x + 2 > 0 (−∞; 1) ∪ (2; +∞).
& " '#
R! $ E(f ) = {y|y ∈ R}
& D(f ) = (−∞; 1) ∪ (2; +∞) E(f ) = (−∞; +∞).
y=
(
1
.
(x + 1)(x − 2)
) " x
(x + 1)(x − 2) = 0
x1 = −1, x2 = 2!
D(f ) = (−∞; −1) ∪ (−1; 2) ∪ (2; +∞)
& D(f ) = (−∞; −1) ∪ (−1; 2) ∪ (2; +∞)
y = √1 1− x2 .
*
x
1 − x2 > 0 D(f ) =
= (−1; 1).
D(f ) = (−1; +1).
!" y = f (x) y = g(x) # $
D(f ) D(g) % &
!" y = f (x) + g(x) ! ' !& x #
!!
' f (x) g(x) #
!" y = f (x) + g(x) D(f ) ∩ D(g)
( !" y = f (x)g(x), y = f (x) − g(x)
f (x)
f (x)
y =
) # !" y =
g(x)
g(x)
& D(f ) ∩ D(g) ∩ {x|g(x) = 0}
√
y = x−1+
1
x−1
y = f (x) + g(x)
√
1
f (x) = x − 1, g(x) =
! "
x−1
D(f ) : x − 1 0 ⇐⇒ D(f ) = [1; +∞),
D(g) : x − 1 = 0 ⇐⇒ D(g) = (−∞; 1) ∪ (1; +∞).
# ! $ % & $ '
"
# (1; +∞)
u = f (x) ( % ! ! ! D(f )
) ! E(f ) y = g(u) ( % ! !
! D(g) E(f ) ⊂ D(g) ) ! E(g)
* " x ∈ D(f ) )%
y ∈ E(g) " x ∈ D(f ) ! u = f (x)
)% u ∈ E(f ) ! y = g(u)
)% y ∈ E(g) %! !
)+! " , ) $ )%! y = g(f (x)). . ! u = f (x) )+!
! y = g(u) (
u = x2 − 3x + 2 y = log2 u "
" y = log2 (x2 − 3x + 2)
u = 1 − x2 y = √1u
/
f (x) = 1 − x2 ⇒ D(f ) = R,
1
E(f ) = (−∞; 1], g(u) = √ ⇒ D(g) = (0; +∞), E(g) = (0; +∞) 0
u
E(f )⊂
1 D(g) #
" {x|1 − x2 > 0} = (−1; 1) & + '
1
# y = √
x ∈ (−1; 1)
1 − x2
X ⊂ R
−x ∈ X x ∈ X
O
!" y = f (x) #
#
f (−x) = f (x) $ x ∈ D(f )
(x; f (x)) (−x; f (−x))
Oy f (−x) = f (x)
D(f ) = (−∞; +∞)
y = x2 + 1
f (−x) = (−x)2 + 1 = x2 + 1 =
f (x)
y = x2 + 1 Oy
!"
y
2
y=x+1
3
2
1
-3 -2 -1
0
-1
1
2
3
x
-2
y = x2 + 1
# $ $ %
& ' $ (
%
y = ϕ(x)
ϕ(x) = f (x) + g(x),
ϕ(x) + ϕ(−x)
f (x) =
! g(x) =
D(f )
2
ϕ(x) − ϕ(−x)
" # y =
2
= x3 + 1 f (x) = 1
g(x) = x3 "
√
$"%& y = x
D(f ) = [0; +∞)
O
=
$"%$ y = x
2
+x
x
' ( ) * D(f ) = (−∞; 0)∪
O" +
,
f (−x) = f (x) f (−x) = −f (x)
x − x2
(−x)2 − x
=
= f (x)
f (−x) =
∪(0; +∞)
−x
x
x2 + x
= f (−x)"
−f (x) = −
x
x2 + x
) y =
x
"
$"%- y = x2 1− 1
' ( ) *
O"
+ ,
f (−x) = f (x) f (−x) = −f (x)"
1
1
= 2
= f (x)"
f (−x) =
2
D(f ) = (−∞; −1) ∪ (−1; 1) ∪ (1; +∞)
)
(−x) − 1
x −1
1
y= 2
x −1
"
y = f (x) T = 0
x − T x + T
f (x) = f (x ± T ) x ∈ D(f )
! " "
# $ % T
T0 : T = n · T0 n ∈ Z, n = 0 &
T0 > 0
' () y = sin x
T0 = 2π
x + 2π ∈ D(f ) x − 2π ∈ D(f ) sin(x ± 2π) = sin x
' (* y = {x}
T0 = 1
x + 1 ∈ D(f ) x − 1 ∈ D(f ) {x + 1} = {x}
+ "
T T
, y = {x} + 1
T = 1 y = {x} y = 1
- u = f (x)
T y = g(f (x)) .
/
, y = sin2 x
T0 = π
$ 0 (1 0 #
2
' ( y = f (x)
T y = Kf (kx + b) + a
T1 = T /|k| k ∈ R
' (3 y = 2 sin(3x + 2)
4 ! 2 y = sin x T = 2π k = 3 5%
T1 y = 2 sin(3x + 2) T1 = 2 3· π
2 T1 = 2·π3
' (6 y = √x
x = 0 T > 0 x−T
T < 0 x + T x = 0
!
" #$
y = x
D(f ) = (−∞; +∞) % x + T ∈ D(f )
x−T ∈ D(f ), x ∈ D(f ) & T0 f (x+T0 ) =
f (x) x + T0 = x '( T0 = 0
' y = x
" ) y = f (x)
M x ∈ D(f )
f (x) M y = f (x)
m
x ∈ D(f ) f (x) m
& y = x2 * m = −2
* + , y = −x4 * +
M = 1 * , y = sin x *
1
−1 sin x 1 , y = x, y = lg(x), y = tg(x), y =
*
x
!
" - y = f (x)
! " X ⊂ D(f ) # x1 ∈ X x2 ∈ X
x1 > x2 f (x1) > f (x2) $ %! #&' x !
#&' y() y = f (x) # ! "
X ⊂ D(f ) # x1 ∈ X x2 ∈ X x1 > x2
f (x1) < f (x2) $ %! #&' x ! ! &' y()
y = f (x) # ! " X ⊂ D(f )
# x1 ∈ X x2 ∈ X x1 > x2
f (x1 ) f (x2 )
y = f (x)
X ⊂ D(f )
x1 ∈ X x2 ∈ X
x1 > x2 f (x1) f (x2)
X
2
y = x (−∞; 0] [0; +∞)
y = √x
D(f ) = [0; +∞) x1 > x2 > 0 ! "
# $ !$ %! √
!
√ " f (x1) >
> f (x2 ) f (x1 ) − f (x2 ) = x1 − x2 & '
!$
√ √( √
√
( x1 − x2 ) · ( x1 + x2 )
x1 − x2
√
√
√
x1 − x2 =
=√
√
√
x + x
x + x
1
2
1
2
x2
√
√
) x1 > x2 ⇔ x1 − x2 > 0 ⇔ √xx1 −
√ > 0 ⇔ x1 − x2 > 0
x2
1+
⇔ f (x1 ) − f (x2 ) > 0 ⇔ f (x1 ) > f (x2 ) &'! !
* !"
(a; x1) (x2; x3) (x4; b) (x1; x2)
(x3 ; x4 )
y
y=f(x)
a
x1
x2
0
x3
x4 b x
f(x1)
y = f (x)
D(f )
E(f )
x ∈ D(f ) y ∈ E(f ).
! x1 ∈ D(f ) x2 ∈ D(f ) x1 = x2,
y ∈ E(f ). " y = x2
! x1 = 1
x2 = −1 y = 1
#$ y = f (x)
x1 = x2
y1 = y2
% y = f (x)
y ∈ E(f ) x ∈ D(f ). &
! ' ' y = f (x)
x = f −1(y). ( ' !
x y &
) * y = f −1(x).
% f −1
' ) f f −1
' f
' ) f −1. +
−1
f f ! ! , !' . *
D(f −1 ) = E(f ); f −1(f (x)) = x x ∈ D(f )/
E(f −1 ) = D(f ); f (f −1(x)) = x x ∈ D(f −1 ).
+
'
0
1!
!
2 !- ' !
' y = x 3 #45
! x y y
x
.
#4 y = f (x)
!
+ 1 ' 1
#44 " y = 2x − 1.
# $ %
y
-1
y=f (x)
y=f(x)
x
0
y=x
x y : y = 2x − 1 ⇐⇒ x =
y+1
. x y y x
2
x+1
. ! " ##
y =
2
y
y=2x-1
y=x
y= x+1
2
1
1
y = 2x − 1
x
y = x2
x ∈ [0; +∞), √
y = x
y
2
y=x (x>0)
y=x
y= x
x
0
y = x2 y =
√
x
! " # "$ % "$
! " &
'
(y = C) (y = xn, n ∈ R)
(y = ax) (y = loga x)
(y = sin x, y = cos x, y = tg x, y = ctg x)
y = arcsin x y = arccos x y = arctg x y = arcctg x
( !
" # # #
" $ $ % &
$ % $"# %
'$
) % "$ * & "
+ , $ & -
" + % -
. * % " /
⎧
⎨ 1 x > 0,
0 x = 0,
y=
⎩
−1 x < 0,
1 x ∈ Q,
y=
⎧ −1 x ∈ I,
⎨ x2 + 2x + 3, x < 0,
3 0 x < 5,
y=
⎩ √
x x 5.
y = kx + b.
D(f ) = (−∞; +∞); k = 0 E(f ) = (−∞; +∞),
b = 0 k > 0
k < 0 k = 0
! " # $%&'
y
y=kx+b, k>0
tg( ϕ )=k
b
y=(x-b)/k, k>0
ϕ
0
y = kx + b
x
k>0
b
− ;0
k
(0; b)
y = kx + b
Ox k = tg ϕ ! " k = 0 #$
b
y = x −
% & &
k
y
b
O
x
ϕ
y= x-b
k
, k 0
* D(f ) = (−∞; +∞),
E(f ) = (0; +∞)
+ y > 0!
a > 1 a < 1 ' , a = 1
y = ax -
& Oy+ (0; 1) #
') a .
y = loga x
y = loga x, a > 0, a = 1
, y = loga x y = ax
&/ +
D(f ) = (0; +∞) E(f ) = (−∞; +∞)
a > 1 a < 1
Ox (1; 0)
! " a
#$ % y = ax
y
y=a x, 01
1
0
1
x
y=log ax, 00
b
x
0
y = f (x) + b
! ! ! !"
" ! Ox b > 0 # b < 0 #
! ! Oy b
$
y = x2 − 1
Oy y = x2
2
y
y=x
2
y=x -1
0
-1
x
1
-1
y = x2 − 1
% y = f (x + a)
!&'
y = f (x) ( ) ) *!
! X = x + a Y = y +
!
Ox −a !" !
* *) Y = f (X)
a > 0 * * ' * !
Ox a x = X − a "
a > 0 y = f (x)
Ox
a a < 0
y = f (x) Ox
|a|
y=f(x+a)
y
y=f(x)
a>0
a
0
x
y = f (x + a)
Oy a > 0 a < 0
! " # Ox " a
$%
y = (x − 2)2
Ox y = x2
y
2
y=x
y=(x-2)
2
x
0
2
y = (x − 2)2
& y = kf (x) k ∈ R " '(
) * y = f (x) k !
Ox
k
y = f (x) k > 1
k Ox Oy 0 < k < 1
1
Ox Oy k −1
k
|k| ! "
Ox # $% −1 k < 0
1
Ox # &'$
|k|
( ) * y = −f (x) "
Ox * y = f (x)
( k > 0
+
Oy )
k k < 0 ,
"
)
% ,
y
π
- 2
- 2π
-1
3
2
1
0
y=-3sin x
π
2
π
y=sin x
2π x
-2
-3
y = −3 sin
x
- * y = f (kx)) k ∈ R) !
y = f (x) k Oy . +
k *
y = f (x) / ) ) ) f (1) = 0) )
X = kx) Y = y, ) * y = f (kx) !
1
kx = 1) x =
k
k > 1 * y = f (x) k Oy
Ox% 0 < k < 1 * y = f (x)
1
Oy Ox% k −1 " +
k
|k|
Oy #
−1 k < 0
1
|k|
Oy
y = f (−x)
y = f (x) Oy
! k > 0 " #
Ox
k $ k < 0 %
& '
y = cos 2x
Oy y = ln(−x)
y = ln x Oy
y = cos x
y
y=ln(-x)
y=ln x
0
-1
x
1
y = ln(−x)
$ "
y = f (kx + b)
!
y = f (x)(
•
y = f (x)
•
y = f (x + b)
& ))
•
y = f (kx+b) * "+
k Oy ,
& & √! "
y = 4 − 5x
-,
(
•
y=
√
x
#
√
•
y = x + 4
Ox√
•
y = −5x + 4
Oy
Oy
!
y
y
y
y= 4-5x
y= x+4
y= x
2
x
2
4/5 x
x
-4
y =
√
4 − 5x
y
y
y= f(x)
y=f(x)
x
x
y = |f (x)|
" # # y = f (x) $
# T # y = K ·f (kx+b)+a #
T1 =
T
%
|k|
& &' (
$
# Ox
# ) * + Ox * $
|k|'
, + |k| % T #
K ) a
# ,# T1 =
T
|k|
y=
√
1
−x + √
.
2+x
! ! "
! y = √−x −x 0 y = √21+ x #
2+x0
2+x=0
⎧
⎨ −x 0
2+x0 ⇔
⎩
2+x=0
(−2; 0].
x0
x > −2
y = arcsin(x + 1)3
! "
(x + 1)3 1
(x + 1)3 −1
! # $
x+11
⇔
x + 1 −1
D(f ) = [−2; 0]
x0
x −2
x
−x
y=
e +e
2
.
! D(f ) =
$ $ %
−x
x
& f (−x) = e 2+ e = f (x),
$'
$'
= (−∞; +∞)
y = x2 − 5x + 6
(−∞; +∞)
! !" !# !
f (−x) = (−x)2 − 5(−x) + 6 = x2 + 5x + 6 = ±f (x)
$ ! !! % %
& % ' ! ! ! " %
(y = x2 + 6) % (y = −5x)
! ( )# ! !! % %
*
T y = 5 sin 3x.
y = sin x ! 2π + !,
! * ' "
2π
y = 5 sin 3x !
3
2π
!
3
- !
y = 3 sin 5x + 4 cos 7x
+ !! . y = 3 sin 5x
2π
T1 =
y = 4 cos 7x / ,
5
2π
T2 =
0 1 1 !
7
2π 4π 6π
,
,
, . . . ! / 1
! 1
5
5
5
2π 4π
,
, . . . ! ) ' " ,
7
7
2 !2 ! 2π 3 ' ,
" 2 "
! T = 2π
4
√
5 y = 1 + x.
√
6 y = 3 1 + x.
1
.
4 − x2
√
= 4 9 − x2 .
√
= 2 + x − x2 .
√
2 − x2
.
=
x
2+x
).
= ln(
2−x
2x
).
= arccos(
1+x
y =
y
y
y
y
y = lg(
y
x2 − 3x + 2
).
x+1
x
y = arcsin(lg( ))
10
−x
e −2 e
√
y = 9 − x2
√
√
y = 1 + x + x2 − 1 − x + x2
y = (x + 1)2 + (x − 1)2
1+x
)
y = lg(
1−x
y = x2 − x + 1
y =
x
º
4
º
3
3
º
º
º
º
! "#$ % & " ' (
" # T )
" # !
y = 2 sin 3x + 7 cos 5x.
√
y = tg x
y = sin2 x
√
y = sin x
º
º
º
y = 2x + 3.
!
x ∈ R r
" x # y # x # y y # x
x 3
y 3
y = 2x + 3 ⇔ x = − ⇔ y = −
2 2
2 2
$ % R $
x 3
y = −
2 2
√
& f (x) = 1 + x2 2f (x) − f 2 (x)
f (x)
' (
− f 2 (x)
√ ( y = 2
2
f (x) √"
√ f (x) = 1 + x √ 2
2
' # y = 2 1+x − (√ 1 + x2 )2 y = 2 1+x − 1 − x2
2
$ 2f (x) − f 2 (x) = 2 1+x − 1 − x2
)
!" ! M(1; 1) #
135◦ Ox
* + ,- y − y0 = k(x − x0) .
. . /( Ox : k = tg(ϕ)
k = tg 135◦ = −1 ' + , " # M, # y − 1 = −(x − 1).
$ y = −x + 2
$%! #% % & #
% # % l1 : 18x + 6y − 17 = 0,
l2 : 14x − 7y + 15 = 0, l3 : 5x + 10y − 9 = 0
* 0 (1 # / A 2 #
# " l1 l2 B 2 # # " l2 l3 C 2
# # " l1 l3 3 . A ABC
k 1 − k2
tg ∠A =
. k1 , k2 2 . " " " l1 l2
1 + k1 k 2
17
*" y l1 l2 l1 : 18x+6y −17 = 0 ⇐⇒ y = −3x+ ;
6
15
l2 : 14x − 7y + 15 = 0 ⇐⇒ y = 2x + ; k1 = −3 k2 = 2
7
−3 − 2
= +1 =⇒ ∠A = 45◦
tg ∠A =
1−6
ABC
A
⎧
⎧
29
17
⎪
⎪
⎨ y = −3x + ,
⎨ x= ,
18x + 6y − 17 = 0,
42
6 ⇔
⇔
15
14x − 7y + 15 = 0
⎪
⎪
⎩ y = 2x +
⎩ y = 2 44 .
7
105
44
29
;2
)
A(
210 105
ΔABC A(1; 1) B(2; 3) C(3; 0)
ΔABC
! "
# $ !
x−1
y−1
=
⇔ y = 2x − 1.
AB :
3−1
2−1
x−2
y−3
=
⇔ y = −3x + 9.
BC :
0−3
3−2
x−1
x 3
y−1
=
⇔y=− + .
AC :
0−1
3−1
2 2
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3
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2
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y
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a
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p
2
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F
p
2
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d
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1
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1
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x
16
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' &)! *
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x2 − 2y 2 + 2x + 12y − 33 = 0
! x "
! y # $ % # &
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(
( '
2
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8
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x
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8
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x2 + 4y2 + 2x − 24y + 21 = 0
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X2 Y 2
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=1
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16
4
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1 3 14 144 1444
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0
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x
x
1
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x→+∞
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x
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b- ε
x
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0
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x→−∞
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./
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b2
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0
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|f (x) − b| |f (x)| − |b| < ε |f (x)| < |b| + ε = C
. y = f (x)
"/ y = f (x)
x→+∞
1
! x → +∞" y = f (x)
(N, +∞)
b = 0
1
f (x) C
lim f (x) = b
( , x→+∞
( '
1
f (x)
lim f (x) = b b = 0
- x→+∞
)
',
|f (x) − b| = |b − f (x)| |b|
− |f (x)| < ε
1
= 1 < 1 = C.
|f (x)| > |b| − ε = 0
f (x) |f (x)| |b| − ε
n
1
yn = 1 +
.
n
!
" # " $% & ' % ()*+
(a + b)n = an + n · an−1 · b +
n · (n − 1) n−2 2
·a
·b +
1·2
n · (n − 1) · (n − 2) n−3 3
·a
· b + · · · + bn .
1·2·3
1
! a = 1, b =
"
n
n
1
1 n · (n − 1) 1
· 2+
yn = 1 +
=1+n· +
n
n
1·2
n
+
+
n · (n − 1) · (n − 2) · · · · · (n − n + 1) 1
n · (n − 1) · (n − 2) 1
· 3 +· · ·+
· n =
1·2·3
n
1 ·2 · 3 · ··· · n
n
1
1
1
1
2
=1+1+
· 1−
+
· 1−
· 1−
+ ···
1·2
n
1·2·3
n
n
1
1
2
n−1
··· +
· 1−
· 1−
· ··· · 1 −
.
1 ·2 · 3 · ··· · n
n
n
n
, " n
2
3
1
1− , 1− , 1−
n
n
n
1 2 3
, ,
n n n
" -%
" % , yn+1 > yn
yn =
%
1
1+
n
n
.
1
yn , 2 , 3
n n
n
yn =
1+
1
n
n
N = L$
%
x → a
! &
lim N (x) = +∞.
x→a
'
−∞
&
(
lim N (x) = −∞.
x→a
)
N (x) M(x)
*
+ !
, !-, # *
#
. ! N (x)
x → a ! N 1(x) " x → a
/ & 0 x → +∞ 1 2
ε > 0
0 ! ! # x
1
1
| < ε
|
3
N (x)
N (x)
- N (x) 4 , *
1
1
|
x > C 5 (! |
ε
N (x)
! # x
!
x→a
lim ϕ(x) = b lim ψ(x) = c
f (x) = ϕ(x) + ψ(x)
x→a
f (x) = ϕ(x) − ψ(x) x → a
lim [ϕ(x) ± ψ(x)] = lim ϕ(x) ± lim ψ(x).
x→a
x→a
x→a
x→a
lim ϕ(x) = b lim ψ(x) = c
x→a
x→a
lim [ϕ(x) + ψ(x)] = lim ϕ(x) + lim ψ(x)
x→a
x→a
lim ϕ(x) = b,
7.6
x→a
lim ψ(x) = c;
⇒
x→a
ϕ(x) = b + α(x),
ψ(x) = c + β(x).
! α(x) β(x) " # x → a
$!
f (x) = ϕ(x) + ψ(x) = [b + α(x)] + [c + β(x)] = (b + c) + [α(x) + β(x)],
lim f (x) = lim [ϕ(x) + ψ(x)] = lim {(b + c) + [α(x) + β(x)]} = b + c.
x→a
x→a
x→a
%
&
lim f (x) = lim [ϕ(x) + ψ(x)] = lim ϕ(x) + lim ψ(x).
x→a
x→a
x→a
' ! (
x→a
lim [ϕ(x) − ψ(x)] = lim ϕ(x) − lim ψ(x).
x→a
x→a
x→a
)
x→a
lim ϕ(x) = b lim ψ(x) = c
x→a
x → a *
( f (x) =
lim [ϕ(x) · ψ(x)] = lim ϕ(x) · lim ψ(x).
x→a
x→a
x→a
ϕ(x) · ψ(x)
+
x→a
lim ϕ(x) = b, lim ψ(x) = c
x→a
lim [ϕ(x) · ψ(x)] = lim ϕ(x) · lim ψ(x)
x→a
x→a
x→a
lim ϕ(x) = b,
x→a
lim ψ(x) = c;
x→a
7.6
⇒
ϕ(x) = b + α(x),
ψ(x) = c + β(x),
α(x) β(x)
x → a
f (x) = ϕ(x) · ψ(x) = [b + α(x)] · [c + β(x)] = (b · c) + [c · α(x) + b · β(x) +
+ α(x) · β(x)].
lim f (x) = lim [ϕ(x)·ψ(x)] = lim {(b·c)+[c·α(x)+b·β(x)+α(x)·β(x)]} =
x→a
x→a
x→a
= b · c + lim [c · α(x) + b · β(x) + α(x) · β(x)] = b · c = lim ϕ(x) · lim ψ(x).
x→a
x→a
x→a
lim [c · α(x) + b · β(x) + α(x) · β(x)] = 0
x→a
! " # # $ $ %
lim [k · ϕ(x)] = k · lim ϕ(x).
&'()
x→a
x→a
* $
'+
&',)
lim [f (x)]n = [lim f (x)]n.
x→a
"
'- '+ !.!
' /
0
x→a
.
%
lim ϕ(x) = b lim ψ(x) = c
x→a
c = 0
x→a
&'')
lim [ϕ(x)/ψ(x)] = lim ϕ(x)/ lim ψ(x).
x→a
'+
x→a
x→a
lim (x2 + 2x − 1)
x→2
1 2 3 $ !
•
4
lim (x2 + 2x − 1) = lim x2 + lim 2x − lim 1.
x→2
x→2
x→2
x→2
•
2
2
2
lim x = [lim x] = 2 = 4.
x→2
•
x→2
! " # # $ $
lim 2x = 2 lim x = 2 · 2 = 4.
x→2
x→2
y = f (x) 0 x
a
x→a
lim f (x) 0
sin x
= 1.
x
π
˘ x
0 < x < AC
2
AB
sin x !
˘
0 < AB < AC
0 < sin x < x
" #
x → 0 sin x $ %
lim sin x = 0 & lim cos x = 1
lim
x→0
x→0
x→0
x
x
lim cos x = lim (1 − 2 sin ) = 1 − 2 lim (sin2 ) = 1 − 2 · 0 = 1.
x→0
x→0
x→0
2
2
2
D
A
1
sin x
tg x
x
cos x B
O
'
sin x
x→0
lim
x
SOAB < S
!
C
SOAB =
S
( )
OAC
< SODC .
cos x sin x
OB · BA
=
,
2
2
OAC
1
1
x
= R 2 x = 12 x = ,
2
2
2
1 tg x
tg x
OC · CD
=
=
,
2
2
2
cos x sin x
x
tg x
< <
.
2
2
2
1
sin x
2
1
x
<
cos x <
sin x
cos x
1
sin x
>
> cos x.
cos x
x
x > 0 x < 0
x → 0 ⇒ lim cos x = 1
x→0
1
cos x
1
1
1
=
= = 1.
lim
x→0 cos x
lim cos x
1
SODC =
x→0
x → 0
%
sin x
'( ! "
x
! " #
$ "
! " &
x → 0
lim
x→0
+
sin x
= 1.
x
sin x
= 1
lim
x→0 x
tg x
),, lim
x→0 x
&)*(
%
- x → 0 %
$ .
+ ' $
sin x
tg x
sin x
1
= lim
= lim
·
· lim cos x = 1 · 1 = 1.
lim
x→0 x
x→0
x→0 x
x→0
x
cos x
5x
x→0
lim
arcsin 3x
x = 0
sin y
0
arcsin 3x = y sin y = 3x x =
0
3
x → 0 y → 0 !
sin y
5·
5x
3 = 5 lim sin y = 5 · 1 = 5 .
= lim
lim
x→0 arcsin 3x
y→0
y
3 y→0 y
3
3
x
1
"# lim 1 +
= e
x→+∞
x
$ % % &'
( )' ' *'% + +
+, %
&'- .
)
"# " - %+ # /
'* " +
"+ +
, %+ ' +,
"+ &'- #
0% % #
- % '* +
- % "+
+ ' ' % ,
1 12 3 4
• 3 4 5 "
yn = n
1 2 3 4
• , , , , · · · . 6 '
2 3 4 5
' -'
n
yn =
n+1
√ √
2, 2 2,
√
2 2 2.
1
3
7
15
2 2 , 2 4 , 2 8 , 2 16 .
!
"" 2n # 2 −1 " 2n −1"
$ yn = 2 2 "
%" yn
+1
y10, yn−1 , yn+1.
yn = 2n
n+3
& ' ()
*+ *)
() "
•
n
n
21
2n − 1
2 ∗ 10 + 1
2 (n − 1) + 1
= ,
yn−1 =
=
,
10 + 3
13
(n − 1) + 3
n+2
2n + 3
2 (n + 1) + 1
=
.
=
(n + 1) + 3
n+4
y10 =
yn+1
%",
0, 2; 0, 23; 0, 233; 0, 2333; . . . .
& $*)
-
yn = 0, 2 + [0, 03 + 0, 003 + 0, 0003 + . . .] = 0, 2 + S.
. - + *+ * * * (
)( ( ( q = 0, 1
b1 =n 0, 03" / + n 0 1n
2 Sn = b1(11 −− qq ) . / * * ()
Sn
b1
. *
S=
1−q
lim {yn } = lim [0, 2 + Sn ] =
n→+∞
n→+∞
0, 2 +
7
0, 03
= .
= 0, 2 +
1 − 0, 1
30
b1
1−q
=
lim
n→+∞
1
2
3
n−1
.
+
+
+
·
·
·
+
n2 n2 n2
n2
0 + 1 + 2 + 3 + . . . + n − 1. ! n
Sn =
0 + (n − 1)
n(n − 1)
a1 + an
·n=
·n=
.
2
2
2
"#
1+2+3+ ...+n −1
1
2
3
n−1
= lim
lim
+
+
+
.
.
.
+
=
n→+∞
n→+∞
n2 n2 n2
n2
n2
n(n − 1)
1
2
= .
= lim
n→+∞
n2
2
$
lim
n→+∞
1 1 1
1
+ + +...+ n
2 4 8
2
.
1 1 1
1
, , . . . n
2 4 8
2
% & n ' '
b1 (1 − q n )
. (
Sn =
1−q
1 1 1
b1 (1 − q n )
1
+ + + . . . + n = lim
=
lim
n→+∞
n→+∞
2 4 8
2
1−q
1
1
(1 − ( )n )
2
2
= 1.
= lim
1
n→+∞
1−
2
)
2n+1 + 3n+1
.
n→+∞
2n + 3n
lim
n → +∞ −→ 3n+1 :
2n+1 + 3n+1
lim
= lim
n→+∞
n→+∞
2n + 3n
2n+1 + 3n+1
3n+1
=
2n + 3n
3n+1
n+1
2
2n+1
+1
+1
n+1
0+1
3
3
=3
= lim
=
lim
= lim
n
n
1
3n
n→+∞ 2
n→+∞ 1
n→+∞ 1
2
1
·
0
+
+
·
+
3
3
3n+1 3n+1
3
3
3
2
n → +∞
3
1.
1
x
−
3;
lim 5
2.
x→3−0
1
x
−
3.
lim 5
x→3+0
1
→ −∞
! " x → 3 − 0# x − 3 → −0
x−3
$ %
#
1
lim 5 x − 3 = 5−∞ =
1
1
= 0.
=
5+∞
+∞
1
→ +∞#
x → 3 + 0# x − 3 → +0
x−3
1
lim 5 x − 3 = 5+∞ = +∞.
x→3−0
& "
#
x→3+0
' ( (
) * +
, %
sin x
=1
lim
x→0 x
lim
x→+∞
1
1+
x
x
1
y
= lim (1 + y) = e.
y→0
sin 5x
x→0
lim
x
! ! "# $
% & ! '
sin 5x
sin 5x · 5
sin 5x
= lim
= 5 lim
= 5 · 1 = 5.
lim
x→0
x→0
x→0 5x
x
x·5
y
( )* 5x = y + x = , x → 0 y → 0 :
5
sin 5x
sin y
sin y
= lim y = 5 lim
= 5.
lim
x→0
y→0
y→0 y
x
5
3πx
2
- lim
x→2
x
)
) ! * $
. ! x → 2
*! ! ! ! ,
! x = 2
3π · 2
3πx
sin
sin
2
2 = sin 3π = 0 = 0.
=
lim
x→2
x
2
2
2
%/
sin
sin x
x→+∞
lim
x
! ! # $
0 ! ! $
sin x
= 1 , x → +∞ $
lim
x→0 x
! *! * 1 lim x = +∞2 !
x→+∞
3 1| sin x| 12
*! * ! *! 4
sin x
= 0.
lim
x→+∞ x
5x
=y
2
lim
x→0
y→0:
5x
2 sin2 y
2 sin2 y
sin2 y
25
2 = lim
lim 2 =
=
2 = lim 4
2
y→0
y→0
x
2 y→0 y
2y
y2
25
5
2
sin y
25
25
25
lim
·1= .
=
=
2 y→0
y
2
2
2y
5
2 sin2
x→0
x=
x→π
lim
1 − sin
x
2
π−x
x = π
π
x
1 − sin
1 − sin
2 =
2 = 1 − 1 = 0,
lim
x→π
π−x
π−π
π−π
0
x−π = y
y → 0
x → π x−π → 0
"
x = y + π
!
y π
y+π
y
x
= sin
= sin
+
= cos ,
2
2
2 2
2
x
y
1 − sin
1 − cos
2 = lim
2.
lim
x→π
y→0
π−x
−y
0
#
$
%
0
y
2 y
= 2 sin
'( 1 − cos
2
4
y
y
1 − cos
2 sin2
2
4.
= lim
lim
y→0
y→0
−y
−y
y
= z y = 4z y → 0 z → 0 :
!
4
y
2 sin2
2
2
4 = lim 2 sin z = − 1 lim sin z =
lim
y→0
z→0 −4z
−y
2 z→0 z
1
1
sin z
· lim sin z = − · 1 · 0 = 0.
= − lim
z→0
2 z→0 z
2
sin
&
lim
x→π
1 − sin
π−x
x
2 = 0.
x→0
lim (cos(mx) − cos(nx))/x2
!
" 1 −0 1 = 00 ! ! #$ ! %&
m−n
m+n
x · sin
x
cos(mx) − cos(nx) = −2 sin
2
2
m−n
m+n
x · sin
x
−2 sin
cos(mx) − cos(nx)
2
2
lim
=
lim
=
x→0
x→0
x2
x2
m+n
m−n
x
x
sin
sin
2
2
· lim
=
= −2 lim
x→0
x
x
x→0
m+n
m−n
m+n
m−n
· sin
x
· sin
x
2
2
2
2
· lim
=
= −2 lim
m+n
m−n
x→0
x→0
·x
·x
2
2
m+n
m−n
x
x
sin
sin
m−n
m+n
2
2
·
= −2 ·
· lim m + n
· lim m − n
=
x→0
x→0
2
2
·x
·x
2
2
n2 − m2
2 2
2
.
= − (m − n ) · 1 · 1 =
4
2
x
3
' x→+∞
lim 1 +
x
! () ! ! &
) ! () *! ! + t = x3 + x = 3t
x → +∞ t = x3 → 0
lim
x→+∞
1+
3
x
x
⎛
⎞
3
1 3
= lim(1 + t) t = ⎝lim(1 + t) t ⎠ = e3 .
t→0
t→0
x→+∞
lim
x−1
x+1
x
!" # $ %
x
x
⎞
⎛
−1
1 x
x
1
+
lim
x · (1 − )
x→+∞
x−1
e−1
x
⎜
x ⎟
x =
= e−2 .
lim
= lim ⎝
=
⎠
1
x→+∞
x→+∞
x+1
e
1
x · (1 + )
lim 1 +
x
x→+∞
x
x
2x + 3
x→+∞
lim
2x − 1
y+1
&# % 2x−1 = y $ x =
$ 2x+3 = y+4
2
' x → +∞ y → +∞
x
y + 1
( y + 1 )
4 2 2
2x + 3
y+4
2
=
lim
= lim
= lim 1 +
x→+∞ 2x − 1
y→+∞
y→+∞
y
y
y
1
y 1 1
4 2
4 2
4
2 2
= lim 1 +
·1 =
· lim 1 +
=
lim 1 +
y→+∞
y→+∞
y→+∞
y
y
y
1
4 2
= (e ) · 1 = e2 .
(
x→+∞
lim (ln(2x + 1) − ln(x + 2))
' x → +∞
)
+∞ − ∞
* %
2x + 1
. + ",
ln(2x + 1) − ln(x + 2) = ln
x+2
* x - *
1
2+
2x + 1
x
= ln
ln
2
x+2
1+
x
⎞
1
⎜
x⎟
lim (ln(2x + 1) − ln(x + 2)) = lim ⎝ln
= ln 2,
2⎠
x→+∞
x→+∞
1+
x
1 2
x → +∞ x x
⎛
2+
! "
#$% 1, 12 , 212 , 213 , · · · .
#&' 2, 32 , 43 , 54 , 65 , · · · .
#&$ 1, 4, 9, 16, 25, · · · .
#&& 21 , 43 , 65 , 87 , · · · .
#&( 1, − 12 , 13 , − 14 , · · · .
#&) n
{yn} = 3n y3 , y5, yn+1
*
#&+
yn
1, 6; 1, 66; 1, 666; 1, 6666; . . .
#&,
lim
n→+∞
#&#
lim
n→+∞
#&-
1 + 3 + 5 + 7 + ... + (2n − 1)
.
n2
1−
1
(−1)n−1
1 1
+ −
+ ··· +
.
3 9 27
3n−1
lim
9n + 8n
.
+ 8n+1
n→+∞ 9n+1
√
2,
√
2,
√
2, . . . .
n sin n!
.
n2 + 1
lim
n→+∞
1
n→4−0
lim 2 n − 4 .
1
n
−
4.
lim 2
n→4+0
x→0 sinx4x .
lim
lim
lim
lim
5x
.
x→0 sin
sin 2x
x→2 sinx x .
x
.
x→0 sin
tg x
x
sin
3.
lim
x→0
x2
lim
cos x
.
lim
π π − 2x
x→
2
sin x − sin a
.
lim
x→a
x−a
x
x
lim
.
x→+∞ x + 1
x
2+x
lim
.
x→0
3−x
x2
2
x +2
lim
.
x→+∞ 2x2 + 1
lim
2
x→0 tg x −x3sin x .
x→+∞
lim x (ln(x + 1) − ln x) .
x→0 ln(1 x− 3x) .
α(x) β(x)
α(x)
=b=0
x → a x→a
lim
β(x)
= +∞
! α(x)
β(x) x → a
α(x)
= 0
lim
x→a β(x)
" α(x)
β(x) x → a
α(x)
= +∞
lim
x→a β(x)
# α(x) β(x)
α(x)
x → a x→a
lim
β(x)
+∞
y = x2 y = 3x x → 0.
$ % &
x2
x
0
= lim = = 0.
x→0 3x
x→0 3
3
y = x2
' y = 3x
lim
'
(
!
y=x
$ % &
2
x → 0
+ x − 6 y = 4 − x2 x → 2.
x2 + x − 6
(x − 2)(x + 3)
x+3
5
= − lim
= − = 0.
= lim
x→2
x→2 −(x − 2)(x + 2)
x→2 x + 2
4 − x2
4
lim
x → 2
cos x
y=
y = x1 x → +∞.
x
!
cos x
lim
x→+∞
x
1
x
= lim cos x.
x→+∞
" cos x x → +∞
x → +∞
# α(x) β(x),
α(x)
= 1!
x→a
lim
β(x)
x → a
" $ x % x = a &'
α(x)
≈ 1 α(x) ≈ β(x) (
x→a
lim
β(x)
& x a ) α(x) β(x)*+
x → a α(x) ∼ β(x)
, "
#
$ $ ! !
α ∼ α1 % β ∼ β1 x → a
lim
x→a
α1 (x)
α(x)
= lim
.
β1 (x) x→a β(x)
- ! α ∼ α1 . β ∼ β1 x → a
- !
lim
x→a
= lim
x→a
α(x)
α(x) α1 (x) β1 (x)
= lim
·
·
=
β(x) x→a β(x) α1 (x) β1 (x)
α(x) α1 (x) β1 (x)
α1 (x)
α1 (x)
·
·
=1·
· 1 = lim
.
x→a
α1 (x) β1 (x) β(x)
β1 (x)
β1 (x)
sin 5x
/ &$ x→
lim
!
0 sin 3x
sin 3x ∼ 3x x → 0
lim
lim
x→0
x→0
sin 5x
sin 3x
= 1 lim
=1
x→0 3x
5x
sin 5x ∼ 5x
sin 5x
5x
5
= lim
= .
sin 3x x→0 3x
3
α(x) β(x)
[α(x) − β(x)]
α(x) β(x)
lim α(x) = lim β(x) = 0 α(x) β(x)
x→a
x→a
x → a γ(x) = α(x) − β(x)
γ(x)
γ(x)
= lim
= 0
lim
α(x) ∼ β(x) x→a
α(x) x→a β(x)
β(x)
(ϕ(x) − ψ(x))
ψ(x)
= lim
= lim 1 −
=
x→a ϕ(x)
x→a
x→a
ϕ(x)
ϕ(x)
lim
= 1 − lim
x→a
ψ(x)
= 1 − 1 = 0.
ϕ(x)
! γ(x) " " #
"! α(x) $ # % !
γ(x)
= 0
lim
x→a
β(x)
&
!
x→a
lim γ(x) = lim α(x) = lim β(x) = 0
x→a
x→a
γ(x), α(x), β(x) x → a ' !" !()
γ(x) * " " # "!
α(x)
β(x)
= lim
= 0.
" # x→a
lim
x→a
+ ,
lim
x→a
γ(x)
γ(x)
γ(x) + α(x) + β(x)
= 1
lim
x→a
γ(x)
x → a - ! γ(x)
γ(x) + α(x) + β(x)
α(x)
β(x)
= lim 1 + lim
+ lim
= 1 + 0 + 0 = 1.
x→a
x→a γ(x)
x→a γ(x)
γ(x)
5x + 6x
x→0
lim
sin 2x
2
.
x → 0 5x + 6x2 ∼ 5x
5x + 6x2
5x
5
= lim
= .
sin 2x ∼ 2x lim
x→0 sin 2x
x→0 2x
2
! " # $ % $ & x → x0 %
' ! !% % " ( "
% ' &
)'*%'& ' ! + ' %" %$ "&" ,
$ "* % ( x → x0
"" % ' & # - + ' !
0
%" ./ %%'&*!
0
∞
∞
0 ' # # / % x → x0 ! (
/ % # # / " ( %
( ! ( / /(" 1 ./ +∞ − ∞
'2 # % $ x → x0 #
# /'& x → x0 %( ./ 0 · ∞
3 ! ( /! $ 2 %
1
lim (1 + α(x)) β(x) ! α(x) β(x) # $ % $
x→x0
x → x0 2 #$/ ( $ % ( % " α(x)
β(x) ' & x → x0 4 ./ 1+∞
$/ ./ 1 + ( / %
%$ 2" % "*! . !
/ % $" . (
!
3x + 5
5 lim
x→+∞ 2x + 7
- / ( / 1 #
# / % $ ./ ∞
∞
x
lim
x→+∞
3
3x + 5
3 + 5/x
= lim
= ,
x→+∞
2x + 7
2 + 7/x
2
x → +∞ x5 x7
!! " # $
!! !! % #
$
&' N (x) M(x)
x → a
N (x)
= 1
lim
x→a M(x)
(! Pn (x) = bnxn + bn−1xn−1 + · · · + b0 !
! x → +∞ !% bnxn
lim
x→+∞
Pn (x)
bn xn + bn−1 xn−1 + · · · + b0
= lim
=1
n
x→+∞
bn x
bn x n
)! % ! x → +∞
!
Qm (x)
am xm + am−1 xm−1 + ... + a0
= lim
=
x→+∞ Pn (x)
x→+∞ bn xn + bn−1 xn−1 + ... + b0
lim
*&+,
am xm
am
=
lim xm−n
x→+∞ bn xn
bn x→+∞
= lim
•
(m < n)
• (m > n)
+∞ −∞
• (m = n)
3
am
bm
8x + 3x − 5
&- x→+∞
lim
4x3 − 2x2 + 3
8x3 + 3x − 5
8x3
=
lim
= 2.
x→+∞ 4x3 − 2x2 + 3
x→+∞ 4x3
lim
5
2
5x − 2x + 3
x→−∞
lim
2x4 + 3x − 5
lim
x→−∞
5x5 − 2x2 + 3
5x5
5x
= lim
= −∞.
= lim
4
4
x→−∞
x→−∞
2x + 3x − 5
2x
2
!
√
3
√
3 x + 5x − 4 + x
" x→+∞
lim √
√
2x4 − 3x + 2 + 2 x
2
6
3
√
√
√ # $ $ $ √x → +∞√3 3 x2 +√5x − 4 ∼ 3 3 x2 = 3x2/3
√
6
x = x1/2 6 2x4 − 3x + 2 ∼ 2x4 = 6 2x√2/3 3 x = x1/3 %
& √x ' & (&
( 3 3 x2 + 5x − 4 ∼ 3x2/3 ) (
√
6
6
2/3
4
& 2x − 3x + 2 ∼ 2x &
√
√
3 3 x2 + 5x − 4 + x
3x2/3
3
= lim √
lim √
= √
.
√
6
6
6
3
4
x→+∞
2x − 3x + 2 + 2 x x→+∞ 2x2/3
2
' *
& $ ' & )+& +* (
• ! (! '
,& )( • & ) ) ( & ' &
& ,&( )+& .( & &
) ! ' $ !
√
√
√
√
√
√
( x + y) · ( x − y) = ( x)2 − ( y)2 = x − y (x 0, y 0),
√
√
√
√
√
√
3
( 3 x ± 3 y) · ( x2 ∓ 3 x · y + 3 y 2 ) = ( 3 x)3 ± ( 3 y)3 =
=x±y
(x 0, y 0).
√
lim ( x2 − 1 − x)
/ x→+∞
x2 + 3x − 1 − x2
3x − 1
√
= lim √
=
x→+∞ ( x2 + 3x − 1 + x)
x→+∞ ( x2 + 3x − 1 + x)
√
3
3x − 1
= .
x2 + 3x − 1 + x ∼ 2x| = lim
= |
x→+∞
2x
2
= lim
N (x) − M(x)
! x N (x) M(x)
"
!#
$ ! ! !
! x → x0 " !
0
" % ! & ! !
0
! (x − x0 ) " !# '
! " !
" x → x0 ! ! (x − x0 )
#!# ! ! #
( !
!
(x − x0 )#
2
x −9
)#*+ x→3
lim 2
x − 3x
, - ! !
! "# .
0
! # ,
0
! / (x − 3)x=3
x2 − 9
(x − 3)(x + 3)
x+3
= lim
= lim
= 2.
x→3 x2 − 3x
x→3
x→3
x(x − 3)
x
lim
!
#
2
5x + 2 sin 2x + tg
)#*0 x→0
lim
arctg 3x + 5x2
2
x
, - 1 x → 0 sin 2x ∼ 2x tg x ∼ x arctg 3x ∼ 3x#
. ! x
3 sin 2x 2 arctg 3x#3! )#+
3) = +∞
lim
x→+∞
⎛
⎞
1
1+ 2
1
x2 + 1
⎜
x ⎟
= lim ⎝
lim
=
3 ⎠ 4
x→+∞ 4x2 − 3
x→+∞
4− 2
x
x2 + 1
4x2 − 3
x+3
lim (x+
x→+∞
1
1
1
= 0.
= ( )+∞ = +∞ =
4
4
+∞
x→+∞
lim
x+8
x−2
x
8
1+
x+8
x
= lim
x→+∞
lim
= 1, lim x = +∞
2
x→+∞
x − 2 x→+∞
1−
x
1+∞
ϕ(x) = 1 + α(x)! "
x
x
x
x+8
x+8
10
−1
lim
= lim 1 +
= lim 1 +
=
x→+∞ x − 2
x→+∞
x→+∞
x−2
x−2
10
·x
x − 2⎫
x
−2
10x
⎪
⎬
lim
10
10
x→+∞
x
− 2 = e10 .
1+
= lim
=e
x→+∞ ⎪
⎪
x−2
⎭
⎩
⎧
⎪
⎨
# " $ % & ' (
%
⎡
⎤8
x
8 8⎥
⎢
lim ⎣ 1 +
⎦
x→+∞
x
lim
x→+∞
x+8
x−2
x
x
8
1+
x
x =
= lim
⎡
x→+∞
2
1−
⎢
x
lim ⎣ 1 +
x→+∞
# '
%
"
10
⎤−2 = e .
− x
−2
2⎥
⎦
x
lim
x→+∞
1+
k
x
x
= ek
x → +∞
∞
x
∞
5x − 4
4
5−
5x − 4
5
x
x
= lim
= ,
= lim
lim
2
x→+∞ 4x + 2
x→+∞ 4x + 2
x→+∞
4
4+
x
x
4 2
x → +∞ !
x x
3x − 4
5x
5
= lim
=
"#$% x→+∞
lim
4x + 2 x→+∞ 4x
4
2
3x + 2x + 1
#& x→+∞
lim
2x2 + 3
x → +∞ ' (
x2
∞
∞
1
2
3x2 + 2x + 1
3+ + 2
3x + 2x + 1
2
x
x x = 3,
= lim
lim
= lim
3
x→+∞
x→+∞
x→+∞
2x2 + 3
2x2 + 3
2
2+ 2
x
x2
2 1 3
x → +∞ ! , 2 , 2 → )
x x x
3x2 + 2x + 1
3x2
3
= lim
"#$% x→+∞
lim
= .
2
x→+∞ 2x2
2x + 3
2
2
2
x − 2x
#* x→−∞
lim
6x + 7
x → −∞
∞
( +
∞
! ' , -
'+ .
x2
x2 − 2x
2
1−
x − 2x
2
1
x
x
= lim
= −∞.
lim
= lim
=
7
x→−∞ 6x + 7
x→−∞ 6x + 7
x→−∞ 6
−0
+
x2
x x2
2
x2 − 2x
x2
6
= lim
= lim
= −∞
x→−∞ 6x + 7
x→−∞ 6x
x→−∞ x
lim
2
5x − 3x + 4
x→+∞
lim √
4x4 + 5
∞
∞
xn!
" n # $ " %&$!
x4 & ! x2
5x2 − 3x + 4
4
3
5− + 2
2
x
x
x
√
= lim
=
x→+∞
4x4 + 5
5
4+ 4
x2
x
5
5−0+0
= .
= √
2
4+0
5x2 − 3x + 4
= lim
lim √
x→+∞
x→+∞
4x4 + 5
'$ $ (" )
(
√
√ $
* *+ 4x4 + 5 ∼ 4x4 = 2x2
5x2 − 3x + 4
5x2 − 3x + 4
5x2
5
√
= lim
= lim
= .
2
4
x→+∞
x→+∞
x→+∞ 2x2
2x
2
4x + 5
P (x)
x→a
lim
! " P (x) Q(x) & "& +$ *&
Q(x)
+ a = 0 +$
, - " $ " x = a . )
0
P (x)
P (a) = Q(a) = 0 ! *
0
Q(x)
$ (x − a)
lim
3
x +1
/ x→−1
lim 2
x +1
x = −1 x3 + 1 =
= −13 + 1 = −1 + 1 = 0 x = −1 #
x2 + 1 = −12 + 1 = 1 + 1 = 2 0!
0
x3 + 1
= = 0.
lim 2
x→−1 x + 1
2
2
x −4
x→2
lim 2
x − 3x + 2
x = 2
0
0
x2 − 4
(x − 2)(x + 2)
x+2
= lim
= lim
= 4.
x→2 x2 − 3x + 2
x→2 (x − 2)(x − 1)
x→2 x − 1
lim
! " ! "
# $ % %
√
1+x−1
& lim √
3
x→0
1+x−1
x = 0
0
' 1 + x = y 6 # !
0
# # % " ( !
x → 0! y → 1!
√
y6 − 1
1+x−1
y3 − 1
.
lim √
=
lim
= lim 2
3
3
x→0
1 + x − 1 y→1 y 6 − 1 y→1 y − 1
#
# ! %
)
$
0
* y = 1+
0
!
y3 − 1
(y − 1)(y 2 + y + 1)
y2 + y + 1
=
lim
=
lim
=
y→1 y 2 − 1
y→1
y→1
(y − 1)(y + 1)
y+1
lim
=
3
1+1+1
= .
1+1
2
) # " #
"
!
!
,
- . % #
%
√
√
√ √
√
√
( x + y)( x − y) = ( x)2 − ( y)2 = x − y
(x 0, y 0)
√
√
√
√
√ √
3
( x ± y)( x2 ∓ 3 xy + 3 y 2 ) = ( 3 x)3 ± ( 3 y)3 = x ± y (x 0, y 0).
√
lim ( x2 − 5x + 6 − x)
x→+∞
x → +∞
+∞ − ∞
√ ! " # "$ %
& ( x2 − 5x + 6 + x) ' " ( )
* &
√
lim ( x2 − 5x + 6 − x) =
x→+∞
√
√
( x2 − 5x + 6 − x)( x2 − 5x + 6 + x)
√
=
= lim
x→+∞
x2 − 5x + 6 + x
x2 − 5x + 6 − x2
6 − 5x
= lim √
= lim √
.
x→+∞
x2 − 5x + 6 + x x→+∞ x2 − 5x + 6 + x
+& & x → +∞ * ! )
∞
&
∞
xn # n , #& % & x
6 − 5x
6 − 5x
x
=
lim √
= lim √
x→+∞
x2 − 5x + 6 + x x→+∞ x2 − 5x + 6 + x
x
6
−5
x
= lim
=
x→+∞
5
6
1− + 2 +1
x x
−5
−5
5
0−5
=
=√
=− .
=√
1+1
2
1−0+0+1
1+1
x→+∞
lim (x −
√
3
x3 + 8x2 )
%
x → +∞
)
+∞ − ∞
! * ! # " &(
# & & - ! ( " &
lim (x −
x→+∞
√
3
x3 + 8x2 ) =
= lim
(x −
√
3
x→+∞
√
x3 + 8x2 )(x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2 )
√
=
x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2
√
x3 − ( 3 x3 + 8x2 )3
√
=
x→+∞ x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2
= lim
x3 − (x3 + 8x2 )
√
=
3
x→+∞ x2 + x x3 + 8x2 + 3 (x3 + 8x2 )2
= lim
−8x2
√
=
3
x→+∞ x2 + x x3 + 8x2 + 3 (x3 + 8x2 )2
= lim
x→+∞
x→+∞
x2 (1 +
= −8 lim
1+
3
x2
=
8
8 2
3
1 + + (1 + ) )
x
x
= −8 lim
1+
1
8
+
x
3
3
8
(1 + )2
x
= −8 ·
8
1
=− .
1+1+1
3
√
3− 5+x
√
x→4
lim
1− 5−x
x = 4
0
0
!
"
!
#
$!
√
→ (3 + 5 + x)# √
% & " ! # $!
→ (1 + 5 − x)
√
√
3+ 5+x
√
√
(3 − 5 + x) ·
3− 5+x
3+ 5+x
√
√
= lim
lim
=
x→4 1 −
√
1+ 5−x
5 − x x→4
√
(1 − 5 − x) ·
1+ 5−x
√
√
√
(3 − 5 + x)(3 + 5 + x)(1 + 5 − x)
√
√
√
=
= lim
x→4 (1 −
5 − x)(1 + 5 − x)(3 + 5 + x)
√
√
(9 − ( 5 + x)2 )(1 + 5 − x)
√
√
=
= lim
x→4 (1 − ( 5 − x)2 )(3 +
5 + x)
√
(9 − (5 + x))(1 + 5 − x)
√
=
= lim
x→4 (1 − (5 − x))(3 +
5 + x)
√
√
(9 − 5 − x))(1 + 5 − x)
(4 − x))(1 + 5 − x)
√
√
= lim
= lim
=
x→4 (1 − 5 + x))(3 +
5 + x) x→4 (−4 + x))(3 + 5 + x)
√
1
1
1+1
1+ 5−x
√
= −1 · = − .
= −1 ·
= −1 · lim
x→4 3 +
3+3
3
3
5+x
+∞−∞
x→3
lim
1
6
−
x − 3 x2 − 9
! " x = 3 #
$ %
& ' ("
+∞ − ∞
lim
6
1
− 2
x−3 x −9
= lim
x−3
1
1
1
= lim
=
= .
(x − 3)(x + 3) x→3 x + 3
3+3
6
x→3
x→3
= lim
x→3
x+3−6
=
(x − 3)(x + 3)
# ' ' ' ( )
! &
( ' (
sin 5x
* x→0
lim
x
sin 5x ∼ 5x x → 0
5x
sin 5x
= lim
=5
lim
x→0
x→0 x
x
x
x→0
lim
tg 5x
tg 5x ∼ 5x x → 0
x
x
1
= lim
=
lim
x→0 tg 5x
x→0 5x
5
1 − cos 5x
x→0
lim
x2
5x
∼2
1 − cos 5x = 2 sin
2
x→0
2
5x
2
2
=2·
25x2
25x2
=
4
2
25 2
x
1 − cos 5x
25
2
lim
=
lim
= .
x→0
x→0 x2
x2
2
α = t2 tg t β = t2 sin2 t t → 0.
α = 5t2 + 2t5 β = 3t2 + 2t3 t → 0.
3
x +1
.
x→−1
lim 2
x +1
x
x→1
lim
2
+ 2x + 5
.
x2 + 1
3
4x − 2x
x→+∞
lim
3x3 − 5
2
2
.
x +x−1
.
! x→+∞
lim
2x + 5
lim
lim
lim
lim
3
x→1 xx −−11 .
2
x + 3x − 10
.
x→2 3x
2 − 5x − 2
2
3x − 2x − 1
.
x→+∞
lim
x3 + 4
x→3 x
2
− 5x + 6
.
x2 − 9
2
+ 10
.
x→2 xx2 −− 7x
8x + 12
√
3
x−1
.
lim √
x→1 4 x − 1
5
(1 + x)3 − 1
.
lim
x→0
x
√
1+x−1
.
lim
x→0
x
√
2x + 1 − 3
√ .
lim √
x→4
x−2− 2
x
lim √
.
x→0 3 1 + x − 1
√
√
1+x− 1−x
.
lim
x→0
x
√
√
lim 3 x + 1 − 3 x .
x→+∞
1
3
.
−
lim
x→1
1 − x 1 − x3
! !
! ! "#$ % $$" &
"' $ "
y = f (x)
x0
•
x0
•
x → x0
• x → x0
x0
! x0 x0
! " " y = ex
x = 1#
" # $ %& ' y = ex
x = 1, & ' ( )*
$
•
y = ex x = 1 ⇒ f (1) = e+
• * lim f (x) = lim ex = e+
x→1
x→1
• , )
x = 1 :
lim f (x) = f (x0 ).
x→x0
lim f (x) = f (1) = e.
x→1
- & y = ex
x = 1
$%#&' "
lim f (x) = f ( lim x),
.
x→x
x→x
x→x
lim x = x0 # ( )
) "
#
0
0
0
x0
x→x
lim f (x) = f (x0 )
−0
x0
Δx = x − x0 Δy = f (x) − f (x0) !"#$
%&
0
lim f (x) = f (x0 ) ⇒ lim [f (x) − f (x0)] = 0 ⇒ lim Δy = 0.
x−x0 →0
x→x0
Δx→0
"' y = f (x)
x0
!"($
lim Δy = 0.
Δx→0
"' y = x3
x
) * & +
Δy
Δy = (x + Δx)3 − x3 = x3 + 3x2 Δx + 3xΔx2 + Δx3 − x3 =
= 3x2 Δx + 3xΔx2 + Δx3 .
,
lim Δy = lim (3x2 Δx + 3xΔx2 + Δx3 ) = 0.
Δx→0
-
Δx→0
y = x3
−∞ < x < +∞
"# !"
# $ % ϕ(x) ψ(x)
x0 # &
x0
%
#
./
.& x0&
lim ϕ(x) = ϕ(x0 )
x→x0
x0
lim ψ(x) = ψ(x0 ).
x→x0
. f (x) = ϕ(x)·ψ(x)
x→x
lim f (x) = f (x0)
0
lim f (x) = lim [ϕ(x) · ψ(x)] = lim ϕ(x) · lim ψ(x) =
x→x0
x→x0
x→x0
x→x0
= ϕ(x0 ) · ψ(x0 ) = f (x0 ).
x
y = xn y = ax
y = sin x y = cos x ! !
(x ∈ R) y = loga x
x > 0 y = tg x
(− π2 +kπ; π2 +kπ) "" xk = (2k+1) π2
(k = 0; ±1; ±2; ...)
# $
u = ϕ(x)
u0 = ϕ(x0 )
x0
x0 y = f (u)
y = f [ϕ(x)]
%
& '
( '
) !*
loga (1 + x)
# + x→0
lim
x
, * - . x → 0
'
00 ( )
1
1
loga (1 + x)
x
= lim
loga (1 + x) = lim loga 1 + x .
x→0
x→0 x
x→0
x
lim
/
x = 0 !
( lim f (x) = f ( lim x))
x→xo
x→xo
lim loga 1 + x
1
x→0
1
x
lim 1 + x
=e
x→0
x
= loga
+
1
x
lim 1 + x
,
x→0
lim
x→0
loga (1 + x)
= loga e.
x
a=e
ln(1 + x)
= ln e = 1.
x
y = ln(1 + x) y = x
x → 0 $
lim
x→0
'
a
x→0
lim
x
(
)
,$ -. $
1=t
%
&
!" #
−1
x
0
0
ax −
*+
"
.
x = loga (t + 1) 0 x → 0 . t → 0
x
a −1
t
1
1
=
.
= lim
= lim
lim
x→0
t→0 loga (t + 1)
t→0 loga (t + 1)
loga (t + 1)
x
lim
t→0
t
t
/
" "
/ "1
loga (1 + t)
= loga e
lim
t→0
t
ax − 1
1
=
= ln a.
lim
x→0
x
loga e
a = e
ex − 1
= ln e = 1,
lim
x→0
x
x
y = e − 1 y = x
x → 0
%
2
!" #
x0
y = f (x)
y = (1 −1 x)2
x = 1
! x = 1, "
# $ !
y
1
0
1
x
y = (1 −1 x)2
% x0 y = f (x)
! x0
lim f (x) = lim f (x) = A
x→x −0
x→x +0
0
0
& " # #
y=
sin x
x
' ( ) * sinx x ! x = 0 !
x = 0 ! $ x → 0
" + )
lim
x→0+0
sin x
sin x
= 1, lim
= 1.
x→0−0
x
x
sin x
x = 0 f (0) = 1
x
f (x) =
sin x
,
x
x = 0;
x = 0
f (0) = 1.
1
0.8
0.6
0.4
0.2
-10
-5
5
10
-0.2
sin x
x
! x0
x0
" y = sin|x|x
# $ % x = 0 &'
( ) *
1
0.5
-10
-5
5
10
-0.5
-1
sin x
|x|
sin x
sin x
= lim
= −1
x→−0 −x
|x|
sin x
sin x
= lim
=1
lim
x→+0 |x|
x→+0 x
lim
x→−0
1
y =
2
(1 − x)
! ! ! x = 1 "
# $ % x = 1 $
&
1
y = sin x1
' ( ) y = sin $
x
x x = 0
*$ + x = 0 #
x → 0
1
% sin # , −1 1 ,
x
- .% / 0
1
− 7
−6
−5
− 4
−
3
− 2
−
1
1
x=±
k∈N
kπ
1
2
4
3
y = sin x1 y = 0
5
6
7
y = f (x)
a b
y = f (x)
[a, b] !
! "
[a, b]
y
y = f (x)
x = a = xΛ
x2 x = b
x
y = f (x)
[a, b] !
x = x1 = a" # ! x2
$ % y = f (x) [a, b]"
& " '
# ! M ! m "
m f (x) M (
) |f (x)| M " " y = f (x)
[a, b]
* y = f (x)
[a, b]
C
+
, -
y = f (x)" ,
[a, b]" . OX " ( #
OX '
x1, x2, x3
y
a
/
y = f (x)
x1
x2
x3
b
! " #
[a, b] f (a) = A f (b) = B
C,
A B
c f (c) = C
y
y = f (b)
y=C
f (b)
f (a)
a
c
x
b
º
y = f (x) [a, b]
x = f −1 (y)
OY
!
" # $
y=
$
3
1+x
.
1+x
x = −1
! !
0
" # $
0
# % (1 + x)! # 1 + x = 0 &"
!
x = 2
! y = 4x2 x = 2 ⇒ f (2) = 16"
#! $ x→2
lim f (x) = lim 4x2 = 16"
x→2
%!
& x = 2
lim f (x) = f (2) = 16.
x→2
'()
x
y
= sin x
*+ $ Δy ,
-
Δy = sin(x + Δx) − sin x = 2 sin
Δx
Δx
=
cos x +
2
2
Δx
sin
Δx
Δx
Δx
2 · cos x + Δx · Δx.
= 2 sin
·
cos x +
=
Δx
2
2
Δx
2
2
Δx
sin
Δx
2
= 1 cos x +
1
. Δx→0
lim
& x
Δx
2
2
⎞
⎛
Δx
sin
⎜
2 cos x + Δx · Δx⎟
lim Δy = lim ⎝
⎠=
Δx
Δx→0
Δx→0
2
2
Δx
sin
Δx
2
· lim cos x +
= lim
· lim Δx = 1 · cos x · 0 = 0.
Δx
Δx→0
Δx→0
Δx→0
2
2
/ y = sin x
−∞ < x < +∞(
'(0
x − 1,
f (x) =
3 − x,
0 x 3,
3 < x 4.
!"# $ % x = 3 ⇒ y = 2 &
%'!
x → 3 :
lim f (x) = 2,
lim f (x) = 0.
x→3−0
x→3+0
( )! x = 3 * %' + ,
' [0, 4] f (x) ' (x = 0) ' (x = 4)
y
2
0
4 x
1
3
-1
2
y = xx −− 25
5
. x = 5 ! ! '
! ) 0/0 / % )!
lim y = lim y = 10.
( )!
x→5−0
x=5
0
x→5+0
*
%'
y = x12
. x = 0 0 %'!
,1 +'
x = 0 ' +∞ ( )! x = 0 1
1 %'
u1 + u2 + u3 + · · · + un + . . . =
+∞
,
un .
n=1
u1, u2, u3, . . . , un, . . .
!
" un #$% &
n' un = f (n)
( ) '
1
11 + 12 + 13 + · · · + n1 + . . .
un = *
n
+ 2 + 6 + 18 + · · · + 2 · 3n−1 + . . .
un = 2 · 3n−1 *
n−1
, 1 − 1 + 1 − 1 + · · · + (−1) + . . .
un = (−1)n−1 *
π
π
π
π
- cos 1 + cos 2 + cos 3 + · · · + cos n + . . . un = cos πn
+ Sn n
n
Sn = u 1 + u 2 + u 3 + · · · + u n =
n
,
uk .
+
k=1
.& $ $ $$ / )
&
1
1
1
1
+
+
+ ··· +
+ ....
1·2 2·3 3·4
n(n + 1)
' 0 )) $ Sn
1& 2 1& / & "
/ ) $ " '
1
1
1
= −
.
n(n + 1)
n n+1
1
1
1
= =1− ;
1·2
2
2
1
1 1 1 1
1
1
+
= − + − =1− ;
S2 =
1·2 2·3
1 2 2 3
3
1
1
1 1 1 1 1 1
1
1
+
+
= − + − + − =1− .
S3 =
1·2 2·3 3·4
1 2 2 3 3 4
4
S1 =
Sn =
=
1
1
1
1
1
+
+
+ ··· +
+
=
1·2 2·3 3·4
(n − 1)n n(n + 1)
1
1
1
1
1
1 1 1 1 1 1
− + − + − + ··· +
− + −
=1−
.
1 2 2 3 3 4
n−1 n n n+1
n+1
lim Sn = lim
n→+∞
n→+∞
1−
!"#$
1
n+1
= 1 − lim
n→+∞
1
= 1.
n+1
2 + 6 + 18 + · · · + 2 · 3n−1 + . . . .
% &
'
S1 = 2, S2 = 2 + 6 = 8,
S3 = 2 + 6 + 18 = 26, . . . ,
(
Sn = 2 + 6 + 18 + · · · + 2 · 3n−1 .
)
S1 = 2 = 3 − 1, S2 = 8 = 32 − 1, S3 = 26 = 33 − 1, . . . , Sn = 3n − 1.
lim Sn = lim (3n − 1) = +∞.
n→+∞
!"#*
n→+∞
1 − 1 + 1 − 1 + · · · + (−1)n−1 + . . . .
|q| < 1
S = 1 b−1 q
|q| > 1 qn → +∞ n → +∞
b1 − b1 q n
= +∞.
n→+∞
1−q
lim Sn = lim
n→+∞
q = 1 !"#$
b1 + b1 + b1 + · · · + b1 + . . . .
% Sn = nb1 b1 = 0 n→+∞
lim Sn = +∞
& q = −1 !"#$
b1 − b 1 + b1 − b1 + . . . .
' Sn = 0 n ( Sn = b1 n ( )
b1 = 0 n→+∞
lim Sn *
|q| < 1
|q| 1
+ , - - )
,.
"#"
u 1 + u2 + u3 + · · · + un + . . .
!"#/
S
au1 + au2 + au3 + · · · + aun + . . . ,
σn
a
!"#0
aS
% 1 Sn n) !"#/
n) !"#0
σn = au1 + au2 + au3 + · · · + aun = a(u1 + u2 + u3 + · · · + un ) = aSn .
23
lim σn = lim aSn = a lim Sn = aS.
n→+∞
n→+∞
n→+∞
!"#0 aS
u 1 + u 2 + u3 + · · · + un + . . . ,
v 1 + v2 + v3 + · · · + vn + . . .
S S̄
(u1 + v1 ) + (u2 + v2 ) + (u3 + v3 ) + · · · + (un + vn ) + . . . ,
S + S̄
n
Sn S̄n σn !"
σn = (u1 + v1 ) + (u2 + v2 ) + (u3 + v3 ) + · · · + (un + vn ) = Sn + S̄n .
# $
% %
lim σn = lim (Sn + S̄n ) = lim Sn + lim S̄n = S + S̄.
n→+∞
n→+∞
n→+∞
n→+∞
!
$ & '
(u1 − v1 ) + (u2 − v2 ) + (u3 − v3 ) + · · · + (un − vn ) + . . .
!"#%$ !"#&$
S − S̄
!"#""$
&
u1 + u2 + u3 + · · · + uk−1 + uk + uk+1 + · · · + un−1 + un + . . .
uk+1 + · · · + un−1 + un + . . . .
(
( !"#"'$
!"#"($ !"#"'$ )
k *) !"#"($
!"#"'$
Sn n % $
Sk ) k *$ (k < n) σn−k
) n − k % $ ("
Sn = u1 + u2 + u3 + · · · + uk + uk+1 + · · · + un ,
Sk = u1 + u2 + u3 + · · · + uk , σn−k = uk+1 + uk+2 + · · · + un .
Sn = Sk + σn−k ,
Sk n
! S n→+∞
lim Sn = S
"# $
lim σn−k = lim (Sn − Sk ) = lim Sn − lim Sk = S − Sk .
n→+∞
n→+∞
n→+∞
n→+∞
% & & σn−k ' n → +∞ (
' !
lim σn−k = σ
' ! σ n→+∞
% $
lim Sn = lim (Sk + σn−k ) = Sk + lim σn−k = Sk + σ,
n→+∞
n→+∞
n→+∞
!
" ' ) * ) ( +
+! !
u1 +u2 +u3 +· · ·+un +. . .
n
un
, ! -
u 1 + u 2 + u3 + · · · + un + . . . ,
( - S . # & &
/(
Sn = u1 + u2 + u3 + · · · + un−1 + un
Sn−1 = u1 + u2 + u3 + · · · + un−1 .
un = Sn − Sn−1
lim un = lim (Sn − Sn−1 ) = lim Sn − lim Sn−1 .
n→+∞
n→+∞
n→+∞
n→+∞
lim Sn−1 = S n → +∞
lim Sn = S
n→+∞
n→+∞
n − 1 → +∞
lim un = S − S = 0
n→+∞
lim un = 0.
n→+∞
n
!
" ! # $ % # #
%
&
! !
1 2 3
n
+ + + ··· +
+ ...
2 3 4
n+1
' ( ) ' "
% n → +∞)
lim un = lim
n→+∞
n→+∞
! # un =
n
n+1
n
1
= lim
= 1.
n + 1 n→+∞ 1 + 1/n
* lim un = 0
n→+∞
# + $# # ! % !
lim un = 0
n→+∞
, ,
1
1
1
1
√ + √ + √ + ··· + √ + ....
-
n
1
2
3
1
lim un = lim √ = 0 . " $ #
n→+∞
n→+∞
n
"
# # %
1
1
1
1
Sn = √ + √ + √ + · · · + √ .
n
1
2
3
1
1
1
1
1
1
/ √ > √ √ > √ √ > √ , . . . # #
n
n
n
1
2
3
1
1
1
1
Sn > √ + √ + √ + · · · + √ ,
n
n
n
n
√
1
Sn > n · √ Sn > n
n
lim Sn = +∞
n→+∞
!
$
" #
un
'
%!
( )
)
!
n→+∞
%
%
lim un = 0
&
!
*+*
un = 2n!
n−1
&
,
1
-
23
1
, +3/*
%!
./0 -
*3/*
4
·
53/* 0 5
03/* 0
· ·
2
+
· · ·...·2
23/* 0 5
$
,
1
2
4
8
2n−1
+ + + + ··· +
+ ··· .
1! 2! 3! 4!
n!
*+0
1+
&
7
8
,
...
9
1 1 1
+ + + ··· .
3 5 7
6
%
%
%!
2 -
2
1
1
*
5
an = a1 + d(n − 1) a1 = 1, d = 2
an = 1 + 2(n − 1) = 2n − 1
1
un =
2n − 1
1+
1 · 4 1 · 4 · 9 1 · 4 · 9 · 16
+
+
+ ··· .
1 · 4 1 · 4 · 7 1 · 4 · 7 · 10
! "
# $ %
n2 " $
&#'
$
& &( '( &
) *a1 = 1, d = 3+ "
)
& ) * '
+ an = 3n − 2
un =
,
1 · 4 · 9 · 16 · · · · n2
1 · 4 · 7 · 10 · · · · (3n − 2)
1
1
1
1
+
+
+ ··· +
+ ....
1 · 12 12 · 23 23 · 34
(11n − 10) · (11n + 1)
! $ #
- '( &. !
(
A\11n+1
B \11n−10
1
=
+
=
(11n − 10) · (11n + 1)
11n − 10
11n + 1
=
A(11n + 1) + B(11n − 10)
.
(11n − 10) · (11n + 1)
/ ' ' ( $
!
1 = A(11n + 1) + B(11n − 10).
A − 10B = 1.
0 ) 12
12A + B = 1.
0 ) 12
%
$ '(
1
1
3 " A = , B = − 4
#
11
11
!
1
1
=
un =
(11n − 10) · (11n + 1)
11
1
1
−
.
11n − 10 11n + 1
u1 =
1
1
1
1−
, u2 =
11
12
11
1
1
1
1
1
−
, u3 =
−
,... .
12 23
11 23 34
1
1
1
1
1
1
1
1
1−
+
−
+
−
+ ···
Sn =
11
12
11 12 23
11 23 34
1
1
1
+
−
=
11 11n − 10 11n + 1
1
1
1
1
1
1
1
1
1−
+
−
+
−
+ ··· +
−
=
11
12 12 23 23 34
11n − 10 11n + 1
1
1
1−
.
=
11
11n + 1
1
1
1
1−
=
S = n→+∞
lim Sn = lim
n→+∞ 11
11n + 1
11
S = 111
2 1 1
1
1
+ + +
+
+ ··· .
3 3 6 12 24
! "! ! b1 = 32 , q = 21 , "# $ %
!
b1
=
S=
1−q
&
2
3
1
1−
2
+∞
,
n=1
5n + 1
.
4n − 1
4
= .
3
lim un = lim
n→+∞
n→+∞
5n + 1
5
= = 0.
4n − 1
4
1
1 1
+ ··· .
1− + −
4 9 16
2 3 4
2
3
4
5
+
+
+
+ ··· .
3
7
11
15
!
1
1
1
1
+
+
+ ··· +
+ ··· .
1·3 3·5 5·7
(2n − 1)(2n + 1)
1
1
1
1
+
+
+ ··· +
+ ··· .
1·2·3 2·3·4 3·4·5
n(n + 1)(n + 2)
1+
1 1 1
+ + + ··· .
2 4 8
"
+∞
,
n=1
n
.
3n − 1
S = n→+∞
lim Sn
!"
# ! $ $
Sn !" n %
! &
$ '(
#
) (
** *
+ (' )
S1 = u1
S2 = u1 +u2 S3 = u1 +u2 +u3 . . . Sn = u1 +u2 +u3 +· · ·+un (
n ) !
! ( ('( (
S 1 < S2 < S 3 < · · · < S n < . . . .
,
* + #
n→+∞
lim Sn = +∞
- + Sn < C
(! n #
) !
** * !
"
u 1 + u 2 + u3 + · · · + un + . . . ,
(U )
v 1 + v 2 + v3 + · · · + vn + . . . .
(V )
u1 v1 , u2 v2 , u3 v3 , . . . , un vn , . . . ,
!
U "
"
Sn
σn
n
Sn = u 1 + u 2 + u 3 + · · · + u n , σ n = v 1 + v 2 + v 3 + · · · + v n .
$
'
lim σn = σ
n→+∞
! !
!
U
Sn σ n
"
% & !
σn < σ !
# !
&
!
U
!
Sn < σ
!
!
V !
Sn < σ
!
u 1 + u 2 + u3 + · · · + un + . . . ,
(U )
v 1 + v 2 + v3 + · · · + vn + . . . .
(V )
$ !
u1 v1 , u2 v2 , u3 v3 , . . . , un vn , . . . ,
"
( " #
# !
#
V
Sn
(
!
σn
Sn = u 1 + u 2 + u 3 + · · · + u n , σ n = v 1 + v 2 + v 3 + · · · + v n .
(
!
Sn σn " V #
lim σn = +∞ *
) !
lim Sn = +∞
n→+∞
!
n→+∞
!
U
#
|q| < 1 |q| 1
1
1
1
1
+
+
+ ··· + p + ...
!!"#
1p 2p 3p
n
p > 1 p 1 p = 1
1 1 1
1
+ + + ··· + + ...,
1 2 3
n
!!$#
% !!"#
&
& '
!!!
1
1
1
1
+ 3 + 4 + ··· +
+ ....
2
2
3
4
(n + 1)n+1
!!(#
% ) * %
1
1
1
1
+
+
+ · · · + n+1 + . . . .
22 23 24
2
!!+#
% (11.6) & &
q = 1/2 < 1 ,
(11.5) & (11.6)
(11.5) -
!!.
√
1
1
1
1
+√
+√
+ ··· +
+ ....
ln 2
ln 3
ln 4
ln(n + 1)
!!/#
% ) * %
1
1
1
1
√ + √ + √ + ··· + √
+ ...,
n
+1
2
3
4
!!0#
(11.7)
(11.8)
√
√
1
1
>√ ,
ln n < n, ln n < n, √
n
ln n
(11.7)
! ! " !
#
$$ %
Un
lim
= k
n→+∞ Vn
n → +∞
$$ % "
+∞
,
#
un = 1 +
n=1
!
# $
1
1 1
+ + ··· +
+ ....
3 5
2n − 1
) )! ! !
1
vn = * ! *
&$$ '(
n! !
n
1
un
2n
− 1 = 1 = 0.
lim
= lim
1
n→+∞ vn
n→+∞
2
n
+ , - *
* ! * &$$ '(
. !
! !
./! ! *
! !
!
# '
¾
$$ 0
% ¾
u 1 + u2 + u3 + · · · + un + . . .
&
&$$ $1(
n
un+1
= ρ,
un
ρ > 1
lim
n→+∞
ρ < 1
#
ρ < 1
un+1
= ρ
n→+∞ un
lim
N
n N
"
#
!
ε
ε > 0
un+1
− ρ < ε $
un
un+1
un+1
− ρ < +ε, ρ − ε <
< ρ + ε.
un
un
un+1
! ρ + ε = q
< q % ρ
un
& '# ε ε #
# # q = ρ + ε < 1 % n N (
uN+1
uN+2
uN+3
< q,
< q,
< q, . . . ,
uN
uN+1
uN+2
−ε <
uN+1 < uN q, uN+2 < uN+1 q < uN q 2 , uN+3 < uN+2 q < uN q 3 , . . . .
) (
uN + uN+1 + uN+2 + uN+3 + . . . ,
2
*
3
uN + uN q + uN q + uN q + . . . .
) + !
|q| < 1
%
+
!
# *
"
+ !
*
, *
!
! -
u1 + u2 + u3 + · · · + uN−1
+ -
#
ρ > 1
un+1
lim
= ρ > 1 $
n→+∞ un
& n N #
/
.
un+1
> 1
un
un+1 > un
n
lim un = 0
n→+∞
lim
n→+∞
n→+∞
lim un = 0
n
un+1
= +∞
un
un+1
>1
un
!
! "
" ρ = 1 !
# $
$
#
$ %
%
"
1
3
5
7
2n − 1
+
+
+
+ ··· +
+ ....
3 32 33 34
3n
# & '
(
2(n + 1) − 1 2n − 1
un+1
3n (2n + 1)
=
lim
=
ρ = lim
= lim
:
n→+∞ un
n→+∞
n−to+∞ 3n+1 (2n − 1)
3n+1
3n
1
2n + 1
2 + 1/n
1
1
lim
=
lim
= .
3 n→+∞ 2n − 1
3 n→+∞ 2 − 1/n
3
ρ = 1/3 < 1
=
)
%
*
# & '
2
4
8
2n
+
+
+ ··· + 4 + ....
1 16 81
n
(
2n+1
un+1
2n+1 n4
2n
=
lim
= lim
:
=
n→+∞ un
n→+∞ (n + 1)4
n−to+∞ (n + 1)4 · 2n
n4
ρ = lim
1
n4
= 2 lim
= 2.
n→+∞ (1 + 1/n)4
n→+∞ (n + 1)4
ρ = 2 > 1
= 2 lim
n
a
un = k (a > 1, k > 1)
n
n+1
a
un+1
an+1 · nk
an
lim
= lim
:
= a.
=
lim
n→+∞ un
n→+∞ (n + 1)k
n→+∞ an · (n + 1)k
nk
n a > 1 ! " #
a
un = k " $ %
n
&
" ρ = 1 '
! # ( &
)
1
1
1
1
√ + √ + √ + ... + √ + ....
n
1
2
3
*
1
1
un+1
:√
=
= lim √
n→+∞ un
n→+∞
n
n+1
√
n
= lim √
= 1.
n→+∞
n+1
ρ = lim
* " #
! +
, & # ρ = 21 < 1 -./
0 # '
! " ," & 1 "&2
3 ¿
u 1 + u2 + u3 + · · · + un + . . .
¿
lim
√
n
n→+∞
un = q
q 1 q = 1
+∞
,
un =
n=1
n
+∞
,
1
1
·
1
+
.
2n
n
n=1
n
√
1
1
1
1
1
· 1+
= < 1.
lim n un = lim n n · 1 +
= lim
n→+∞
n→+∞
n→+∞ 2
2
n
n
2
!
" # #$ !
% &
' & (
"
&
+∞
,
n=2
un =
ln n
ln 2 ln 3
+
+ ··· +
+ ··· .
2
3
n
) & !
* & !
1
vn = " un + (#&
n
+∞
. 1
ln n
1
> -
vn ,
n → +∞
!
n
n
n=1 n
.
+∞
,
n=0
un =
+∞
,
n=0
1
.
4 · 2n − 3
+∞
.
n=0
+∞
.
1
n
2
n=0
vn =
1
q =
2
!"
1
n−3
un
2n
1
1
4
·
2
= lim
lim
= lim
= lim
= .
3
1
n→+∞ vn
n→+∞
n→+∞ 4 · 2n − 3
n→+∞
4
4− n
2n
2
#
%
+∞
.
n=0
un =
+∞
.
n=0
1
4 · 2n − 3
q =
+∞
.
n=0
1
2
1
2n
$
&
''(
+∞
,
un =
n=0
+∞
,
n=1
2n2
1
.
− 3n
+∞
.
+∞
.
1
vn =
2
n=1
n=1 n
)''(* !" !
1
2 − 3n
un
1
2n
= lim
= = 0.
lim
1
n→+∞ vn
n→+∞
2
n2
#
+∞
.
n=1
un =
'',
+∞
,
n=1
+∞
.
n=1
1
2n2 − 3n
um =
+∞
,
1
sin .
n
n=1
+∞
.
n=1
vn =
+∞
.
n=1
1
n2
+
=
+∞
.
1
n
n=1
+∞
.
n=1
vn =
1
sin
un
sin m
n
= 1 = 0.
lim
= lim
= lim
1
n→+∞ vn
n→+∞
m→ 0 m
n
+∞
+∞
.
.
1
! un = sin
n
n=1
n=1
""#
+∞
,
n=1
2 · 5 · 8 . . . (3n − 1)
.
1 · 5 · 9 . . . (4n − 3)
$ n + 1 %
un+1 =
2 · 5 · 8 . . . (3n − 1)(3(n + 1) − 1)
=
1 · 5 · 9 . . . (4n − 3)(4(n + 1) − 3)
3n + 2
2 · 5 · 8 . . . (3n − 1) · (3n + 2)
= un
.
1 · 5 · 9 . . . (4n − 3) · (4n + 1)
4n + 1
un+1
3n
3
= < 1
lim
= lim
n→+∞ un
n→+∞ 4n + 1
4
&%
""'
+∞
,
n!
n=1
(
en
!
.
un+1
(n + 1)!/en+1
(n + 1)! ·en
= lim
=
lim
=
n→+∞ un
n→+∞
n→+∞
n!/en
n! · en+1
lim
= lim
n→+∞
n+1
= +∞ > 1.
e
!
)
+∞
,
n=1
n+1
2n − 1
n
.
-
n
√
n+1
n+1
1
n
n
= < 1.
lim
un = lim
= lim
n→+∞
n→+∞
n→+∞ 2n − 1
2n − 1
2
+∞
.
1
.
n=2 ln n
+∞
. 2n
.
n
n=0 5 + 1
+∞
.
1
.
2
n=2 n ln n
+∞
.
1
tg .
n
n=1
10n
.
n=1 n!
+∞
.
n
.
n/2
3
n=1
+∞
.
+∞
.
n=1
2n
3n + 1
n
.
! "# $ " !
% &
' $ " "% ! " "
&
(% # ! ) *
" ) "% "% " (
! #
+ !
& !
1
1
1
1
1
1
1
1
1
1
− − + + − − + + − · · · + (−1)n(n−1)/2 2 + . . . .
12 22 32 42 52 62 72 82 92
n
!
!
"# u1, u2, . . . , un, . . . #
! $
! # %
u1 − u2 + u3 − u4 + · · · + (−1)n−1 un + . . . .
&'( ')
* !
+!#
'( '
u1 > u2 > u3 > · · · > un > . . .
n→+∞
lim un = 0
! " # ! $ % ! ! #
* ,
S2m = u1 − u2 + u3 − u4 + · · · + u2m−1 − u2m .
-
%
S2m = (u1 − u2 ) + (u3 − u4 ) + · · · + (u2m−1 − u2m ).
.
# #
# S2m
m
/ $ S2m # %
u1 − [(u2 − u3 ) + (u4 − u5 ) + · · · + (u2m−2 − u2m−1 ) + u2m ].
- ! # ! 0 1
S2m < u1 # m . #
S2m m
1 ! - S2m !
|u1 | + |u2 | + |u3 | + · · · + |un | + . . . ,
u1 + |u1 | u2 + |u2 |
un + |un |
+
+ ··· +
+ ... .
2
2
2
!
un > 0 : |un | = un
un < 0 : |un | = −un
"
un + |un |
|un | + |un |
=
= |un |;
2
2
un + (−un )
un + |un |
=
= 0.
2
2
#
% &
" '
" $
% (# ) %
*+ % &
) % &
1
2
$
|u1 | |u2 |
|un |
+
+ ··· +
+ ... .
2
2
2
,&
, # %
u2 + |u2 | |u2 |
un + |un | |un |
u1 + |u1 | |u1 |
−
+
−
+· · ·+
−
+. . . .
2
2
2
2
2
2
-
+ %
. )
2·
/
)+ %
un
un + |un | |un |
−
=2·
= un .
2
2
2
% %
+ %
1
1
1
1
1
1
−
−
+
+
−
−... .
12 22 32 42 52 62
0
1
1
1
1
+
+
+
+ ... .
12 22 32 42
p = 2 > 1 !
"
$ !
# #
1 1 1
1
+ − + · · · + (−1)n−1 · + . . . ,
2 3 4
n
%&' ()
# *+ ,-#
1−
1+
1
1 1 1
+ + + ··· + + ...,
2 3 4
n
%&' ()
!
.
%&' /) %&' ()
%&' /)
%&' ()
0 1 #
&' & u1 +u2+u3 +· · ·+un+. . .
|u1 | + |u2 | +
+ |u3 | + · · · + |un | + . . .
2
&' ' u1 +u2+u3 +· · ·+un+. . .
|u1 | + |u2 | +
+ |u3 | + · · · + |un | + . . .
! " # $
!
% ! ! &
' ( &
) ! $ * )
!&
' ) + $ &
$ $
, ! &
- !
$ ! $
$ !)
. $ &
$
/ $ !) "
1 1 1 1 1 1 1 1
+ − + − + − + − ...,
2 3 4 5 6 7 8 9
S
1−
$ $ 0
1 2 0 0 &
0 3 ! 1 $
1 1 1 1 1 1
1
1
1
−
+ − ... .
1− − + − − + −
4
2 4 3 6 8 5 10 12 7
* $ " Sn 4 5
σn 60
1
7
1
1 1 1
= , S4 = 1 − + − = ,
2
2
2 3 4
12
37
1 1 1 1 1
S6 = 1 − + − + − = , . . . ;
2 3 4 5 6
60
1
7
1 1
1 1 1 1 1
σ3 = 1 − − = , σ6 = 1 − − + − − = ,
2 4
4
2 4 3 6 8
24
S2 = 1 −
1
1
1
37
7
+ −
−
=
,....
24 5 10 12
120
σ3 = 0, 5S2 , σ6 = 0, 5S4 , σ9 = 0, 5S6 , . . .
σ3m = 0, 5S2m lim S2m = S lim σ3m =
σ9 =
m→+∞
= 0, 5 lim S2m = 0, 5S
m→+∞
m→+∞
!"
0, 5S
#
lim σ3m+1 = lim
m→+∞
m→+∞
lim σ3m+2 = lim
m→+∞
m→+∞
σ3m +
1
2m + 1
= 0, 5S
1
1
−
= 0, 5S.
σ3m +
2m + 1 4m + 2
lim σn %
$ n→+∞
n & !"
' ( ) *"
) +
, +
-"
. ) S
Sn = u1 +u2 +u3 +· · ·+un n → +∞ S = n→+∞
lim Sn
'( / n 0
S ≈ Sn ,
1"
) n # 2
0 ) 1" )
3
S n
u1 + u2 + u3 + · · · + un + un+1 + un+2 + . . . .
Sn n
n n
! " # "
|u1 | + |u2 | + |u3 | + · · · + |un | + . . .
$
% n $&
rn = un+1 + un+2 + un+3 + . . . ; rn = |un+1 | + |un+2 | + |un+3 | . . . .
p &
|un+1 + un+2 + · · · + un+p | |un+1 | + |un+2 | + · · · + |un+p |.
' p → +∞# "
lim |un+1 + un+2 + · · · + un+p | lim |un+1 | + |un+2 | + · · · + |un+p |,
p→+∞
p→+∞
|rn | # "
( r3
rn |
sin 1 sin 2 sin 3
sin n
+ 2 + 3 + ··· + n + ....
2
2
2
2
% ) & * # # #
sin 1 > 0, sin 2 > 0, sin 3 > 0, sin 4 < 0, sin 5 < 0, sin 6 < 0, sin 7 > 0, . . . .
%
sin 1 sin 2 sin 3
+
+ · · · + sin n + . . . .
+
2 22 23
2n
sin n
1
2n 2n # , "
+
- " , " ,
1
1
1
1
+
+
+ ··· + n + ....
2 22 23
2
'
. " , # ,
" # , " , " r3, r3 , r3#
|r3 | < r3 < r3
|r3 | < r3 =
1
1
1
1
1
1/24
= .
+ 5 + 6 + ··· + n + ··· =
4
2
2
2
2
1 − 1/2
8
!"# $
"
!"
" %
& ' ' ( (
1
1
1
1
− + − · · · + (−1)n−1
+ ....
1! 3! 5!
(2n − 1)!
!
"
ΔS = |S − Sn | = |rn | un+1 .
#$ % &# & ' ((
& & % ## %% |rn | un+1 0, 01 #
1
0, 01.
(2n + 1)!
)
%%
%
# & *+
1
1
S ≈ S2 = − ≈ 1 − 0, 17 = 0, 83.
1! 3!
) ' '
1−
1
1
1
1
1
− 2 + 3 − 4 − 5 + ··· .
2 2
2
2
2
, %
1+
- #&
#' %#& &# %
1
1
1
1
1
+
+
+
+
+ ··· .
2 22 23 24 25
.
&
%'/
&
1
q = < 1 # %# 2
%
% #
0
+∞
,
(−1)n+1
5n + 6
n=1
.
! " #
$ % &
1
= 0,
• ' lim |un | = 0
n→+∞
lim
n→+∞
5n + 6
1
1
1
>
>
> ··· .
11
16
21
• ' |un | |un+1 |
( $ % &
(
1
" un =
#
"
(−1)n+1
un =
5n + 6
5n + 6
)
+∞
,
(−1)n−1 · n
n=1
10n + 9
.
*
n
=
% & n→+∞
lim |un | = 0 * $ lim
n→+∞
=
10n + 9
1
= 0
10
( $
+
+∞
,
n=1
(−1)n+1
3 · 5 · 7...(2n + 1)
.
2 · 5 · 8...(3n − 1)
, *
"
3 · 5 · 7...(2n + 1) · (2n + 3)
|un+1 |
2n + 3
2
2 · 5 · 8...(3n − 1) · (3n + 2)
= lim
= lim
= < 1.
lim
n→+∞ |un |
n→+∞
n→+∞ 3n + 2
3 · 5 · 7...(2n + 1)
3
2 · 5 · 8...(3n − 1)
+∞
,
sin nα
.
(ln
10)n
n=1
! " # $ % sin nα & $ %
' | sin nα| 1 ( ' )*
! ++ + vn = (ln 110)n
,!"
lim
√
n
n→+∞
-
vn = lim
n→+∞
n
1
1
1
=
< 1.
= lim
(ln 10)n n→+∞ ln 10
ln 10
- + ) + *
. . (−1)n
+∞
n−1
! " /% *
( % '
' " S = |S − Sn| = |rn| |un+1| 0, 001
n=1
|un+1 | =
0+" #1222
1
1
0, 001 =
⇒ n + 1 1000 ⇒ n 999.
n+1
1000
3 1 − 214 − 314 + 414 − 514 − 614 + · · ·
+∞
.
4
(−1)n+1
.
n=1 ln(n + 1)
2
+∞
.
(−1)n+1 · (n + 1)
.
(n3 + 1)
n=1
+∞
.
(−1)n
n=1
+∞
.
3n + 1
2n + 1
n
.
1 · 4 · 7...(3n − 2)
.
7 · 9 · 11...(2n + 5)
+∞
.
1
(−1)n+1 1 + n .
10
n=1
(−1)n
n=1
+∞
.
(−1)n n2−n .
n=1
+∞
.
n
n
n=1 (2n + 1) · 5
+∞
.
cos 5n
.
n+1
5
n=1
! "
#$%
$ & #$% y = f (x)
M0 (x0 , y0) !
'($ M0 (x0, y0 ) &$ )# $ #$*
% +
& $ $ &*
$ , " "
& $ " $ " (" &"," )#
#$% - ./0 , $ $ M(x, y) " M0
M $$ 1 " " " $ M M0 & + "
x−x0 = Δx → 0. ' M0 + $ ($
& $) M0 " + 23
454
M0 (x0 , y0 )
M0 M
M(x, y) M0 (x0, y0 )
MP
=
M0M k = M
0P
= tg ϕ
Δy
.
kT = tg α = lim
Δx→0 Δx
!
M0(x0, y0) "
# $ y = f (x)
M0 (x0, y0) %
& !
" ! ! ' s = s(t). ( t
y
T
N
M(x,y)
M 0(x0 ;y 0)
ϕ
O
P
α
x
x0
x
s(t)
t+ Δt s(t+ Δt).
Δt Δs = s(t + Δt) − s(t).
v
Δt
v =
Δs
Δt
!
" # t $ # %
Δt # & %
' Δt.
v
v Δt → 0
v = lim
Δt→0
Δs
.
Δt
t
!
(# ) $ %
)*
' $ +
'*
* , )* )
#) &) )%
-. . $
'
(a; b) y = f (x).
x ∈ (a; b).
y = f (x). x
Δx Δx y
Δy. ! "
#
Δy
Δy → 0
Δx → 0. $ % & Δx Δx → 0
'" 00 .
Δy
Δx→0
lim
Δx
() ) y = f (x) x
Δy
Δx
Δx → 0.
f (x).
*%
y
f (x) = lim
= f (x)
x
Δy
f (x + Δx) − f (x)
= lim
.
Δx Δx→0
Δx
f (x) "
Δx→0
, $
dy df (x)
,
.
dx
dx
# x0
() +
$ -
y , y (x),
! %
-
$
dy
df (x0 )
|x=x0 =
.
dx
dx
() .
* $
"%
# $ , %
y = y(x) y (x) $ yx % x = x(y)
x(y) $ xy
/ % 0
# # #( 1 # 2 #
$# x0 " # 2# #
x0 () (
y |x=x0 = y (x0 ) = f (x0 ) =
t s
t
y = x2.
! " #$
% &' Δy "
Δy = (x + Δx)2 − x2 = 2xΔx + Δx2 .
% Δx $ Δx → 0,
Δy
2xΔx + Δx2
= lim
= 2x.
f (x) = lim
Δx→0 Δx
Δx→0
Δx
( )% &'*
+* &' y = x2
x = 0, 5 '
Δy
( * lim
%
Δx→0 Δx
) Δx , * * &
x0
Δy
y − y0
= lim
f (x0 ) = lim
.
Δx→0 Δx
Δx→0 x − x0
- $ x0 +
). x0 ), x → x0 − 0
x → x0 + 0 ' /
0
y = f (x) x0
f (x0 + 0) = lim
x→x0 +0
Δy
.
Δx
y = f (x)
x0
Δy
.
x→x0 −0 Δx
f (x0 − 0) = lim
( f (x) %
x0 *
. %
, .* "
Δy
Δy
= lim
.
lim
x→x0 −0 Δx
x→x0 +0 Δx
f (x) x0
+∞ −∞
x0
+∞ −∞
!"# y = f (x)
!"$ y = f (x)
(a; b)
f (a + 0) f (b − 0)
[a; b]
% &
!"! y = f (x)
' ( ) x Δx = 0. *
+ &
Δy =
)
lim Δy = lim
Δx→0
Δx→0
Δy
Δx.
Δx
Δy
lim Δx = f (x) · 0 = 0,
Δx Δx→0
, (
*
-
! . (−∞; +∞) y = |x|
+ x = 0, +
(
f (−0) = −1, f (+0) = 1.
√
3
x
1
y = √
x = 0,
3
3 x2
! ! ! "
! Δx " # ! !
! $ !
(−∞; +∞)
y =
y = C %
y = C !
! ! !$ "
" #
Δy
0
= lim
= 0.
(C) = lim
'()
Δx→0 Δx
Δx→0 Δx
y = xn * !
$ '+
&
!
(xn ) = n · xn−1 .
'(,
- ! '(, !
! " &! ! & ! .
√
1
1
n = : ( x) = (x1/2 ) = 1/2 · x−1/2 = √ ,
'(/
2
2 x
1
1
= (x−1 ) = −x−2 = − 2 .
n = −1 :
'(0
x
x
y = ax
Δy = ax+Δx − ax = ax (aΔx − 1).
%1
!
!! 1 0
ax − 1
= ln a,
lim
x→0
x
Δy
ax (aΔx − 1)
= lim
= ax ln a.
Δx→0 Δx
Δx→0
Δx
3 ! !$ a = e
(ax ) = lim
(ex ) = ex .
'('2
'(''
y = log
Δy = loga (x + Δx) − loga x = loga
a
x
x + Δx
Δx
= loga 1 +
.
x
x
Δx
loga 1 +
x
.
(loga x) = lim
Δx→0
Δx
loga 1 +
Δx
x
1
=
x
Δx
loga 1 +
Δx
x
Δx
x
Δx → 0,
lim
x→0
' (
x
Δx Δx
1
.
= loga 1 +
x
x
loga (1 + x)
1
= loga e =
,
x
ln a
1
.
x ln a
!"#$"%&
1
.
x
y = sin x y = cos x
!"#$"#&
(loga x) =
a=e
(
(ln x) =
Δy = sin(x + Δx) − sin x = 2 sin
2 sin
(sin x) = lim
Δx→0
) *
lim
Δx→0
Δx
Δx
· cos x +
.
2
2
Δx
Δx
· cos x +
2
2
.
Δx
Δx
2 lim cos x + Δx ,
Δx Δx→0
2
2
sin
(sin x) = cos x,
(cos x) = − sin x.
!
" # $
%
2x sin 2x
.
lim
x→0 1 − cos2 x
√
x+9−3
.
& lim
x→0
x x
x
lim
.
x→+∞ 1 + x
1
1
1
+ ... + n−1 .
lim 1 + +
n→+∞
4 16
4
%
!
%
%
%
%
%
+∞
.
n
n(n + 1)
(−1)n
.
'
n=1 (n + 3)!
.
n=1
+∞
.
( ) *
" + ,
lim
x→0
2x sin 2x
lim
x→0 1 − cos2 x
2x sin 2x
4x sin x cos x
x
= lim
lim cos x = 4 · 1 · 1 = 4.
= 4 lim
2
2
x→0
x→0
1 − cos x
sin x x→0
sin x
√
lim
x→0
x+9−3
.
x
√
√
√
x+9−3
x+9−3
x+9+3
= lim
·√
lim
=
x→0
x→0
x
x
x+9+3
√
( x + 9)2 − 32
x+9−9
1
1
= lim √
= lim √
= lim √
= .
x→0 x( x + 9 + 3)
x→0 x( x + 9 + 3)
x→0
6
x+9+3
lim
x→+∞
x
1+x
x
.
x
−x
x
1+x
= lim
=
x→+∞ 1 + x
x→+∞
x
−x
x −1
1
1
1
1+
= lim 1 +
= lim
= e−1 = .
x→+∞
x→+∞
x
x
e
lim
lim
n→+∞
1+
1
1
1
+
+ ... + n−1
4 16
4
.
1 1
1
4 16
1
! " q = n → +∞ "
4
b1
S =
# !
1−q
4
1
1
1
1
b1
= .
+ ... + n−1 =
=
lim 1 + +
1
n→+∞
4 16
4
1−q
3
1−
4
+∞
.
n
.
n(n + 1)
n=1
n
1
= lim
= 1 = 0,
1
n(n + 1) x→+∞
1+
n
lim Un = lim
x→+∞
x→+∞
" % &
lim |Un | = lim
n→+∞
1
=0
(n + 3)!
' (
lim
n→+∞
(−1)n
.
n=1 (n + 3)!
|Un | > |Un+1 |.
+∞
.
1
(
n=1 (n + 3)!
Un+1
1/(n + 4)!
1
= lim
= 0 < 1,
= lim
n→+∞ 1/(n + 3)!
n→+∞ n + 4
Un
" ! "#$
+∞
.
! "#$
n→+∞
"
)
#
*"& y = f (x) !
x0
+ ,"& y = f (x) ! x0
- ! .
/+ ,"& x → x0 .
+ lim f (x) = f (x0 )
x→x0
x→0
0 lim ctg2
x
5x
· tg2 .
8
4
2
2
z→a za4 −− az4 .
lim
x+a
1
lim 1 +
.
x→+∞
x−1
(−1)n−1
1 1 1
lim 1 − + − + · · · +
.
n→+∞
2 4 8
2n−1
+∞
.
√
n=1
n
.
5n − 1
. √(−1)
.
2n + 9
n=1
+∞
n
3
#
#
! " # $
% &$ ' ! (
%
)' *
+ '
% '
% , % ' '
u(x) v(x)
(u + v) = u + v .
- .
/ % * / x % y(x) = u(x) + v(x)
+ Δx &
Δy = (u(x + Δx) − u(x)) + (v(x + Δx) − v(x)) = Δu + Δv.
0 *
Δy
Δu + Δv
Δu
Δv
= lim
= lim
+ lim
= u + v .
y = lim
Δx→0 Δx
Δx→0
Δx→0 Δx
Δx→0 Δx
Δx
u(x) v(x)
(u · v) = u · v + v · u.
! " " x #" $
% Δx.
y(x) = u(x) · v(x)
Δy = (u + Δu) · (v + Δv) − u · v.
&! %! % ' %"
Δy = v · Δu + u · Δv + Δu · Δv.
(! '
(u · v) = lim
Δx→0
v · Δu + u · Δv + Δu · Δv
.
Δx
) ! ' %! % * %
! ! " +
lim
Δx→0
Δu
·v
Δx
= u · v,
Δv
· u = v · u,
Δx→0
Δx
Δu
· Δv = u · 0 = 0.
lim
Δx→0 Δx
lim
,
lim Δv = 0,
Δx→0
## $" #" $ v " %!
(-! ' %" " #""
(c · u(x)) = c u(x) + c · u (x) = c · u (x).
(u − v) = u − v ,
!
!
" #
(−v) = −v .
(u · v · w) = u · v · w + u · v · w + u · v · w .
u(x) v(x)
x v(x) = 0,
u
v
$
Δx.
u v − v u
.
v2
$ % %
Δy =
)!
=
x
&% '
y(x) =
u(x)
v(x)
(
u(x + Δx) u(x)
−
.
v(x + Δx) v(x)
#
u(x + Δx) = u + Δu;
* %
v(x + Δx) = v + Δv;
vΔu − uΔv
u + Δu u
− =
.
v + Δv
v
v(v + Δv)
Δx,
%!
Δy =
+
!
Δx → 0
lim
Δx→0
lim Δv = 0
Δx→0
,
# %!
#
lim
Δx→0
&& ' %
/! , # !
Δu
= u ,
Δx
, # !
Δv
= v
Δx
&% '
v
% -
(
%. & %%
,
" "
! ,0 %!
Cv
C
=− 2 ,
v
v
u
C
=
u
.
C
,
y = tg x
y = ctg x
!
(tg x) =
sin x
cos x
=
1
(sin x) cos x − (cos x) sin x
=
.
2
cos x
cos2 x
" ! #
(ctg x) = −
1
.
sin2 x
$
%! & & ' ( ( & & '
& & & &)*
&&
y = y(x)
+&&& & ( , '
- &. *& -
& ' / ' - x = x(y) & & &
(a; b)
&& & , & y 0 ! & ) x (y),
& ) ) / #& & )*&, & x ( ' - y = y(x) && ) y (x) .
y (x) =
1
x (y)
yx =
1
.
xy
1
2 ' - x = x(y)
&&' & &
& && ' - y = y(x) *& &
&& 3 !& x *& & Δx = 0. 2!
' - y = y(x) *& & Δy = 0. & & #& &&
lim Δy = 0.
Δx→0
2!
Δy
=
Δx
1
1
= .
Δx
x (y)
lim
Δy→0 Δy
arcsin x arctg x
4 ' - x = sin y (y ∈ [−π/2; π/2]) ( ' y = arcsin x.
+
& & (−π/2; π/2). 5. y (x) = lim
Δx→0
x = cos y = 0 0 & &.
yx =
1
1
1
.
=
=
xy
cos y
1 − sin2 y
sin y = x,
(arcsin x) = √
1
.
1 − x2
! " arctg x
yx =
1
1
1
1
= cos2 y =
=
=1:
.
xy
cos2 y
1 + tg2 y
1 + x2
#
arccos x arcctg x
$ % &!% %'( )(* )
) % & ! "
arcsin x + arccos x =
+
&
π
π
, arctg x + arcctg x = .
2
2
" ! , !-& (π/2) = 0 *.
1
(arccos x) = − √
,
1 − x2
(arcctg x) = −
1
.
1 + x2
/
0! y = y(u) u = u(x). 1 y 2 ! " & .
* x, & u 3 &2! %* !& y = y(u) = y(u(x)).
4 * - !'5 6 , - !'(
&
ux
u = u(x)
x y = y(u)
yu
u y = y(u(x)) x
yx
yx = yu · ux .
7
x Δx u y
Δu Δy. Δx → 0
Δu
"
Δy
Δy Δu
=
·
.
Δx
Δu Δx
! u = u(x) !
Δx → 0 Δu → 0.
lim
Δx→0
Δy
Δy
Δu
= lim
· lim
,
Δx Δu→0 Δu Δx→0 Δx
#$% $&'
( #$% $&'
Δu Δx → 0
$% $ y = sin3 x.
) * +
y = u3 , u = sin x, y = (u3 )u · (sin x)x = 3u2 · cos x = 3 sin2 x · cos x.
$% &
) * +
y = sin x3.
y = sin u, u = x3 , y = (sin u)u · (x3 )x = cos u · 3x2 = 3x2 cos x3 .
! , ! #-. '
! / - ! - 0
! # ' - *
! y = sin5 √x 1 ! 0
y x - * +
• √
x2
• sin x2 √
• sin x
/ - 0
u /
, " "
√
u = sin x y = u5 . 3 #$% $&'+
√
√
√
yx = (u5 )u · (sin x)x = 5 sin4 x · (sin x)x .
sin √x
√ x
y = sin u, u = x!
√
√
1
1
(sin x)x = cos u · √ = cos x · √ .
2 x
2 x
"# #
(sin5
√
x)x = 5 sin4
√
1
x · cos x · √ .
2 x
u #
√
# $ %
& '$ ! $
$(
(sin3 x) = 3 sin2 x(sin x) = 3 sin2 x cos x ,
(sin x3 ) = cos x3 · (x3 ) = 3x2 cos x3 .
& ( )* % ' $ % +
' ,
(arcsin 5x) =
1
· 5,
1 − (5x)2
−1
1
·
( arcctg 3x) = √
· 3,
2 arctg 3x 1 + (3x)2
1
1 −1
cos
= − sin · 2 ,
x
x x
tg ln x
1
1
tg ln x
2
,
=2
· ln 2 ·
2 ln x x
cos
√
(ctg3 2x − x2 ) =
√
−1
1
√
· √
· (2 − 2x).
3ctg2 2x − x2 ·
2
2
sin 2x − x 2 2x − x2
# )* %( $ +
#$ $ -
u = u(x).
1. (C) = 0,
√
1
2. (un ) = nun−1 u , ( u) = √ u ,
2 u
3. (au ) = au ln a · u , (eu ) = eu · u ,
u
u
4. (loga u) =
, (ln u) = ,
u ln a
u
5. (sin u) = cos u · u ,
6. (cos u) = − sin u · u ,
u
,
7. (tg u) =
cos2 u
u
8. (ctg u) = − 2 ,
sin u
u
,
9. (arcsin u) = √
1 − u2
u
,
10. (arccos u) = − √
1 − u2
u
,
11. (arctg u) =
1 + u2
u
12.(arcctg u) = −
1 + u2
1
u
= − 2,
u
u
! " # y = f (x)!
$ # " !
# $ " %
" & ! " ! '
# & '!
% " " #
y=2
sin2 x3
· arctg
√
√
tg 3 x
.
x+ 2
ln (x + 3)
√
2 3
2 3
y = 2sin x ln 2 · 2 sin x3 · cos x3 · 3x2 · arctg x + 2sin x
1
1
√ +
1+x2 x
ln(x + 3) √
1
√
−2
tg 3 x
3
2
x3 x
x+3
=
+
ln4 (x + 3)
√
1
2 3
+
= 2sin x 3x2 ln 2 · sin 2x3 · arctg x + √
2 x(1 + x)
√
1
ln(x + 3)
2 tg 3 x
√
+ 3
.
−
√
x+3
ln (x + 3) 3 3 x2 · cos2 3 x
ln2 (x + 3)
1
cos2
√
3
!
y = x3
• f (x + Δx) = (x + Δx)3 ,
• Δy = f (x+Δx)−f (x) = (x+Δx)3 −x3 = 3x2 Δx+3xΔx2 +Δx3 ,
Δy
= 3x2 + 3xΔx + Δx2 ,
•
Δx
Δy
= lim (3x2 + 3xΔx + Δx2 ) = 3x2 .
• y = lim
Δx→0 Δx
Δx→0
√
" y = x.
√
• f (x + Δx) = x + Δx,
√
√
• Δy = f√
(x + Δx) − f (x) =√ x + Δx − x, √
√
√
√
x + Δx − x
x + Δx − x
x + Δx + x
Δy
=
=
·√
•
√ =
Δx
Δx
Δx
x + Δx + x
1
=√
√ ,
x + Δx + x
• y = lim
Δx→0
Δy
1
1
= lim √
√ = √ .
Δx Δx→0 x + Δx + x
2 x
!
"#
$%&'(
%)& y = x5.
" # * y = (x5) = 5x5−1 = 5x4.
√
%)) y = x3.
√
" # * y = ( x3) = (x3/7) = 3/7x3/7−1 = √3 4 .
7
7
7
7 x
%)+ y = x12 .
" # *
y =
1
x2
%), y = x −x22
" # *
=−
y =
1
3
+ 2√ .
2
x
x x
= (x−2 ) = −2x−3 = −
√
x
2
.
x3
.
√
x−2 x
= (x−1 − 2x−3/2 ) = −x−2 + 3x−5/2 =
x2
%)' y = ax−5.
" # * y = (ax−5) = a(x−5) = −5ax−6 = 5a
.
x6
%)- y = √x.
" # * y = ( √x) = (x1/n ) = 1/nx1/n−1 = √1 n−1 .
n
n
n
%). y =
"#
√
3
n x
x.
√
* y = ( 3 x) = (x1/6) = 1/6x−5/6 = √61 5 .
6 x
√
3 5
%)%/ y = x x.
√
16x2 5 x
√
5
.
y = (x3 x) = (x16/5) = 16/5x11/5 =
5
y = sin x + cos x.
y = (sin x + cos x) = (sin x) + (cos x) = cos x − sin x.
y = tgxx .
y =
tg x
x
x
− tg x
2x
(tg x) x − x tg x
x − sin x · cos x
cos
.
=
=
=
2
2
x
x
x cos2 x
! y = ctg x · arccos x.
y = (ctg x · arccos x) = (ctg x) arccos x + (arccos x) ctg x =
arccos x
ctg x
.
=−
−√
sin2 x
1 − x2
"# $% # &
y = log2 x · 2x.
y = (log2 x) 2x + (2x ) log2 x =
$' '
2x
+ 2x · ln 2 · log2 x.
x ln 2
( y = lne x .
x
(ex ) ln x − (ln x) ex
ex (x ln x − 1)
=
.
y = (
ln2 x
x ln2 x
) % %' %* ' #
+ ,
- % & %& &
$
%* ' #
. y = cos3 x.
2
y = 3/ cos
·
(− sin x)
=
01 x2
/ 01 2
$% *
$% *
2
= −3 sin x cos x.
y =
y =
y =
√
tg x.
√
3
1
√
·
2 tg x
/ 01 2
1
1
√
.
=
2x
2 x tg x
cos
2
cos
/ 01 2
arctg x − (arcsin x)3 .
1
1
1
y =
− 3(arcsin x)2 · √
.
3
2 1 + x2
1 − x2
3 arctg x
y = lg sin x.
1
· cos x.
ln 10 · sin x
y = arcctg(ln x) + ln(arctg x).
y =
y = −
1
1
1
1
+
.
1 + ln2 x x arctg x 1 + x2
y = (e5x − ctg 4x)5.
y = 5(e5x − ctg 4x)4 · 5e5x +
4
.
sin2 4x
y = cos e3x .
y = − sin e3x · e3x · 3 = −3e3x sin e3x .
√
y = arctg −x.
y =
1
1
1
√ .
· (−1) = −
· √
1 − x 2 −x
2(1 − x) −x
y = ln
(x − 2)5
.
(x + 1)3
(x − 2)5
3
5
−
.
) = (5 ln(x − 2) − 3 ln(x + 1)) =
y = (ln
3
(x + 1)
x−2 x+1
y = 2arcsin 3x + (1 − arccos 3x)2.
y = (2arcsin 3x + (1 − arccos 3x)2 ) =
1
1
· 3 + 2(1 − arccos 3x) √
· 3.
= 2arcsin 3x ln 2 √
2
1 − 9x
1 − 9x2
y = x1 .
y = cos x.
y = √x.
y = x(1 − x2).
y = sin x + 3 cos x.
y = x arctg x.
sin x
.
y = log
x
3
3
y = arcctg x + x ln x − tgxx .
y = x5 − 4x3 + 2x − 3.
√
y = x2 x2.
+3
.
y = x2 2x
− 5x + 5
x
y=
.
2 + ex
y = arcsin x + arccos x.
3
6
y = √axa2 ++ bb2 .
y = x ctg x.
2
x−x
.
y = (1 + x ) arctg
2
3
y = x3 ln x − x3 .
y = (x2 − 2x + 2)ex.
y = ln(ex − 5 sin x − 4 arcsin x).
1
1
cos(5x2 ) − cos x2 .
20
4
y = arctg ln x.
1
y = ln arcsin x + ln2 x + arcsin ln x.
2
y = −
!" # !$
!
# ! " %
!"
& ' !
% " % !
( !"
!
$
) ' % " *
y = f (x)
ln y = ln f (x).
x, y = f (x)*
1
(ln y)x = (ln f (x))x =⇒ · y = (ln f (x))x,
y
y = y(ln f (x)) = f (x)(ln f (x)).
+ $
, -
y = f (x)
!
" "
. /$( !
% !
" %
! , 0- )%! ) % *
1
n
y = xn =⇒ ln y = n ln x =⇒ y = =⇒ y = (xn ) = nxn−1 .
y
x
u = u(x) v = v(x)
y = u(x)v(x)
!" #
$
y = u(x)v(x) =⇒ ln y = v ln u =⇒
v
v
1
=⇒ y = v ln u + u =⇒ y = (uv ) = uv v ln u + u .
y
u
u
%&% y = (sin x)cos x.
' (
$
y = (sin xcos x ) = sin xcos x
cos2 x
− sin x ln sin x .
sin x
) #
*
# +
+
cos x
%&, y = sintg xx√
' (
$
2
4
√
3
ln x2
arcsin x3
.
√
3
sin xcos x ln x2
√
ln y = ln
= ln sin xcos x +
4
2tg x arcsin x3
√
√
4
3
+ ln 2 ln x − ln 2tg x − ln arcsin x3 =
1
1
1
= cos x ln sin x + ln 2 + ln ln x − tg x ln 2 − ln arcsin x3 =⇒
3
3
4
1 cos2 x
1
ln 2
3x2
√
y =
− sin x ln sin x +
−
−
=⇒
2
y
sin x
3x ln x cos x 4 arcsin x3 1 − x6
√
3
sin xcos x ln x2
√
·
y =
4
2tg x arcsin x3
2
1
ln 2
3x2
cos x
√
.
− sin x ln sin x +
−
−
·
sin x
3x ln x cos2 x 4 arcsin x3 1 − x6
-# " *
# * #
t
y = y(t),
x = x(t).
x = x(t) y = y(t)
t ! "
Δx → 0, Δt → 0.
#
yx
Δy
Δy/Δt
Δy
= lim
= lim
= lim
Δx→0 Δx
Δt→0 Δx/Δt
Δx→0 Δt
3
Δx
y
= t .
Δt→0 Δt
xt
lim
$ % #
#
yx =
yt
y (t)
= .
xt
x (t)
&'()*
'(
x = a(t − sin t),
y = a(1 − cos t),
+ , &'()*
yx
t
t
2 sin cos
a(1 − cos t)
sin t
2
2 = ctg t .
=
=
=
a(t − sin t)
1 − cos t
2
2 t
2 sin
2
! # %
# #
- F (x, y) = 0.
x y
x
y = yx ,
y
x sin y − y 2 ln x = 0.
!
"
(x sin y − y 2 ln x)x = sin y + x cos y · y − 2yy ln x −
# $
y =
y2
= 0.
x
"
2
y − x sin y
.
x(x cos y − 2y ln x)
%
&
'
(
) * T
y = f (x) M0 (x0 ; y0 ) +
,-.
/ 01 L $ 2 0
M0 (x0 ; y0 )
)
k
'"
y − y0 = k(x − x0 ). 3 *
y = f (x)
M0 (x0 ; y0 ),
' )
kT = y (x0 ) = f (x0 ). 4 y −f (x0 ) = f (x0 )(x−x0 )
* T 1 "
y = f (x0 ) + f (x0 )(x − x0 ).
+ '.
5 L1 L2
$
)
! k1 · k2 = −1. 4
)
N M0 (x0 ; y0 )
y = f (x) 1 * kN
1
1
=−
kT
f (x0 )
N M0 (x0 ; y0 )
1
kN = −
*
y = f (x) "
y = f (x0 ) −
1
(x − x0 ).
f (x0 )
+ .
θ λ1 λ2
T1 T2 λ1 λ2 ,
M0 (x0; y0).
y
T2
22
T1
Mo( x o ; y o )
α1
O
θ
α2
θ = α2 − α1 .
tg θ = tg(α2 − α1 ) =
x
tg α2 − tg α1
y2 − y1
=
.
1 + tg α2 · tg α1
1 + y1 · y2 M0
θ = arctg
21
y2 − y1
.
1 + y1 · y2 x=x0
!!
!!
M0 (2; 4).
" # $ x0 = 2, y0 = f (x0) = 4. % !& !'
( ( T $
y = x2
N $
y = f (x0 ) + f (x0 )(x − x0 ) =⇒ y = 4x − 4
y = f (x0 ) −
1
f (x
x 9
(x − x0 ) =⇒ y = − + .
4 2
0)
!) θ
2
λ1 : y = (x − 2)2
λ2 : y = −4 + 6x − x .
√
ds
1
s = t,
v= = √
dt
t
2
Q = a(1 + be−kt ).
dQ
= −abke−kt . !
dt
" Q #$ % " Q = a(1 +
dQ
−kt
+be ) abe−kt = Q − a.
= k(a − Q).
dt
& ' % %% (
' & ) & &
* & "* & + &"* % $
% $ ,* ) $
y = (sin x)arctg x.
ln y = ln(sin x)arctg x =⇒ ln y = arctg x · ln sin x.
#%% ) & '
1 ln sin x
·y =
+ arctg x · ctg x.
y
1 + x2
- . ) & ' y = (sin x)arctg x ,
$
ln sin x
+ arctg x · ctg x .
1 + x2
√
x2 (x − 1)3 2x + 3
√
y=
.
(3x − 4)2 4 3x + 2
y = (sin x)arctg x
/
1
1
ln y = 2 ln x + 3 ln(x − 1) + ln(2x + 3) − 2 ln(3x − 4) − ln(3x + 2) ⇒
2
4
3
2
2·3
3
1 2
+
−
−
⇒
(ln y)x = y = +
y
√x x −1 2(2x + 3) 3x − 4 4(3x + 2)
2
3
3
1
6
3
x (x − 1) 2x + 3 2
√
y =
+
+
−
−
.
(3x − 4)2 4 3x + 2 x x − 1 2x + 3 3x − 4 4(3x + 2)
y2 cos x = a2 sin 3x.
! " # # y $ x
2y · y cos x − y 2 sin x = 3a2 cos 3x ⇒ y =
3a2 cos 3x + y 2 sin x
.
2y cos x
% x3 + y3 − 3axy = 0.
!
" # # y $ x
3x2 + 3y 2 y − 3a(y + xy ) = 0 ⇒ y =
&$
x2 − ay
.
ax − y 2
$ #
x = cos2 t,
y = sin2 t.
' ( ( $ )
yx =
(sin2 t)t
= −1.
(cos2 t)t
( # * # #
( + $
,
f (x) = y = x3 − 2x2 + 3 x0 = 1.
- # x0 = 1 . #
/
$
f (x0 ) = y0 = f (1) = 2,
f (x) = 3x2 − 4x, y (x0 ) = y0 = f (1) = −1.
s
s1 = 100 + 5t s2 = t2/2.
s1 = s2 ⇒ 100 + 5t = t2 /2 ⇒ t2 − 10t − 200 = 0.
t = 20.
ds1
ds2
= 5 !, v2 (t) =
= t ⇒ v2 (20) = 20 !.
v1 (t) =
dt
dt
" # $ v2 − v1 = 15 !.
% ! " # $%&
' & Q = 2t2 +3t+1. (
) *
!' (
N
" J =
dQ
= 4t + 3. & t = 3 J = 15
dt
( + ), y = xsin x.
x3 · sin x
.
) ( + ), y =
ln x · arctg x
* ( yx x3 + ln y − x2 ey = 0.
+ ( yx x2 + y 2 = 1.
( yx + )
⎧
3at
⎪
⎨x =
,
1 + t3
2
⎪
⎩y = 3at .
1 + t3
, T
x2 + 2xy2 + 3y4 = 6 M(1; −1).
− 4x
-
3
2
- y = x3 − x2 −
"++)& .
y = x2 x + y = 2
s t s = 14 t4 − 4t3 + 16t2,
t = t0
t = tk !"
#
s $ t $
! % ! mv2/2 &
s = 1+t+t2 ,
!
" #
$ #%
!
& "
" & y = f (x)
(a; b)' # ( ) *
Δy
.
f (x) = lim
Δx→0 Δx
+
&) # ' %
$
Δy
= f (x) + α(Δx),
Δx
α(Δx) , $ & Δx → 0. "(
!
Δy = f (x)Δx + α(Δx)Δx
. / 0
# $ 1 1' )
1
* Δx (
!
' f (x) ) &- ' '
x' && & $ $ & '
Δx.
Δx = dx,
!
y = f (x) $
dy = y · dx = f (x)dx.
" #
x0
dy = f (x0 )dx.
% y = f (x) x
√
& y = sin x x = π4
' (
√
√
cos x
· dx.
dy = d sin x = ( sin x) dx = √
2 sin x
) x = π/4 dy = √1 dx.
2 2
* !
4
dy
,
dx
y = f (x)
y =
+
x (
,# $
# # ) - +
. #
)- x / . M # y = f (x)
0 1 x 2 Δx. . # x + Δx $
N ) M - MP. 3 $
Δx = dx = MK Δy = N K, y = tg ∠P MK.
- MP K
MK · tg ∠P MK = y dx = P K = dy.
x
x + Δx.
y
MK= Δ x
PK=dy
NK= Δy
N
M
x
O
P
K
x+Δx x
1.dC = 0;
2.d(u + v) = du + dv;
3.d(uv) = v · du + u · dv;
4.d(cu) = cdu;
u
v · du − u · dv
.
v2
u v !
" #! !
5.d
v
=
x
d(uv) = (uv)dx = (u v + v u)dx = v · u dx + u · v dx = v · du + u · dv.
$
dy = ydx
x
!
x ! u"
x
dy = y dx.
y = y(x), x = x(u).
! "
#
$ %& (16.8) (16.10) "'
( & " ' ) *
)
dy = yu · du = yx · (xu · du) = yx · dx.
+ & ' , " *
, '- " & y = f (u(v(x))) "
" ).& " "& &/
dy = f (u)du = f (u)u(v)dv = f (u)u(v)v (x)dx.
0
sin x
.
x
$ % /
x2
e
arcsin2 x3
dex = ex dx2 = ex 2xdx = 2xex dx
dx
2 3
d arcsin x = 2 arcsin x3 d arcsin x3 = 2 arcsin x3 √
dx3 =
1 − x6
2 arcsin x3 2
6x2 arcsin x3
= √
3x dx = √
dx
6
6
1−x
1 − x
sin x
x cos x − sin x
sin x
=
dx =
dx
d
x
x
x2
2
2
2
2
1 " )2& 3& 3*
"2' 0 2 '- *
! "
0 - "' '
2
3
6x arcsin x
√
1 − x6
x cos x − sin x
x2
2xex
2
2xex dx = ex 2x dx = ex dex2 = dex
6x arcsin x
dx3
2 arcsin x3 2
√
dx = √
3x dx = 2 arcsin x3 √
=
1 − x6
1 − x6
1 − x6
x cos x − sin x
dx =
= 2 arcsin x3 d arcsin x3 = d arcsin2 x3 =
x2
sin x
sin x
=
dx = d
x
x
2
2
2
2
2
3
! ! " !
# $ ! % &
! &
$ ! $ %$
! ! & "!
'((
e sinx x
x2
) ex * & x ∈ (−∞; +∞)
sin x
& x ∈ (−∞; 0) ∪ (0; +∞)
x
%$# # & F (x)
sin x
dx $ %
Φ(x) &# dF (x) = ex dx dΦ(x) =
x
% "! &! # #
sin x
ex
+ ,
x
! " ! +, ! & "!
&# #
2
2
2
- $
% %$%
! & ! Δx.
." ! ! &# Δx ! ! / !
Δy ≈ dy.
0'('12
3 !& 0'(42 !
f (x + Δx) ≈ dy + f (x) ≈ f (x) + y dx.
0'('52
! "# " # "
$ sin 29◦.
% & ' () sin 29◦
! y = f (x) = sin x. * +#
! ! #
x = 30◦ = π/6. * x + Δx 29◦ , )
" ,)#
x + Δx =
π
29π
⇒ Δx = −
≈ −0, 01745.
180
180
& )'
sin(x + Δx) ≈ sin x + (sin x) Δx, sin(x + Δx) ≈ sin x + Δx cos x.
-) .) x = π/6 = 0, 52359#
Δx = −0, 01745#
sin 29◦ ≈ sin 30◦ − cos 30◦ · 0, 01745 ≈ 0, 4849.
√
3
/ 8, 05.
%√ & ' () √8, 05 !
y = x x + Δx = 8, 05. 0+) x = 8, Δx = 0, 05.
1 )'
√
√
√
√
√
1
x + Δx ≈ x + ( x) · Δx,
x + Δx ≈ x + √ · Δx.
2
3
3
3
3
3
3
3
3
3 x
-) .) x Δx, #
√
3
8 = 2,
0, 05
3
≈ 2, 0041.
8, 05 ≈ 2 +
3·4
- ) ) ! ! y = f (x)
2 ) ! !
y
f (x)
f (x) = (f (x)).
y = x
5
y = (x5 ) = 5x4 , y = (5x4 ) = 20x3 .
N
n
n − 1.
y (n) = f (n) (x) = (f (n−1) (x)).
n y = sin x
πn
.
y = cos x, y = − sin x, y (3) = − cos x, . . . , y (n) = sin x +
2
! " #
! " # ! " ! " $%
& ' % ##
! ! !
d2 y = d2 f (x) = d(dy) = d(y dx) = (y dx) dx = y dxdx = y dx2 .
% #
( (y dx) )' % dx x
dx dx $ % ) dx2 .
* n! ! ! n
n − 1.
y = sin 3x
dn y = dn f (x) = d(dn−1 y) = y (n) dxn .
"
dy = y · dx = 3 cos 3x · dx,
d2 y = y · dx2 = −9 sin 3x · dx2 ,
d3 y = y (3) · dx3 = −27 cos 3x · dx3 .
tg 46◦
y = f (x) = tg x.
x = 45◦ = π/4. x + Δx 46◦ .
x + Δx =
π
46π
⇒ Δx =
≈ 0, 01745.
180
180
!"#$"%&
Δx
tg(x + Δx) ≈ tg x + 2 .
cos x
' ( ) x = π/4 = 0, 78538 Δx =
= 0, 01745
tg(x + Δx) ≈ tg x + (tg x) Δx,
tg 46◦ ≈ tg 45◦ +
0, 01745
≈ 1 + 2 · 0, 01745 ≈ 1, 0349.
cos2 45◦
tg 46◦ ( ) *+ ",%--$
√
"#$"" 70.
√
√ 70
y = x x + Δx = 70. . x = 64, Δx = 6.
!"#$"%& /
3
3
3
√
√
√
3
x + Δx ≈ 3 x + ( 3 x) · Δx,
' (
0
)
√
3
√
3
70
70 ≈ 4 +
√
√
1
3
x + Δx ≈ 3 x + √
· Δx.
3
3 x2
√
x Δx, 3 64 = 4,
6
= 4, 125.
3 · 16
( 10−3
1"2"$
3 4 ) 4
"#$"2 y = x2√x.
"#$"% y = x arctg x.
"#$"1 y = sinx x .
"#$"- ctg x ln sin x
y = (1 + x2 ) arctg x.
n
y = 1 +1 x .
2 x
y = x 2e
y − 2y + y = ex ?
d3y.
x4
, d4 y.
y =
2−x
y = x2 e−x ,
ln 1, 02.
√
35.
" arctg 1, 05.
!
5
#$ % & %$ & ' (
) *+ ,- * . $ /
0 12 3 *
4 $ 3 50 $
1+
3 2 %$ 0 2 + $3
3 2 %/
$
/
y = f (x)
(a; b)
ξ ∈ (a; b) ! "
# $
% & ξ ' &
$
•
•
f (ξ) = lim
f (ξ + Δx) − f (ξ)
.
Δx
f (ξ) = M
Δx→0
$%& (a; b)
'% ! Δx
! " #
f (ξ) f (ξ + Δx)
( )
− f (ξ)
0 ⇒ f (ξ) 0;
Δx > 0, f (ξ + Δx)
Δx
*
− f (ξ)
0 ⇒ f (ξ) 0.
+
Δx < 0, f (ξ + Δx)
Δx
, ξ %- %
Δx. !% % % (17.2) (17.3)
% f (ξ) = 0. ,
*
y = f (x) :
• [a; b]
• (a; b)
•
f (a) = f (b) = 0.
(a; b) !
"
#
ξ ! f (ξ) = 0.
, $%& ( .
. ! ". ". M m
/ M = m, $%&
% !
% M = m. ,. '0 M
% ,. ' # %
ξ, &0 $%& % 1
( 2 f (ξ) = 0. ,
ξ OX
f (a) = f (b) = 0.
y = f (x) :
• [a; b]
• (a; b)
! "
#
f (b) − f (a) = f (ξ)(b − a).
! " # AB $%
ξ$
yx − f (a)
f (b) − f (a)
f (b) − f (a)
=
, ! : yx = f (a) +
(x − a).
x−a
b−a
b−a
& '! y = f (x) yx ( F (x)
F (x) = y − yx = f (x) − f (a) −
f (b) − f (a)
(x − a).
b−a
,! - F (x) !
*+ !
[a; b] ! # &
. F (x) :
• # [a; b]/
• ,,-! (a; b) 01
f (b) − f (a)
b−a
(a; b), 1 !3
F (x) = f (x) −
!3 !
• F (a) = F (b) = 0
)
2
! f (x)/
4 5 ! # & ! [a; b] .6
" (# ξ . F (ξ) = 0. 7
2 "
f (b) − f (a)
= 0,
F (ξ) = f (ξ) −
b−a
! ! 8
y
B
A
D
0
x
a
c
b
f (b) − f (a)
b−a
! "
y = f (x) c # $ %!
f (ξ) =
AB
# ! & '(%!
)*+ (a; b) !' , - ' . !
/ ,'. '(%! '
! "
# " $ % &
' ( ( [a; b]
) * ! # )
( ! ) +
! " #
, ( f (x) ϕ(x) - [a; b]
* #! ϕ(x) = 0*
(a; b) ! ξ *
f (ξ)
f (b) − f (a)
=
.
0
ϕ (ξ)
ϕ(b) − ϕ(a)
F (x) = f (x) − f (a) −
!"
f (b) − f (a)
· (ϕ(x) − ϕ(a)).
ϕ(b) − ϕ(a)
#
.
$ 00 ∞
∞
%&"'
f (x) ϕ(x)
x0
! "
x → x0
# # " $% &
'# x → x0 '# & &
( " )
f (x)
f (x)
= lim .
x→x0 ϕ(x)
x→x0 ϕ (x)
lim
(%&"%)*
+," - , "
0
0
%&"% *
lim
. , /
x→0
sin 3x
.
x
sin 3x
(sin 3x)
3 cos 3x
= lim
= 3.
= lim
x→0
x→0
x→0
x
x
1
lim
lim
x→0
1 − cos x
.
x2
1 − cos x
(1 − cos x)
sin x
.
=
lim
= lim
x→0
x→0
x→0 2x
x2
(x2 )
lim
!
" #
sin x
(sin x)
cos x
1
1
= lim
= lim cos x = .
= lim
x→0 2x
x→0 (2x)
x→0
2
2 x→0
2
lim
∞
∞
lim
x→+∞
x3
.
ex
x3
3x2
6x
6
= lim x = lim x = lim x = 0.
x
x→+∞ e
x→+∞ e
x→+∞ e
x→+∞ e
lim
! ! " #
$ !
x → +∞
% & ' ( ! ) * +
,
lim
x→+∞
lim
x→+∞
x + sin x
.
x
x + sin x
(x + sin x)
1 + cos x
= lim
.
= lim
x→+∞
x→+∞
x
x
1
lim cos x.
x→+∞
! "
x ! #$
+∞ − ∞
% ! & "
.
' 00 ∞
∞
()$*
lim
π
−0
2
x→
1
− tg x .
cos x
+ , & x → π2 !
- . +∞ − ∞.
/ - & +
1
sin x
1 − sin x
1
− tg x =
−
=
.
cos x
cos x cos x
cos x
x → π2 ! &
-$ 0 ' +∞ − ∞
' 00 . +
lim
π
x→
2
1 − sin x
− cos x
= lim
= 0.
π − sin x
cos x
x→
2
0 · +∞
% ! &
.
' 00 ∞
∞
()$1
lim x ln x.
x→+0
1
ln x
x
= lim
lim x ln x = lim
= − lim x = 0.
1
x→+0
x→+0
x→+0 1
x→+0
− 2
x
x
! !
" #
1
$%#%
+∞
n
lim (1 + axm )b/x , m > 0, n > 0.
x→0
n
y = (1 + axm )b/x ⇒ ln y =
b ln(1 + axm )
.
xn
& ' "
0
.
x → 0# '
0
(
abm
(b ln(1 + axm ))
abmxm−1
lim xm−n =
lim ln y = lim
= lim
=
n
m
n−1
x→0
x→0
x→0
(x
)
(1
+
ax
)nx
n
⎧
⎪
⎪0, m > n,
⎪
⎨ab, m = n,
=
) *# +#,⎪
+∞, m < n, ab > 0,
⎪
⎪
⎩
−∞, m < n, ab < 0.
⎧
e0 = 1, m > n,
⎪
⎪
⎪ ab
⎨
e , m = n,
n
lim (1 + axm )b/x =
*$%#$$x→0
⎪
e+∞ = +∞, m < n, ab > 0,
⎪
⎪
⎩ −∞
= 0, m < n, ab < 0.
e
x→0
& ' ! a = b = 1 m = n ' " '
lim (1 + x)1/x = e.
*$%#$.x→0
0
0
lim xx .
x→+0
y = xx .
x
ln y = ln x = x ln x. !"# ln y $
% " &# ! ' #
lim y = lim xx = 1.
x→+0
x→+0
+∞
0
(
lim (tg x)cos x .
x→π/2−0
y = (tg x)cos x .
ln sin x − ln cos x
ln y = cos x ln tg x =
. ' ln sin x − ln cos x#
1
cos x
1
'
% % x → π/2 − 0.
cos x
)
(ln sin x − ln cos x)
= lim
x→π/2−0
x→π/2−0
1
cos x
*% %% ln y % " &#
lim
lim
+
cos x sin x
+
sin x cos x
·
cos2 x
= 0.
sin x
lim (tg x)cos x = 1.
y=
x→π/2−0
x→π/2−0
ϕ(x) → 0, ψ(x) → +∞ x → x0
lim ϕ(x)ψ(x) = 0
x→x0
0+∞
,- ' # "
ln y = ψ(x) ln ϕ(x) → −∞, y → 0.
y = f (x) = x3 + 4x2 − 7x − 10
[−1; 2]
•
[−1; 2]
• !""
• #$! % " $ % f (−1) = 0
f (2) = 0
& ' ! !
!( "
)
x3 −3x+c = 0
[0; 1].
y = x3 − 3x + c
% [0; 1] ! * +!
! a, b ∈ (0; 1) ⇒ f (a) − f (b) = 0. ,!
(a; b) ! - $ ! ( %!
f (b) − f (a)
=
# f (a) = f (0) = c, f (b) = f (1) = c−2,
b−a
= −2 = 0. .!- % * ! $!
- ! $ %$ ( % [0; 1]
/ 01 * 0 23)1
+ ! ! -%- %
% %- !
4
lim
x→0
ln x
ctg x
∞
∞
.
ln x
(ln x)
sin2 x
= lim
.
= − lim
x→0 ctg x
x→0 (ctg x)
x→0
x
lim
0
+ !- ! -
0
* !
sin2 x
sin x
= lim
· sin x = 1 · 0 = 0.
x→0
x→0 x
x
lim
lim
x→0
lim
x→0
1
1
−
sin2 x x2
ln x
= 0.
ctg x
(
+∞ − ∞) .
lim
x→0
1
1
−
sin2 x x2
= lim
x→0
x2 − sin2 x
x2 sin2 x
!
0
.
0
" # $
! % & % ! $% x4
x2 sin2 x ∼ x4 .
lim
x→0
1
1
−
sin2 x x2
= lim
x→0
#
lim
x→0
x2 − sin2 x
x4
1
1
−
sin2 x x2
= lim
x→0
!
0
.
0
2x − sin 2x
=
4x3
1 − cos 2x
2 sin2 x
1
= lim
= .
2
x→0
x→0
6x
6x2
3
= lim
'
2
lim (cos 2x)3/x (
x→0
1+∞) .
#() #
2
lim ln(cos 2x)3/x = lim
x→0
*
x→0
3 ln cos 2x
tg 2x
= −6.
= −6 lim
x→0 2x
x2
2
lim (cos 2x)3/x = e−6 .
x→0
+
lim (tg x)sin x (
x→+0
00) .
x→1
lim x1/x .
x→+∞
lim ln x ln(x − 1).
x→1
lim (1 − x) tg
πx
.
2
! " #
'
$ #%
& ' (
$ #
u1 (x) + u2 (x) + ... + un (x) + ... =
+∞
,
un (x).
) * +
n=1
*
! " x
" #
,
! x # %
' x0 %
# !& # ' '
!
*- $ x = x0
%& " %
#
*. ' ( % %
# " ! # %
/ '
& # n
'
n
,
Sn (x) =
uk (x) = u1 (x) + u2 (x) + ... + un (x)
) *-+
k=1
! # ! x& # 0!
x
Sn(x) n → +∞ ! " "
Sn(x)
S(x)
S(x)−Sn(x) n
+∞
.
rn(x) =
uk (x)
# " $ $
$
lim un (x) = 0
%&' ()
n→+∞
"
lim rn (x) = 0.
%&' *)
n→+∞
$
% && *) $
$ %&' &)+
k=n+1
u (x)
n+1
lim
< 1.
n→+∞
un (x)
%&' ,)
! x %&' ,)
-
$ "
. $ "
%&' ,) " "
u (x)
n+1
lim
%&' /)
= 1.
n→+∞
un (x)
! %&' /)
0 %&' &)
&' &
+∞
,
n=1
1
.
n(x + 3)n
un =
1
1
, un+1 =
.
n(x + 3)n
(n + 1)(x + 3)n+1
u
n(x + 3)n
1
n+1
< 1.
lim
= lim
= lim
n→+∞
n→+∞ (n + 1)(x + 3)n+1
n→+∞ |x + 3|
un
|x + 3| > 1
• x = −2 :
⇒ x > −2 x < −4
x = −2 x = −4 :
+∞
+∞
.
. 1
1
=
n
n(x
+
3)
n=1
n=1 n
+∞
.
+∞
.
1
1
=
n
n
n=1 n(x + 3)
n=1 n(−1)
!
" " #$%
& $ &
• x = −4 :
+∞
,
1
,
n
n=1
!
+∞
.
1
' $ $
n
n=1 n(x + 3)
x ∈ (−∞, −4] ∪ (−2, +∞)
()*
+∞
.
√
n 3 cosn x.
n=1
√
√
3
un = n 3 cosn x un+1 = (n + 1) cosn+1 x
3
(n + 1)√
un+1
√
cosn+1 x
√
= lim
lim
= lim | 3 cos x| .
3
n
n→+∞ un
n→+∞
n→+∞
n cos x
+ " %& " x xk = kπ
k = 0, ±1, .. . %& k " &
+∞
. √
• " , k
n 3 cosn xk = 1 + 2 + 3 + ...
•
k=1
+∞
.
" , k
k=1
√
n 3 cosn xk = −1 + 2 − 3 + ... + (−1)n n + ... .
+∞ √
. n cosn x
n=1
xk = kπ, k = 0, ±1, ...
n
+∞
.
5−x
1
3n + 2 8x − 3
3
n=0
! " #$ % &
' ()
un+1 (x)
< 1.
lim
n→+∞ un (x)
" $
' ()
n
n+1
5−x
5−x
1
1
un (x) =
, un+1 =
3n + 2 8x − 3
3n + 5 8x − 3
un+1 (x)
= lim 3n + 2 5 − x < 1.
lim
n→+∞
n→+∞
un (x)
3n + 5 8x − 3
*
lim
n→+∞
3n + 2
= 1,
3n + 5
5−x
8x − 3 < 1
$!
⎧
5−x
⎪
⎪
> −1,
⎪
⎨ 8x − 3
⎪
⎪
⎪
⎩ 5 − x < 1.
8x − 3
⇒
−1<
⎧
5−x
⎪
⎪
+ 1 > 0,
⎪
⎨ 8x − 3
⎪
⎪
⎪
⎩ 5 − x − 1 < 0.
8x − 3
5−x
< 1.
8x − 3
⇒
⎧
7x + 2
⎪
⎪
> 0,
⎪
⎨ 8x − 3
⎪
⎪
⎪
⎩ 9x − 8 > 0.
8x − 3
+
, $
2 4 8
, +∞ -$
x ∈ −∞, − 7
9
& $.&
$.$ %!
8
⇒
9
+∞
.
1
.
3n
+2
n=0
n
+∞
. (−1)
2
⇒
• x = − ⇒
7
n=0 3n + 2
• x =
x∈
2 5 8
, +∞
−∞, −
7
9
!! "
" ! ! "#
! " $ ! %
! % ! !
!
&' (
D ! "
# $ % $
# $ # $ "
! "
) ! * ! #
% ! +* ! #
% +* ) ! #
, !
- !
* " ! +*
!
&' & & '
D # # $ %
&' ( ( $ . sinn2nx #
n=1
# x ∈ (−∞, +∞)
+∞
sin nx
2
n
1
x
n
+∞
. 1
!
2
+∞
.
sin nx
n2
n=1
n=1
2
n
x!
!"
D
!# ! "
#$
D
S(x) %
S (x)
% & % %
' % D
$% & '(
) * + ' !
!, (
+∞
,
an (x − x0 )n = a0 + a1 (x − x0 ) + ... + an (x − x0 )n + ...,
n=0
% #
(x − x0) #$
a1 an , . . . #
- x0 = 0, ) ! *
+∞
,
n=0
) ! *
a0 a1 an
n )
an xn = a0 + a1 x + ... + an xn + ...,
) !.*
' / !
0
1 ! 1
%' / %
) !.* ' %'
1
2
-R
0
3
R
n+1
un+1
= lim an+1 x = |x| lim an+1 < 1.
lim
n
n→+∞
n→+∞
n→+∞
un
an x
an
a
n
|x| < lim
= R.
n→+∞ an+1
an
R = n→+∞
lim
an+1
|x| < R
|x| > R (−R; R)
! " x = ±R #$%
$ &$ ' "$( $ & " ##( & & %
#$ $ ) x = −R x = R * & ##
) $ # $ (−R; R) $ +$
# #( " $ ( $ #( # #, %
+ [−R; R] ! - ' " # $ (−R; R) )
*#
" $ # *' & ##' $ ) & x = 0
. " # R = 0 .
- & *( x ∈ (−∞; +∞) " "
R = +∞
n ! ! "
+∞
.
√x n .
n=1
xn
xn+1
1
1
un = √ un+1 = √
an = √ an+1 = √
n
n
n+1
n+1
√
an
n+1
= lim
√
R = lim
=1
n→+∞ an+1
n→+∞
n
(−1; 1)
+∞
+∞
. xn
. 1
√ =
√
• x = 1
+∞
.
1
α
n=1 n
n
n=1
n=1
n
α 1
• x = −1
+∞
.
+∞
. (−1)n
xn
√ =
√
n n=1
n
n=1
!" ! #
$ % % [−1; 1)
&' (
an =
(n + 1)!
=
n!
= lim (n + 1) = +∞.
+∞
.
xn
.
n=1 n!
1
1
an+1 =
R =
n!
(n + 1)!
lim
n→+∞
an
=
an+1
lim
n→+∞
n→+∞
$ % % x ∈ (−∞; +∞)
! ) %
(n + 1)! = n!(n + 1)
* + %
&' ,
(−R; R) r < R !
" # $ [−r; r]
# %
• & ' ' $
( (
$ ( $$ " (
) " "
$ $( $ (
( ( (
• & " ( *$
+ $ $ "
•
an
,
R = lim
n→+∞ an+1
(x0 − R; x0 + R),
un+1
an+1 (x − x0 )n+1
an+1
0 > 0 "
4 x3
0 < x < +∞
• #
√
$%& y = 2x − 3 x2.
' ( ) *
√
2 x−2
2
√
.
• + , y = f (x) = 2 − √ =
x
x
• )
•
4
4
3
3
3
f (x) = 0,
√
23x−2
√
= 0,
3
x
√
3
x = 1,
3
x = 1.
x = 1
- x = 0
x = 0 #
)
−∞ < x < 0;
•
0 < x < 1;
1 < x < +∞.
. / " ,
√ )
2 x−2
0 −∞ < x < 0 y = √x > 0
3
3
2(−1 − 1)
= 4 > 0
y (−1) =
−1
* "
√
0 0 < x < 1 y = 2 √x x− 2 < 0
" < 0
> 0 * " 1
√
0 1 < x < +∞ y = 2 √x x− 2 > 0
> 0
* "
3
3
3
3
x = 0
x −∞; −3 −3; 0
0; +∞
y
0
>0
27
y
− 3
e
!"
# $ y = x3 − 3x2.
# % y = x(1 + √x)
# & y = 1 + √x.
√
# ' y = x2 − x.
3
1
y = e x − x.
# (
3
# (( y = x x+2 4
º
) * +
, - ., .,
+ /0 . 1 .
0 . 2 0
# 3 * *
4 . *5 6
. 7
8 . .
(x − x0 )2k+1
x > x0 :
x > x0 x < x0
f (2k+1) (ξ)
(x − x0 )2k+1 > 0,
(2k + 1)!
x < x0 :
f (2k+1) (ξ)
(x − x0 )2k+1 < 0.
(2k + 1)!
x0
! f (x) < f (x0) x < x0
f (x) > f (x0 ) x > x0 " # $
% f (x)
&
f (2k+1) (x0 ) < 0
x0 $ % f (x)
f (x) = x4
' ( ) * + + f (x) = 4x3 . ,
x = 0 x = 0 %
- x = 0 + . f (0) = 12x2 |x=0 = 0,
f (0) = 24x|x=0 = 0 f (4) (0) = 24 > 0
$ % f (x) = x4 x = 0 #
f (x) = x5
' ( ) -
f (0) = f (0) = f (0) = f (4) (0) = 0 f (5) (0) = 120 > 0
# $ % f (x) = x5
+ / x
, +
0 y = x2k
x = 0 +
0 y = x2k+1
(−∞; +∞) + $ %
1 +
%
! # . $
)
x0 f (x),
x0 f (x) = 0
f (x0) < 0 x0
f (x0) > 0 x0
f (x)
y = f (x) = x3 − 4x + 2
x1 = − √2
2
x2 = √
3
f (x) = 6x
3
! " # $
# % f
2
x1 = − √
3
12
2
√
= √ > 0
f
3
3
2
−√
3
12
= − √ < 0.
3
& $ '(
2
x2 = √
3
(
#
)
f (x) =
√
3
!
x2 − x3 .
"
*# +% % '
$ x = 23 '
2
,- f 23 = − 2
= − - < 0 $ (
4
5
5
9 3 x4 (1 − x)
'
x=
2
3
$
2
1
3
3
√
3
f (x) = x2 − x3
93
. / 0 ' ,1)- $$ 2 0 '(
# [a; b] / $ 0 ' %
% (a; b) 0 % x = a x = b 3 (
# ' ' #
# [a; b] % $
•
xk (a; b)
f (x) f (xk ),
f (x) f (a) f (b),
•
• M m
f (xk ) f (a) f (b)
xk
!"# M m f (x) = x3 − 4x + 2
[0; 1] , [−1; 1] , [−2; 2]
$ % & ' !(" ' ) '
(0; 1)
'
f (x) * f (0) = 2, f (1) = −1
+ M = f (0) = 2, m = f (1) = −1 * *
[0; 1]
, (−1; 1)
f (−1) = 5, f (1) = −1 M = f (−1) = 5, m = f (1) = −1 $
' [−2; 2] , (−2; 2)
2
2
x1 = − √ x2 = √
3
3
2
2
−√
≈ 5, 0792 √
≈ −1, 0792 +
3
3
*
f(−2) = 2 f (2) = 2
2
M = f − √
3
2
m=f √
* [−2; 2]
3
, ' ' *
!""
y = f (x) (a; b)
! "
# x0 $ % &&'(
!"!
y = f (x) (a; b)
y
yT
f(xo)
y
T
Mo
y=f(x)
x
a
0
x
x
0
b
x0
y
y=f(x)
y
T
yT
f(xo)
Mo
x
0
a
xo
x b
x0 !
"# "
$ (x0 − δ; x0 + δ),
!
x0
" " %
"
y = f (x)
(a; x0 ) (x0 ; b) x0
&" '"( y = f (x) ) " !
" f (x) * * (a; b) + * *
f (x) < 0 ' '"(
)
" f (x) > 0 % "
x ∈ (−∞; +∞)
x ∈ (−∞; +∞)
y = 2k(2k − 1)x(2k−2) > 0
!
y = x(2k−1)
y = 2(k − 1)(2k − 2)x(2k−3) < 0
x < 0 y > 0 x > 0
" x < 0 x > 0
#
y = x3 − 4x + 2
y = 6x < 0 x < 0
y = 6x > 0 x > 0 $" % $ "&
"' x < 0 ' x > 0 ( )*
+
√
y =
3
x2 − x3
2
f (x) = − 4
'
9 x (1 − x)5
" " x = 1 ' f (x) < 0 x < 1
f (x) > 0 x > 1 $" % $ "&
"' x < 1 ' x > 1 ( *
,
3
x0
y = f (x) f (x0) f (x0) = 0.
- " % ' ' .
/ " ' 0 "0 "
"
1
2 " " f (x) = 0 1 3 " 4
"' " '
5 "
1 3 " % "
$ ' ' 1 "
6 ! x0 "
y = f (x)
#
$ (x0 − δ; x0 + δ)
x0 " f (x) > 0 f (x) < 0 $
x ∈ (x0 − δ; x0)
x ∈ (x0; x0 + δ) f (x) < 0
x0 y = f (x)
!
x0
f (x) = 0 ! "
"
# x0 $
"# #
$ ## # ! % & ' # !
( y = x2k #
# ) % ( y = x(2k+1) #
* x = 0 + x = 0
y = x3 − 4x + 2 # √,
- # & y = x2 − x3 + #
*
. x = 1 .( .
f (x) . x = 0 '
f (x) ( #*
/
√
01 y = x3 − 4x + 2 y = x2 − x3 1
" 1 01 y = x3 − 4x + 2
f (x) > 0
3
3
x
y
0
−∞; 0
0
√
" 1 01 y = x2 − x3
y = f (x)
3
x
y
0
y = f (x)
−∞; 0
0
f (0) = 2 > 0. " #
- 13 < x < +∞ f (x) < 0 !
f ( 23 ) < 0. #
1
• + x =
3
x = 13 "! #
/ !
•
−∞;
x
y
1
3
>0
y = f (x)
y = x(x + √x).
1
3
1
; +∞
3
0
+∞
>0
$ ! $
y = f (x)
)*+ y = x3ex.
, - "
• .
y = f (x) = x2 ex (3 + x), y = (x3 + 6x2 + 6x)ex .
•
/ $ x = −3
x = 0 0 ' '
f (−3) = (−27 + 54 − 18)e−3 =
9
> 0.
e3
/
f (0) = 0 .!'
"
f (0) = (x3 + 9x2 + 18x + 6)ex |x=0 = 6 > 0.
1 x = 0
2
f (0) > 0
•
f (x) = 0, (x3 + 6x2 + 6x)ex = 0, ex = 0, x(x2 + 6x + 6) = 0, x1 = 0,
√
√
x2 + 6x + 6 = 0, x2,3 = −3 ± 9 − 6 = −3 ± 3.
√
√
√
−∞ < x < −3 − 3; −3 − 3 < x < −3 + 3;
−3 +
√
3 < x < 0; 0 < x < +∞.
√
" "
• !
# −∞ < x < −3 − 3
f (x) = (x3 + 6x2 + 6x)ex < 0 ex > 0 $
(x3 +6x2 +6x) < 0 x < 0 % $
√
√
# −3 − 3 < x < −3 + 3
9
3
2
x
f (x) = (x + 6x + 6x)e > 0 f (−3) = 3 > 0
e
$
$
√
# −3 + 3 < x < 0
f (x) =
3
2
x
= (x + 6x + 6x)e < 0 f (−1) = (−1 + 6 −
1
− 6)e−1 = − < 0 $
e
# 0 < x < +∞
f (x) =
3
2
x
= (x + 6x + 6x)e > 0 f (1) = (1 + 6 + 6)e =
$ √
= 13e > 0 $
• & √ " x = −3 − 3 ≈ −4, 732;
x = −3 + 3 ≈ −1, 268 x = 0
$
%
√
x
−∞; −3 − 3
y
4
x !! ! "# $ !
! %! 2x + x2 = 0,
&!" "# $ y = 2x y = −x2 '# ! (
!
ξ2 = 2 ξ3 = 4
) ! ! # # ! &!" &
* #
(a, b)
f (x)
x = a x = b f (x)
+ ! , ! !
! ! # #! f (x) = 0, &!
&!" - ! ! "# $
! ! ! .! #!
x3 − 4x + 2 = 0.
/
x3 − 4x + 2 = 0
y
y=ψ(x)= x
2
y=ϕ( x)=2
y=ϕ( x)= 2
x
x
x
ξ1 0 ξ2 ξ3
y=-x 2
x2 = 2x
y = x3 − 4x + 2
y → −∞ x → −∞ y → +∞ x → +∞,
x3 − 4x + 2 = 0 ξ1 ∈ (−∞; − √2 )
2 2
ξ2 ∈ (− √ ; √ )
3 3
2
ξ3 ∈ ( √ ; +∞).
3
3
!"
# !$
" ! # % & %
& % ' ( ) f (−3) = −13 < 0
f (−2) = 2 > 0 ) f (0) = 2 > 0 f (1) = −1 < 0 !)
f (1) = −1 < 0 f (2) = 2 > 0' * ! ξ1 ∈ (−3; −2) ξ2 ∈ (0; 1)
ξ3 ∈ (1; 2)'
+ ! ! ,-'./ &
(a; b). + # ) #0 % x̄ ∈ (a; b).
+ ! % ) ξ ∈ (a; b), )
1)# ,.2'3/ % [ξ; x̄],
|f (ξ) − f (x̄)| = |f (ζ)||ξ − x̄|,
ζ ∈ (ξ, x̄). ξ
f (ξ) = 0
|f (x̄)| = |f (ζ)||ξ − x̄|.
m1
f (x)
[a; b], |f (x̄)| m1 |ξ − x̄|
f (x̄)
|ξ − x̄|
.
m1
! "
x̄ ξ.
#$!
x̄ = 0, 5
%
x3 − 4x + 2 = 0.
& ' x̄ ∈ (0; 1),
( ξ2 . )
f (x̄) = f (0,5) = 0, 125,
f (x) = 3x − 4 f (0) = −4 f (1) =
= −1 m1 = |f (x)| [0; 1] = 1.
|ξ − 0,5| 0,125. * #+
$! x̄ = 0, 5 ξ #
0,125, , 0, 375 ξ 0,625
2
- .( , . +
/ #$ / . "
- " , / / "+
0 / $ .
1 .# . , / +
.
1 2
/ /' , +
3 . /
(a; b). ! f (a) < 0" f (b) > 0. #$
$ [a; b] f (a + b −2 a ). % & $
" ξ = a + b −2 a , f (a + b −2 a ) > 0, ξ ∈ (a; a + b −2 a );
f (a + b −2 a ) < 0, ξ ∈ (a + b −2 a ; b). '
() $ " * " + ,
" * $) )
+ $ "
() " +
-
|b − a|
|x − ξ| <
,
.
2
* n/ +" x 0 $ n) +
,$ - . 1 2 $ !*
2 3 $ $ +"
2 = 8 2 2 = 16 > 10.
4
n
n
n
3
4
x3 − 4x + 2
(0; 1).
= 0,
# 2 1 5 $
"
! $
6 ( !
$
7" )
$
$ 8
7
72 * . 2
* "
" 2 |x − ξ| 2
ξ2 , f (0) = 2 > 0
f (1) = −1 < 0,
x1 = 0, 5,
f (0, 5) = 0, 125 > 0 ⇒ ξ2 ∈ (0, 5; 1, 0).
x2 = 0, 75, f (0, 75) = 0, 753 − 3 + 2 < 0 ⇒
ξ2 ∈ (0, 5; 0, 75).
f (xn ),
x3 = 0, 625; f (0, 625) = 0, 6253 − 2, 5 + 2 < 0 ⇒ ξ3 ∈ (0, 5; 0, 625).
x4 = 0, 5625
1
=
ξ
x4
16
= 0, 0625.
n
!"# (a; b)
$ f (x) f (x) % & ' ( ) %
y = f (x) ξ * ( +, "-.# *
h = ξ − xn , xn − n *
f (ξ) = f (xn ) + f (xn )h + O(h2 ) = 0,
!/#
$ % ξ , !"#
!/# O(h2 ), & h hn = xn+1−xn, xn+1
% * % ξ :
f (xn ) + f (xn )hn = f (xn ) + f (xn )(xn+1 − xn ) = 0.
xn+1 = xn −
f (xn )
, n = 0, 1, 2, . . .
f (xn )
!.#
( 0 %
y
T
y=f(x)
yn
xn
ξ
xn
0
x n+1
1 "!" ' &
, y = f (x) :
x n+1
x
(xn ; yn )
yT = f (xn ) + f (xn )(x − xn ).
% T
xn+1, yT (xn+1) = 0.
! " # " ξ !
y = f (x) $ $ ! (xn ; yn )
%&'
f (x) [a; b]
xn f (xn),
xn+1, !"#$%
ξ.
( ! $ ) xn
* f (xn) > 0, ! $ * + '&'+
f (x) > 0 + ! + xn+1 ξ. !
x̄n, ! $ f (x̄n) < 0,
! $+ ! $ y = f (x) ! x̄n
! x̄n+1, $ , ξ, xn,
! $+ x̄n+1
, + ! + $
- , , ! f (x) = 0
x̄ = xn+1 ! * .%&&/
|xn+1 − ξ|
|f (xn+1 )|
.
m1
0 , * 1$ .'2%/
xn+1
n=2
f (ζ)
f (xn+1 ) = f (xn) + f (xn )(xn+1 − xn ) +
(xn+1 − xn )2 ,
2
ζ ∈ (xn , xn+1 ) ! ! +
.%&3/ 1
|f (xn+1 )| =
M2 !
|f (ζ)|
M2
(xn+1 − xn )2
(xn+1 − xn )2 ,
2
2
, |f (x)| [a; b],
|xn+1 − ξ|
M2
(xn+1 − xn )2 .
2m1
! +
.%&4/
5 .%&4/ ! +
! , !
!" ! " $
x1 = −3 −
x0 = −3,
13
= −2, 4348,
23
x2 = −2, 4348 −
(−2, 4348)3 + 4 · 2, 4348 + 2
= −2, 2415,
3 · (−2, 4348)2 − 4
x3 = −2, 2415 −
(−2, 2415)3 + 4 · 2, 2415 + 2
= −2, 2151.
3 · (−2, 2415)2 − 4
!" # $
18
(2, 2415 − 2, 2151)2 < 10−3 .
2·8
% & 10−3 ξ1 = −2, 215.
ξ3 ∈ (1; 2) :
|x3 − ξ1 |
f (1) = −1, f (1) = −1, f (1) = 6,
f (2) = 2, f (2) = 8, f (2) = 12.
x0 = 2.
x1 = 2 −
2
= 1, 75,
8
x2 = 1, 75 −
1, 753 − 4 · 1, 75 + 2
= 1, 6811,
3 · 1, 752 − 4
x3 = 1, 6811 −
1.68113 − 4 · 1, 6811 + 2
= 1, 675.
3 · 1, 68112 − 4
(1; 2)
!"#$ [1, 5; 2]. % & '
M2 = f (2) = 12, m1 = f (1, 5) = 2, 75.
(
12
(1, 681 − 1, 675)2 < 10−3 .
2 · 2, 75
) * 10−3 ξ3 = 1, 675.
|x3 − ξ| 2
f (x) =
x = √a.
! "
xm − a
1
a
xn+1 = xn − n m−1 = ((m − 1)xn + m−1 ), n = 0, 1, 2, . . . #
mxn
m
xn
$ % x = √a, m = 2
1
a
xn+1 = (xn + ), n = 0, 1, 2, . . .
&'
2
x
= xm − a = 0
10−4 .
m
(
√
n
5
) " * ! a = 5 &' + x0 = 2
,
1
5
1
a
1
x1 = (x0 + ) = (2 + ) = 2, 25,
2
x0
2
2
5
1
a
1
) = 2, 23611,
x2 = (x1 + ) = (2, 25 +
2
x1
2
2, 25
5
1
a
1
x3 = (x2 + ) = (2, 23611 +
) = 2, 23607.
2
x2
2
2, 2361
x3 = 2, 2361 - .
2 - 10−4.
/0
) . 1 3
40 54 . !
√
& y = 1 − x3
3
y=
y=
1 − x3
x2
3
3
(x − 1)2
6
M m
y = (x + 1)2 − (x − 1)2 [−1; 1] [−2; 0] [−2; 2]
3
3
(x + 1)2 −
R
y =
√
3
1 − x3
•
D(y) = (−∞; +∞)
•
!"
!"
• #! ! $ Oy : x = 0 y = 1 $ Ox : y = 0
x = 1
• #! %& &' (
) ( & (& % ( (
*
√
3
1 − x3
1
1
3
= lim (
k = lim
− 1) = − lim (1 − 3 )1/3 = −1,
3
x→+∞
x→+∞
x→+∞
x
x
x
√
1
3
b = lim ( 1 − x3 + x) = lim (−x(1 − 3 )1/3 + x) =
x→+∞
x→+∞
x
1x
1
1
1
= lim −x +
+
x
=
lim
= 0.
+
x
·
O
+
O
3
6
2
x→+∞
x→+∞
3x
x
3x
x5
+
• ,' ( % $
y =
( y = −x
1 (−3x2)
x2
= −
.
3
3
3
2
3 (1 − x )
(1 − x3 )2
- ! ! x = 0
3
y = −
x = 1 . %
(1 − x3 )2 2x −
3
x2 2(−3x2)
√
3 3 1 − x3
(1 − x3 )4
=
2x
(1 − x3 )2x + 2x4
= −
3
3
3
5
(1 − x )
(1 − x3 )5
( / (& ! !
=−
•
x −∞; 0
y
0
y
∪
•
0
0
0
0; 1
0
±+∞
0
+∞
max
√
3
4
>0
∪
∪
• '
( %
) *+,.
" %
" %
/ /
[a; b]
0
$+ ,
y =
3
(x + 1)2 −
3
(x − 1)2
M
[−1; 1] [−2; 0] [−2; 2]
y
1
-1
0
-1
1
y =
x
3
(x + 1)2 −
3
(x − 1)2
•
(−1; 1)
√
√ y(−1) = − 4 = m
3
y(+1) =
3
4=M
(−2; 0)
! y(−2) =
y(0) = 0 " [−2; 0]
√
m = y(−1) = − 4 M = y(0) = 0
•
(−2; √
2) √
ymax = y(1) = 3 ymin = y(−1)
√ = − 4 #
√
$ y(−2)
=
1
−
9
%
y(2)
=
9
−
1
&
√
√
m = ymin = y(−1) = − 4 M = ymax = y(1) = 4
'( ) R
*
r
+ h " , - .
h2
/() 0 12 V = πr2 h * r2 + = R2
•
x = −1
√
1− 39
3
4
3
3
3
3
h2
r =R −
4
2
2
3
4
*
2
r
h
h 3
V = π R2 −
4
3h2
dV
2
=π R −
= 0
dh
4
2
V ,
- 12
2R
h= √
3
r=
R2 −
h2
=
4
R
h
O
2r
= R
2
3
V =0
h = 0
r = R h = 2R r = 0
2R
2
h = √ ;r = R
3
3
y
=
y
= x2 e−x
x + arctg x
!" $
!" %
x2 − 4x
− 4x + 8
!" #
x2
M m
y = x2e−x [0; 1]! [0; 3]! [−1; 2]
" # a #
$ ! $ $
!" &
!" '(
!
⎛
a11
⎜ a21
A = (aij ) = ⎜
⎝
m × n
a12
a22
...
...
⎞
a1n
a2n ⎟
⎟
⎠,
am1 am2 . . . amn
m × n aij
" !#
$ aij % & ' ( )
'* + ' *
, -
. i , "i = 1, 2, ..., m)
j , - "j = 1, 2, ..., n) & -'+*
' -& A, B
. &( (
&( -& *
'* *
•
+ + & -
"m = n# ' &*
•
+ + & - "m = n#
' . +/ + / -
'
0
A=
a11 a12
a21 a22
i = j
(a11 , a22, . . . , ann )
•
aij = 0 i = j
⎛
⎞
•
0 ...
a11 0
⎜ 0 a22 0 . . .
⎜
⎜ 0
0 a33 . . .
⎜
⎝
0
•
0
0
0
0 ⎟
⎟
0 ⎟
⎟
⎠
. . . ann
!
E
⎛
⎞
1 0 0 ... 0
⎜ 0 1 0 ... 0 ⎟
⎜
⎟
0 0 1 ... 0 ⎟
E=⎜
⎜
⎟
⎝ ⎠
0 0 0 ... 1
"
# "
• $
• $ %
•
A = (a11 , a12 , . . . , a1n )
•
$ % &
&
⎛
⎞
a11
⎜ a21 ⎟
⎟
A=⎜
⎝ ⎠
am1
•
$ B #
A ! A
AT
a11 a12 a13
a
a
a23
⎛ 21 22 ⎞
a11 a21
AT = ⎝ a12 a22 ⎠
a13 a23
A=
(A = B)
aij = bij
A
B
!
!
"
(m × n)
C # !
"
A B$
A
B #
B=
%
$
2 4 1
3 0 5
A + B = C, cij = aij + bij .
1 2 3
! %
A=
2 4 5
.
&
" #
1 2 3
2 4 5
'(
&
+
2 4 1
3 0 5
=
A + B = B + A,
3 6 4
5 4 10
.
) )*
(A + B) + C = A + (B + C).
)* (
( &
) +
A + 0 = A.
3 −1
=
−1 2
1 · 3 + 1 · (−1) 1 · (−1) + 1 · 2
2 1
=
=
3 · 3 + 1 · (−1) 3 · (−1) + 1 · 2
8 −1
BA =
1 1
3 1
AB = BA.
AB
= BA
A(BC) = (AC)B
(A + B)C = AC + BC.
!
!
1 1
"! A = 1 1
B=
1
1
−1 −1
AB =
1 1
1 1
1
1
0 0
·
=
−1 −1
0 0
a11 a12
a21 a22
A=
.
a11 a22 − a21 a12
a
a
|A| = 11 12
a21 a22
.
! "#
" $ %
a
a
|A| = 11 12
a21 a22
= a11 a22 − a21 a12 .
&
' a11 , a12 , a21 , a22 " (
& !
) %
2 5
3 −4
2 5
3 −4
.
= 2 · (−4) − 5 · 3 = −23.
* +
• ,
-
a11 a12
a21 a22
•
.
* .
! "+ .
#
a11 a12
a21 a22
•
a11 a21
=
a12 a22
,
= − a21 a22
a11 a12
.
"
•
a11 a12
a11 ka12
a21 ka22 = k a21 a22 .
!"
!#
•
! $
" % "
%" # #
•
a11 + λa12 a12
a21 + λa22 a22
a11 a12
=
a21 a22
.
& % !
#
' ⎛! ($
⎞
a11
⎜ a21
⎜
A=⎝
...
an1
a12
a22
...
an2
...
...
...
...
|A| =
a12
a22
···
an2
···
···
···
···
a1n
a2n ⎟
⎟.
... ⎠
ann
)*#+
.
)*#.
)*#, n
! "
- % %
a11
a21
···
an1
a1n
a2n
···
ann
/ " % 0
1 $ " $ % $ #
)*#2 #
n $ (n − 1)
%
& "
M
M
a
ij
12
!"#$ M
12
a21
a
= 31
···
an1
n
a23
a33
···
an3
···
···
···
···
12
a2n
a3n
···
ann
%
&
!"'(
) a A
* i j &
+ +
, -
- .
A = (−1) M .
!"/$
0 . A = (−1) M = M 1 A = (−1) M = −M
2 -
.
,
a A
|A| =
+ i
!"3$
,
|A| =
a A
+ j.
!"'($
4 - 5
ij
i+j
ij
11
1+1
11
11
12
ij
ij
1+2
12
12
n
ij
ij
ij
ij
j=1
n
i=1
! n
" #
!"3 !"'(
n 6
n − 1 5 !"3$ !"'($
! n n
n − 1
!
a11 a12 a13
|A| = a21 a22 a23
a31 a32 a33
" # |A| = a11 A11 +a12 A12 +a13 A13, "
! $% |A| = a12 A12 + a22 A22 + a32 A32 .
aa1121 aa1222
A11 = a22 A12 = −a21
a
a
A21 = −a12 A22 = a11 11 12
a21 a22
! "!! #
& $ % % %
−1 −2 −10
9
10 .
|A| = 1
1
2
0
' ( ) #
*
* " # (i = 3) :
+
|A| = a31 A31 + a32 A32 + a33 A33 .
, !$ * - .
+
−2 −10
= (−2)10 − 9(−10) = −20 + 90 = 70;
A31 = (−1) M31 = 1
9
10
−1 −10
= −[(−1)10 − 1(−10)] = 0;
A32 = (−1)3+2 M32 = −1
1
10
−1 −2
= (−1)9 − 1(−2) = −9 + 2 = −7.
A33 = (−1)3+3 M33 = 1
1
9
3+1
,
#
!$ -
+
|A| = 1 · 70 + 2 · 0 + 0 · 7 = 70.
!"
!
"
# $
! " #
# $
$
a11 A21 + a12 A22 + a13 A23 = 0%
a12 A11 + a22 A21 + a32 A31 = 0.
& ' $ $
(
)
$*
a11 A21 + a12 A22 + a13 A23 = a11 (−M21 ) + a12 M22 + a13 (−M23 ) =
a12 a13
a11 a13
a11 a12
=
= −a11 ·
+ a12 ·
− a13 ·
a32 a33
a31 a33
a31 a32
= −a11 (a12 a33 − a13 a32 ) + a12 (a11 a33 − a13 a31 ) − a13 (a11 a32 − a12 a31 ) =
= −a11 a12 a33 + a11 a13 a32 + a12 a11 a33 −
−a12 a13 a31 − a13 a11 a32 + a13 a12 a31 = 0.
+
$ $ (
, -. )
$ /
& )
(
, 0 .*
−1 −2 −10
9
10 1
|A| = 1
1
2
0
2 3 * & ) )
$ ( ) ) $
0 0 −10
10
|A| = 0 7
1 2
0
!
|A| = a11 A11 + a21 A21 + a31 A31
0 −10
= 70.
= 1
7
10
" # $
n
n − 1 %
" n
&
'⎛
⎞ ⎛
⎞
1
5 3
3 2 4
A = ⎝ 2 −1 0 ⎠ B = ⎝ 2 3 −2 ⎠
−1 2 2
4 0 2
( ) *
+
" + ) cij = aij + bij
⎞
⎞ ⎛
4 7 7
1+3
5+2 3+4
C = A + B = ⎝ 2 + 2 −1 + 3 0 − 2 ⎠ = ⎝ 4 2 −2 ⎠ .
3 2 4
−1 + 4 2 + 0 2 + 2
⎛
!"#$
A=
( ) %
0 5
4 1
,B =
1 3
2 −2
.
C = α · A ⇒ cij = α · aij
0 5
0 10
1 3
5 15
2A = 2 ·
=
, 5B = 5 ·
=
4 1
8 2
2 −2
10 −10
0 + 5 10 + 15
5 25
D = 2A + 5B =
=
.
8 + 10 2 − 10
18 −8
⎛
⎞
A · B B · A
⎛
⎞
2 4 0
2 1 0
A = ⎝ 2 0 4 ⎠ , B = ⎝ 1 −1 2 ⎠ .
1 2 3
3 2 1
i j !
" i
# ⎛ j $ #
⎞
2 · 2 + 4 · 1 + 0 · 3 2 · 1 + 4 · (−1) + 0 · 2 2 · 0 + 4 · 2 + 0 · 1
%& A · B = ⎝ 2 · 2 + 0 · 1 + 4 · 3 2 · 1 + 0 · (−1) + 4 · 2 2 · 0 + 0 · 2 + 4 · 1 ⎠ =
1 · 2 + 2 · 1 + 3 · 3 1 · 1 + 2 · (−1) + 3 · 2 1 · 0 + 2 · 2 + 3 · 1
⎛
⎞
8 −2 8
= ⎝ 16 10 4 ⎠ .
13 5 7
⎛
2·2+1·2+0·1
¾µ B·A = ⎝ 1 · 2 + (−1) · 2 + 2 · 1
3·2+2·2+1·1
⎞
2·0+1·4+0·3
1 · 0 + (−1) · 4 + 2 · 3 ⎠ =
3·0+2·4+1·3
2·4+1·0+0·2
1 · 4 + (−1) · 0 + 2 · 2
3·4+2·0+1·2
⎛
⎞
6 8 4
= ⎝ 2 8 2 ⎠.
11 14 11
' !( ) ! ( *
* (! ! A · B = B · A
A=
2 −1 3 5
4 0 1 2
⎛
−2
⎜ −1
⎜
,B = ⎝
4
3
⎞
1 0
5 −2 ⎟
⎟.
0 1 ⎠
1 −1
+ * !" !$(
" # $ ( ( !* $ *
* !
' ! *
, * ' # '
!#
2 · (−2) − 1 · (−1) + 3 · 4 + 5 · 3 2 · 1 + (−1) · 5 + 3 · 0+
A·B =
4 · (−2) + 0 · (−1) + 1 · 4 + 2 · 3 4 · 1 + 0 · 5 + 1 · 0+
5 · 1 + 2 · 0 + (−1) · (−2) + 3 · 1 + 5 · (−1)
2 · 1 + 4 · 0 + 0 · (−2) + 1 · 1 + 2 · (−1)
=
24 2 0
2 6 −1
.
B · A
! M13 M32
A13 A32
⎛
⎞
2 −1 0
A = ⎝ −1 2 3 ⎠ .
4
1 3
" # $ %
&' ij
&' ( (
i j ) Aij =
= (−1)i+j Mij
M13
−1 2
=
4 1
= −1 · 1 − 4 · 2 = −1 − 8 = −9.
A13 = (−1)1+3 M13 = (−1)4 · (−9) = −9.
2 0
= 2 · 3 − (−1) · 0 = 6.
M32 =
−1 3
A32 = (−1)3+2 M32 = (−1)5 · 6 = −6.
*
5 6 3
|A| = 0 1 0
7 4 5
.
" # $ +
,
|A| = a21 A21 + a22 A22 + a23 A23 = −a21 M21 + a22 M22 − a23 M23 =
6 3
+1· 5 3 −0· 5 6 =
= −0 ·
4 5
7 5
7 4
5 3
= 5 · 5 − 3 · 7 = 25 − 21 = 4.
=
7 5
|A| =
4
2 1 4
1 −2 0 3
−2 −3 2 1
3
2 0 1
.
|A| = a13 A13 +a23 A23 +a33 A33 +a43 A43 = 1·A13 +0·A23 +2·A33 +0·A43 =
= A13 + 2A33 = M13 + 2M33 .
M13 M33 ! " #
!
1 −2 3
−3 1
− (−2) · −2 1 + 3 · −2 −3 =
M13 = −2 −3 1 = 1 ·
3
3 1
2
2
1
3
2 1
= 1(−3 · 1 − 2 · 1) + 2(−2 · 1 − 3 · 1) + 3(−2 · 2 − 3 · (−3)) = −5 − 10 + 15 = 0.
4 2 4
−2 −3
− 2 · 1 3 + 4 · 1 −2 =
M33 = 1 −2 3 = 4 ·
3 2
2
1
3 1
3 2 1
= 4(−2 · 1 − 2 · 3) − 2(1 · 1 − 3 · 3) + 4(1 · 2 − 3 · (−2)) = −32 + 16 + 32 = 16.
|A| = M13 + 2M33 = 0 + 2 · 16 = 32
! $ % $ #
! $ !
• & ! !!$'( #
! $ ) −2 :
4
2 1 4
1 −2 0 3
.
|A| =
−10 −7 0 −7
3
2 0 1
•
1
|A| = 1 · (−1)1+3 −10
3
−2 3 1 −2 3
−7 −7 = −10 −7 −7 .
2
1
2
1 3
•
−3 :
1
0
0
|A| = −10 −27 23
3
8 −8
•
.
−27 23
= 8 −27 23 = 8(27 − 23) = 32.
|A| = 1 · (−1)1+1
1 −1
8 −8
!
0 −1 5
3 4 7 = 12.
x x 8
" #
$ % "$
#
%
4 7
− (−1) · 3 7 + 5 · 3 4 = 12 ⇒ 24 − 7x + 5(3x − 4x) = 12
0 ·
x x
x 8
x 8
⇒ 24 − 12x = 12 ⇒ x = 1.
&
A=
2 4
−1 3
1 0
B=
.
2 3
'(
⎛
⎞
5 8 4
A = ⎝ 3 2 5 ⎠ .
7 6 0
A
B
! "
''
⎛ D = 3A + 5E
⎞ E !
! A = ⎝
1 1 2
0 1 4 ⎠.
−1 2 3
⎛
1 1 2
3 1 ⎠.
4 1 1
A2 A = ⎝ 1
⎞
⎛ ⎞
1 3 2
2
A = ⎝ 2 0 4 ⎠ B = ⎝ 1 ⎠ .
1 2 0
3
⎛
⎞
A · B
M12 M22
⎛A12 A22 ⎞
1
3 1
A = ⎝ −2 −1 2 ⎠ .
0
1 3
1 0 4
|A| = −2 2 3 .
1 0 5
|A| =
3
2 1
0
2 3
−2 −3 2
4
1 2
2
0
1
4
1 1 0
4 x 6 = 0.
3x 1 4
.
! A
AA−1 = E.
'
$% & A
A
−1
A−1
" !#
A−1 A = E.
( )
!
A
A ! "" # $
"
* ( % &
) A−1 (% & &
A % % + |A| = 0
* |A| = 0
A
|AA−1 | = |A||A−1 | = 0.
, % & |AA−1| = |E| = 1
*
&
(
⎛
⎞
a11 a12 a13
A = ⎝ a21 a22 a23 ⎠
a31 a32 a33
- % +
a11 a12 a13
|A| = a21 a22 a23
a31 a32 a33
= 0.
Aij aij
A−1 A
• ! B A "# $ %
aij Aij $"
|A| " A&
⎛
⎞
A11 /|A| A12 /|A| A13 /|A|
B = ⎝ A21 /|A| A22 /|A| A23 /|A| ⎠ .
A31 /|A| A32 /|A| A33 /|A|
•
' B T (
B )
⎛
⎞
A11 /|A| A21 /|A| A31 /|A|
B = ⎝ A12 /|A| A22 /|A| A32 /|A| ⎠ .
A13 /|A| A23 /|A| A33 /|A|
T
B T # A
*
⎞
⎞⎛
a11 a12 a13
A11 /|A| A21 /|A| A31 /|A|
AB = ⎝ a21 a22 a23 ⎠ ⎝ A12 /|A| A22 /|A| A32 /|A| ⎠ =
a31 a32 a33
A13 /|A| A23 /|A| A33 /|A|
⎛
T
⎛
⎜
=⎝
a11 A11 +a12 A12 +a13 A13
|A|
a21 A11 +a22 A12 +a23 A13
|A|
a31 A11 +a32 A12 +a33 A13
|A|
a11 A21 +a12 A22 +a13 A23
|A|
a21 A21 +a22 A22 +a23 A23
|A|
a31 A21 +a32 A22 +a33 A23
|A|
a11 A31 +a12 A32 +a13 A33
|A|
a21 A31 +a22 A32 +a23 A33
|A|
a31 A31 +a32 A32 +a33 A33
|A|
⎞
⎟
⎠=
⎞
1 0 0
= ⎝ 0 1 0 ⎠ = E,
0 0 1
⎛
# " |A| + %
" , "- " + # %
# + ,
# + , ,
AB T = E
A
−1
B T = A−1
⎛ A
11
⎜ |A|
⎜ A
⎜ 12
=⎜
⎜ |A|
⎝ A13
|A|
A21
|A|
A22
|A|
A23
|A|
A31
|A|
A32
|A|
A33
|A|
⎞
⎟
⎟
⎟
⎟.
⎟
⎠
|A| = 0
⎛
A−1
A11 A21
1 ⎜
A12 A22
⎜
=
|A| ⎝ . . . . . .
A1n A2n
⎞
. . . An1
. . . An2 ⎟
⎟
... ... ⎠
. . . Ann
⎛
"#%
⎞
3 2 2
A = ⎝ 1 3 1 ⎠
5 3 4
!"#"$
& ' ( ) A
3 2 2
|A| = 1 3 1
5 3 4
= 27 + 2 − 24 = 5.
*+ , ,
Aij = (−1)i+j · Mij (
3 1
A11 =
3 4
1
A12 = −
5
1 3
A13 =
5 3
= 9, A21 = − 2
3
3
1
= 1, A22 = 5
4
= −12, A23 = −
2
4
2
4
3
5
= −2, A31 = 2
3
= 2, A32 = − 3
1
3
2
= 1, A33 =
3
1
2
= −4,
1
2
= −1,
1
2
= 7.
3
⎛
A−1
9
5
⎜
⎜
⎜
⎜ 1
=⎜
⎜ 5
⎜
⎜
⎝ −12
5
−2
5
2
5
1
5
⎞
−4
5 ⎟
⎟
⎟
−1 ⎟
⎟.
5 ⎟
⎟
⎟
7 ⎠
5
⎛
a11
⎜ a21
⎜
⎜
A=⎜
⎜ ai1
⎝
a12
a22
. . . a1k
. . . a2k
ai2
. . . aik
⎞
. . . a1n
. . . a2n ⎟
⎟
⎟
⎟,
. . . ain ⎟
⎠
am1 am2 . . . amk . . . amn
m n ! "# $ %
! k k & ! ' !%
!( $ k
) ' k% ! A $! %
# ! ' # $ # !
! $ !( k k
) !
⎛
⎞
2 3
4 5
A = ⎝ 0 −2 3 1 ⎠ ,
0 2
2 4
# '!
$ %
2
3 4
0 −2 3
0
2 2
' !# !%
# # #
!
A *
3 4
−2 3 " ! ! + %
! ) ! $ !
! ! , ' !
A r
A
r
r ! "
A
r(A)
"
⎛
⎞
1
2 5 3
⎜ 0
1 7 4 ⎟
⎟.
A=⎜
⎝ 0
0 0 0 ⎠
0 −1 0 0
#
1
2 5 3
0
1 7 4
=0
0
0 0 0
0 −1 0 0
$
! %
1
2 5
1 7 = 7 = 0. &
M34 = 0
0 −1 0
' r(A) = 3
(
$
)
! (*
+
! !, • * $ . /
$
* 0
•
$ . /
$
!, . /
*
* 0
•
. /
0
•
!
"
#
$
" $ $
%&%
⎛
⎞
2 3
5 −3 −2
3 −1 −3 ⎠ .
A=⎝ 3 4
5 6 −1
3 −5
' (
) * % ! + $
+
⎛
⎞
1 1 −2
2 −1
3 −1 −3 ⎠ .
A1 = ⎝ 3 4
5 6 −1
3 −5
* ! $ ! A1
#
, &
⎛
⎞
1 1 −2
2 −1
9 −7
0 ⎠.
A2 = ⎝ 0 1
0 1
9 −7
0
* $ !
A2
⎛
⎞
1 1 −2
2 −1
9 −7
0 ⎠.
A3 = ⎝ 0 1
0 0
0
0
0
- A3 !
1 1 −2
2 −1
A4 =
0 1
9 −7
0
! . $
!
A # r(A) = 2
/ $ $ 0
! !
⎛
⎞
1 2 3 4
A = ⎝ 2 4 6 8 ⎠.
3 6 9 12
!
"
# $ %
&
' ( r(A) = 1
⎛
⎞
3 5 7
A = ⎝ 1 2 3 ⎠.
1 3 5
(
)*
+ )* , -"
- ). ' / * "
! )* 0 ' % )& "
- * 1 !
% )& $ '
• ( $
- " 2"
*
⎛
⎞
⎛
⎞
⎛
⎞
3 5 7
3+1 5+3 7+5
4 8 12
⎝ 1 2 3 ⎠ ∼ ⎝ 1
2
3 ⎠ = ⎝ 1 2 3 ⎠.
1 3 5
1
3
5
1 3 5
• *
3 % $
⎞
3 5 7
⎝ 1 2 3 ⎠
1 3 5
⎛
1
& "
4
⎞
⎛
1 2 3
∼ ⎝ 1 2 3 ⎠.
1 3 5
• 4* 0 '
0
*
-
⎞
3 5 7
⎝ 1 2 3 ⎠
1 3 5
⎛
⎞
0 0 0
⎝ 1 2 3 ⎠.
1 3 5
⎛
•
∼
⎛
⎞
3 5 7
⎝ 1 2 3 ⎠
1 3 5
∼
1 2 3
1 3 5
.
1 2
1 3 = 3 − 2 = 1 = 0
!
" #
⎛
2 −4 3
1
⎜ 1 −2 1 −4
A=⎜
⎝ 0 1 −1 3
4 −7 4 −4
⎞
0
2 ⎟
⎟.
1 ⎠
5
$ % $ &' (
• )
%
⎛
2 −4 3
1
⎜ 1 −2 1 −4
⎜
A=⎝
0 1 −1 3
4 −7 4 −4
•
⎞
0
2 ⎟
⎟
1 ⎠
5
⎛
∼
1 −2 1 −4
⎜ 2 −4 3
1
⎜
⎝ 0 1 −1 3
4 −7 4 −4
⎞
2
0 ⎟
⎟.
1 ⎠
5
! *
& &
+( &
, (−2)
(−4) '
⎛
⎞
1 −2 1 4 −2
⎜ 0 0
1 9 −4 ⎟
⎟
A∼⎜
⎝ 0 1 −1 3 1 ⎠ .
0 1
0 12 −3
•
⎛
⎞
1 −2 1 4 −2
⎜ 0 1 −1 3 1 ⎟
⎟.
A∼⎜
⎝ 0 0
1 9 −4 ⎠
0 1
0 12 −3
•
⎛
⎞
1 −2 1 4 −2
⎜ 0 1 −1 3 1 ⎟
⎟.
A∼⎜
⎝ 0 0
1 9 −4 ⎠
0 0
1 9 −4
•
!
⎛
⎞
⎛
⎞
1 −2 1 4 −2
1 −2 1 4 −2
⎜ 0 1 −1 3 1 ⎟
⎟ ∼ ⎝ 0 1 −1 3 1 ⎠ .
A∼⎜
⎝ 0 0
1 9 −4 ⎠
0 0
1 9 −4
0 0
0 0 0
•
" # $ %&
# '
%& & (
1 −2 1
1 −1
= 1 = 0.
M = 0 1 −1 = 1
0 1
0 0
1
) & #
*+ ,
⎛
⎞
1 2 0
A = ⎝ 3 2 1 ⎠.
0 1 2
" $ ( -
1 2 0
|A| = 3 2 1
0 1 2
= −9.
|A| = 0
A11 = (−1)
2 1
1 2
1+1
= 3,
A12 = (−1)
3 1
0 2
1+2
= −6,
3 2
= 3.
A13 = (−1)1+3
0 1
2 0
= −4,
A21 = (−1)2+1
1 2
A23 = (−1)2+3
1 0
= 2,
A22 = (−1)2+2
0 2
1 2
= −1.
0 1
2 0
= 2,
A31 = (−1)3+1
2 1
A33
!
1 0
= −1,
A32 = (−1)3+2
3 1
3+3 1 2
= −4.
= (−1)
3 2
⎛
⎞
−3/9 6/9 −3/9
B = ⎝ 4/9 −2/9 1/9 ⎠ .
−2/9 1/9
4/9
" # $
⎛
⎞
−1/3 4/9 −2/9
A−1 = ⎝ 2/3 −2/9 1/9 ⎠ .
−1/3 1/9
4/9
%&&
⎞
⎛
1 0 0 0 5
A = ⎝ 0 0 0 0 0 ⎠.
2 0 0 0 11
⎛
⎛
⎛
⎞
4 3 2 2
A = ⎝ 0 2 1 1 ⎠.
0 0 3 3
⎞
1 2 3 6
A = ⎝ 2 3 1 6 ⎠.
3 1 2 6
⎞
0 2 0 0
A = ⎝ 1 0 0 4 ⎠.
0 0 3 0
⎞
10 20 −30
A = ⎝ 0 10 20 ⎠ .
0 0 10
⎛
⎞
1 2
2
A = ⎝ 2 1 −2 ⎠ .
2 −2 1
⎛
! " #
$ %
&
m '" !
( x1 , x2 , . . . , xn :
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
a11 x1 + a12 x2 + · · · + a1k xk + · · · + a1n xn = c1 ,
a21 x1 + a22 x2 + · · · + a2k xk + · · · + a2n xn = c2 ,
ai1 x1 + ai2 x2 + · · · + aik xk + · · · + ain xn = ci ,
am1 x1 + am2 x2 + · · · + amk xk + · · · + amn xn = cm .
n
) *
aij
x !
" # $
# %
& # &
" # $
#! %
# &
#! #
# &
'
#
# & # &
& # &
( # %
# )
$ # $ *
$ #
#%
# # #*
• # $ $ %
*
• #
$ $
# # #
•
•
+ # # %
#
, # # $ $
#
# - # # # %
$ & , #
# # %
&
.# # # $
%
#
#
# # #
#
$
## # - %
# #
.# A # %
$ /26.10)
⎛
⎞
a11 a12 . . . a1n
⎜ a21 a22 . . . a2n ⎟
⎟
A=⎜
⎝
⎠
am1 am2 . . . amn
X
⎛
⎞
x1
⎜ x2 ⎟
⎜
⎟
X=⎝
⎠
xn
⎛
⎞
c1
⎜ c2 ⎟
⎟
C=⎜
⎝ ... ⎠
cn
!
A·X
"
⎞
a11 x1 + a12 x2 + · · · + a1n xn
a21 x1 + a22 x2 + · · · + a2n xn ⎟
⎟
⎟
⎟
ai1 x1 + ai2 x2 + · · · + ain xn ⎟
⎠
⎛
⎜
⎜
⎜
A·X =⎜
⎜
⎝
#
am1 x1 + am2 x2 + · · · + amn xn
$%
& &
'
"
(
A · X = C.
)
&
. ,
(
% * &
( ,
*
+,
-
"
/
⎛
⎞
a11 a12 . . . a1n
⎜ a21 a22 . . . a2n ⎟
⎟
A=⎜
⎝ ºººººººººººººº ⎠
am1 am2 . . . amn
⎛
⎞
a11 a12 . . . a1n c1
⎜ a21 a22 . . . a2n c2 ⎟
⎟
B=⎜
⎝ ⎠,
am1 am2 . . . amn cm
r(A) = r(B)
"
r(B) = n
A
B
r(A) = r(B) < n
$
"
! &
' $
# (
•
! %#
# !
r(A) =
! #
" $ $
!
* $
A:
|A| = 0
!
A·X = C
)
A−1
(
A−1 · (A · X) = A−1 · C.
•
+
%
% ! &
$ $
%)
(A−1 · A) · X = A−1 · C.
•
, & & &
A−1 · A = E
-
E · X = X
#
)
X = A−1 · C.
.
⎧
⎨ 3x1 + 2x2 + 2x3 = 5,
x1 + 3x2 + x3 = 0,
⎩
5x1 + 3x2 + 4x3 = 10.
⎞
⎛ ⎞
⎛ ⎞ ⎛
x1
5
3 2 2
X = ⎝ x2 ⎠
C = ⎝ 0 ⎠.
AX = C A = ⎝ 1 3 1 ⎠
10
5 3 4
x3
A−1
⎛
⎞
9/5 − 2/5 − 4/5
−1
2/5 − 1/5 ⎠ .
A = ⎝ 1/5
−12/5 1/5
7/5
⎞
⎞ ⎛
⎞ ⎛
⎛
1
5
9/5 − 2/5 − 4/5
2/5 − 1/5 ⎠ · ⎝ 0 ⎠ = ⎝ −1 ⎠ .
X = ⎝ 1/5
2
10
−12/5 1/5
7/5
!" #
#$%
x1 = 1% x2 = −1% x3 = 2 & ' $() %
* $#"
+, $ n $ n
A·
- ./
X = C % # ' |A| = 0
#$"+
⎛ A
An1
11 A21
...
⎜ |A| |A|
|A|
⎜ A A
An2
12 22
⎜
...
⎜
−1
X = A ·C = ⎜ |A| |A|
|A|
⎜ ºººººººººººººººº
⎜
⎝ A1n A2n
Ann
...
|A| |A|
|A|
0#
⎞
x1
⎜ x2 ⎟
1
⎟
⎜
⎝ ⎠ = |A|
xn
⎛
⎛
⎞
⎛
⎞
⎟⎛
c1
A11 A21 . . . An1
⎟
⎟⎜
⎟
1 ⎜
⎟ ⎜ c2 ⎟
⎜ A12 A22 . . . An2
·
=
⎟·⎝
ºº ⎠
⎟
|A| ⎝ º º º º º º º º º º º º º º º
⎟
cn
A1n A2n . . . Ann
⎠
⎞
A11 c1 + A21 c2 + · · · + An1 cn
⎜ A12 c1 + A22 c2 + · · · + An2 cn ⎟
⎟
·⎜
⎝ ⎠.
A1n c1 + A2n c2 + · · · + Ann cn
⎞⎛
⎞
c1
⎟ ⎜ c2 ⎟
⎟·⎜
⎟
⎠ ⎝ ºº ⎠.
cn
⎧
1
⎪
⎪
x1 =
(A11 c1 + A21 c2 + · · · + An1 cn )
⎪
⎪
|A|
⎪
⎪
⎪
1
⎨
x2 =
(A12 c1 + A22 c2 + · · · + An2 cn )
|A|
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
⎩ xn =
(A1n c1 + A2n c2 + · · · + Ann cn )
|A|
! " # $% $!$
# !& '
# |A| ' & % ' %
! ! ! & &
!' !'(
c1 a12 . . . a1n
c2 a22 . . . a2n
,
Δx1 = A11 c1 + A21 c2 + · · · + An1 cn =
cn an2 . . . ann
a11 c1 . . . a1n
a c . . . a2n
Δx2 = A12 c1 + A22 c2 + · · · + An2 cn = 21 2
an1 cn . . . ann
) * !' & + " ,
(
x1 =
Δ x1
Δ x2
Δ xn
, x2 =
, . . . , xn =
.
|A|
|A|
|A|
- ' # + . ½½ # # #
+ /
⎧
⎨ 3x1 + 2x2 + 2x3 = 5,
x1 + 3x2 + x3 = 0,
⎩
5x1 + 3x2 + 4x3 = 10.
½½
x1
! a11 = 0"
# $
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
a1k
a1n
c1
a12
x2 + · · · +
xk + · · · +
xn =
,
a11
a11
a11
a11
a21 x1 + a22 x2 + · · · + a2k xk + · · · + a2n xn = c2 ,
x1 +
ai1 x1 + ai2 x2 + · · · + aik xk + · · · + ain xn = ci ,
%&'(&)
am1 x1 + am2 x2 + · · · + amk xk + · · · + amn xn = cm .
* #
%26.12) a21
+
a31
#
, #
# $
⎧
x1 + a12 x2 + · · · + a1k xk + · · · + a1n xn = c1 ,
⎪
⎪
⎪
a22 x2 + · · · + a2k xk + · · · + a2n xn = c2 ,
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
+ #
ai2 x2 + · · · + aik xk + · · · + ain xn = ci ,
%&'(-)
am2 x2 + · · · + amk xk + · · · + amn xn = cm .
$
a1k =
a1k
,
a11
aik = aik −
c1 =
c1
,
a11
ci = ci −
a1k
ai1 ;
a11
c1
ai1 ;
a11
i = 2, 3, . . . , n;
i = 2, 3, . . . , m.
. #
%26.13) !
a22 "
# a32 , . . . , ai2, . . . , am2
/
0 ! /
!
.
! A %&'(&)
1 ! B
r(A) = r(B).
xr+1 , . . . , xn
αr+1 , αr+2, . . . , αn
x1 , x2 , . . . , xr αr+1 , αr+2
. . . , αn
! "#
$ %
! "
! &
⎧
⎨ 3x1 + 2x2 + 2x3 = 5,
x1 + 3x2 + x3 = 0,
⎩
5x1 + 3x2 + 4x3 = 10.
' % ( ) % *
$
•
% (
⎧
⎨
x1 + 3x2 + x3 = 0,
3x1 + 2x2 + 2x3 = 5,
⎩ 5x + 3x + 4x = 10.
1
2
3
•
+$ &
$ , (
⎧
⎨ x1 + 3x2 + x3 = 0,
−7x2 − x3 = 5,
⎩
−12x2 − x3 = 10.
•
$-
x2
x3
⎧
⎨ x1 + x3 + 3x2 = 0,
x3 + 7x2 = −5,
⎩ −x − 12x = 10.
3
2
•
)$
⎧
⎨ x1 + x3 + 3x2 = 0,
x3 + 7x2 = −5,
⎩
−5x2 = −5.
x2 = −1
x3 = −5 − 7x2 = 2
x1 = −3x2 − x3 = (−3)(−1) − 2 = 3 − 2 = 1.
!" #
$%& !
"
⎧
⎨ x1 + 5x2 + 4x3 + 3x4 = 1,
2x1 − x2 + 2x3 − x4 = 0,
⎩
5x1 + 3x2 + 8x3 + x4 = 1.
' ! ( ! " )
*
#
⎛
1
5 4
⎝ 2 −1 2
5
3 8
⎛
1
∼⎝ 0
0
⎞ ⎛
1
5
4
3 1
−1 0 ⎠ ∼ ⎝ 0 −11 −6
1 1
0 −22 −12
⎞ ⎛
5 4 3 1
1 5 4
11 6 7 2 ⎠ ∼ ⎝ 0 11 6
11 6 7 2
0 0 0
1 5 4 3 1
∼
.
0 11 6 7 2
⎞
3
1
−7 −2 ⎠ ∼
−14 −4
⎞
3 1
7 2 ⎠∼
0 0
+ ! *) *
*) ,,)
*)
r(A) = r(B) = 2,
!"" # " "
x1 + 5x2 + 4x3 + 3x4 = 1,
11x2 + 6x3 + 7x4 = 2.
$ # x1
x3 x4 # %
#
x2
⎧
14
1
2
⎪
⎪
⎨ x1 = 11 − 11 x3 + 11 x4 ,
⎪
⎪
6
2
7
⎩
− x3 − x4 .
x2 =
11 11
11
x3 = 1, x4 = 1 # x1 = −1, x2 = −1
! " #
&'(
⎧
2x1 + x2 + x3 − x4 = 8,
3x1 + 2x3 + 5x4 = 12,
x1 − x2 + x3 = 4,
⎪
⎪
⎩ 8x + x + 5x + 3x = 10.
1
2
3
4
⎪
⎪
⎨
) * " #"
+ % # " ! # #
, "
-
⎞ ⎛
⎞
⎛
x1 x2 x3 x4
⎜ 2
1 1 −1
⎜
⎜ 3
0 2
5
⎜
⎝ 1 −1 1
0
8
1 5
3
⎛
x2 x1 x3 x4
⎜ 1 2 1 −1
⎜
5
∼⎜
⎜ 0 3 2
⎝ 0 3 2 −1
0 6 4
4
x2 x1 x3
⎟ ⎜ 1 2 1
⎟ ⎜
⎟∼⎜ 0 3 2
⎟ ⎜
⎠ ⎝ −1 1 1
1 8 5
⎞ ⎛
x2 x1 x3
⎜
8 ⎟
⎟ ⎜ 1 2 1
⎜
12 ⎟
⎟∼⎜ 0 3 2
12 ⎠ ⎝ 0 0 0
0 0 0
2
8
12
4
10
x4
−1 8 ⎟
⎟
5 12 ⎟
⎟∼
0 4 ⎠
3 10
⎞
x4
−1
8 ⎟
⎟
5
12 ⎟
⎟∼
0 ⎠
6
−6 −22
⎛
x2 x1 x3 x4
⎜ 1 2 1 −1
8
⎜
12
0
3
2
5
∼⎜
⎜
⎝ 0 0 0
0
1
0 0 0
0 −22
⎧
⎪
⎪
⎨
⎞
⎟
⎟
⎟.
⎟
⎠
x2 + 2x1 + x3 − x4 = 8,
3x1 + 2x3 + 5x4 = 12,
x4 = 0,
⎪
⎪
⎩ 0 · x + 0 · x + 0 · x + 0 · x = −22.
2
1
3
4
⎧
⎨ x + 2y + z = 0,
2x + y + z = 1,
⎩
x + 3y + z = 0.
! " #
⎛
⎞
1 2 1
A = ⎝ 2 1 1 ⎠,
1 3 1
⎛
⎞
0
C = ⎝ 1 ⎠,
0
⎛
⎞
x
X = ⎝ y ⎠.
z
$%
&' (
' )"
⎛
A−1
*
X=A
⎞
−2
1
1
0
1 ⎠.
= ⎝ −1
5 −1 −3
⎛
−1
⎞ ⎛ ⎞ ⎛
⎞
−2
1
1
0
1
0
1 ⎠· ⎝ 1 ⎠ = ⎝ 0 ⎠.
· C = ⎝ −1
5 −1 −3
0
−1
$ " x = 1 y = 0 z = −1
⎧
⎨
x + 2y − z = 2,
2x − 3y + 2z = 2,
⎩
3x + y + z = 8.
1
|A| = 2
3
1
Δy = 2
3
2 −1
−3
2 = −8,
1
1
2 −1
2
2 = −16,
8
1
2
2 −1
2 −3
2 = −8.
8
1
1
1
2 2
2 −3 2 = −24.
3
1 8
x=
Δx =
Δz =
−16
−24
−8
= 1, y =
= 2, z =
= 3.
−8
−8
−8
⎧
2x1 + x2 − x3 = 1,
3x1 + 2x2 − 2x3 = 1,
⎩
x1 − x2 + 2x3 = 5.
⎨
! "#$ %
⎞ ⎛
⎞
⎛
1 −1
2 5
2
1 −1 1
⎝ 3
2 −2 1 ⎠ ∼ ⎝ 2
1 −1 1 ⎠ ∼
1 −1
2 5
3
2 −2 1
⎛
⎞ ⎛
⎞
1 −2
2
1 −1
2
5
5
3 −5 −9 ⎠ ∼ ⎝ 0
6 −10 −18 ⎠ ∼
∼⎝ 0
0
5 −8 −14
0
5 −8 −14
⎞ ⎛
⎞
⎛
1 −1
2
1 −1
2
5
5
1 −2 −4 ⎠ ∼ ⎝ 0
1 −2 −4 ⎠ ∼
∼⎝ 0
0
5 −8 −14
0
0
2
6
⎛
⎞ ⎛
⎞
1 −1
2
5
&$
⎠
1 −2 −4 ⎠ ⎝
∼⎝ 0
0
0
1
3
⎧
⎨ x1 − x2 + 2x3 = 5,
x2 − 2x3 = −4,
⎩
x3 = 3,
x3 = 3, x2 = −4 + 6 = 2, x1 = 5 + x2 − 2x3 = 1.
x1 = 1 x2 = 2 x3 = 3
!" #
⎧
⎨ x1 + 2x2 + 4x3 − x4 − 3x5 = 7,
2x1 + x3 + x5 = 4,
⎩
x2 + 2x4 − x5 = 6.
$
% & '
( )
*
⎞ ⎛
⎞
1
2
4 −1 −3
1 2 4 −1 −3 7
7
⎝ 2 0 1
0
1 4 ⎠ ∼ ⎝ 0 −4 −7
2
7 −10 ⎠ ∼
6
0 1 0
2 −1 6
0
1
0
2 −1
⎛
⎞ ⎛
⎞
1 2
4 −1 −3 7
1
2
4 −1 −3
7
6 ⎠∼⎝ 0 1
1
0
2 −1
0
2 −1 6 ⎠ .
∼⎝ 0
0 −4 −7
2
7 −10
0 0 −7 10
3 14
⎛
+
&, )
• & !
•
& &
• )
& - & #
.
'
⎧
⎨ x1 + 2x2 + 4x3 − x4 − 3x5 = 7,
x2 + 2x4 − x5 = 6,
⎩
−7x3 + 10x4 + 3x5 = 14,
. &
)
⎧
⎨ x1 + 2x2 + 4x3 = 7 + x4 + 3x5 ,
x2 = 6 − 2x4 + x5 ,
⎩
−7x3 = 14 − 10x4 − 3x5 .
&,
x
x x x
1
2
3
x4
⎧
3
10
⎪
⎪
⎨ x3 = −2 + 7 x4 + 7 x5 ,
x2 = 6 − 2x4 + x5 ,
⎪
⎪
⎩ x = 3 − 5x − 5x .
1
4
5
7
7
5
x x
x x x x x
x x x
!" #
$# % x = 0 x = 0 x = −2 x = 6
x = 3 x = 7 x = 7 x = 11 x = −1 x = −7
&' (
⎧
4
1
5
2
3
4
1
2
5
3
5
1
5
4
3
2
4
3
2
1
x − y − z = 5,
2x + y + 3z = 3,
⎩
x − 4y − 6z = 7.
⎨
) % * % +
" ⎞ ⎛ ,
⎞ ⎛
⎞
⎛
1 −1
1
1 −1 1
1 −1
1 5
5
5
⎝ 2
1
3 3 ⎠∼⎝ 0
3
5 −7 ⎠ ∼ ⎝ 0
3 5 −7 ⎠ .
1 −4 −6 7
0 −3 −5
0
0 0 −5
2
+
• - . " −2 & /
& "
• 0 1 / " 1
"
• *
& 1 " /
/ +⎧
x − y − z = 5,
3y + 5z = −7,
⎩
0x + 0y + 0z = −5.
⎨
!" "" /
% /
!
"
" # $%&
B
_
a
A
a
' "
# (& ! # )&
* AB. + , * - " "
* . a, , " / 0 .
a
|a|
$
0
|0| = 0 1
2 , - -
-
* , "
3 a b
! " "
# $ a b
4 a b
#
$ a b
% a b
#
$ a ↑↓ b
a b
#
$ a = b
BC = AD AB ↑↓ CD
B
C
A
D
! ! " !
# a + b !
" # $% b !
" " & !
a ' ()
_
b
_
a
_ _
a+b
_
b
$ %!& ' a + b (
a " ) "
b
* % (
'+ b ! a
, , - .#/ a + b
a
a (−b) a b !"#
_
b
_
-b
_ _
a-b
_
a
$ O % OACB OC
"# # %
a + b BA
"# # % a − b
A
_
a
O
C
_ _
a-b
__
a+b
_
b
B
&' a
λ c a
|c| = |λ| · |a| a λ > 0,
a λ < 0
&'&2a a
a, − 12 a
a a
( # −a % %
a $ λ = −1) −a = (−1)a
λa = aλ
λ1 (λ2 a) = (λ1 λ2 )a
λ(a + b) = λa + λb
$ !
% a
b = λa
a = λb
b
!
"#
& b = λa a = λb b a
' !
& b a a = 0 b = λa ( ' ) '
b = 0 0 = 0 · a & a = 0 '
a = 0 = 0 · b
%
" a * ea
#
a + ! ' !
a = |a|ea
a
.
ea =
|a|
l
el
O
_
a
O
_
el
ϕ
1
l
a b
ϕ
!"
a
_
a
b
ϕ
_
b
ϕ
0 ϕ π
!" a l
a el
!# ! " l
# # α$ # %&! ' # #
l$ l ! AB l
$ # ! (
l AB = x2 − x1.
$ %
__
AB
_
el
α A
A1
x1
B
β
B1
x2
l
& ϕ
AB '( l
!"" x2 > x1 ) l AB ' ϕ
) l AB * ' AB ⊥ l (ϕ = 90◦) ) l AB = 0
AB l
AB l
el
AB l
l AB = !l AB · el = A1 B1 .
"
! a l #$ a $#%
#$ $ $& #%$ #
!l a = |a| cos ϕ.
' (
! a l (# )# &
# * #$ # # $ & a
% O +
B
a
X
O
B1
l
, -#$ !l a = x−0 = x .( #$&
x
& OBB1 : cos ϕ =
$ x = |a| cos ϕ !l a = |a| cos ϕ
|a|
! $##/ $##
!l (a + b) = !l a + !l b.‘
+
' (
!$ AC = AB +BC 0& !l AB = x2 −x1 !l BC =
= x3 − x2 !l AC = x3 − x1 1 2#
!l AC = x3 − x1 = (x2 − x1 ) + (x3 − x2 ) = !l AB + !l BC
0#
)& & #/3
B
C
A
X 1 X2
O A1
B2
X3
C3
e
l (λa) = λ l a.
!"
#
$ λ > 0
a % l & ϕ
λa % l & ϕ λ < 0
λa % l & π − ϕ
# & (λ > 0) l (λa) = |λa| cos ϕ =
= |λ||a| cos ϕ = λ l a
#
& (λ < 0)
l (λa) = |λa| cos(π − ϕ) = |λ||a| cos(π − ϕ) = −λ|a|(− cos ϕ) =
= λ|a| cos ϕ = λ l a
'
# l (a − b) = l (a + (−1)b) = l a + l (−1)b =
= l a + (−1) l b = l a − l b
(
⎧
⎨ 2x1 + 3x2 + 2x3 = 9,
x1 + 2x2 − 3x3 = 14,
⎩
3x1 + 4x2 + x3 = 16.
⎧
2x − x2 + x3 + 2x4 + 3x5 = 2,
⎪
⎪
⎨ 1
6x1 − 3x2 + 2x3 + 4x4 + 5x5 = 3,
6x1 − 3x2 + 4x3 + 8x4 + 13x5 = 9,
⎪
⎪
⎩ 4x − 2x + x + x + 2x = 1.
1
2
3
4
5
⎛
⎧
2x + x2 − x3 = 4,
⎪
⎪
⎨ 1
−x1 − 3x2 + 2x3 = 3,
3x1 + 4x2 − 3x3 = 1,
⎪
⎪
⎩ 4x + 7x − 5x = −1.
1
2
3
⎧
⎨ 2x1 + 3x2 + 2x3 = 9,
x1 + 2x2 − 3x3 = 14,
⎩
3x1 + 4x2 + x3 = 16.
⎛
2 3
A = ⎝ 1 2
3 4
⎛
⎞
x1
X = ⎝ x2 ⎠
x3
⎞
2
3 ⎠
1
⎞
9
B = ⎝ 14 ⎠ ! "
16
# ! X = A−1 B $%! & A−1
&' A
!" "( A
2 3 2
2 −3
− 3 1 −3 + 2 1 2 =
|A| = 1 2 −3 = 2
3 4
3 1
4
1
3 4 1
= 2 · 14 − 3 · 10 + 2 · (−2) = 28 − 30 − 4 = −6.
2 −3
3 2
= 5,
A11 =
= 14, A21 = −
4 1
4 1
3 2
1 −3
= −10,
= −13, A12 = −
A31 =
3 1
2 −3
2 2
= −4, A32 = − 2 2 = 8,
A22 =
1 −3
3 1
1 2
= −2, A23 = − 2 2 = 1,
A13 =
3 4
3 4
2 3
= 1,
A33 =
1 2
⎛
⎞
14
5 −13
1
A−1 = − 6 ⎝ −10 −4 8 ⎠ .
−2 1
1
⎛
⎞
⎛
⎞⎛
⎞
x1
14
5 −13
9
1
X = ⎝ x2 ⎠ = A−1B = − 6 ⎝ −10 −4 8 ⎠ ⎝ 14 ⎠ =
x3
−2 1
1
16
⎛
⎞
⎛
⎞ ⎛
⎞
126 + 70 − 208
−12
2
1
1
= − ⎝ −90 − 56 + 128 ⎠ = − ⎝ −18 ⎠ = ⎝ 3 ⎠ .
6
6
18 + 14 + 16
12
−2
! " # $
" % & %"
⎛
2 3 2 9
⎝ 1 2 −3 14
3 4 1 16
⎛
6
∼⎝ 0
6
⎞
⎛
⎞ ⎛
2 3 2 9
2
⎠ ∼ ⎝ 2 4 −6 28 ⎠ ∼ ⎝ 0
3 4 1 16
3
⎞ ⎛
2 3
2
9 6 27
1 −8 19 ⎠ ∼ ⎝ 0 1 −8
0 −1 −4
8 2 32
⎛
⎞
2 3 2
9
∼ ⎝ 0 1 −8 19 ⎠ .
0 0 −12 24
' ( " " ) %
• * +&
⎞
3 2 9
1 −8 19 ⎠ ∼
4 1 16
⎞
9
19 ⎠ ∼
5
•
•
•
•
⎧
⎨ 2x1 +3x2 +2x3 = 9,
−8x3 = 19,
x2
⎩
−12x3 = 24,
!
9 − 3x2 − 2x3
=
x3 = −2 x2 = 19 + 8x3 = 19 − 16 = 3 x1 =
2
9−9+4
= 2.
2
"
⎧
2x − x2 + x3 + 2x4 + 3x5 = 2,
⎪
⎪
⎨ 1
6x1 − 3x2 + 2x3 + 4x4 + 5x5 = 3,
6x1 − 3x2 + 4x3 + 8x4 + 13x5 = 9,
⎪
⎪
⎩ 4x − 2x + x + x + 2x = 1.
1
2
3
4
5
# $
⎛
⎜
⎜
⎜
⎜
⎝
⎛
x1
2
6
6
4
1
⎜ 0
∼⎜
⎝ 0
0
⎛
x3
⎜ 1
⎜
∼⎝
0
0
!
⎞
⎛
x3
⎜
2 ⎟
⎟ ⎜ 1
⎜
3 ⎟
⎟∼⎜ 2
9 ⎠ ⎝ 4
1
1
⎞ ⎛
2 −1 2
3
2
1
⎜ 0
2 −1 0 −1 −1 ⎟
⎟∼⎜
−2 1
0
1
1 ⎠ ⎝ 0
2 −1 −1 −1 −1
0
⎞ ⎛
x1 x2 x4 x5
x3
⎜
2 −1 2
3
2 ⎟
⎟∼⎜ 1
2 −1 0 −1 −1 ⎠ ⎝ 0
0
2 −1 −1 −1 −1
x2 x3 x4 x5
−1 1 2 3
−3 2 4 5
−3 4 8 13
−2 1 1 2
x1
2
6
6
4
x2 x4 x5
−1 2 3
−3 4 5
−3 8 13
−2 1 2
⎞
2
3
9
1
⎟
⎟
⎟∼
⎟
⎠
⎞
−1 2
3
2
−1 0 −1 −1 ⎟
⎟∼
−1 0 −1 −1 ⎠
−1 −1 −1 −1
⎞
x2 x1 x4 x5
−1 2
2
3
2 ⎟
⎟∼
1 −2 0
1
1 ⎠
−1 2 −1 −1 −1
2
2
2
2
⎛
⎞
x3 x2 x1 x4 x5
⎜ 1 −1 2
2 3 2 ⎟
⎟
∼⎜
⎝ 0 1 −2 0 1 1 ⎠ ⇒ x4 = 0.
0 0
0 −1 0 0
•
•
!"
# ! $ !
• # (−1)
•
•
• %
& ! $ $ x2 x3 x4 x1
x5
x2 = 1 + 2x1 − x5 x3 = 2 + x2 − 2x1 − 3x5 = 3 − 4x5 %
' xn = x5 = 0 ! (
⎛
⎜
⎜
X=⎜
⎜
⎝
0
1
3
0
0
⎞
⎟
⎟
⎟.
⎟
⎠
)%#
⎧
2x + x2 − x3 = 4,
⎪
⎪
⎨ 1
−x1 − 3x2 + 2x3 = 3,
3x1 + 4x2 − 3x3 = 1,
⎪
⎪
⎩ 4x + 7x − 5x = −1.
1
2
3
* (
⎛
⎜
⎜
∼⎜
⎜
⎝
x1 x2
2
1
−1 −3
3
4
4
7
x3
−1 4
2
3
−3 1
−5 −1
⎞
⎛
⎞
1 3 −2 −3
⎟
⎟ ⎜ 2 1 −1 4 ⎟
⎟∼⎜
⎟
⎟ ⎝ 3 4 −3 1 ⎠ ∼
⎠
4 7 −5 −1
⎛
⎞
⎛
⎞
1 3 −2 −3
1 3 −2 −3
⎜ 0 −5 3 10 ⎟
⎟ ⎝ 0 −5 3 10 ⎠ ∼
∼⎜
⎝ 0 −5 3 10 ⎠ ∼
0 −5 3 11
0 −5 3 11
⎞
1 3 −2 −3
∼ ⎝ 0 −5 3 10 ⎠ .
0 0
0
1
⎛
•
•
•
• !
" 0 · x1 + 0 · x2 + 0 · x3 = 1
!!
# !
$!
⎧
⎨ 2x1 − x2 + x3 = 2,
3x1 + 2x2 + 2x3 = −2,
⎩
x1 − 2x2 + x3 = 1.
$!%
⎧
⎪
x1 + 5x2 − 2x3 − 3x4 = 1,
⎪
⎪
⎪
⎨ 7x1 + 2x2 − 3x3 − 4x4 = 3,
x1 + x2 + x3 + x4 = 5,
⎪
⎪
2x1 + 3x2 + 2x3 − 3x4 = 4,
⎪
⎪
⎩ x − x − x − x = −2.
1
2
3
4
a
! "#$% & '
(
a
) "#$
a = OM = OP + P M = OP + OM3 = (OM1 + OM2 ) + OM3 .
* OM1 OM2 OM3 + a
* , !-$ .% / OM1 =
= (Ox a · i OM2 = (Oy a · j OM3 = (Oz a · k (Ox a = ax
(Oy a = ay (Oz a = az /
!-0 "%
a = ax · i + ay · j + az · k.
ax ay az
a ! "
# a = (ax; ay ; az )
-0 " $ rM % M(a1 ; a2; a3)
OM % % &
' %
* , -0 "
rM = OM = (a1; a2; a3 ).
AB
A(x1 ; y1 ; z1 ) ' 1 B(x2 ; y2 ; z2 ) ) '
(Ox AB = x2 − x1 (Oy AB = y2 − y1 (Oz AB =
AB / ax = x2 − x1
= z2 − z1 . ( 2
ay = y2 − y1 az = z2 − z1 +
/
!-0 -%
AB = (x2 − x1 ) · i + (y2 − y1 ) · j + (z2 − z1 ) · k.
3 AB
' , /
AB = rB − rA .
AB = OB − OA = rB − rA .
|M1 M |
|M1 M |
= λ M1 M = λMM2
|MM2 |
|MM2 |
(x − x1 )i + (y − y1 )j + (z − z1 )k = λ((x2 − x)i + (y2 − y)j + (z2 − z)k)
=
M1
r
r1
M
r2
O
M2
= λ(x2 − x), y − y1 = λ(y2 − y), z − z1 = λ(z2 − z)
z
x=
x1 + λx2
,
1+λ
y=
y1 + λy2
,
1+λ
!
Oxy
x − x1 =
x, y
z1 + λz2
.
1+λ
M1 M2 !" #
z=
$ !%
!& '
M1(1; 2; 3) M2(2; 1; 1)
M1M2
( )
*
M1 M2 λ =
!+
x=
-
1+2
2+1
3+1
= 1, 5, y =
= 1, 5, z =
= 2
2
2
2
M(1, 5; 1, 5; 2)
. #
0
!
% '
*
1 2 !
#
α β
γ
#
#
345 . + #
!
M1 M
= 1
MM2
%
!
/
! /
"
/
cos α cos β
cos γ
6
a = ax · i + ay · j + az · k 7#
ax = 6Ox a =
= |a| · cos α, ay = 6Oy a = |a| · cos β , az = 6Oz a = |a| · cos γ
ay
az
ax
; cos β =
; cos γ =
.
cos α =
|a|
|a|
|a|
z
γ
β
O
y
α
x
|a|
ax
ay
cos α = 2
; cos β = 2
;
ax + a2y + a2z
ax + a2y + a2z
az
cos γ = 2
.
ax + a2y + a2z
! "
cos2 α + cos2 β + cos2 γ = 1.
#$
% & '( ) α β γ *
+ ! , #$
-) + *&) ) "
ea ) *. )
ea = cos α · i + cos β · j + cos γ · k.
##
% *. a !
, /#
'( '
0 ) &+ ) +
1) 2 "
"
)
3 AB
A(1; 2; −2) B(2; −1; 0)
AB AB = (2 − 1)i + (−1 − 2)j + (0 − (−2))k = i − 3j + 2k
√
3
2
AB
1
|AB| = 12 + (−3)2 + 22 = 14 eAB =
= √ i − √ j + √ k
|AB|
14
1
3
2
cos α = √ cos β = − √ cos γ = √
14
14
14
14
14
a = ax i + ay j + az k b = bxi + by j +
!" b = λa a =
bz k
λb # $
%&'
bx = λax by = λay bz = λaz ax = λbx ay = λby az = λbz
# λ # (' $
bx
by
bz
=
=
ax
ay
az
ax
ay
az
=
= .
bx
by
bz
a b
) *" +
, -# $ -
-
-' - ' -
.
*/ a = (1; 3; 5)
b = (2; 6; 0)
1
3
5
= =
2
6
0
5 = λ · 0
1
$$ λ =
2
a b
*0 a = (1; 3; 0)
b = (2; 6; 0)
%- λ
a b
1
3
0
= =
2
6
0
0 = λ·0
- # 1
. 2$
a b
a · b = |a| · |b| · cos ϕ.
a
b
b
a
a b
a · b
(a, b)!
!a b = |b| cos ϕ " #
" $% & ' a · b = |a| · !a b
'
a·b
.
!a b =
|a|
a · b = b · a $
( ) ' a · b = |a| · |b| · cos ϕ = |b| · |a| · cos ϕ = b · a
λ(a · b) = (λa) · b = a · (λb) * $
( ) λ > 0' λ(a · b) = λ|a| · |b| · cos ϕ =
= |λa| · |b| · cos ϕ = (λa) · b + )$ & #
, &
a(b + c) = a · b + a · c
- $
( ) )
*
& ' a(b + c) = |a| · !a (b + c) = |a|(!a b + !a c) = |a| · !a b +
+ |a| · !a c = a · b + a · c
( , & -$ $. -$ $
-.
& & -$ . #
) '
a ⊥ b,
|a| = |0,
|b| = 0
⇐⇒
a · b = 0.
/
( )
a ⊥ b , ϕ = (a;
b) = 90◦ =⇒ cos ϕ = 0
a · b = |a| · |b| · cos ϕ = 0 0 -
a · b = 0 |a| · |b| · cos ϕ = 0 =⇒
% 1 |a| = |0
|b| = 0
cos ϕ = 0 =⇒ ϕ = 90◦ a ⊥ b
a · b > 0 ⇐⇒ (a;
b)
a · b < 0 ⇐⇒ (a; b)
! " " "! " #
a2 = |a|2 .
$
2
◦
2
%"&! a = a·a = |a|·|a|·cos 0 = |a| '
$
√
a2 = |a|.
) "" "&!*
√
x2
= |x|
(
$ (a − 2b)2 |a| = 1 |b| = 2 (a;
b) = 60◦
2
+ , ! (a−2b)2 = a2 −4a·b+4b = |a|2 −4|a|·|b|·cos 60◦ +4|b|2 =
1
= 1 − 4 · 1 · 2 · + 4 · 4 = 13
2
-" (a − 2b)2 = 13
% ! " .! ! # "! " * " /
2
" * ! i2 = j 2 = k = 1 i·j = i·k =
= j · k = 0
0 a = ax · i + ay · j + az · k b = bx · i + by · j + bz · k a · b =
2
2
= (ax · i + ay · j + az · k)(bx · i + by · j + bz · k) = ax · bx · i + ay · by · j +
2
+ az · bz · k + ax · by · i · j + ... + ay · bz · j · k = ax · bx + ay · by + az · bz
- !&!
a · b = ax · bx + ay · by + az · bz .
(
a = (1; 0; 2) b = (−2; 1; 3)
+ , ! a · b = 1 · (−2) + 0 · 1 + 2 · 3 = 4 > 0 =⇒ # (a;
b)
1"
" " 2 ! # "!
! ! " * " " ! 3*
! 3 " ! & "!"
ax · bx + ay · by + az · bz = 0.
4
a = (1; −3; m)
b = (2; 1; 4)!
1
a ⊥ b ⇐⇒ a·b = 0 ⇒ 1·2+(−3)·1+4m = 0 ⇒ m =
4
1
a⊥b m=
4
! "
# $
cos ϕ =
ax bx + ay by + az bz
a·b
.
= 2
ax + a2y + a2z · b2x + b2y + b2z
|a| · |b|
%
& ' ( cos ϕ ( ) ( *
% ' ' )' ) '
+ + , - (
(
.
a = (1; 2; 3)
b = (−2; −1; −1)
/ " + %
cos ϕ = √
7
=− √
2 21
12
1 · (−2) + 2 · (−1) + 3 · (−1)
−7
√ =
=√
2
2
2
2
2
14 · 6
+ 2 + 3 · (−2) + (−1) + (−1)
$ 0 ) ( '
( ( cos ϕ(0 ≤ ϕ ≤ π) /
+' cos ϕ < 0
$ *1 + *2 ( ϕ 3)4
) ' ( *2
7
cos ϕ = − √ ≈ −0, 764 =⇒ ϕ ≈ 2, 44
2 21
cos ϕ = −
7
√ ,
2 21
ϕ ≈ 2, 44
ABCD O
AB = a AD = b.
!" # CD CB CO BD
CD = −a |CD| = |AB| = |a| CD ↑↓ a CB = −b
1
1
1
CO = CA = (−AC) = − (a + b) AC = a + b
2
2
2
O CA = −AC BD = BC + CD =
= AD + CD = b − a
B
C
_
a
A
_
b
O
D
AC = b
BC AN KO
ABC O
AB = a
a b
B
A
M
N
a O
A
C
b K
BC = BA + AC = −a + b = b − a AN = AB + BN =
1
1
1
1
1
1
1
= a + BC = a + (b − a) = a + b − a = a + b = (a + b)
2
2
2
2
2
2
2
ABC
1
1
ABA C AN = AA = (a + b)
2
2
KO !
1
1
2:1,
! " # KO = |KB| KO = KB =
3
3
1
1 1
1 1
1
1
1
1
= (KA + AB) = (− AC + a) = (− b + a) = (a − b) = a − b
3
3 2
3 2
3
2
3
6
$ % ! CN , BO, CO, OM
a−b
b − 2a
a − 2b
a − 2b
BO =
CO =
OM =
& CN =
2
3
3
6
O
AB = a AF = b. !
" " # ! $% & DE OB OC AD BC CF
' # !
B
C
_
a
A
D
_
b
O
F
E
( & DE = −a OB = −b OC = a AD = 2(a + b) BC = a + b
CF = −2a
ABC CA = a CB = b M N
AB ! ) CM *
B
N
M
A
b
a
C
1
1
AB = b−a AM = AB = (b−a) CM = CA+AM =
3
3
1
2a + b
= a + (b − a) =
3
3
AC = c
ABC AM
) AM
+ , AB
= b
|BM|
|BM |
|AB|
|b|
|b|
⇒
,
=
=
=
|c|
|MC|
|AC|
|BC|
|b| + |c|
|BC| = |BM | + |MC|
BC = c − b
|b|
(c − b) ⇒,
|b| + |c|
BM =
⇒ AM = AB + BM = b +
|b|
|b| · c + |c| · b
(c − b) =
.
|b| + |c|
|b| + |c|
B
b
A
M
C
c
! " ABC r1 r2 r3
r M
O
B
_
r2
_
r
_
O
_
r1
M
D
r3
A
C
1
1
BC = r3 − r2 BD = BC = (r3 − r2 )
2
2
r3 − r2
1
+r 2 −r1 = (r3 +r2 −2r1 )
BC AB = r2 −r 1 AD = BD +AB =
2
2
2
1
1
AM = AD = (r3 + r2 − 2r1 ) ⇒ r = OM = r1 + AM = r1 + (r3 + r2 −
3
3
3
1
− 2r1 ) = (r1 + r2 + r3 )
3
a = 3i + 4j + 5k
√
√
32 + 42 + 52 = 50 = 5 2,
√
√
√
3 2
4 2
5 2
3
; cos β =
; cos γ =
.
cos α = √ =
10
10
10
5 2
|a| =
√
ABCD
A(3; 2; −2) B(4; 4; 1) C(−1; 2; 0) D(−3; −2; −6)
! " # $%
% ! # &'( (
) %#!
a = i + 3j − k
√
|a| = 12 + 32 + (−1)2 = 11
! '! * & $ +
'! a '
ea =
a
1
3
1
= √ i + √ j − √ k.
|a|
11
11
11
) '! '! '! a,
−ea
1
3
1
−ea = − √ i − √ j + √ k.
11
11
11
, a · b
a = 2i − 3j + k b = −i + j + 3k
+
a · b = 2 · (−1) + (−3) · 1 + 1 · 3 = −2.
a · b < 0 a b
a − b l
|a| = 2 |b| = 1 a b
l
π/3
π/4
√
l a = |a| · cos(π/3) = 2 · (1/2)√= 1 l b = |b| ·
2/2 l (a − b) = l a − l b = 1 − 2/2
m a(2; 3; 5)
· cos(π/4) =
b(−2; 1; m)
! " ! # $ !"
%
1
a · b = 0 ⇒ 2 · (−2) + 3 · 1 + 5 · m = 0 ⇒ m = .
5
& (3a − 2b) · (2a + 3b) |a| = 1 |b| = 2
a ⊥ b
a ⊥ b ⇒ a · b = 0 a2 = |a|2 = 1 b = |b|2 = 4 '
2
2
2
(3a − 2b)(2a + 3b) = 6a − 6b + 9ab − 4ab = 6 · 1 − 6 · 4 = −18.
( a = (−1; 0; 3)
b = (2; 1; 0)
) ! ! !
*
√
ab
2
2
−1 · 2 + 0 + 0
√
=−
.
cos ϕ =
√
=−
=
5
|a||b|
50
(−1)2 + 32 22 + 12
6 √ 7
√
2
2
ϕ = arccos −
= π − arccos
.
5
5
+ a b
|a| = 2 |b| = 3
(2a − b) ⊥ (a + 2b)
ABC M BC
AB = a AC = c AM
ABC r1 = i +
+2j + 3k r2 = 3i + 2j + k r3 = i + 4j + k
ABC ! " #
A(1; −5; 3)
B AB = i − j + 5k
" $
%&' " %(
2a − 3b a = −i + 2k b = 2i + j − 2k
" %&'
) " # ) AB A(2; −3; 1)
B(1; −1; −1)
O ! " % (
ABC AO + BO + CO = 0
" $ & 2a − 3b # l
a b !
* +&' #& l , 2π/3 3π/4
" $ & - b(3a−2b) a = 2i+j−3k
b = j − k . " #+ " $ / ( 0
1" ! " , %
& AB CD A(1; 2; 1) B(−2; 0; 2) C(−3; 1; 2) D(2; 5; 0)
- a+b(2a − 3b) a = (1; −2; −3)
b = (−3; 2; 1)
- ( + " ( s (2a−b) ⊥
a = (0,5; 2; 5) b = (s; 2; −1)
⊥ (a + b)
, ( ( AC BD %
) " ( A B C D
, B ABC % )
" ( 2 A B C
a, b, c
a, b, c c
b ! a
" !
# ! $ a ! b
% a, b, c ! ! !
& a, b, c
! '
! "
!
( !
a ! b c
)
*% c ! )
c ⊥ a c ⊥ b+
,% ! c - !)
|c| = |a| · |b| · sin (a;
b);
.% a, b, c ! !
!
a b
a × b
[a, b]
a
!
"#
a × b $
b
_
b
_ _
a xb
_
a
a × b = −b × a % &
' $
# a×b b×a
( |a| · |b| · sin (a;
b) = |b| · |a| · sin (b;
a)
)# $
λ(a × b) = (λa) × b %" &
' ) * $ λ > 0(
λ(a × b)
)
a × b (λa) × b λ > 0
) '# *
) ( |λ(a × b)| =
= |λ| · |a × b| = λ · |a| · |b| · sin (a;
b), |(λa) × b| = |λa| · |b| · sin (a;
b) =
= λ|a| · |b| · sin (a; b)
+, $ $ $ λ < 0
- a × (b + c) = a × b + a × c % &
.
/ 0
a b,
|a| = |0,
|b| = 0
⇐⇒ a × b = 0.
! a b ϕ = (a;
b) = 0◦ =⇒ sin ϕ = 0
a×b = 0 |a×b| = 0 " a×b = 0 |a|·|b|·sin (a;
b) = 0 =⇒
#! $ % |a| = |0 |b| = 0
8b) = 0 (a;
sin(a;
b) = 0 (a;
b) = π a b
& ' ! a × a = 0
( ! ! )
* +!
, -
# .&/
_
b
_
a
! |a × b| = |a| · |b| · sin (a;
b)
0!
- ! ! 1
. (2i+ j − k) × j − i × (j − 2k)
2 1 % !
(2i+j −k)×j−i×(j−2k) = 2i×j +j ×j −k×j −i×j +2i×k =
= i × j + 2i × k + j × k
# i × j |i × j| = |i| · |j| · sin 90◦ = 1
k .&3 i × j = k 4
i × k = −j j × k = i 5
i × j + 2i × k + j × k = k − 2j + i = i − 2j + k
6 i − 2j + k
_
k
_
j
_
i
i j j
i×i=j×j =k×k = 0
i × j = k,
j × i = −k,
j × k = i,
k × i = j,
k × j = −i,
i × k = −j.
!
"#$%
a × b = (ax · i + ay · j + az · k) × (bx · i + by · j + bz · k) =
= ax by · i × j + ax bz · i × k + ay bx · j × i + ay bz · j × k+
+ az bx · k × i + az by · k × j = ax by · k − ax bz · j − ay bx · k+
+ ay bz · i + az bx · j − az by · i = (ay bz − az by )i−
− (ax bz − az bx )j + (ax by − ay bx )k.
&
ay az
ax az
ax ay
· k.
·i−
·j+
a×b=
by bz
bx bz
bx by
'"#$()
!
" # ! $%&
' $ ()*
L
c
d=a b
ϕ
_
b
_
a
+ # ' d = a × b ,
,
a b
- |a × b| = S .
! (a b c) = (a × b) · c = |a × b| · |c| · cos ϕ
π
ϕ / d c 0 ()* $/ ϕ <
2
+ ' h = |c| cos ϕ
1 #- (a b c) = |a × b| · |c| · cos ϕ = S · h = V $%& ,
π
$/& ()* 2 ϕ >
2
h = −|c| · cos ϕ cos ϕ < 0 a · b · c = −V 1 # V = |(a b c)|
* 3
A(2; −1; −1) B(5; −1; 2) C(3; 0; 3)
D(6; 0; −1)
4 ! - 4 DA = (−4; −1; 0) DB =
(−1; −1; 3) DC = (−3; 0; −2) . ! # ,
$%& $ DA DB DC !#
#! $%& ' $-
a b c
(a × b) × c (a × b) × c =
= a × (b × c) = a × b × c a × b × c = c × a × b = b × c × a a × b × c =
= −b× a × c = −c× b × a = −a× c × b = ! " #
$ i × j × k % i × j = k ! &
'( )# i × j × k = 0 *
&
a × b × c × d
$ + ,
'( )
a = 2i − 3j + k b = −i + j + 3k
-
a×b
$ . !'( /#
i
j
a × b = 2 −3
−1 1
k
1
3
= i −3 1
1
3
2
1
−3
−j 2
+
k
−1 3
−1 1 =
= i(−9 − 1) − j(6 + 1) + k(2 − 3) = −10i − 7j − k.
j × i + 3j × k − 5k × i + (3i + 5j − k) × (i − 6j + 5k).
i×j = k i×k = −j j ×k = i
! "
−i × j + 3j × k + 5i × k + 3i × i − 18i × j + 15i × k + 5j × i − 30j × j+
+25j × k − k × i + 6k · j − 5k · k = −k + 3i − 5j + 3 · 0 − 18k − 15j − 5k−
−30 · 0 + 25i − j − 6i − 5 · 0 = 22i − 21j − 24k.
A(1; 1; 1) B(1; 2; 3) C(−1; 2; 1)
ABC
ABC
AB AC
!
1
SABC = AB × AC .
2
"# $ AB AC
AB(0; 1; 2), AC(−2; 1; 0),
i
j k
1 2
0
0
2
1
AB × AC = 0
−j
+k
=
1 2 = i
1 0
−2 0
−2 1
−2 1 0
= −2i − 4j + 2k.
%$& ! ' ( )*+
√
√
AB × AC = (−2)2 + (−4)2 + 22 = 24 = 2 6.
SABC =
, &
√
1 √
· 2 6 = 6
2
-
p − q 2p + q p q
60◦
"# &
(p − q) × (2p + q) = 2p × p + p × q − 2q × p − q × q =
= 2 · 0 + p × q + 2p × q − 0 = 3p × q.
. & # $
&
S = |3p × q| = 3|p| · |q| · sin 60◦ = 3 · 1 · 1 ·
√
√
3 3
3
=
.
2
2
CD ΔABC !"
/
CD = h
AB
1
SΔABC = |AB| · h
2
√
!" SΔABC = 6!
# AB = (0; 1; 2)
|AB| =
√
1√
6=
5h
2
√
12 + 22 =
√
5,
$
√
√
2 30
2 6
.
h= √ =
5
5
!%
a = (2; −1; 5) b = (−2; 3; 0)
a
b, & $
i
j
k
−1 5
a × b = 2
−1 5 = i
3
0
−2 3
0
2
5
−1
−j 2
+
k
−2 0
−2 3 =
= −15i − 10j + 4k !
' (
a×b
−15i − 10j + 4k
15
10
4
= √
= −√
i− √
j+√
k.
|a × b|
152 + 102 + 42
341
341
341
e ) &
−e!
e=
#
)
!*
a = (1; −1; 2) b = (0; 1; 2) c = (2; 0; 1)
+ !,- $
1 −1 2
1 2
2 = 1 ·
(abc) = 0 1
0 1
2 0
1
+1· 0 2
2 1
= 1 − 4 − 4 = −7.
+2· 0 1
2 0
=
A(2; 2; 2), B(4; 3; 3), C(4; 5; 4), D(5; 5; 6)
!" #
V $ " % & !!!
V $ ' AB AC AD ("$! V =
1
= S H V = S H S S H ) $$ !
3
* $+ # !!! ' *+ #$ ,$!
$ S = 21 S -!* ! !
!* !!! . ! V = 16 V /#$!
# $!#' AB = (2; 1; 1) AC = (2; 3; 2)
AD = (3; 3, 4) " $
2 1 1
(AB AC AD) = 2 3 2
3 3 4
=2· 3 2
3 4
−1· 2 2
3 4
+1· 2 3
3 3
=
= 2 · 6 − 1 · 2 − 1 · 3 = 7.
! V = 61 V = 16 |AB AC CD| = 76
A
0 #$ # #$ !
!! " H 1 !# V = S H (!+ '2+
S !* $+ !!! #$!
BC ×BD /#+ 3 # !
! BC = (0; 2; 1), BD = (1; 2; 3)
i j k
BC × BD = 0 2 1
1 2 3
=i· 2 1
2 3
−j· 0 1
1 3
= 4i + j − 2k.
√
S = |BC × BD| = 42 + 12 + (−2)2 = 21.
√
7
7 = 21H ! H = √ .
21
+k· 0 2
1 2
4 +
=
N
.
Mo ( x o ; yo ; z o)
M(x;y;z)
−y0 )j+(z −z0 )k
r = OM r0 = OM0
M
M0 M = r − r0
!
M0
(r − r0 ) · N = 0.
$ %
"#
&
'( &
)
*
"
%
A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0.
& M
+(
"
M0 M ⊥ N
"#
M0 (1; 2; 3) N = (2; −1; 1)
,
-
"
$ )
A = 2 B = −1 C = 1 .
2(x − 1) − 1(y − 2) + 1(z − 3) = 0
2x − y + z − 3 = 0
/
2x − y + z − 3 = 0
"# )
r · N + D = 0,
0
"
D = −r0 · N
$
'
%
(
()
#
Ax + By + Cz + D = 0.
"
A = 0
A B C
%
# $
!
&'
D
+ B(y − 0) + C(z − 0) = 0.
A x+
A
(
N = (A; B; C)
)
!
*
M0 −
D
; 0; 0
A
+
! " # " $
#
! %
,)
% *
'
*
M1 (x1 ; y1 ; z1 ) M2 (x2 ; y2 ; z2 ) M3 (x3 ; y3 ; z3 )
+-+
./ r1 , r2 , r3 , r 0 ! )
M(x; y; z)
M1
M
r
- r1 r M2
2
* !
M3
r3
O
,
= M1 M2 × M1 M3
+
)/
)
'
(r − r1 )((r2 − r1 ) × (r3 − r1 )) = 0.
,
N =
!
((r − r1 )(r3 − r1 )(r2 − r1 )) = 0.
!"
# $% & ' " (
) &
x − x1 y − y1 z − z1
x2 − x1 y2 − y1 z2 − z1
x3 − x1 y3 − y1 z3 − z1
= 0.
!
!*"
+ !" && , ' A B C
%-' )
. && , D = 0 (
' O(0; 0; 0) % %
Ax + By + Cz = 0.
. && , A = 0 ')
Ox
N = (0; B; C; ) i = (1; 0; 0)"
By + Cz + D = 0 Ox
. && , ' A = 0 D = 0 By + Cz = 0
( Ox Ox (
. && , ' A = 0 B = 0 ')
Oz
N = (0; 0; C) k = (0; 0; 1)
Cz + D = 0 Oxy
Oz
. && , ' A = B = D = 0
z = 0 ) % Oxy
( "
Oxy
ϕ
α1
α2
ϕ
_
N2
_
N1
α1 : A1 x + B1 y + C1 z + D1 = 0,
α2 : A2 x + B2 y + C2 z + D2 = 0,
N 1 = (A1 ; B1 ; C1 )
N 2 = (A2 ; B2 ; C2 )
A 1 A 2 + B1 B2 + C 1 C 2
N1 · N2
cos ϕ = = 2
.
N 1 · N 2
A1 + B12 + C12 · A22 + B22 + C22
+1=0
x − y + 3z = 0
! "
2x + y − z +
#
2 · 1 + 1 · (−1) − 1 · 3
2
⇒
cos ϕ =
= −√
2
2
2
2
2
2
66
2 + 1 + (−1) · 1 + (−1) + 3
2
⇒ ϕ = π − arccos( √ ) ≈ 1, 82.
66
2
$ cos ϕ = − √ .
66
! "" " #!$
#!
%
" " !
& "
"
"
! "" " ' !$
A1 · A2 + B1 · B2 + C1 · C2 = 0.
' !
2x−3y+5 = 0
mx + 7y − 6 = 0
A1
B1
C1
=
=
,
A2
B2
C2
0
= .
( ) $ % " #!$ m2 = −3
7
0
14
=⇒ m = − % *
() m2 = −3
7
3
2x − 3y + 5 = 0 − 143 x + 7y − 6 = 0 +
2x − 3y + 5 = − 143 x + 7y − 6 "" " ,
- $ m = − 143
% M0(x0; y0; z0) α : Ax + By + Cz +
, " d M0 α
" M0 M1 α '. ! " "
/ $
|Axo + Byo + Czo + D|
√
d=
.
''!
2
2
2
+D = 0
A +B +C
M0
_
N
M1
α
N
α : N = (A; B; C) M1 (x1 ; y1 ; z1 ) d = M1 M0
M1M0 M1M0 = (x0 − x1)ı + (y0 − y1)j + (z0 −
−z1 )k N M1 M0 ϕ = 0 ϕ = π !
" "
N · M1 M0 = N · M1 M0 · cos ϕ = ± N · M1 M0 .
' ( &)%)
N · M1 M0 = A(x0 − x1 ) + B(y0 − y1 ) + C(z0 − z1 ).
#$%&
#$%#
* M1 ∈ α + Ax1 + By1 + Cz1 + D = 0 =⇒ −Ax1 −
− By1 − Cz1 = D N · M1 M0 = Ax0 + By
0+
1 − Cz1 =
√ Cz0 − Ax1 − By
= Ax0 +By0 +Cz0 +D , " N = A2 + B 2 + C 2 , M1 M0 = d
( #$%& #$%#
√
± A2 + B 2 + C 2 · d = Ax0 + By0 + Cz0 + D,
d=±
Ax0 + By0 + Cz0 + D
√
A2 + B 2 + C 2
d ≥ 0 ( #$%% - ( (
"" " . #$#
d=
r 0 · N + D
|N |
.
#$%/
0 1 ! #$/
/2
Ax + By + C = 0,
A2 + B 2 = 0
N = (A; B)
!
A1 x+B1 y+C1 = 0 A2 x+B2 y+C2 = 0
!
"
#
N1 · N2
A 1 A 2 + B1 B2
cos ϕ = = 2
.
N 1 · N 2
A1 + B12 · A22 + B22
$%&'()
*"
" +
" " N1 N2 #
B1
C1
A1
=
=
,
A2
B2
C2
"
$%&',)
N1 ⊥N2 #
A1 · A2 + B1 · B2 = 0.
$%&'-)
. M0 (x0 ; y0 ) Ax + By + C = 0
Oxy / $%&'0)#
d=
|Ax0 + By0 + C|
√
.
A2 + B 2
$%&'0)
1 " /
"
%&'
2x − 3y + z − 1 = 0
M1 (1, 0, −1)
2 +
M2 (1, 1, 1)
α
:
.2
# 3 M1 +
# 2 − 1 − 1 = 0 M1 ∈ α 1 M2
! # 2 − 3 + 1 − 1 = 0 M2 ∈
/ α
%&4
.2
# 3 /
$%&'') #
|2 − 3 + 1 − 1|
1
d=
=√ .
2
2
2
14
2 + (−3) + 1
M2
α
M0(2; 1; 1) 2x − 2y + 2z + 1 = 0
! " ! M0 # ! $ %
&
N = (2, −2, 2) '
& ( !
2(x − 2) − 2(y − 2) + 2(z − 1) = 0.
x − y + z − 2 = 0.
)
M0(2; 0; 3) N = (2; 2; −2)
*
!
2(x − 2) + 2(y − 0) − 2(z − 3) = 0 + x + y − z + 1 = 0
,
O(0; 0; 0) M1(−4; 2; −1) M2(−2; −4; 3)
* - !
x
x−0
y
z
y
−
0
z − 0
−4 − 0 2 − 0
−1 = 0 ⇔
−1 − 0 = 0 ⇔ −4 2
−2 −4 3
−2 − 0 −4 − 0 3 − 0
2
−4 −1
−1
+ z −4 2 = 0 ⇔
⇔ x ·
−
y
·
−4 3
−2 3
−2 −4
⇔ x(6 − 4) − y(−12 − 2) + z(16 + 4) = 0 ⇔ x + 7y + 10z = 0.
M1 (2; 0; −1) M2 (1; −1; 3)
3x + 2y − z − 5 = 0
*
! " ! # ! $ '
% N & M1 M2
N1 = (3; 2; −1)
M1 M2 × N1 . &
M1 M2 = (−1, −1, 4) &
i
N = M1 M2 ×N1 = −1
3
= −7i + 11j + k
j
−1
2
k
4
−1
= i· −1
2
−1
4
−j·
−1
3
−1 −1
4
+k·
−1
3
2
=
M1
N
−7(x − 2) + 11(y − 0) + 1(z + 1) = 0 − 7x + 11y + z + 15 = 0.
α1
α2 : 2x + y = 0
!"
# N1 = (1, −1, 2) N2 = (2, 1, 0)
$ % &'
x − y + 2z − 3 = 0
cos ϕ =
N1 N2
1
1 · 2 + (−1) · 1 + 2 · 0
=√ .
√
=
|N1 ||N2 |
30
12 + (−1)2 + 22 · 22 + 12 + 02
&
M0(0, 2, 1) a = i + j + k b = i + j − k
! " ( )
i j k
N = a × b = 1 1 1
1 1 −1
= −2i + 2j
(
=i· 1 1
1 −1
−j· 1 1
1 −1
+k· 1 1
1 1
=
$ % *'
x − y + 2 = 0
+
AB
A(−7; 2; −2), B(3; 4; 10)
!"
, Mo - AB
. $ %*& ' xo = −72+ 3 = −2 yo = 2 +2 4 = 3
−2 + 10
= 4 ( ) /
zo =
2
AB = (10; 2; 12)
10(x+2)+2(y−3)+12(z−4)
5x + y + 6z − 17 = 0
0 !
α1 : 2x + y − z + 3 = 0 α2 : −2x − y + z − 5 = 0
−2(x − 0) + 2(y − 2) = 0
M α1
α2
M xo = 0 yo = 0 zo = 3 M(0, 0, 3)
! "#$%%&
|3 − 5|
2
d=
=√ .
2
2
2
6
(−2) + (−1) + 1
#$%% M
AB 1 : 2 x − 3y + z − 6 = 0
A(2, 0, 1) B(−1, 3, 1)
#$%' A AB
A(−1, 2, −3) B(0, 2, 4)
#$%#
M1 (2, −5, 0) M2 (6, 0, 2) x + 5y +
+2z − 10 = 0
( )
*+ , -
-
- . /
" %01& "#%%& ! "#%'&
N1 · r + D1 = 0,
N2 · r + D2 = 0.
A1 x + B1 y + C1 z + D1 = 0,
A2 x + B2 y + C2 z + D2 = 0.
"#%%&
"#%'&
3
2 3
N 1 = (A1; B1; C1), N 2 = (A2; B2; C2), N 1
"#%%& "#%'& . . .
∦ N 2,
α1
l
α2
N1 N 2
__
N2
__
N1
M0 (x0 ; y0 ; z0 )
s = (m; n; p),
s
!
M(x; y; z) l " #$% &
OM = OM0 + M0 M . ' M0 M s =⇒ M0 M = s · t, (
t ∈ (−∞; +∞). ) M0 M
r0 = OM0 r = OM , &
r = r0 + t · s.
" %
z
_
s
M
l
M0
ro
r
0
y
x
M ∈ l.
t
r
*
r = OM = xi + yj + zk, r0 = OM0 = x0 i + y0 j + z0 k, ts =
= tmi + tnj + tpk.
⎧
⎨ x = x0 + tm,
y = y0 + tn,
⎩
z = z0 + tp.
M0 ! s. "
t x; y; z M(x; y; z)
! t ! "# $
t=
! &' &
y − y0
z − z0
x − x0
, t=
, t=
.
m
n
p
x − x0
y − y0
z − z0
=
=
,
m
n
p
%
#
( ) % * $
⎧
x − x1
y − y1
⎪
⎨
=
,
m
n
y
−
y
z
−
z
1
1
⎪
=
.
⎩
n
p
z − z1
x − x1
=
m
p
+
, *
-
, . ! ) % ! & & $
) ! +)
. & . ! * Oz, /
Ox 0 1 0
x − x1
y − y1
z − z1
=
=
,
0
n
p
s = (0; n; p).
s
x = x1 .
y = y1 ).
x − x1
y − y1
z − z1
=
=
0
0
p
Oz, s ⊥ Ox
s ⊥ Ox
s ⊥ Oy (x = x1,
4x − y − z + 12 = 0,
y − z − 2 = 0.
! " # $ %#
$ M0 &'$ s $
#( M0 $ ) *%
) + + z = 0, $ ) *
4x − y + 12 = 0,
y − 2 = 0,
⇔
5
x0 = − ,
2
y0 = 2.
,
- % ) M0(− 52 ; 2; 0) ∈ l $
#( &' s ) +
) # N 1 = (4; −1; −1) N 2 = (0; 1; −1)
# # $ ( &'
$ # + s l ./
s = N 1 × N 2 , s ⊥ N 1 , s ⊥ N 2 .
i j
k
0 s = 4 −1 −1
0 1 −1
= i(1 + 1) − j(−4) + k(4) = 2i + 4j + 4k.
1 + $ &
5
2 = y − 2 = z.
2
4
4
x+
5
2 = y−2 = z
2
4
4
x+
5
x+
2 = y − 2 = z = t,
t :
2
4
4
⎧
5
⎪
x
+
⎪
⎧
⎪
5
⎪
2 =t,
⎪
⎪
⎨
⎨ x = − + 2t,
2
2
⇔
y−2
y = 2 + 4t,
⎪
⎪
= t,
⎪
⎩
⎪
4
⎪
z = 4t.
z
⎪
⎩
= t.
4
⎧
5
⎪
⎨ x = − + 2t,
2
y = 2 + 4t,
⎪
⎩
z = 4t.
! " t
⎧
5
⎪
x+
⎧
⎪
⎪
5
⎪
5
2,
⎪
⎪
⎨ x = − + 2t,
⎨ t=
x+
y−2
z
2
2
2
y = 2 + 4t, ⇐⇒ ⎪ t = y − 2 , ⇐⇒ 2 = 4 = 4 .
⎪
⎩
⎪
⎪
4z
⎪
z = 4t.
⎪
⎩
t= .
4
⎧
5
⎪
x
+
⎪
⎨
2 = y − 2,
2x + 5 = y − 2,
2x − y + 7 = 0,
⇐⇒
⇐⇒
2
4
y
−
2
=
z.
y − z − 2 = 0.
⎪
z
y−2
⎪
⎩
= .
4
4
2x − y + 7 = 0,
y − z − 2 = 0.
!"#
$ %
!"# &
!'# (
! !
M0 (− 52 ; 2; 0) M1 (−1; 5; 3)
!"# !$# %
!"# !$# M0M1 .
% & ' ' &( '
M0 (x0; y0; z0) M1 (x1; y1; z1),
(
s = M0 M1 = (x1 − x0 )i + (y1 − y0 )j + (z1 − z0 )k. )
' &( '
x − x0
y − y0
z − z0
=
=
.
!!*#
x1 − x0
y1 − y0
z1 − z0
!+
M0 (− 52 ; 2; 0) M1(−1; 5; 3).
, - . / !!*#
.
5
5
x+
2 = y − 2 = z ⇐⇒
2 = y − 2 = z.
3
5
5−2
3
3
3
−1 +
2
2
x+
% (& ' & 0&
⎧ '.
5
⎪
x+
⎪
⎪
⎪
2 = y − 2,
⎨
3
3
⇐⇒
⎪
2
⎪
⎪
y−2
z
⎪
⎩
= ,
3
3
2x + 5 = y − 2,
⇐⇒
y − 2 = z,
2x − y + 7 = 0,
y − z − 2 = 0.
1 !$#
2 .
x − x1
y − y1
z − z1
=
=
,
m1
n1
p1
x − x2
y − y2
z − z2
l2 :
=
=
.
m2
n2
p2
l1 :
!!!#
s1 = (m1; n1; p1) s2 = (m2 ; n2; p2) :
cos ϕ =
s1 · s2
m1 m2 + n1 n2 + p1 p2
= 2
.
|s1 | · |s2 |
m1 + n21 + p21 m22 + n22 + p22
! "
m1
n1
p1
=
=
m
n
p
2
2
2
# $ %
& & ' $
&
"
m1 · m2 + n1 · n2 + p1 · p2 = 0.
(
%
l α"
y − y0
z − z0
x − x0
=
=
,
m
n
p
α : Ax + By + Cz + D = 0.
)
*
+& P (x1; y1; z1 ) & l
α# ' , & & -
( # ' x, y, z * #
& ' t1 #
) * "
l:
A(x0 + mt) + B(y0 + nt) + C(z0 + pt) + D = 0 =⇒
Ax0 + By0 + Cz0 + D
.
Am + Bn + Cp
.
% ' $ ' - ( #
"
x1 = x0 + t1 m, y1 = y0 + t1 n, z = z0 + t1 p.
/
=⇒ t1 = −
!" "
#
r = r0 + t · s,
r · N + D = 0.
$ r %
#
t1
t1 = −
r0 · N + D
.
N ·s
&% ϕ '
( ) %
% ' ! s
( N
#
cos(N ; s) = cos(90◦ − ϕ) = sin ϕ.
N
l
S
α
ϕ
p
* %
+,+-
#
Am + Bn + Cp
cos(N ; s) = √
.
A2 + B 2 + C 2 m2 + n2 + p2
sin ϕ = √
A2
Am + Bn + Cp
.
+ B 2 + C 2 m2 + n2 + p2
!"#
s # N
Am + Bn + Cm = 0
$%
& '
# (
!"# s #
N
m
n
p
=
= .
$
A
B
C
)
l,
l
!"
' * +"
A1 x + B1 y + C1 z + D1 = 0,
A2 x + B2 y + C2 z + D2 = 0.
$$
,& ( "( *
! $$
(A1 x + B1 y + C1 z + D1 ) + λ(A2 x + B2 y + C2 z + D2 ) = 0,
$
# λ !+- - * −∞
+∞.
#
$ !+ *
λ !
%) # A = A1 + λA2 , B = B1 + λB2, C = C1 + λC2, D = D1 + λD2
. ( * ! $$ * x; y; z
!" $$ +"! &
$ -& +( - !
! M0 (x0; y0; z0) ∈/ α2, A2x0 + B2 y0 +
+C2 z0 + D2 = 0 "
λ0
M0 !
α2 A2 x + B2 y + C2 z + D2 = 0)
A1 x0 + B1 y0 + C1 z0 + D1 + λ0 (A2 x0 + B2 y0 + C2 z0 + D2 ) = 0 =⇒
=⇒ λ0 =
A1 x0 + B1 y0 + C1 z0 + D1
.
A2 x0 + B2 y0 + C2 z0 + D2
# λ = λ0
$ %& " !
! M0 λ0 '
$( ) M0 , '
& $" " M0 ∈/ α2 ), !
! % "* ( '
&!& '
% %
# ! M1 (x1; y1; z1) M2(x2; y2; z2 ) '
!& S 1(m1 ; n1; p1) S 2(m2; n2; p2) S1 ∦ S2
+ "* h
'
( % M1 M2 S 1 S 2
! '
$ & % ,-
S
M2 1
S2
M1
S1
S2
z = 0 Oxy
z = 0
l
M0 (0; −1; 2), M1 (1; 1; 1)
l:
x + y − z + 3 = 0,
2x − y − 1 = 0.
! M0
−1 − 2 + 3 = 0,
M0 ∈ l.
1 − 1 = 0,
" M1
−1 − 1 + 3 = 0,
2 − 1 − 1 = 0,
M1 ∈/ l.
" #
$% " & x0 = 0
y0 = −1 z0 = 2 ' M0 ( ) #
$%* +
, & s = N1 × N2 * N1 = (1, 1, −1) N2 = (2, −1, 0)
i j
s = N1 × N2 = 1 1
2 −1
1
+k
2
-
k
1
1 −1
−1
−
j
−1 = i
2 0
−1
0
0
1
= −i − 2j − 3k.
−1
x
y+1
z−2
=
=
−1
−2
−3
+
$
x
y+1
z−2
=
=
.
1
2
3
.
- %
t
x
y+1
z−2
=
=
= t.
1
2
3
⎧ x
⎪
⎧
= t,
⎪
⎪
⎨ y1 + 1
⎨ x = t,
= t, ⇔
y = −1 + 2t,
2
⎪
⎩ z = 2 + 3t.
⎪
⎪
⎩ z − 2 = t;
3
l1 :
y−1
z+2
x
=
=
,
2
3
2
l1 l2
l2 :
y+1
z−1
x
=
=
.
−3
1
0
s1 = (2, 3, 2) s2 = (−3, 1, 0) !" #
cos ϕ =
s1 · s2
2 · (−3) + 3 · 1 + 2 · 0
3
.
=√
= −√
2
2
2
2
2
|s1 ||s2 |
170
2 + 3 + 2 · (−3) + 1
$
l:
y+1
z−1
x
=
=
;
2
3
1
l α
α:
3x − y + 2z − 1 = 0.
% &
' s = (2, 3, 1) (% N = (3, −1, 2)
' )"
s·N
2 · 3 + 3 · (−1) + 1 · 2
5
=√
= ,
14
|s||N |
22 + 32 + 12 · 32 + (−1)2 + 22
5
.
ϕ = arcsin
14
sin ϕ =
* l
+ ' l , $
#
α
⎧ x
⎧
⎪
= t,
⎪
⎪
⎨ x = 2t,
⎨ y2 + 1
x
y+1
z−1
= t, ⇔
y = −1 + 3t,
=
=
=t⇔
⎩
⎪
3
2
3
1
⎪
z = 1 + t.
⎪
⎩ z − 1 = t;
1
t
2
3 · 2t − (−1 + 3t) + 2(1 + t) − 1 = 0 ⇒ t = − .
5
⎧
2
4
⎪
⎪
x
=
2
−
=− ,
⎪
⎪
⎪
5 5
⎪
⎨
11
2
=− ,
y = −1 + 3 −
⎪
5
5
⎪
⎪
⎪
3
2
⎪
⎪
= .
⎩ z =1+ −
5
5
4 11 3
M − 5 ; − 5 ; 5
!" #
x2 = y −3 1 = z1 M(1, 1, 2).
$ % & %' ( M0
M1
M, M0 , M1 ) M0 ( '
M0 (0, 1, 0) *
& M1 %
⎧ x
⎪
⎧
= t,
⎪
⎪
⎨ 2
⎨ x = 2t,
x
y−1
z
y−1
y = 1 + 3t,
=
= =t⇔
= t, ⇔
⎪
⎩ z = t.
2
3
1
⎪
⎪ z 3
⎩
= t;
1
(' t ∈/ 0, t = 1
M1 x1 = 2 y1 = 4 z1 = 1 M1(2, 4, 1)
+, M M1 M2
-!. /0
x − 1 y − 1 z − 2,
0 − 1 1 − 1 0 − 2,
2 − 1 4 − 1 1 − 2,
x − 1 y − 1 z − 2,
= 0 ⇔ −1
0
−2,
1
3
−1.
=0⇔
0 −2
⇔ (x − 1)
3 −1
− (y − 1) −1 −2 + (z − 2) −1 0
1
1
−1
3
⇔ 2x − y − z + 1 = 0.
=0⇔
! " "
#$
⎧
⎪
⎨ x = y − 1,
2
3
y−1
z ⇔
⎪
⎩
=
3
1
3x − 2y + 2 = 0,
y − 3z − 1 = 0,
% " $
3x − 2y + 2 + λ(y − 3z − 1) = 0.
& λ " #
'" M :
M(1; 1; 2)
1
3 − 2 + 2 + λ(1 − 6 − 1) = 0 ⇔ λ = .
2
( " # " # )
λ = 21 $
1
3x − 2y + 2 + (y − 3z − 1) = 0 ⇔ 2x − y − z − 1 = 0.
2
*
ABC,
C(3; 4; 0)
A
A(1; 2; −1) B(1; −2; −2)
+ $ % ,,
A BC $
1 + 3 −2 + 4 −2 + 0
;
;
, M(2; 1; 1).
M
2
2
2
( " AM # " #
" - ./$
"
x−1
y−2
z+1
x−1
y−2
z+1
=
=
⇔
=
=
.
2−1
1−2
−1 + 1
1
−1
0
0 ## 1 -234/$
d=
(2 − 1)2 + (1 − 2)2 + (−1 + 1)2 =
√
2.
x + y + z + 1 = 0,
x − y − z − 1 = 0.
M(2, 1, 3)
x + 2y − z + 5 = 0
l : x −5 2 = y −1 3 = z +2 1
α : x + 4y − 3z + 1 = 0
n
n
! " ! # $%& ' ' !
! ( ) a = (a1 , a2 , a3 ) * ! !
'! & ' ! ' !) a = (a1 , a2 ) +! !"
, " x1 , x2 , x3 - a = (x1 ; x2 )
& a = (x1 ; x2 ; x3 ) ( ./
n !
$ n x1, ..., xn
n!
" n! #
a = (x1 ; ...; xn) $ x1 , ..., xn
- & !0& !"
& ! ' !.) 0 = (0; 0; ...; 0)
- (−x1 ; −x2 ; ...; −xn) 0 ' '!
a = (x1 ; x2 ; ...; xn) 0 −a
$ %
⎧
a x + a12 x2 + ... + a1n xn = b1 ,
⎪
⎪
⎨ 11 1
a21 x1 + a22 x2 + ... + a2n xn = b2 ,
.................................................
⎪
⎪
⎩ a x + a x + ... + a x = b .
m1 1
m2 2
mn n
m
x1 = α1; x2 = α2 ; ...; xn = αn
n x = (α1 ; α2 ; ...; αn)
b1; b2; ...; bm m
b = (b1 ; b2 ; ...; bm)
a = (x1; x2; ...; xm)
b = (y1 ; y2 ; ...; ym) a = b
xi = yi(i = 1, 2, ..., n)
! " #
n
b = (y1 ; y2 ; ...; yn)
a = (x1 ; x2 ; ...; xn)
c = a + b = (x1 + y1 ; x2 + y2 ; ...; xn + yn ),
$ %&
d = a − b = (x1 − y1 ; x2 − y2 ; ...; xn − yn ).
"
a = (x1 ; x2 ; ...; xn)
$ &
λ
λa = {λx1 ; λx2 ; ...; λxn}.
$ &
' n (
!
) n
! "
# " $ "
Rn
(1)
(1)
(1)
(2)
(2)
(2)
* a1 = (x1 ; x2 ; . . . ; xn ), a2 = (x1 ; x2 ; . . . ; xn ), ...,
(k)
(k)
(k)
ak = (x1 ; x2 ; ...; xn )
λ1 ; λ2 ; ...; λk
!
λ1 a1 + λ2 a2 + ... + λk ak = 0.
$ )&
λ1, λ2, ..., λn
a a1 , a2, ..., an
a = λ1 a1 + λ2 a2 + ... + λn an = 0.
λ1 , λ2 , ..., λn e1 =
(1; 0; 0; ...; 0) e2 = (0; 1; 0; ...; 0), ..., en = (0; 0; ...; 0; 1)
a = λ1 a1 +λ2 a2 +...+λnan
!
⎧
x = λ1 · 1 + λ2 · 0 + ... + λ2 · 0,
⎪
⎪
⎨ 1
x2 = λ1 · 0 + λ2 · 1 + ... + λ2 · 0,
.................................................
⎪
⎪
⎩ x = λ · 0 + λ · 0 + ... + λ · 1.
n
1
2
2
"
# λ1 = x1 , λ2 = x2 , ..., λn = xn $
%
Rn
& n% ' a = (x1 ; x2 ; ...; xn) x1 ; x2 ; ...; xn
& &(& ' e1 , e2 , ..., en
"
# n%
& & &
' ) *' ' %
+ # * ' %
, -&! . /
0 & n% ! ,
-( & ' %
& & ' , '
- /1 a = (x1 ; x2 ; ...; xn)
b = (y1 ; y2 ; ...; yn) a · b = (x1 + y1 ; x2 + y2 ; ...; xn + yn ) 2 & 3
# * ' ,
- / & 4! !
5 #
Q
& ' #&
0 & 6 &
*
# x y x1 x2 y1 y2
i j 7 e1 e2
%
8 & &(6
x1 x2
y1 y2
y1 = a11 x1 + a12 x2 ,
y2 = a21 x1 + a22 x2 .
' a11 , a12 , a21 , a22 % &
M Q x1 x2
N
y1 y2
! N
M
M
Q L
λ
!"#$%
Q
" #
$ $ x1 x2 % #
" # Ox1 x2 %
& # '
A=
a11 a12
a21 a22
,
# ((' )
%
y1
x1
* ' %' X = x1 , Y = y1 ,
# $ + ( ,
Y = AX.
!
- & ' A
'
( |A| = aa11 aa12 ) (
21 22
*
( ! ++
% ' ( |A| = 0
|A| = 0
(
. (( % # #
$ x1 x2 . ( ,
y1 a12
y2 a22
a
a
= 22 y1 − 12 y2 .
x1 =
|A|
|A|
a
a
11
12
a21 a22
a11 y1
a21 y2
a
a
= − 21 y1 + 11 y2 .
x2 =
|A|
|A|
a11 a12
a21 a22
⎧
a22
a12
⎪
y1 −
y2 ,
⎨ x1 =
|A|
|A|
a
a
21
11
⎪
y2 .
⎩ x 2 = − y1 +
|A|
|A|
N (y1 ; y2)
M(x1; x2) M,
N ! "" #
Q
"" $ % & % % A'
A−1 =
a22 / |A| − a12 / |A|
−a21 / |A| + a11 / |A|
.
( " &
$ ) % A−1'
! A−1A =
A−1 Y = A−1 AX.
10
= E EX = X
01
A−1Y
= X.
*+
X = A−1 Y.
,
y1 = x1 ,
y2 = x2
% E =
**
10
01
.
y1 = 2x1 + 3x2 ,
y2 = 3x1 + 5x2
*
23
|A| =
35
= 1
A =
23
35
!"!# x1
%
x1 = 5y1 − 3y2 ,
x2 = −3y1 + 2y2 .
A−1 =
x2 $
5 −3
−3
2
y1 = 2 · 1 +
M(1; 2) N
+3 · 2 = 8 y2 = 3 · 1 + 5 · 2 = 13 L : x1 + 2x2 − 2 = 0
λ
x1 + 2x2 − 2 = 0
x1
x2 y1
y2
!" #$
(5y1 − 3y2 ) + 2(−3y1 + 2y2 ) − 2 = 0,
− y1 + y2 − 2 = 0.
& '
(
$ )
A=
23
46
y1 = 2x1 + 3x2 ,
y2 = 4x1 + 6x2
23
,
|A| =
46
* %
)
)
+
!,# ) *
M 2x1 +3x2 = 0
y1 = 2x1 + 3x2 y2 = 4x1 + 6x2 = 2(2x1 + 3x2 ) = 2 · 0 = 0
(
y1 = x1 ,
y2 = −x2
10
0 −1
M(x1 ; x2 ) N
Ox1
M(1; 2) N (1; −2)
A =
OM M(x1; x2) : OM = x1e1 + x2e2
ON
N ! M : ON = y1 e1 + y2 e2 "
x1 , x2 y1 , y2 # $
!% &
' ! ( Ox1x2x3
"% "% " )
⎛
⎛
⎞
⎞
⎞
y1
a11 a12 a13
x1
X = ⎝ x2 ⎠ , Y = ⎝ y2 ⎠ , A = ⎝ a21 a22 a23 ⎠ ,
x3
y3
a31 a32 a33
⎛
% * (|A| = 0) (
* % + ,-)
Y = AX, X = A−1 Y.
&& ( ( !
! ( ( ! ( . (
/ ( Q ( *
e1 e2 0 Ox1 x2
!% Ox1 x2
e1 e2 0 (
,1+
'!2 ( Q ( ! M x1, x2 . 2
% x1, x2 . 0 !
% % 3 #* !
OM
M ( * % ! e1 , e2
e1 , e2 )
Ox1 x2
OM = x1 e1 + x2 e2 , OM = x1 e1 + x2 e2 .
4 !
x1 e1 + x2 e2 = x1 e1 + x2 e2 .
,5
x2
x,1
x’2
.M
e2
,
e2
,
e1
α
0
x1
e1
α11 = e1 e1 ; α12 = e1 e2 ; α21 = e2 e1 ; α22 = e2 e2 .
! " #
x1 = d11 x1 + d12 x2 ,
x2 = d21 x1 + d22 x2 .
$!
% " $! & ' " #
(
L=
d11 d12
d21 d22
)!
* (+
, (# ( X = xx12 X = xx1 . -
2
$! + ' .
X = LX .
L
! "
d d
# L = d d ,
$ L %
L L &$ ' $
L $
T
11
21
12
22
T
d11 d21
d12 d22
LT L =
d11 d12
=
d21 d22
10
=
= E.
01
() *+
, '
- $
! α ! " $ . () */
!$ .
)
0" " Ox x
1 () + Q Y = AX $
M(x ; x ) $ N (y ; y )
- $" X Y $
(X = LX ) 1 " $
.
Y = L ALX .
() *2
3! $ M(x ; x )
$ N (y ; y ) 1
a a
1 A = a a :
Y = A X ! 1
' A = L AL
4
! $ 5
!
d211 + d221 d11 d12 + d21 d22
d12 d11 + d22 d21 d212 + d222
LT
L
LT = L−1 .
=
1 2
1
2
1
2
1
−1
1
2
−1
11
21
12
22
2
x1 x2
! "! # $#
" "! $% " %
&' () $) " $ " x1, x2,
$ "
"
*+ ! ' " %
* a12 = a21 , '
F (x1 , x2 ) = a11 x21 + 2a12 x1 x2 + a22 x22 .
F (x1 , x2 ) = (a11 x1 + a12 x2 )x1 + (a21 x1 + a22 x2 )x2 .
,( A = aa1121 aa1222 ( "
- ( $( X = xx12 %
( " XT = (x1; x2)! !
" %
+ )
F (x1 , x2 ) = XT AX.
.
* ) / ) ! $ (
(
0 " ) x1 x2 "" " "
) " Ox1 x2 1 %
" Ox1 x2 *) " "
+ $
$ .2
x1 = a11 x1 + a12 x2 ,
x2 = a21 x1 + a22 x2 .
)
( $ L =
$ .2 + ) %#
) X =
x1
x2
X = LX .
x1
.
, X =
x2
a11 a21
a12 a22
3
.
x1 x2
! x1 x2"
x1 x2# F (x1, x2)
$ % & ! # %
Ox1 x2 " %
& F (x1, x2)
!
" " % '
(& #
2
2
F (x1 , x2 ) = λ1 x1 + λ2 x2 ,
& !
)
( ! % ! % '
& * + ,' X = (x1 x2)
-+ % " ( #
XT = X T L−1 .
.
$ XT X
! . #
F (x1 , x2 ) = X T (L−1 AL)X .
/ %
, A ( #
A =
Ox1x2 " % '
λ1 0
0 λ2
.
/ 0 + " ,
+ '
$ 0
F (x1, x2) ! 1
2 "
% % ! " '
% , L % ! 1
λ1 0
0 λ2
3 % 0 +
L
2 " ,
L
λ1 0
0 λ2
= L−1 AL.
, L#
= LL−1 AL = EAL = AL.
% !
L
λ1 0
0 λ2
= AL.
λ1 0
0 λ2
=
α11 α12
α21 α22
AL =
=
a11 a12
a21 a22
λ1 0
0 λ2
=
α11 α12
α21 α22
α11 λ1 α12 λ2
α21 λ1 α22 λ2
=
a11 α11 + a12 α21 a11 α12 + a12 α22
a12 α11 + a22 α21 a21 α12 + a22 α22
α11 λ1 α12 λ2
α21 λ1 α22 λ2
=
,
a11 α11 + a12 α21 a11 α12 + a12 α22
a12 α11 + a22 α21 a21 α12 + a22 α22
.
α11 λ1 = a11 α11 + a12 α21 ,
α21 λ1 = a12 α11 + a22 α21 .
α12 λ2 = a11 α12 + a12 α22 ,
α22 λ2 = a21 α12 + a22 α22 .
α11 (a11 − λ1 ) + α21 a12 = 0,
α11 a21 + α21 (a22 − λ1 ) = 0;
α12 (a11 − λ1 ) + α22 a12 = 0,
α12 a21 + α22 (a22 − λ1 ) = 0.
!"" α11
α12 α21 α22 # $ %&'
!# $ ( )
* # '
+ & $ !# ,
a11 − λ1 a12
a21 a22 − λ1
= 0,
a11 − λ2 a12
a21 a22 − λ2
= 0.
λ1 λ2 ) '
a11 − λ a12
-
a21 a22 − λ = 0,
λ2 − (a11 + a22 )λ + (a11 a22 − a12 a21 ) = 0.
.
λ1 , λ2 λ3
a11 − λ a12 a13
a21 a22 − λ a23
a31 a32 a33 − λ
= 0.
2
2
F (x1 , x2 ) = λ1 x1 + λ2 x2
λ1 0
,
0 λ2
!"#$%& Ox1 x2
A
' (
y1 = λ1 x1 + 0 · x2 ,
y2 = 0 · x1 + λ2 x2 .
y1 = λ1 x1 ,
y2 = λ2 x2 .
!"#$")&
*
Ox1 x2 M1 (1; 0)
M2 (0; 1)$ + OM1 = e1 OM2 = e2 $ , !"#$")&
M1 M2 - Q1 (λ1 ; 0)
Q2 (0; λ2 )$ , OQ1 = λ1 e1 OM1 = e1
OQ2 = λ2 e2 OM2 = e2 $
. - !"#$")& e1 e2
λ1 e1 λ2 e2 $
"#$))
r
λ1r r
λ
/ e1 e2
- !"#$")&$
, $
"#$% !
y1 , y2 x1, x2 " "
# $
% N (y1; y2) "
" M(x1; x2)&
α11 (1 − (−2)) + α21 · 3 = 0,
α11 · 3 + α21 (1 − (−2)) = 0;
α12 (1 − 4) + α22 · 3 = 0,
α12 · 3 + α22 (1 − 4) = 0.
α11 = −α21 , α22 = α12 α21 = 1
α11 = −1 α12 = 1 = α22
L=
−1 1
1 1
,
!
x = −x + y ,
y = x + y .
"# $
% &
(y − x )2 + 6(y 2 − x 2 ) + (x + y )2 + 6y − x + 2(x + y ) − 1 = 0 ⇔
⇔ −4x 2 + 8y 2 − 4x + 8y − 1 = 0
' & %
# ( # xy # ) # #
# * ! + #(
& & $ #) ,
&
1
1
2
2
2
2
−4x +8y −4x +8y −1 = 0 ⇔ −4(x +x + )+8(y +y + ) = 2 ⇔
4
4
1
1
(y + )2 (x + )2
1 2
1 2
2
2 = 1.
⇔ −2(x + ) + 4(y + ) = 1 ⇔
−
1
1
2
2
4
2
1
1
- X = x + , Y = y + , . ,
2
2
Y 2 X2
. X, Y # ( 1 − 1 = 1
4
2
' # # ) ) & ,
# # # /* #
,
# & ( # % %
# #
Y2
X2
4
2
0 ( 1 − 1 = 1
xy − 2x − 3y + 6 = 0
1
a11 = 0 a12 = a21 =
2
−α 1
2 = 0 ⇔
1
−α
2
a22 = 0
1
1
1
= 0 ⇒ α1 = α2 = − .
4
2
2
! "# ! $# %
⇔ α2 −
⎧ 1
⎪
⎪
⎨ − 2 α11 +
⎪
⎪ 1
⎩
α11 −
2
1
α21 = 0,
2
1
α21 = 0;
2
⎧ 1
⎪
⎪
⎨ 2 α12 +
⎪
⎪ 1
⎩
α11 −
2
1
α22 = 0,
2
1
α22 = 0,
2
,
α11 = α21 , α22 = −α12 .
&% α11 = α21
= α12= 1 α22 = −1 ' ( %) *
1 1
L =
+ ! $# %)
1 −1
x = x + y ,
y = x − y .
, -
%
(x +y )(x −y )−2(x +y )−3(x −y )+6 = 0 ⇔ x 2 −y 2 −5x +y +6 = 0
%) - %*
% *
xy . %
25
2
% (x − 5x + ) −
4
1
5
1
−(y 2 −y + ) = 0 ⇔ (x − )2 −(y − )2 = 0. /
4
2
2
5
1
X = x − Y = y − X Y % *
2
2
% % X 2 − Y 2 = 0 ⇔ (X − Y )(X + Y ) = 0
) % + Y = X Y = −X
. % . %
0 %12 % Y = X Y = −X.
AB
2a − 3b A(1; 2; 0) B(−2; 2; 1) a = (2; 1; 3) b = (0; 2; 1)
ΔABC A(1; 2; 3) B(2; −1; 0)
C(1; 1; −1)
l
l:
4x − y − z + 12 = 0,
y − z − 2 = 0.
M(1; 0; 0) α :
x + y + z + 1 = 0.
A(1; 2; 0) B(−2; 2; 1) a = (2; 1; 3) b = (0; 2; 1)
2a−3b AB
=
AB(2a − 3b)
AB = (−3; 0; 1)
|2a − 3b|
2a − 3b = (4; −4; 3)
9
−12 + 0 + 3
= − √ ≈ −1,4
2a−3b AB = √
16 + 16 + 9
41
A(1; 2; 3) B(2; −1; 0) C(1; 1; −1)
1
SΔABC = |AB × AC|. AB = (1; −3; −3), AC = (0; −1; −4).
2
i j
k
AB × AC = 1 −3 −3 = 9i + 4j − k.
0 −1 −4
√
98
1√
SΔABC =
≈ 4,9.
81 + 16 + 1 =
2
2
¿¾º¿
l:
4x − y − z + 12 = 0,
cos α =?, cos β =?, cos γ =?
y − z − 2 = 0.
S = 2i + 4j +√4k
! "! |S| = 4 + 16 + 16 = 6
eS =
S
1
2
2
= i + j + k.
3
3
3
|S|
# ! "
$ l
1
cos α = ,
3
2
cos β = ,
3
% & M(1; 0; 0) α :
2
cos γ = .
3
x + y + z + 1 = 0.
"
l$ ' " ( " M
α )* + !
N = (1; 1; 1)$ " l
⎧
⎨ x = 1 + t,
y = t,
l=
⎩
z = t.
" " l α
1+t+t+t+1=0⇒t=−
2
3
,
1
2
2
x= , y=− , z=− .
3
3
3
1 2 2
! M ; − ; −
3 3 3
⇒
% - a = (4; −2; −4) b = (6; −3; 2)
(2a − 3b) · (a + 2b)
% . A B C D
A(3; 1; 1) B(−2; 1; −2) C(−3; −1; 0) D(2; 0; 17)
% /
y−1
z−2
y+2
z−3
x+1
x
=
=
=
=
−1
2
−1
3
2
−3
% 0 ! "
M(0; 1; 2) x −2 1 = y1 = z +0 1
!"
"
#
$%
" %
!
& & !
& &
''( z
(x; y) x
x = Re z y
y = Im z
)''(*
# 1 = (1; 0) i = (0; 1)
& ! , & x = (x; 0) %
iy = (0; y) & . ! % &
, ! # i & ! & & &/
i2 = −1.
)''0*
z = (x; y)
+ , & !
- %
&
+
''0 z = x + iy
! "
)'''*
1 &, & ! ,
& ! & ,
+ !&" z1 = x1 + iy1 z2 = x2 + iy2 & ! &/
z = x + iy.
•
z1 = x1 + iy1 z2 = x2 + iy2
"
z1 = z2 ⇐⇒ x1 = x2 , y1 = y2 .
)''2*
•
z1 ± z2 = (x1 ± x2 ) + i(y1 ± y2 ).
•
!
z1 · z2 = (x1 + iy1 )(x2 + iy2 ) = x1 x2 − y1 y2 + i(x1 y2 + x2 y1 ).
•
" z = x−iy #
z = x + iy # $%
zz = (x + iy)(x − iy) = x2 + y 2 .
& '#%
#
z1
• ( z =
)
z2
) * # ' zz1
2
z2% # % #
z1
x1 x2 + y1 y2 x2 y1 − x1 y2
z1 · z2
(x1 + iy1 )(x2 − iy2 )
=
=
=
+i
.
z2
z2 · z2
(x2 + iy2 )(x2 − iy2 )
x22 + y22
x22 + y22
+% %
z1 = 2 + 3i z2 = 3 − 2i
z1 + z2 = 5 + i, z1 − z2 = −1 + 5i,
z1 z2 = 6 + 6 + i(9 − 4) = 12 + 5i,
z1
(2 + 3i)(3 + 2i)
6 − 6 + i(9 + 4)
2 + 3i
=
=
= i.
=
z2
3 − 2i
(3 − 2i)(3 + 2i)
9+4
2 x2 + px + q = 0
D = p4 − q < 0
z 2 + pz + q = 0
2
p4 − q < 0
z1,2 = α ± βi,
!
"
p
α=− β=
2
q−
p2
4
# $ $ %&
'
z 2 + pz + q = (z − z1 )(z − z2 ) = ((z − α) − βi)((z − α) + βi).
( z2 + 8z + 25 = 0
) * √ + , - ! +
z1,2 = −4± 16 − 25 = −4±3i z 2 +8z +25 = ((z +4)−3i)((z +4)+3i).
z 2 + pz + q
. / 0 & Re z = x "
& z Ox 0 Im z = y Oy
Oxy 1
y=Im z
z
y
z
r
ϕ
x
x=Re z
z = x + iy
Oxy
Oxy
z
z z
z
!
z = ∞
z
!"# $
z O z
% % z# % z
&'%& &' % z
z % N %& O
z %
N
N
z
z
y
0
z
x
(% ) |z|
% )
*
" !#$
|z| = r =
%
r z.
x2 + y 2 .
z
r z Ox
⎧
y
⎪
arctg
x > 0,
⎪
⎪
⎪
x
⎪
y
⎪
⎨ π + arctg
x < 0,
x
ϕo = arg z =
π
⎪
y > 0, x = 0,
⎪
⎪
⎪ 2π
⎪
⎪
⎩ −
y < 0, x = 0.
2
√ √
√
!
√
z1 = 1+i 3" z2 = −1+i 3" z3 = −1−i 3 z4 = 1−i 3 #
! $
y
x √
3
π
= z3
z1 ϕo1 = arctg
1√ 3
4
− 3
= π.
ϕo3 = π + arctg
−1
3
z2 z4
√ !
√
3
2π o
π
− 3
o
=
= − $#%
"# ϕ2 = π + arctg
ϕ4 = arctg
−1
3
1
3
√
! &! ! r = x2 + y 2 = 1 + 3 = 2. '
% ( # # r = 2
) * "# +
z1
z3
y
2
z2
-2
-1
z1
3
0
1
2
- 3
z3
-2
z4
x
z
R
R
!
" # # $
|z| < R,
%
&
|z| = R,
'
( ! $ ) z0
*+ &
|z − z0 | = R
*
|z| > R.
y
1111111
0000000
0000000
1111111
R
0000000
1111111
z
0000000
1111111
0000000
1111111
1111111
0000000
0000000
1111111
0000000
1111111
0000000
1111111
R
0000000
1111111
0000000
1111111
0
0000000
1111111
0000000
1111111
0000000
1111111
0
|z| < R
x
|z − z0 | < R.
" # ,# $
|z − z0 | < R,
-
&
|z − z0 | > R.
.
( % - /
# *+
z
!"#
$ % & z
2πk k ∈ Z ' $ $ $(
ϕ = Arg z
ϕ = Arg z = arg z + 2πk = ϕo + 2πk, k ∈ Z,
)*
k = 0
+ ,-
x = r cos ϕ, y = r sin ϕ
)
. $ r
ϕ
z = r(cos ϕ + i sin ϕ),
))
/( 0 )
1
2. .
3
4
eiϕ = cos ϕ + i sin ϕ.
)
,
z = reiϕ ,
)5
!
"
6 )) )5 /
/ z
.
5 # " "
$ z% &
' √( z1 = 3) z2 = −3) z3 = 3i) * z4 = −3i
+ z5 = 3 − i 3
7 8 9 : /$
. ϕ◦ = 0
)) )5 0 ) )* z1 = 3 = 3(cos 2πk +
+i sin 2πk) = 3ei2πk .
ϕo = π
z2 = −3 = 3(cos(2k + 1)π + i sin(2k + 1)π) = 3ei(2k+1)π .
π
x = 0 y > 0 → ϕo =
2
1
1
π + i sin 2k +
π) = 3i(2k+1/2)π .
z3 = 3i = 3(cos 2k +
2
2
y < 0 → ϕo = −
π
2
1
1
1
π + i sin 2k −
π = 3i(2k− 2 )π .
z4 = −3i = 3 cos 2k −
2
2
√
z5 = 3 − i 3
√
− 3
o
= − π6
ϕ = arctg
3
√ 2k− 1 π
√
1
1
6
π + i sin 2k −
π = 2 3e
z5 = 2 3 cos 2k −
.
6
6
!
! # !!$% &
! '
z1 = r1 eiϕ1 z2 = r2 eiϕ2
"
z = reiϕ = z1 z2 = r1 r2 ei(ϕ1 +ϕ2 ) = r1 r2 (cos(ϕ1 + ϕ2 ) + i sin(ϕ1 + ϕ2 ),
('
z1
r1
r1
z = reiϕ =
= ei(ϕ1 −ϕ2 ) = (cos(ϕ1 − ϕ2 ) + i sin(ϕ1 − ϕ2 )) (' )
z2
r2
r2
r = r1 r2
ϕ = ϕ1 + ϕ2
r1
r =
r2
ϕ = ϕ1 − ϕ2 .
* !! z = reiϕ
z n = (reiϕ )n = rn einϕ .
sin ϕ cos ϕ eiϕ
! " # " 2π $
%&' " # n $ ϕ
( ) ! ϕo = arg z.
*) " ) " ( n "
+ +$ $ ($ ) ,
" - ",
. ϕ = Arg z = arg z + 2πk. "
n, " " z = reiϕ "
(r(cos ϕ + i sin ϕ))n = rn (cos nϕ + i sin nϕ).
√
n
z=r
1/n
ϕo + 2πk
√
ϕo + 2πk
ϕo + 2πk
n
+ i sin
.
e
= n r cos
n
n
i
k
"
√
0, 1, 2, ..., n − 1
/
" n
z
z
√
r n
0 k " ,
n
n
( 1 n − 1 + " !
2 n, "
z n − a = 0,
√
z = n a
-
" ,
a = reiϕ & " n
)$ " /
3 ! "#
"
√
z = 3 − i 3.
4 1√ 5 6 √
z = 3 − i 3 ) 1 " 7 ) 2 3
− π6 8 "
√
√
π
(3 − i 3)6 = (2 3e−i 6 )6 =
= 26 33 ei11π = 1728(cos π − i sin π) = 1728(−1 − i0) = −1728,
1/6 √ √ (−1+12k)π
√
√
π
6
12
6
36
3 − i 3 = 2 3e−i( 6 +2πk)
= 2 3ei
.
√
6
k 3 − i 3
√
√
√ −i π
√
11
6
6
12
12
36
z1 = ( 3 − i 3)1 = 12e
, z2 = ( 3 − i 3)2 = 12ei 36 π ,
√
√
√
√
23
35
6
6
12
12
z3 = ( 3 − i 3)3 = 12ei 36 π , z4 = ( 3 − i 3)4 = 12ei 36 π ,
√
√
√
√
47
59
6
6
12
12
z5 = ( 3 − i 3)5 = 12ei 36 π , z6 = ( 3 − i 3)6 = 12ei 36 π .
!" #$ % %" "%"$ z "& '
( √
)* ( "$*)% " * %&"$) '
" 12 12 " +
y
z3
z2
11π
36
z4
12
12
z1
12
z5
x
z6
12
z4 + 1 = 0.
√
, (
z 4 + 1 = 0 → z 4 = −1 → z = 4 −1.
"%)% -.* $/ $ )* "$$ )* "
arg z = π,
−1 = 1ei(π+2πk)
" $ )
z = ei
π+2πk
4
.
k
π
π
π
+ i sin =
4
4
3π
3π
+ i sin
= cos
4
4
5π
5π
+ i sin
= cos
4
4
7π
7π
+ i sin
= cos
4
4
z1 = ei 4 = cos
z2 = ei
3π
4
z3 = ei
5π
4
z4 = ei
7π
4
√
2
(1 + i),
2 √
2
=
(−1 + i),
2
√
2
=
(−1 − i),
√2
2
=
(1 − i).
2
z
! "#$!
y
1
z1
z2
π
4
1
-1
z3
-1
x
z4
!"
z .
% & z5 = 32 = 25 ' ( )
z1 = 2! *
)
! "+$!
, ' - . !/$ )
- 01 sin nx cos nx
' sin x cos x.
!/ sin 2x cos 2x sin x cos x.
z 5 − 32 = 0
y
2 z2
z3
2π
5
-2
2
z1
x
z4
-2 z 5
2
n = 2 : (cos ϕ + i sin ϕ) = cos2 ϕ + 2i sin ϕ cos ϕ − sin2 ϕ =
= cos 2ϕ + i sin 2ϕ.
! "#
"
cos 2ϕ = cos2 ϕ − sin2 ϕ sin 2ϕ = 2 sin ϕ cos ϕ.
$ % &' ' ! '
& %( !
)
z1 = −3 + 4i z2 = 4 − 2i z1
z1 /z2
z2 z1 + z2 z1 − z2 z1 · z2
*% + ,
z1 + z2 = 1 + 2i, z1 − z2 = −7 + 6i,
z1 z2 = (−3 + 4i)(4 − 2i) = −12 + 8 + i(6 + 16) = −4 + 22i,
z1
−3 + 4i 2 + i
−6 − 4 + i(8 − 3)
1
−3 + 4i
=
·
=
= −1 + i.
=
z2
4 − 2i
2(2 − i) 2 + i
2(4 + 1)
2
1
= (4 − 2i) −1 + i =
2
1
z1 = z2 −1 + i
2
= −(2 − i)2 = −(4 − 4i − 1) = −3 + 4i.
z1 z2 z1 +z2 z1 −z2 14 z1 z2 zz1
2
y
6
z1 z2
z1
-7
-5
z 1z 2
4
4
z1 z2
z1 2
z2
-3 -1
-2
1
4
z2
x
z2 + 4z + 13 = 0.
! " # $ z1,2 = −2 ±
= 2 ± 3i.
√
4 − 13 =
% %$ &
' & "$
z = 1 + i z = 1 − i
! "
π
1
= ,
1
4
π
arg z = arg(1 − i) = arctg(−1) = − ,
4
arg z = arg(1 + i) = arctg
"
|z| =
√
1+1=
√
2,
|z| =
√
2.
z=
z=
√
√
π
√
π
π
,
2 cos + i sin
4
4
√
π
π
.
= 2 cos − i sin
4
4
2e 4 i =
π
2e− 4 i
π
2 < |z| < 4 2 < arg z < π
2 < |z| < 4
! " # ! $ $
! $ " % $ $ r = 2
r = 4 &''
y
4
2
-4
-2
0
2
4
x
-2
-4
π2 < arg z < π ! $(
) *# + * *
# ! $ $ r = 2 r = 4
$ #" &''
# " !
#! # " !
, (1 + i)4
! "
!"#
$% m = 4, a = 1 b = i"
4·3 2 4·3·2 3 4·3·2·1 4
i +
i +
i =
1·2
1·2·3
1·2·3·4
= 1 + 4i − 6 − 4i + 1 = −4.
(1 + i)4 = 1 + 4i +
& ' 1 + i $ % ( $%
) & ** *# $
k =0: 1+i =
√ πi √
π
2e 4 = 2(cos + i sin π4 ).
4
+( ** ,-" ** ,."
√
(1 + i)4 = ( 2)4 eπi = 4(cos π + i sin π) = −4,
# # & &
** / √1 + i
4
0 $' $$( # $$ &
$ $ **# ** , "
√
4
1$ #
'
1+i =
k
4
√ i ( π +2πk)
2e 4 4
.
& 2# #,#*# ) & &)
√
√
√
π
π
π
8
8
4
+ i sin
,
1 + i = 2ei 16 = 2 cos
16
16
1
√
√
√
9π
9π
9π
8
8
+ i sin
,
z2 = 4 1 + i = 2ei 16 = 2 cos
2
16
16
√
√
√
17π
17π
17π
8
8
4
i
16
,
1 + i = 2e
= 2 cos
z3 =
+ i sin
3
16
16
√
√
√
25π
25π
25π
8
8
z4 = 4 1 + i = 2ei 16 = 2 cos
+ i sin
.
16
16
4
√
3 $ & $ 8 2
z1 =
) $ -."
** -
z 3 + 27 = 0
z
y
8
z2
2
z1
x
π
16
0
z3
8
2
z4
√
→z=
3
−27.
z3 = −27
→
z2 = −3 z2 = −3
arg(−3) = −π
!
r = 3 " #$%& ' (
π
5π
z1 z3 arg z1 = arg z3 =
|z| = 3
3
3
y
3
z2
-3
x
3
0
-3
z1
z3
) * ' sin 3ϕ cos 3ϕ sin ϕ
cos ϕ
n = 3 :
3
(cos ϕ + i sin ϕ) = cos 3ϕ + i sin 3ϕ → cos3 ϕ + 3i cos2 ϕ sin ϕ+
+ 3i2 sin2 ϕ cos ϕ + i3 sin3 ϕ = cos3 ϕ − 3 sin2 ϕ cos ϕ+
+ i(3cos2 ϕ sin ϕ − sin3 ϕ) = cos 3ϕ + i sin 3ϕ.
cos 3ϕ = cos3 ϕ − 3 sin2 ϕ cos ϕ sin3ϕ = 3 cos2 ϕ sin ϕ − sin3 ϕ.
z1 = 4 + 3i z2 = 2 − i z1 z2
z1
z1 + z2 z1 − z2 z1 z2
z2
! z2 + 2z + 10 = 0
2
3z + 2z + 4 = 0
"# √
z1 = i z2 = −2 z3 = 1 + i 3
√
"" (1 + i 3)3 !
" z6 + 1 = 0 z3 − i = 0
"
$% & ' ( ) &
) ( ) * ' +
* %* * + ,- (
%*- ' &
R(x) =
Qm (x)
,
Pn (x)
."
Pn (x) = an xn + an−1 xn−1 + an−2 xn−2 + · · · + a1 x + a0 =
Qm (x) = bm xm +bm−1 xm−1 +bm−2 xm−2 +· · ·+b1 x+b0 =
n
,
k=0
m
,
ak xk ,
bk xk .
k=0
!
!
R(x) =
x4 + 5x3 − 6x + 5
.
x3 + 2x2 − 1
" # $ "% % &
x + 3 −6x2 − 5x + 8&
R(x) = x + 3 +
−6x2 − 5x + 8
.
x3 + 2x2 − 1
' Pn (x) % x − a
&
(
! "# $ % &'
x − a Pn(x) x = a.
( '
P (x) = 3x9 − 2x5 + 3x2 + 4x − 8 x + 1
Pn (x)
" # $
) a = −1. *+
$
9
P (−1) = 3(−1) − 2(−1)5 + 3(−1)2 + 4(−1) − 8 =
= −3 + 2 + 3 − 4 − 8 = −10.
x2 −5x − 6 = 0,
−3 2
x1 = 1, x2 = −3, x3 = 2.
= 5(x − 1)(x − 2)(x − 2)(x − 3)º
5x4 −4x3 +115x2 −140x+60 =
x3 + x = x(x − i)(x + i)
! " #
$
Pn (x) = an (x − a)k1 (x − b)k2 ...(x − p)ks ,
%
" a, b...p
# " ! "! n
& a, b...p '" # (
a ) " k1 b ) " k2 p ) (
" ks
! "
α ± βi * " $ " α + βi " (
# " k "! + # " α − βi "
, # "
% -, "
% " (x−(α+βi))k
,
! "" (x − (α − βi))k
- ""'. ! " "(
!
! # " /$
(x − (α + βi))k (x − (α − βi))k = ((x − α) − βi)k ((x − α) + βi)k =
= ((x − α)2 + β 2 )2 = (x2 − 2αx + α2 + β 2 )k = (x2 + px + q)k ,
p = −2α, q = α2 + β 2
"
# +# x2 + px + q
0
"
*" 1" ! "2 $ " (
# " " ,22 0 !"(
" '. $
Pn (x) = an (x − a)k1 (x − b)k2 ...(x2 + p1 x + q1 )s1 (x2 + p2 x + q2 )s2 ....
* 2 n = k1 + k2 + · · · + ks + 2(s1 + s2 + · · · + sm ).
! "
A
,
x−a
A
II.
(n = 2, 3, ...),
(x − a)n
Mx + N
III. 2
(D = p2 − 4q < 0),
x + px + q
Mx + N
IV. 2
(D = p2 − 4q < 0, n = 2, 3...)
(x + px + q)n
I.
# $ %
$% % &
(x)
QPm(x)
n
Pn (x) = (x − a)k · · · (x2 + px + q)l · · ·
! "
!
Qm (x)
A1
A2
Ak
=
+
+ ··· +
+ ···
2
Pn (x)
x − a (x − a)
(x − a)k
M1 x + N1
M2 x + N2
Ml x + Nl
+
+ ··· + 2
,··· ,
x2 + px + q (x2 + px + q)2
(x + px + q)l
Ai,
Bi , Mi , Ni (i = 1, 2, ...)
#
' ! ()
* ' ! () %
a
+% * ' ! () %
%
$ , ,,
# $ ,,, ,-
Pn (x)
1
1
A = 1, B = − , C = .
5
5
! ""# ! $%&'(
1
1
x2 + 2x − 6
1
5
= −
+ 5 .
x3 + x2 − 6x
x x+3 x−2
$%&)(
%&'
*
3x2 + 5
.
(x − 1)2 (x2 + 2x + 5)
A
B
Mx + N
3x2 + 5
=
+
.
+
(x − 1)2 (x2 + 2x + 5)
x − 1 (x − 1)2 x2 + 2x + 5
-
$%&+,(
3x2 + 5
=
(x − 1)2 (x2 + 2x + 5)
A(x − 1)(x2 + 2x + 5) + Bx(x2 + 2x + 5) + (Mx + N )(x − 1)2
=
.
(x − 1)2 (x2 + 2x + 5)
.
x2 + 5 = A(x3 + x2 + 3x − 5) + B(x2 + 2x + 5)+
+M(x3 − 2x2 + x) + N (x2 − 2x + 1)
x2 + 5 = (A + M)x3 + (A + B − 2M + N )x2 +
+(3A + 2B + M − 2N )x + (−5A + 5B + N ).
. ""# ! !/ ./ x
⎧
x3 : A + M = 0,
⎪
⎪
⎪
⎨x2 : A + B − 2M + N = 3,
⎪
x : 3A + 2B + M − 2N = 0,
⎪
⎪
⎩
− 5A + 5B + N = 5.
-! !
A = 1 , B = 1, M = − 1 , N = 5
4
4
4
!
"#$
5
1
1
− x+
3x2 + 5
1
4
4 .
= 4 +
+
(x − 1)2 (x2 + 2x + 5)
x − 1 (x − 1)2 x2 + 2x + 5
% & $ #!
$ "' $ ( # '
P (x) ≡ Q(x) )* $ +# +$ !
$ x = a : P (a) = Q(a) ( a ! + #
, $" $ - .
-. + + ' / $ ' - "
$ x + . #
& $ * ( $ ( # Pn (x) -0
+$ $ + # #
+ . +# *"- # $ !
-$ +$ 1
!"
2 2 + $ .
* "#
x2 + 2x − 6 = A(x + 3)(x − 2) + Bx(x − 2) + Cx(x + 3).
, $ ' +#- x
x = 0 : −6 = −6A =⇒ A = 1,
1
x = 2 : 2 = 10C =⇒ C = ,
5
1
x = −3 : −3 = 15B =⇒ B = − .
5
3- "#$
2$ $ $ $ - + '- +$!
- $ ' $ ' + ) + #- - +
$ ( #
x4 1+ 1 .
x4 +1
1
x1 = √ (1+i),
1
x2 = √ (−1 + i),
2
1
x3 = − √ (1 + i),
2
1
x4 = √ (1 − i).
2
2
1
1
1
1
√
√
√
√
−i
−i
x +1= x−
x+
·
2
2
2
2
1
1
1
1
x − √ + i√ .
· x + √ + i√
2
2
2
2
4
! "# " ! "# "
1
1
1
1
− i√
x− √
+ i√ ·
x− √
2
2
2
2
1
1
1
1
·
x+ √
− i√
x+ √
+ i√
=
2
2
2
2
7 6
7
6
2
2
1
i2
i2
1
x+ √
=
−
−
=
x− √
2
2
2
2
x4 + 1 =
=
1 1
1 1
2
2
x2 + √ x + +
=
x2 − √ x + +
2 2
2 2
2
2
√
√
= (x2 − 2x + 1)(x2 + 2x + 1).
$ %" & '()* ! ! +, .%% ,!
1
Ax + B
Cx + D
√
√
=
+
=
x4 + 1
x2 − 2x + 1 x2 + 2x + 1
√
√
(Ax + B)(x2 + 2x + 1) + (Cx + D)(x2 − 2x + 1)
=
x4 + 1
!
√
√
Ax3 + A 2x2 + Ax + Bx2 + B 2x + B + Cx3 −
√
√
−C 2x2 + Cx + Dx2 − D 2x + D = 1.
& '(//*
x
A B C D
⎧
A + C = 0,
⎪
⎪
⎪
⎨A√2 + B − C √2 + D = 0,
√
√
⎪
A + B 2 + C − D 2 = 0,
⎪
⎪
⎩
B + D = 1.
C A D
C = −A D = 1 − B
B
√
√
√
A 2 + B + A 2 + 1 − B = 0 ⇒ 2A 2 = −1 ⇒
√
√
1
2
2
⇒ A=− √ =−
⇒ C = −A =
,
4
4
2 2
√
√
√
A+B 2−A− 2+B 2=0 ⇒
√
√
1
1
2B 2 = 2 ⇒ B =
⇒ D= .
2
2
A, B,
& ' x4 1+ 1 (
C, D
!"#$$%
√
√
2 1
2 1
+
+
−
1
4√ 2 +
4√ 2 .
=
4
x +1
x2 − 2x + 1 x2 + 2x + 1
!"#$)%
!"#$
P4(x) = x4 + 1
! "
# "$% & " $ " '
" " " x2 +
+px + q D < 0% ()
x4 + 1 = (x2 + p1 x + q1 )(x2 + p2 x + q2 ) =
= x 4 + p 1 x 3 + q1 x 2 + p 2 x 3 + p 1 p 2 x 2 + p 2 q 1 x + q 2 x 2 + p 1 q 2 x + q 1 q 2 .
* " "$ ! " &$ x &"
&" " &"% " $ $ $
p1 p2 q1 q2
⎧
√
⎪
p + p2 = 0 ⇒ p2 = −p1 ⇒ p2 = − 2,
⎪
⎪ 1
⎪
⎪
p1 p2 = −q1 − q2 = −2 ⇒
⎪
⎪q1 + p1 p2 + q2 = 0 ⇒
⎪
⎨⇒ p2 = 2 ⇒ p = √2,
1
1
⎪
p2 q1 + p1 q2 = 0 ⇒ −p1 q1 + p1 q2 = p1 (q2 − q1 ) = 0 ⇒ q1 = q2 ,
⎪
⎪
⎪
⎪
p1 = 0,
⎪
⎪
⎪
⎩q q = 1 ⇒ q 2 = q 2 = 1 ⇒ q = q = 1.
1 2
1
2
1
2
q1 = q2 = −1,
2
p = −2,
!"# $%&
'
*
2x
2
x4 + 1
()
+ ()
√
x4 + 1 = x4 + 2x2 + 1 − 2x2 = (x2 + 1)2 − ( 2x)2 =
√
√
= (x2 + 1 − 2x)(x2 + 1 + 2x).
!# $
" !#
%
*
-
!
,
& !
! !
3
2
2
x + 2x − x + 1 | x + x − 1
'
−−−−−−−
| x+ 1
x3 + x2 − x
− − − − − − −−
(x2 + 1
x2 + x − 1
− − − − −−
−x + 2
x3 + 2x2 − x + 1
x2 + x − 1
!
x3 + 2x2 − x + 1
−x + 2
=x+1+ 2
.
x2 + x − 1
x +x−1
!"
x3 (x
& ' (
x3 (x
#$"#
& ' (
(x2
x2 + 1
.
− 1)2
#$"%
Ex + F
.
+ 2x + 5)2
x2 + 1
E
A B
C
D
+
= + 2+ 3+
.
2
− 1)
x x
x
(x − 1) (x − 1)2
x+5
.
x(x + 3)(x2 + 2x + 5)2
x+5
B
Cx + D
A
+
+
= +
x(x + 3)(x2 + 2x + 5)2
x x + 3 x2 + 2x + 5
) !
' ! )
* !"
+3
.
#$"$ x(x +x1)(x
− 2)
A
B
C
+3
= +
+
.
& ' ( x(x +x1)(x
− 2)
x x+1 x−2
! * )
* (
x + 3 = A(x + 1)(x − 2) + Bx(x − 2) + Cx(x + 1).
+ ⎧ * ) * x(
3
⎪
⎪
x = 0 ⇒ 3 = −2A ⇒ A = − ;
⎪
⎪
2
⎨
2
x = −1 ⇒ 2 = 3B ⇒ B = ;
⎪
3
⎪
⎪
⎪
⎩ x = 2 ⇒ 5 = 6C ⇒ C = 5 .
6
3
2
5
x+3
=− +
+
.
x(x + 1)(x − 2)
2x 3(x + 1) 6(x − 2)
x+5
.
x2 (x − 1)
A B
C
x+5
= + 2+
.
x2 (x − 1)
x x
x−1
x + 5 = Ax(x − 1) + B(x − 1) + Cx2 .
x = 0 ⇒ 5 = −B ⇒ B = −5;
x = 1 ⇒ 6 = C ⇒ C = 6;
x = −1 ⇒ 4 = 2A − 2B + C ⇒ A = −6.
x+5
6
5
6
=− − 2+
.
− 1)
x x
x−1
x2 (x
!"
! #$%% & ' (' #"
$%% &
)
2x + 1
.
+ 2x + 5)
x(x2
2x + 1
A
Bx + C
= + 2
.
x(x2 + 2x + 5)
x x + 2x + 5
* ! + #
+
2x + 1 = A(x2 + 2x + 5) + x(Bx + C).
# # #
,
' + '
2x + 1 = (A + B)x2 + (2A + C)x + 5A.
* , #$%% & ' #' , x
⎧
⎪
⎨A + B = 0,
2A + C = 2,
⎪
⎩5A = 1.
$ !
1
8
1
A= , B=− , C= .
5
5
5
2x + 1
1
−x + 8
=
+
.
x(x2 + 2x + 5)
5x 5(x2 + 2x + 5)
x−3
.
x2 (x2 + 1)
x−3
A B
Cx + D
= + 2+ 2
.
x2 (x2 + 1)
x x
x +1
x − 3 = Ax(x2 + 1) + B(x2 + 1) + x2 (Cx + D).
x − 3 = (A + C)x3 + (B + D)x2 + Ax + (A + B),
⎧
A + C = 0,
⎪
⎪
⎪
⎨B + D = 0,
⎪
A = 1,
⎪
⎪
⎩
A + B = −3,
A = 1 B = −4 C = −1 D = 4
x−3
1
4
−x + 4
= − 2+ 2
.
x2 (x2 + 1)
x x
x +1
x5 − x4 + 2x + 3
x2 − 1
! ""! " #$% & ' (() *
+ " "$ & " &! "$,
-
x+5
.
x4 (x + 1)
x
.
(x2 + 5)2 (x + 3)
./
! ""! "%0 " "$ ",*
, (() & "% &! % ,
x
.
(x − 1)(x + 2)
x2 (x
x+3
.
+ 1)
!!" # $#
!!"
x+5
.
− 5)
x2 (x
x(x2
x+4
.
+ 4x + 8)
un =
S=2
(−1)n+1
n2
x ∈ R
n
n+1
un = 4n−1
1
2
1
4
|r2 | 0, 01
− 2a12 e
2
.
3
− x12 . − sin x. 3 √31x2 . 1 − 3x2. cos x −
x −xex
x log3 x−sin x
x
−3 sin x. 1+x
x ln 3 cos
. 2+e
.
2 + arctg x.
(2+ex )2
x ln 3 log23 x
1
x−sin x cos x
4
2
0. − 1+x
+
ln
x
+
1
+
.
5x
−
12x
+
2.
2
(x cos x)2
2 −6x+25
5
√6ax .
83 x5/3 . −2x
.
ctg x − sinx2 x .
2
2
(x −5x+5)
a2 +b2
√
x
2
(e −5 cos x) 1−x −4
√
x arctg x. 3x2 ln x. x2 ex . (ex −5
.
sin x−4 arcsin x) 1−x2
x
1
1√
2
2
2 (sin(5x ) + sin(x )). x(1+ln2 x) . arcsin x 1−x2 .
sin x−1
y = xsin x cos
.
x ln x + sin xx
x3 ·sin x
3
1
1
.
ln x·arctg x x + ctg x − x ln x − arctg x(1+x
2)
3
(2xey −3x2 )y
.
1−x2 yey
t(2−t )
.
− xy .
T : x − 4y − 5 = 0, N : 4x + y − 3 = 0.
1−2t3
15
10
( 3; − 2 , −2; 3 .
θ1 = − arctg 0, 6; θ2 = arctg 3.
t0 = 8, t1 = 0, t2 = 4, t3 = 8. 181, 5
√
x
x
dx. x cos x−sin
52 x xdx. arctg x + 1+x
dx.
2
x2
n n!
(−1)
2x
2 arctg x + 1+x2 . (1+x)n+1 .
−e−x (x2 − 6x +
284
3
4
≈ 0, 81
+6)dx . (2−x)5 dx .
y < 0 x ∈ (−∞; 0) ∪ (0; +∞),
y = f (x) x = − 12 −
y = f (x) x < − 12
4
x > − 21 , y (0) y > 0 x ∈ (−∞; 0) (0; ∞)
m = f (−1) = −4, M = f (0) = 0.
m = f (−2) = −20, M = f (4) = 16.
!
"
#
x = 0 x+y =41. $
x = − 12 %
D(y) = (−∞; 0) (0; ∞) 4&
' x = 0 y = x () (−∞; 0)4 (2; ∞)
(0; 2) ymin = y(2) = 3 * (−∞; 0) (0; ∞)
$ + , -.
) arccos k1 arctg hd .
y = 1 ymin = y(2) = −1 &
y = 0 x → +∞ ymin = y(0) = 0 ymax = y(2) = e44
y = x + π2 x → +∞
y = x − π2 x → −∞ & /
0 [0; 1]
M = y(1) = 1e 1 m = y(0) = 0 [0; 3] M = y(2) = e42 m = y(0) = 0
[−1; 2] M = y(−1) = e m = y(0) = 0
*2' a2
⎛
⎞
⎛
⎞
−4 −8 −4
8 3 6
3 4
1 6
⎝ −3 −1 −5 ⎠ . ⎝ 0 8 12 ⎠ .
−7 −6 1
−3 6 14
⎛
⎞
⎛
⎞
M12 = −6
10 6 5
11
M22 = 3
⎝ 8 11 6 ⎠ . ⎝ 16 ⎠ .
A12 = 6
9 8 10
4
A22 = 3.
/
⎛
⎞
1/10 −1/5 7/10
1/10 −1/5 ⎠
2 3 3 3 ⎝ 0
0
0
1/10
⎛
⎞
1 2
2
19 · ⎝ 2 1 −2 ⎠
2 −2 1
x = 2 y = 3 z = −2 x = −3, y = 2, z = −1
x1 = 1, x2 = 5, x3 = 2
x1 = 2 x2 = −1 x3 = −3
x1 = 25 + 35 x3 x2 = 14 + 34 x3
x4 =
7
20
+
13
x
20 3
AM = 5a+c
. B(2; −6; 8). 2a − 3b = −8i − 3j + 10k.
6
e1,2 = ±(− 13 i + 23 j − 23 k), cos α = − 13 , cos β = 23 , cos γ = − 23
√
√
√
3 22−2 . 4 2. √− 2. 7, − 152
cos(AC; BD) = − 9√2310 . cos B = 2 1010 .
√
20i + 14j + 2k. 25 3. {ha =
{±(−2i + j + 3k)} {ha = 1; hb =
12
√7 .
11
x−1
1
◦
x + 7z + 22 = 0.
√
42
,
3
√
3 5
,
5
hb =
hc =
√
70
}.
5
√
3 2
}.
2
2y − 5z + 10 = 0.
x
= y+1
= z−1
.
= y−1
= z−2
. α = 90◦ ,
3
1
−1
2
1
β = 45 , γ = 135◦ M (1; −1; 4). 11x − 17y − 19z + 10 = 0.
200
5x = − 52 (y − 1) = −(z − 2)
6 + 2i, 2 + 4i, 11 + 2i, 1 + 2i
x+1
−1
=
y−1
2
=
z−2
−3
−1 ± 3i,
√
−1±i 11
3
i( π3 +2πk)
π
ei( 2 +2πk) = cos π2 + i sin π2 2ei(π+2πk) = 2(cos
π + i sin π)
2e
√
√
± 3±i
i± 3
π
π
, −i, 2 .
= 2 cos 3 + i sin 3 ±i,
2
=
E
+ C. Ax + xB2 + xC3 + xD4 + x+1
.
x3 − x2 + x − 1 + 3x+2
x2 −1
Ax+B
Cx+D
E
1
2
2
3
2
+
+
.
+
.
−
+
+ x+1
.
x2 +5
(x2 +5)2
x+3
3(x−1)
3(x+2)
x
x2
2
1
2
1
x+4
− 5x − x2 + 5(x−5) . 2x − 2(x2 +4x+8) .
y
2
1
x
1
y = (x − 1)3 + 2
y
1
x
-2
0
y = x−2
x+2
y
1
x
-1
0
y =
√
−x
y
x
0
y = ln(1 − x)
1
− π
3
1
π
− 12
y
0 π
6
π
12
2π
3
x
−1
y = 5 sin(2x − π/3)
y
0
x
y = x21+1
y
0
x
y = x2 + x1
y
1
−2π
−π
0
−1
π
2π
y = sin1 x
3π x
y
−5π/2 −3π/2 −π/2
π/2
1
3π/2
5π/2
0
x
y = lg(cos x)
x
0
r = ϕ
y
π
x
0
r = ϕπ
y
r
ϕ
0
1
2
x
r = 2 cos ϕ
y
1
-1
1
0
x
-1
r = 1
0
x
3
r = 3 cos 4ϕ
y
x
0
-1
1
y = | lg |x||
y
1
0
x
y = 2|x|
y
1
-1
x
0
1
-1
y = x · |x|
y
0
x
3
y = x3 − 3x2
y
O
x
y = x(1 +
√
x)
y
x
O
y = 1 +
√
x
y
4/27
x
0
1
8/27 0,5
y =
√
3
x2 − x
y
1
0
x
1
y = e x − x
1
y
3
-2
0
2
x
y = (x3 + 4)/x2
y
1
4
2
x
−1
2 −4x
y = x2x−4x+8
y
4/e 2
2
x
y = x2 e−x
y
π/2
x
−π/2
y = x + arctg x
Aa −
Gg −
Mm −
Ss −
Y y −
Aα −
Zζ −
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Âèêòîð Ãåîðãèåâè÷ ÇÓÁÊÎÂ,
Âëàäèìèð Àíàòîëüåâè÷ ËßÕÎÂÑÊÈÉ,
Àíàòîëèé Èâàíîâè÷ ÌÀÐÒÛÍÅÍÊÎ,
Âåíèàìèí Áîðèñîâè÷ ÌÈÍÎÑÖÅÂ
ÊÓÐÑ ÌÀÒÅÌÀÒÈÊÈ
ÄËß ÒÅÕÍÈ×ÅÑÊÈÕ ÂÛÑØÈÕ
Ó×ÅÁÍÛÕ ÇÀÂÅÄÅÍÈÉ
×ÀÑÒÜ 1
Àíàëèòè÷åñêàÿ ãåîìåòðèÿ.
Ïðåäåëû è ðÿäû.
Ôóíêöèè è ïðîèçâîäíûå.
Ëèíåéíàÿ è âåêòîðíàÿ àëãåáðà
Ïîä ðåä. Â. Á. Ìèíîñöåâà, Å. À. Ïóøêàðÿ
Ó÷åáíîå ïîñîáèå
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ËÐ ¹ 065466 îò 21.10.97
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îò 21.04.2010 ã., âûäàí ÖÃÑÝÍ â ÑÏá
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