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Индивидуальные задания для студентов первого курса по теме Производные и Графики функций 1

1Красным цветом выделены необязательные задания.

Раздел 1. ПРОИЗВОДНЫЕ

Задание • 1. Найти f′(a) непосредственно по определению.

1.1.	f(x) = sin 5x,				a = 0						1.19.	f(x) =	1			,								a = 2

													(x 1)2
1.2.	f(x) = cos 9x,				a = 0						1.20.	f(x) =	1							,						a = 0
1.2.	f(x) = cos 9x,				a = 0						1.20.	f(x) =			2					,						a = 0
													(2x + 3)
1.3.	f(x) = tg x,			a =		π					1.21.	f(x) = e 4x,			a = 0
1.3.	f(x) = tg x,			a =							1.21.	f(x) = e 4x,			a = 0
					4
1.4.	f(x) = ctg x,				a =		π				1.22.	f(x) =	1						,						a = 0
1.4.	f(x) = ctg x,				a =						1.22.	f(x) =							,						a = 0
					4								2 + sin x
1.5.	f(x) = 3x2,			a = 0							1.23.	f(x) =	1												,			a = 0

																	2
													sin(x +				2						)
													sin(x +										)
															3
1.6.	f(x) = (x 2)3, a = 0										1.24.	f(x) =	1	,	a = e

													ln x
1.7.	f(x) = x3,		a = 0								1.25.	f(x) =	1													,		a = 0


													x2 + 2x + 3
1.8.	f(x) = e2x,			a = 0							1.26.	f(x) = x sin x, a = 0
1.9.	f(x) = ln 2x,				a = 1						1.27.	f(x) =	1									, a =							π

													2x sin x																2
1.10.	f(x) = 5x,		a = 0								1.28.	f(x) = (x 1)4,												a = 0
1.11.	f(x) = 3x x2, a = 0										1.29.	f(x) =	1								,					a = 0

													(2x 1)4
1.12.	f(x) = 2 sin x x2,									a = 0	1.30.	f(x) = sin3 x,			a = 0
1.13.	f(x) = 2 tg(x 3),								a = 3		1.31.	f(x) = x2 + 2x 3,																a = 1
1.14.	f(x) = sin 2(x + 1),									a = 1	1.32.	f(x) = cos(x + 1),																a = 1
1.15.	f(x) = (x 4)2, a = 2										1.33.	f(x) = (2x 3)3,														a = 1
1.16.	f(x) = 5 x,			a = 1							1.34.	f(x) = ex+2,			a =												2
		1						π
1.17.	f(x) =	1	,		a =			π			1.35.	f(x) = lg(x 2),													a = 4

		sin x						2
1.18.	f(x) =	1		,	a = 0						1.36.	f(x) =	2					,							a = 1
1.18.	f(x) =			,	a = 0						1.36.	f(x) =						,							a = 1
		cos x											x(x + 1)

Задание • 2. Написать уравнение нормали (в вариантах 1 18) или касательной (в вариантах 19 36) к данной кривой в точке с абсциссой x0.

2.1.

y =

x0 = 2

2.19.

y = 2px + 3px,

x0 = 16

y = 2p

y = 3x2 2x + 1, x0 = 1

16x,

2.2.

2.20.

x0 = 4

2.3.

2.21.

x17 + 2

y = 4

x + 1 + 3, x0 = 7

y =

, x0 = 1

2.4.

2.22.

x5 + 1

y = 2

x 4 x, x0 = 64

y = x2 + 3, x0 = 2

y = px3 35,

2.5.

x0 = 16

2.23.

y = 2x2 + 2x2 ,

x0 = 1

, x0 = 3

2x3 + 1

= 2

2.6.

y = 3(x + 1) +

2.24.

y =

x + 1

5x3 1

2.7.

y =

2(x2 + 1)

x0 = 1

2.25.

y = 4x4 + 3x3 2x + 1, x0 = 2

3(x4 + 2)

y =

5(x6 + 1)

2.8.

= 4

2.26.

y =

= 1

px + 1

2x2 + 1

y = 2x + 3px,

2.9.

x0 = 16

2.27.

y =

1 +

x0 = 1

2.10.

y = 2x3 + x2 x + 1, x0 = 4

2.28.

y = 5x2 0.1x + 7,

x0 = 1

2.11.

, x0 = 1

2.29.

y = 5

x 21, x0 = 32

y = 2 x + 3x

2.12.

2 + x2

2.30.

y = 3 + x3 , x0 = 1

y = 2x

2.13.

2.31.

4 x 11, x0 = 4

y = 3 x + 2 +

, x0

= 7

y =

x + 7 x, x0 = 1

x + 2

2.14.

y =

8 x3

= 1

2.32.

y =

x7 2

8 + x3

2x2 + 1

y = 5x2 2x + 3, x0 = 1

y = 5p

+ x2,

2.15.

2.33.

x 2

x0 = 6

x15 + 9

2x + 1

= 1

2.16.

y =

x0 = 1

2.34.

y =

2x4

x6 + x2

2.17.

y = x3 4x2 + 1, x0 = 1

2.35.

y = x4 + 2x3 8,

x0 = 2

2.18.

y = 0.2x5 3x,

x0 = 2

2.36.

y = 2p

+ x3,

x0 = 1

Задание • 3. Составить уравнение касательной (в вариантах 1 18) или нормали (в вариантах 19 36) к данной кривой в точке, соответствующей значению параметра t = t0.

3.1.

x(t) = a cos t,

y(t) = a sin t,

x(t) =

3.2.

2 cos t,

y(t) = 2 sin t,

3.3. x(t) = t4

y(t) = t

4t , t0 = 2

3.4.

(

) = 0

x t

.2t

y(t) = 2t + 2, t0 = 2

3.5.

x(t) = ln tg t,

y(t) = ctg t,

t0 =

3.6.

2 + t2

x(t) =

y(t) =

3 t

t0 = 1

3.7.

( ) = t

x t

y(t) =

, t0 = 3

t + 1

x t

3.8.

(

) = sin 3

y(t) = cos t,

t0 =

3.9.

x(t) = 2t

y(t) = 0.1t + 5, t0 = 3

3.10.

x(t) = 23t

+ t 1,

y(t) = t

1, t0 = 2

3.11.

x(t) = sin

2t,

y(t) = cos

t0 =

3.12.

x(t) = 3 ctg t,

y(t) = 2 sin t + cos t, t0 =

x(t) =

y(t) = t2 + 1, t0 = 2

x(t) =

t2 + 1

y(t) = 2t, t0 = 1

x(t) = sin t + cos t,

y(t) = 2 tg t, t0 =

x(t) = 3t cos t,

y(t) = 3t sin t, t0 =

x(t) = sin3 t 1,

y(t) = cos3 t 1, t0 =

x(t) = ln tg t,

y(t) = cos2 t, t0 =

x(t) =

y(t) = t2, t0 = 2

x(t) =

t3 + 1

y(t) = t3 + 2, t0 = 1

x(t) = 4 cos t,

y(t) = 3 sin t, t0 =

4π

x(t) = t t2,

y(t) = t2 2t3, t0 = 3

x(t) =

+ 4t,

y(t) =

, t0 = 1

t + 1

x(t) = 3t2 + 1,
2
y(t) = t2 + 1	, t0	= 2

x(t) = t t3,

y(t) = t2 + 2t4, t0 = 1

x(t) = t3 1, y(t) = t2, t0 = 2

x(t) = ln 2t,
y(t) = 2 ln(4t 1),	t0 =		1

			2
x(t) = et + 1,
y(t) = cos t + 2, t0 = 0
x(t) = 3 4 cos t,		π
y(t) = 1 + 2 tg t, t0 =		π

		4
x(t) = a cos t,
y(t) = b sin t, t0 =	π

	3

3.31.

x(t) = 3t4 2t2,

y(t) = 2t + 5, t0 = 1

3.32.

x(t) = sin 2t + t,

y(t) = 2 cos t + 2t, t0 = 0

3.33.

(

) = t1

+ 1

x t

t + 1

2, t0

= 1

y(t) = t3

3.34.

x(t) = 2 + 2 sin t,

y(t) = 3 cos 2t,

t0 =

3.35.

(

) =

x t

+ 2t2

y(t) =

, t0 = 3

2+ 2

3.36.

x(t) = 34t

+ 1,

1, t0

= 2

y(t) = t

Задание • 4. Найти дифференциал функции.

	1	+ arctg p
4.1. y = ln			x
	x2 + 1

4.2. y = x arcsin x + x3 + 1

4.3.y = e2x(cos 3x + 2 sin 3x)

4.4.y = arcsin e x + ln cos x

p p

4.5. y = x2 ln 5 + x2 5 + x2

4.6.y = sin x ln ctg x 2 ln ctg x2

√

4.7.y = 5 x 1 x + 5

4.8.y = cos ln x + 2 sin ln x

x2 + 1

4.9. y = arctg

+ 1

4.10.

y = arccos

+ x

4.11. y = lg p

+ x tg x

sin x

+ x

4.12. y = log2

x2 + 1

4.13. y = arctg ln x + earctg x

4.14. y = p1 + x2 + arcsin

x + 1

4.15. y = ln 2 + 3x2 (x + 1) arcsin x

4.16. y =	√3
4.16. y =	√3	2x2+		3
		x2	1

	3
4.17. y = ln(e2x + pe3x 1) + arctg e3x
x2	1

4.18. y = sin x2 + 4 + arcsin x2 + 4

4.19. y = arctg(2 + tg 2x)
4.20. y = px	(1 + px) arcsin	1
3	3	1 + p

			x2
		3

4.21. y = arctg(ln(3 + x2))

4.22. y = 3x2 + arccos

x2 + 3

+ 1

+ 2

4.23. y = lg

px2

4.24. y = x arctg

+ ln cos x

4.25.

+ 11

arctg x2 + 11

y =

4.26. y = cos(ex2 + 1) ln(ex2 + 1)

4.27. y = cos(2 ln(x4 + 1)) sin(2 ln(x4 + 1))

4.28. y = √ 3 1 x

4 cos2(x + 1)

4.29. y =

sin x

x2 + 1

x2 + 7

4.30.

x2 1

1 2)e

y = ( x

4.31.

ln x

y = log

px2

+ 8

2 arcsin

sin x

4.32. y = arctg2

ln x

x2 + 2

4.33. y =

sin3(x2 3)

+ x lg(x3 + 4)

px3 2

4.34. y = ln arcsin(x3 1) + x sin p

4.35. y =

x4 1

sin(ex

+ x2)

p2x + 2

4.36. y = arcctg sin √5

x + 1

2 lg px3

x 2

Задание • 5. Вычислить приближенно с помощью дифференциала.

5.1.

5.19.

y = p

x = 31.98

y =

+ 1,

x = 2.001

5.2. y = arcsin x,

x = 0.04

5.20. y =

, x = 0.01

4 + tg x

5.3. y = arctg x,

x = 1.02

x = 7.82

5.21. y = px,

5.4. y = p

2x + 3

x = 2.97

5.22. y =

x = 1.04

px3

5.23. y = x4,

5.5. y =

x = 7.99

x = 4.01

5.6. y = x5,

x = 2.02

5.24. y = ln(7x + 1), x = 0.03

5.7. y = p

5.25. y = ex,

1 + sin x,

x = 0.02

x = 0.01

5.8. y = p

x = 0.001

5.26. y =

x = 4.001

4 + tg x,

p3x + 4

5.27. y = p

5.9. y = sin x,

x = 0.251π

2x,

x = 8.002

1003

5.28. y = (x + 1)4,

5.10. y = cos x,

x =

x = 3.01

3000

5.11.

5.29.

y =

2x 2,

x = 5.001

y =

1 + sin x + tg x, x = 0.001

5.12. y = p3x + 4,

x = 3.999

5.30. y = p2x2 + 5x + 9, x = 2.001

5.13. y = tg x,

x = 0.253π

5.31. y = 2 tg π(x 2), x = 2.01

5.14. y =

x = 0.01

5.32. y =

x = 4.97

1 + 2x

x 1

x = 32.003

5.33. y = px2 + 3x,

x = 0.99

5.15. y = px3,

5.16. y = p

5.34. y = ln(x2 3), x = 1.98

2x2 + 3x + 2

x = 1.97

x = 0.01

5.35. y = (x + 1)3,

x = 1.005

5.17. y = p2x + cos x + 7,

5.18. y = √8x + tg

πx

, x = 1.01

5.36. y = px2 + 2,

x = 5.02

Задание • 6. Найти y′.

6.1. y =

2(3x3 + 4x2

x 2)

15 1 + x

(2x2 1)p

6.2. y =

1 + x2

3x3

6.3. y =

x4 8x2

2(x2 4)

6.4. y =

2x2

x 1

3 2 + 4x

(1 + x8)p

6.5. y =

1 + x2

12x12

6.6. y =

2 1 3x

(x2 6)√

6.7. y =

(4 + x2)3

120x5

6.8. y =

6x3

4+ 3x3

6.9.y = √

x 3 (2 + x3)2

√

6.10. y =

(1 + x2)2

6.11. y =

+ x3 2

p1 x3

4 + x

6.12. y =

(x 2)

24x2

1 + x2

6.13. y =

2 1 + 2x

6.14. y =

x 1(3x + 2)

√

4x2

6.15. y =

(13x3

+ x2)3

6.16. y =

x6 + 8x3 128

p8 x3

6.17. y =

2x + 3(x 2)

6.18. y = (1

x2)√5 x3 +

6.19. y =

(2x3 + 3)p

x2 3

9x3

y =

x 1

6.20.

+ 5

+ 5)

6.21. y =

(2x + 1)

√

6.22. y = 2

1 + px

y =

p 1

6.23.(x + 2) x2 + 4x + 5 p

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

6.25.

y =

3√

(

x + 2

1)2

x + 1)(x

+ 2x + 7

y =

(x + 7) x

6.26.

xx + 1

6.27.y = x2 + 3x + 5

+ 2

6.28. y =

6.29. y =

(x + 3)

2x 1

2x + 7

3x + p

6.30. y =

x2 + 2

x2 + p

6.31. y =

1 2x2 x3

y =

1 + p

6.32.

x 3

2 + x

6.33. y =

(1 x)p

3x2 + x

6.34. y =

(x3 5)px + 2

6.35. y =

(√

x 4

1 3

6.36. y =

(x + 2)(4

5x)

2x x2

Задание • 7. Найти y′.

7.1. y =

2 (x

+ x

+ 2)

7.2. y = p

+ arccos e2x

2 + 3e2x

7.3.

px p

+ 10x 30

y = 2e

(

+ 60

x + 30)

p p p

7.4. y = ln( 1 ex 1) ln 1 ex x + (x 1) 1 ex

7.5.y = arctg(e x ex)

7.6.y = ((x2 1) cos x + (x 1)3 sin 2x)ex

(		)
7.7. y = ecos x	x2	1
			sin x

7.8. y = ln		ex + 1 + p
		ex + 1 + p			1 e2x e3x
					1 e2x e3x
		ex 1 + p1 e2x e3x
		ex 1 + p1 e2x e3x
7.9.	px		p 3		4
	5		4		p
	5			x 2 x + 1)
	y = 5e (
	y = 5e (

(√ )

3 a

7.10. y =

arctg e

7.11. y =

x2 + 1

7.12. y =

+ x

e 3 + 1

7.13. y = x2 log2(2x + 1)

arctg2

2 2 arctg 2

7.14. y = ln(arcsin e x + ex) + p

1 + ex

7.15.

3 ln(√e

y = 3 arctg

e 3

+ 1(1 + e

)) + 3x

7.16. y = x +

+ 2 ln(3x + 3)

+ 1

7.17. y = 1 + ln(1 + ex + 1 + ex)

7.18. y = e2x(2 sin 3x 3 cos 3x)(e3 e2)

7.19. y = 13e3x(3 sin 3x)

7.20. y = e3x(3 sin 2x 2 cos 2x)(e3 e2)

7.21. y =

arcctg

e2x 2

35h2x

log2(e3x 5x)

7.22. y =

23x

7.23.

y =

ln 3

1 arcsin 3

1 1 + 3x

7.24. y =

ln 3

3 1

7.25. y = 4 arctg e4x +

ln(e4x

+ 1)

7.26. y =

√arcsin5 e3x

7.27. y =

8e2x + 7ex + 6

3(ex + 1)2

7.28. y = p

arccos3 e4x

7.29. y =

a cos 26x + 26 sin 26x

2(a2 + 3b2)

7.30. y = p

+ log2

+ 1

e2x + 1

e + 1 + 2

7.31. y = ln3

(e3x +

)

x 3

7.32. y = sin3(ln

+ e2x 3) +

5 + x

7.33. y =

5 cos3

(2x + log (4x

5))

√

ln(2x

33)

7.34. y = arcsin

x−1

log

5x)

42 3x + 34 3x

7.35.y = arccos(x2 32)

7.36.y = arcctg2(ex3 + log4(x4 x2 + 1))

8.26. y = ln 4

Задание • 8. Найти y′.

sin 1

8.1. y = lg x lg x

1 8.2. y = ln ln cos(1 x2 )

1 8.3. y = log2(arcsin p )

3 x

8.4.y = lg xp10 p1 + x3

x 10 + 1 + x3

8.5. y = arctg(ln 1 + ex)

1 8.6. y = log3 p3 1 + x3

8.7. y = ln sin x + 1 x 1

8.8. y = log3 log2 sin x

8.9. y = lg tg

x 2

x + 2

8.10. y = ae√

3 x

p3 x + p3

8.11. y = ln2 sin (

)

8.12. y = log2 1 + x2

x2 + a2

8.13. y = lg x2 a2

8.14. y = log3 pax4 1

8.15. y = lg(ax + a2x2 + a)

8.16.y = 3 x 2 ln(3 3 x)

8.17.y = ln3 ln2 ln x

	sin x	+ p
			3
2		p
8.18. y = lg	cos x2
				3

8.19. y = ln(e2x 3 ex + 1)

8.20. y = ln( ax2 + b2 + ax3)

√

8.21. y = arctg 1 + ln(1 + ex)

	ln(	1 + tg x + 3 tg x)
8.22. y =		p
			3

8.23.y = x3 (sin ln x cos ln x)

8.24.y = log11 log6 ctg x5

8.25.y = log2 log3 tg x2

√

1 + 4x

sin3 x

8.27.y = ln5(sin 5x + 1)

8.28.y = lg3(x3 + sin 3x)

8.29.y = ln( x2 + 1 + 3 x2 + 1)

8.30. y =

x + a ln(

x + a

8.31. y =

log22 cos (π

)

8.32. y = lg

2 x + sin x

8.33. y = arcsin ln

x + 1

x 1

√3

8.34. y = log3

1 + 3x

2 x

8.35. y = ln3 sin log2 x + 1 2

8.36. y = cos2 ln 3 x 8

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