Высшая математика. Том 2
.pdf' 1: y = (x − 1)13 .
2 D(y) = R.
& -
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(x − 1) |
− 2 |
3 |
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(x − 1)53 . |
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*. 3.32.
( ) y = 3 x − 1
% x=1. % -
. , x = 1. ; (1; +?), (–?; 1).3
' 2: y = e− x2 .
2 D(y) = R, y′ = −2xe− x2 .
y′′ = −2e− x2 + 4x2e− x2 = 2e− x2 (2x2 − 1).
:
, ’( : 2x2 – 1 = 0.
*. 3.33.
( ) y = e− x2
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= ± 1 . , − 1 |
, + 1 . ; |
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− 1 ; 1 |
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−∞;− 1 |
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;+∞ .3 |
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3.39. & ' ( ( 8 ) |
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% . |
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= f(x), - |
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M , |
*. 3.33. |
< |
#
%
# # .
1 # % , ( #-
# #, -
# (.
161
0- δ M ,
, - δ M .
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( !$ & ' ( ( |
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) x> x0 #- # =f(x) # % |
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# , # |
x→x |
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= ∞ # |
x→ x −0 |
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= ∞ |
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lim f |
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lim |
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lim f |
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= ∞ . , , - x = x |
% - |
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x→x +0 |
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x→x |
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. , - x |
= x |
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lim |
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= ∞ . |
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, , = f(x) - % , - f(x) 6? # x6 x0–0 # x6x0+0, x=x0.
, ( = f(x)
# x = x0, # % -
( ). , %
x = x0.
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' 1: |
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y = x + |
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1 |
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2 |
lim |
x + |
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= −∞ |
, lim |
x + |
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= +∞ , |
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x→2−0 |
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x − 2 |
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x→2+0 |
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x − 2 |
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% x = 2 .3 |
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' 2: |
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1 |
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y = e x . |
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1 |
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1 |
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−∞ |
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2 lim e x |
= +∞ , |
lim e |
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= e |
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= 0 |
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e |
∞ |
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x→0+0 |
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x→0−0 |
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' x = 0 – .3
! & ' ( (
– , - =f(x) % -
, # =kx+b. ( % k b.
4 ' 14. ' =kx+b x6+?
= f(x) , |
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k = lim |
f |
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b = lim |
f (x)− kx . |
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x→+∞ |
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x→+∞ |
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< x 6 –?.
162
*. 3.34. $ |
*. 3.35. ' |
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& # .
5 #" 6 1. , %, - -
. ', - # %
# # % , %.
5 #" 6 2. 4 , k = 0 = b % -
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lim f |
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= b # lim f |
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= b . |
x→+∞ |
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x→−∞ |
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5 #" 6 3. ! ( k b # x 6 +? x 6 –?, , x 6 +?
x6 –?.
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' 1: y = |
x2 + 2x − 1 |
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. |
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x |
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x2 + 2x − 1 |
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x2 |
+ 2x − 1 |
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= −∞ . |
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2 $ |
lim |
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= − |
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= +∞ , |
lim |
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x→0−0 |
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x→0+0 |
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x = 0 – .
' k = lim x2 + 2x − 1 = 1,
x→+∞ x2
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2 |
+ 2x − 1 |
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b = lim |
x |
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− x |
= 2 . |
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x |
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x→+∞ |
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' x 6 –? % k b. , y= x+2 % -
. 3
' 2: y = e–x sin x + x.
2 ; # , , -
%. ' :
163
) x 6+? |
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k = lim |
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e− x sin x + x |
sin x |
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= lim |
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+ 1 = 1, |
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x |
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x→+∞ |
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x→+∞ xe |
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b = lim |
(e |
− x |
sin x + x − x)= lim |
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sin x |
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= 0 . |
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e |
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x→+∞ |
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x→+∞ |
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, x 6 +? = . |
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#) x 6–? |
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e− x sin x + x |
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− x |
sin x |
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k = lim |
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= lim e |
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+ 1 = ∞ . |
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x |
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x→−∞ |
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x→−∞ |
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− x |
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& " lim |
e |
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= − lim e− x = −e∞ = −∞ . |
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x→−∞ |
x |
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x→−∞ |
, x 6 –? %.3
' 3: y = x – 2arctg x.
2 ; # , , -
%. |
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' |
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) x 6+? |
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2 arctg x |
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k = lim |
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x − 2 arctg x |
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= lim 1 |
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= 1. |
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x |
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x→+∞ |
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x→+∞ |
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b = lim |
( |
x − 2 arctg x − x |
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= −2 lim |
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arctg x |
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= −π . |
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x→+∞ |
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x→+∞ ( |
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' y = x– , x 6+?. |
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#) x 6–? |
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k = lim |
x − 2 arctg x |
= 1,b = −2 lim |
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arctg x |
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= π . |
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x→−∞ |
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x |
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x→−∞ |
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, x 6 –? y = x+ ,.3
3.40. " # !$ " % 3.15
/ ’ , # (
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y = (x2 + 1) |
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1. |
y = (17 − x2 ) (4x − 5). |
2. |
4x2 − 3. |
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3. |
y = (x3 − 4x) (3x2 − 4). |
4. |
y = (4x2 + 9) |
(4x + 8). |
5. |
y = (4x3 + 3x2 − 8x − 2) (2 − 3x2 ). |
6. |
y = (x2 − 3) |
3x2 − 2 . |
7. |
y = (2x2 − 6) (x − 2). |
8. |
y = (2x3 + 2x2 − 3x −1) (2 − 4x2 ). |
164
9. y = (x3 − 5x) (5 − 3x2 ). |
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10. |
y = (2x2 − 6x + 4) (3x − 2). |
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11. |
y = (2 − x2 ) |
9x2 − 4 . |
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12. |
y = (4x3 − 3x) |
(4x2 − 1). |
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13. |
y = (3x2 − 7) |
(2x + 1). |
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14. |
y = (x2 |
+ 16) |
9x2 − 8 . |
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15. |
y = (x3 |
+ 3x2 − 2x − 2) |
(2 − 3x2 ). |
16. |
y = (21 − x2 ) |
(7x + 9). |
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17. |
y = (2x2 − 1) |
x2 − 2 . |
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18. |
y = (2x3 − 3x2 − 2x +1) (1− 3x2 ). |
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19. |
y = (x2 − 11) |
(4x − 3). |
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20. |
y = (2x2 − 9) |
x2 − 1. |
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21. |
y = (x3 |
− 2x2 − 3x + 2) |
(1 − x2 ). |
22. |
y = (x2 |
+ 2x − 1) |
(2x + 1). |
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23. |
y = (x3 |
+ x2 − 3x − 1) (2x2 − 2). |
24. |
y = (x2 |
+ 6x + 9) |
(x + 4). |
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25. |
y = (3x2 − 10) |
4x2 − 1. |
26. |
y = (x2 |
− 2x + 2) |
(x + 3). |
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27. |
y = (2x3 + 2x2 − 9x − 3) |
(2x2 − 3). |
28. |
y = (3x2 − 10) |
(3 − 2x). |
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29. |
y = (− x2 − 4x + 13) (4x + 3). |
30. |
y = (−8 − x2 ) |
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x2 − 4 . |
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31. |
y = (9 − 10x2 ) |
4x2 − 1. |
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3.41. 5 8 !$ & ' &! 6 )# * + 7# " 8 ) "
1. & # D f (x).
2. &'% .
0- f (− x) = f (x), f (x) % . ! -
- ( Oy).
0- f (− x) = − f (x), f (x) % . !
- . 3. &'% .
0- f (x + T ) = f (x) T>0, f (x) %
. ! %
… , [–2T; –T], [–T; 0], [0; T], [T; 2T], … . , #
#- -
( .
4. &
# ( ). :
– # % f ′(x) , #
, f ′(x) = 0, ± ∞ # %;
165
– % , #
: - f ′(x) > 0 , %, - f ′(x)< 0 , # %;
–- % (
# x0D), x0 – : - -
% « » « » – , - « »« » – . 0- # %
, %.
5. & .
:
– # % f ′′(x) , : f ′′(x) = 0, ± ∞ # %;
– % ,
: - f ′′(x)< 0 , , - f ′′(x) > 0 , -
;
–- % (- -
# x0D), f ′′(x) = 0, ± ∞ # %, x0 – -
.
6.& .
) $ : , |
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# x0 D , ( - D (− ∞;∞), % |
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lim f |
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/# |
lim |
f |
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x→x −0 |
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x→x |
+0 |
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k |
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k |
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0- # , x = xk |
– |
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y = f (x). |
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#) ': - |
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lim |
f (x) |
= k lim |
f (x)− kx = b , |
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x |
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x→±∞ |
x→±∞ |
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y = f (x) ( - |
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y = kx + b |
– |
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k = 0 , b = lim f (x), y = b – ). |
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x→±∞ |
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5 #" 6 1. < x 6+? |
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x 6–? - |
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5 #" 6 2. ' #
. 7. .% .
!
' y = x3 − 4 x2
21. # : x ≠ 0 # D(y) = (−∞;0) (0;+∞ )
2.; , , :
166
y(−x) = (− x)3 − 4 = − x3 − 4 ≠ ± y(x)
(− x)2 x2
3.; % .
4.#.
: :
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y′ = |
(x3 − 4)′ x2 − |
(x3 − 4) |
(x |
2 )′ |
3x4 |
− 2x(x3 − |
4) |
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3x4 − 2x4 + 8x |
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x3 + 8 |
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= |
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4 |
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1 : |
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(−2)3 + 8 |
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x = −2 , y′ |
= |
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x3 + 8 |
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= 0 |
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x3 |
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(−2)3 |
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x = 0 , y′ |
= |
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x3 |
+ 8 |
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8 |
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= ∞ |
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x3 |
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x |
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− 2 |
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1 % x (−∞;−2) (0;+∞)− y′ > 0 |
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1 # % x (−2;0)− y′ < 0 |
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, : |
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(x0; y0 ) = (−2;−3)– |
% « » « » ( x0 #-
x0 D ).
5.. , .
: :
y′′ = |
(x3 + 8)′ x3 − (x3 + 8) (x3 )′ |
= |
3x2 x3 |
− 3x2 (x3 + 8) |
= |
3x5 − 3x5 − 24x2 |
= |
−24 |
. |
x6 |
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x6 |
x4 |
x4 |
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1 : x = 0 , y′′ = −24 = −24 = −∞ |
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x4 |
0 |
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1 x (−∞;0) (0;+∞)− y′′ < 0 . |
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167
, %, % -
#
x0 D
6. <. |
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) : , |
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# x0 D , ( - D (−∞;∞), % |
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lim |
f (x) = lim |
x3 − 4 |
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− 0(x < 0) x |
2 |
→ 0 + |
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−λ |
= −∞ . |
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x |
2 = x → 0 |
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0 = |
0 + 0 |
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x→xk −0 |
x→0−0 |
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, x = x0 = 0 – -
#) : y = kx + b – , :
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f (x) |
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(x3 − 4) |
x2 |
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1 − |
4 |
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1 |
± 0 |
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k = lim |
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x3 |
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= lim |
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= lim |
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= |
= 1. |
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x→±∞ |
x |
x→±∞ |
x |
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x→±∞ |
1 |
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b = lim |
f (x)− k x = lim |
x3 − 4 |
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x3 − 4 − x3 |
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−4 |
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− x |
= lim |
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= lim |
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2 |
= 0 . |
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2 |
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2 |
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x→±∞ |
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x→±∞ |
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x→±∞ |
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x→±∞ x |
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' (
% y = x – .
.
y(0) = x3 − 4 = ∞ – # - x2
" %
0 = x3 − 4 0 = x3 − 4 x = 3 4 – x2
# " (1,58; 0)
7. & |
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# % |
*. 3.36. |
! |
y = |
x |
3 |
− 4 |
. |
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x2 |
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. 3.36 |
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y = x3 − 4 # Maxima.3 x2
3.42. " # !$ " % 3.16
/ ’ , # (
.
' # .
168
1. |
y = (x3 + 4) x2 . |
3. |
y = 2 (x2 + 2x). |
5. |
y = 12x (9 + x2 ). |
7. |
y = (4 − x3 ) x2 . |
9. |
y = (2x3 + 1) x2 . |
11. |
y = x2 |
( |
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) |
2 . |
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x − 1 |
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13. |
y = (12 − 3x2 ) |
(x2 + 12). |
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15. |
y = −8x |
(x2 + 4). |
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17. |
y = (3x4 + 1) |
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x3 . |
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19. |
y = 8(x − 1) |
(x + 1)2 . |
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21. |
y = 4 (x2 + 2x − 3). |
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23. |
y = (x2 + 2x − 7) (x2 + 2x − 3). |
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25. |
y = − (x (x + 2))2 . |
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27. |
y = 4(x + 1)2 |
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(x2 + 2x + 4). |
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29. |
y = (x2 − 6x + 9) (x − 1)2 . |
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31. |
y = (x3 − 4) |
x2 . |
2. |
y = (x2 − x + 1) |
(x − 1). |
4. |
y = 4x2 (3 + x2 ). |
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6. |
y = (x2 − 3x + 3) |
(x − 1). |
8. |
y = (x2 − 4x + 1) |
(x − 4). |
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( |
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) |
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x2 . |
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10. |
y = |
( |
x − 1 2 |
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12. |
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) |
2 . |
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y = |
1 + 1 x |
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(x2 − 2x+13). |
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14. |
y = (9 + 6x− 3x2 ) |
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16. |
y = |
( |
( |
x |
) |
|
( |
x + |
)) |
2 . |
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− 1 |
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1 |
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18. |
y = 4x |
(x + 1)2 . |
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20. |
y = (1 − 2x3 ) |
x2 . |
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22. |
y = 4 (3 + 2x − x2 ). |
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24. |
y = 1 (x4 − 1). |
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26. |
y = (x3 − 32) |
x2 . |
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28. |
y = ( |
3x − 2) |
|
x3 . |
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30. |
y = (x3 − 27x + 54) |
x3 . |
3.43. " # !$ " % 3.17
/ ’ , # (
.
' # .
|
y = (2x + 3)e |
−2 |
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x+1 |
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e2( x+1) |
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1. |
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( |
) . |
2. |
y = |
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. |
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2 |
( |
) |
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x + 1 |
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3. |
y = 3ln |
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x |
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− 1. |
4. |
y = (3 − x)ex−2 . |
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x − |
3 |
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5. |
y = |
e2− x |
. |
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6. |
y = ln |
x |
+ 1. |
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2 − x |
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x + 2 |
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169
7. y = (x − 2)e3− x .
9. y = 3 − 3ln x . x + 4
2(x+2)
11. y = e( ).
2 x + 2
13. y = (2x + 5)e−2( x+2) .
15. y = 2ln |
x |
− 1. |
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x + 1 |
−2(x+2)
17. y = − e ( ).
2 x + 2
19. |
y = (2x − 1)e2(1− x) . |
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21. |
y = 2ln |
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x |
− 3. |
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x − 4 |
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23. |
y = |
ex+3 |
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. |
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x + |
3 |
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25. |
y = − (2x + 3)e2( x+2) . |
27. y = ln x − 5 + 2. x
29.y = ex−3 . x − 3
31. y = 2ln x − 1 + 1. x
2( x−1)
8. y = e( ).
2 x − 1
10. y = − (2x + 1)e2( x+1) .
12. |
y = ln |
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x |
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− 2. |
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x − |
2 |
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14. |
y = |
e3− x |
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3 − x |
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16. |
y = (4 − x)ex−3 . |
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18. |
y = 2ln |
x + 3 |
− 3. |
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x |
20.y = − e−( x+2) . x + 2
22. y = − (x + 1)ex+2 .
24. y = ln |
x |
− 1. |
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x + 5 |
−2( x−1)
26. y = − e ( ).
2 x − 1
28. y = (x + 4)e−( x+3) .
30. y = ln x + 6 − 1. x
3.44. " # !$ " % 3.18
/ ’ , # (
.
' #
1. |
y = 3 (2 − x)(x2 − 4x + 1). |
2. |
y = −3 (x + 3)(x2 + 6x + 6). |
3. |
y = 3 (x + 2)(x2 + 4x + 1). |
4. |
y = 3 (x + 1)(x2 + 2x − 2). |
170