Задача 2. Составить уравнение нормали (в вариантах 2.1 – 2.12) или уравнени касательной (в вариантах 2.13 – 2.31) к данной кривой в точке с абсциссой x0 .

2.1. y = (4x − x2 )

4,

x

= 2.

0

2.3. y = x − x3, x = −1.

0

2.5. y = x +

=1.

x3 ,

x

0

2.7. y = 1+

,

x

x

= 4.

1−

x

0

2.9. y = 2x2 − 3x +1,

x

=1.

0

2.11. y =

− 33

,

x

= 64.

x

x

0

2.13. y = 2x

2 + 3,

x

= −1.

0

2.15. y = 2x +

1

,

x

=1.

x

0

2.17. y =

x5

+1

,

x

=1.

x4 +1

0

2.19. y = 3(3

− 2

),

x0

= 1.

x

x

2.21. y = x (x2 +1),

x

= −2.

0

2.23. y = 2x (x2 +1)

,

x =1.

0

2.25. y =

1+ 3x2

,

x

=1.

3 + x2

0

2.27. y = 34

−

,

x

=1.

x

x

0

2.29. y = x2

10 + 3,

x

= 2.

0

2.31. y = 63

−164

3,

x

=1.

x

x

0

2.2. y = 2x2 + 3x −1,

x

= −2.

0

2.4. y = x2 + 8

− 32,

x

= 4.

x

0

2.6. y = 3

x2

− 20,

x

= −8.

0

2.8. y = 84

− 70,

x

=16.

x

0

2.10. y = (x2 − 3x + 6)

x2 ,

x

= 3.

0

2.12. y = (x3 + 2)

(x3 − 2),

x

= 2.

0

2.14. y =

x29 + 6

,

x

=1.

x4 +1

0

2.16. y = −2(x8 + 2)

(3(x4 +1)),

x0 =1.

2.18. y =

x16 + 9

,

x

=1.

1− 5x2

0

2.20. y =1 (3x + 2),

x0

= 2.

2.22. y = (x2 − 3x + 3)

3,

x

= 3.

0

2.24. y = −2(3

+ 3

),

x0

= 1.

x

x

2.26. y =14

−153

+ 2,

x

=1.

x

x

0

2.28. y = (3x − 2x3 )

3,

x

=1.

0

2.30. y = (x2 − 2x − 3)

4,

x

= 4.

0

Задача 3. Найти дифференциал dy .

34

3.1. y = xarcsin(1 x) + ln

x +

,

x2 -1

x > 0.

3.2. y = tg(2arccos

), x > 0.

1- 2x2

3.4. y = x2 arctg

−

3.3. y =

1+ 2x

- ln

x +

1+ 2x

.

x2 −1

x2 −1.

3.5. y = arccos(1

),

1+ 2x2

x > 0.

3.6. y = xln

x +

x2 + 3

-

x2 + 3.

3.7. y = arctg(sh x) + (sh x)lnch x.

3.8. y = arccos((x2 -1) (x2

)).

2

3.9. y = ln(cos2 x +

).

3.10. y = ln(x +

)-

1+ cos4 x

1+ x2

1+ x2 arctg x.

ln

x

1

x2

3.12. y = ln(ex +

)+ arcsinex .

3.11. y =

-

ln

e2x -1

2

2

2

1+ x

1+ x

3.13. y = x

+ a arcsin(x 2).

3.14. y = lntg(x 2) - x sin x.

4 - x2

3.15. y = 2x + ln

sin x + 2cos x

.

3.16. y =

-

tg3 x 3.

ctg x

3.17. y = ln

x +

x2 +1

.

3.18. y = 3

x + 2

.

2x

x - 2

x2 -1

3.20. y = ln

x2 -1

1

3.19. y = arctg

.

-

.

x2 -1

æ

x

x

ö

3.21. y = arctg

3.22. y = ln

2x +

x

2

+ x

ç tg

+1÷.

2

+1

.

2

è

ø

3.24. y = ex (cos2x + 2sin 2x).

3.23. y = ln

cos

+

tg

.

x

x

x

æ

1

ö

3.25. y = x(sinln x - cosln x).

3.26. y = ç

x -1

-

÷e2 x−1 .

2

è

ø

x

3.27. y = cos x ×lntg x - lntg

.

3.28. y =

3 + x2

- xln

x +

3 + x2

.

2

3.29. y =

− (1+ x)arctg

3.30. y = xarctg x − ln

.

1+ x2

.

x

x

3.31. y = xx2 -1 + ln x + x2 -1 .

35

Задача 4. Вычислить приближенно с помощью дифференциала.

4.1. y = 3

x = 7,76.

x

,

4.2. y = 3

x3 + 7x,

x = 1,012.

4.3. y = (x +

)

x = 0,98.

4.4. y = 3

x = 27,54.

5 − x2

2,

,

x

4.6. y = 3

x = 0,97.

4.5. y = arcsin x,

x = 0,08.

x2 + 2x + 5,

4.7. y = 3

x = 26,46.

4.8. y =

x = 1,97.

,

x2 + x + 3,

x

4.9. y = x11, x =1,021.

4.10. y = 3

,

x =1,21.

x

4.11. y = x21,

x = 0,998.

4.12. y = 3

x = 1,03.

x2 ,

4.13. y = x6 ,

x = 2,01.

4.14. y = 3

,

x = 8,24.

x

4.15. y = x7 ,

x =1,996.

4.16. y = 3

,

x = 7,64.

x

4.17. y =

x = 2,56.

4.18. y = 1

2x2 + x +1, x = 1,016.

4x −1,

4.19. y = 3

,

x = 8,36.

4.20. y =1

, x = 4,16.

x

x

4.21. y = x7 ,

x = 2,002.

4.22. y =

x =1,78.

4x − 3,

4.23. y =

x = 0,98.

4.24. y = x5 ,

x = 2,997.

x3 ,

4.25. y = 5

x = 1,03.

4.26. y = x4 ,

x = 3,998.

x2 ,

4.27. y =

, x = 0,01.

4.28. y = 3

,

x = 0,01.

1+ x + sin x

3x + cos x

4.29. y = 4

x = 1,02.

4.30. y =

x = 1,97.

2x − sin(π x 2)

,

x2 + 5,

4.31. y =1

x =1,58.

2x +1,

Задача 5. Найти производную.

2(3x3 + 4x2 − x − 2)

(2x2 −1)

5.1. y =

.

5.2. y =

1+ x2

3x3

.

15 1+ x

5.3. y =

x4 − 8x2

5.4. y =

2x2 − x −1

.

.

2(x2 − 4)

3

2 + 4x

36

	(1+ x8 )
5.5. y =			1+ x8		.
		12x12
	(x2	- 6)
5.7. y =				(4 + x2 )3		.
		120x5

5.9. y = 4 + 3x3 .

x 3(2 + x3 )2

5.11. y = x6 + x3 - 2.

1- x3

5.13. y =		1+ x2
					.
2		1+ 2x2

5.15. y =		(1+ x2 )3		.
		3x3

5.17.y = 2x + 3(x - 2).

x2

	(2x2	+ 3)
5.19. y =			x2 - 3
		9x3	.

5.21.y = (2x +1)x2 - x .

x2

5.23. y =			1								.
		(x + 2)
						x2 + 4x + 5
5.25. y =	3× 3			(x +1)
								.
				(x -1)2
		x					.
5.27. y =			x +1
		x2 + x +1

5.6. y =			x2			.
2		1-		3x4

	(x2	- 8)
5.8. y =			x2 - 8
		6x3	.

5.10.y = 3 (1+ x34 )2 .

x32

	(x2	- 2)
5.12. y =			4 + x2		.
		24x3

5.14. y = x -1(3x + 2).

4x2

5.16. y = x6 + 8x3 -128.

8 - x3

5.18. y = (1- x2 )5x3 + 1x .

5.20. y =		x −1
										.
	(x2 + 5)
								x2 + 5
		1-					.
5.22. y = 2				x
		1+
				x

5.24.y = 3 3x2 + x +1.

x+1

5.26. y =			x + 7
							.
	6		x2 + 2x + 7
5.28. y =		x2 + 2			.
	2	1- x4

37