CalcI Derivatives Assignments

CALCULUS I Assignment Problems

Derivatives

Paul Dawkins

Calculus I

© 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

Table of Contents Preface ............................................................................................................................................. 1 Derivatives ...................................................................................................................................... 1

Introduction .............................................................................................................................................. 1 The Definition of the Derivative .............................................................................................................. 2 Interpretations of the Derivative ............................................................................................................. 4 Differentiation Formulas .......................................................................................................................... 8 Product and Quotient Rule ..................................................................................................................... 11 Derivatives of Trig Functions ................................................................................................................ 12 Derivatives of Exponential and Logarithm Functions ......................................................................... 15 Derivatives of Inverse Trig Functions ................................................................................................... 16 Derivatives of Hyperbolic Functions ..................................................................................................... 17 Chain Rule ............................................................................................................................................... 17 Implicit Differentiation........................................................................................................................... 21 Related Rates .......................................................................................................................................... 24 Higher Order Derivatives ....................................................................................................................... 27 Logarithmic Differentiation ................................................................................................................... 29

Preface Here are a set of problems for my Calculus I notes. These problems do not have any solutions available on this site. These are intended mostly for instructors who might want a set of problems to assign for turning in. I try to put up both practice problems (with solutions available) and these problems at the same time so that both will be available to anyone who wishes to use them. As with the set of practice problems I write these as I get the time and some sections will have only a few problems at this point and others won’t have any problems in them yet. Rest assured that I’m always trying to get more problems written but this site has been written and maintained in my spare time and so I usually cannot devote as much time as I’d like to the site.

Derivatives

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems.

http://tutorial.math.lamar.edu/terms.aspx

Calculus I

© 2007 Paul Dawkins 2 http://tutorial.math.lamar.edu/terms.aspx

Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. The Definition of the Derivative Interpretation of the Derivative Differentiation Formulas Product and Quotient Rule Derivatives of Trig Functions Derivatives of Exponential and Logarithm Functions Derivatives of Inverse Trig Functions Derivatives of Hyperbolic Functions Chain Rule Implicit Differentiation Related Rates Higher Order Derivatives Logarithmic Differentiation

The Definition of the Derivative Use the definition of the derivative to find the derivative of the following functions. 1. ( ) 10g x =

2. ( ) 8T y = −

3. ( ) 5 7f x x= +

4. ( ) 1 12Q t t= −

5. ( ) 2 3f z z= +

6. ( ) 2 8 20R w w w= − +

7. ( ) 26V t t t= −

8. ( ) 22 8 10Q t t t= − +


Calculus I


9. ( ) 21 10 7g z z z= + −

10. ( ) 35f x x x= −

11. ( ) 32 9 5Y t t t= + +

12. ( ) 3 22Z x x x x= − −

13. ( ) 23

f tt

=−

14. ( ) 21xg x

x+

=−

15. ( )2

2tQ t

t=

+

16. ( ) 8f w w= +

17. ( ) 14 3V t t= +

18. ( ) 2 5G x x= −

19. ( ) 1 4Q t t= +

20. ( ) 2 1f x x= +

21. ( ) 1W tt

=

22. ( ) 41

g xx

=−

23. ( )f x x x= +


Calculus I


24. ( ) 1f x xx

= +

Interpretations of the Derivative

For problems 1 – 3 use the graph of the function, ( )f x , estimate the value of ( )f a′ for the

given values of a. 1. (a) 5a = − (b) 1a =

2. (a) 2a = − (b) 3a =

3. (a) 3a = − (b) 4a =


Calculus I


For problems 4 – 6 sketch the graph of a function that satisfies the given conditions. 4. ( )7 5f − = , ( )7 3f ′ − = − , ( )4 1f = − , ( )4 1f ′ =

5. ( )1 2f = , ( )1 4f ′ = , ( )6 2f = , ( )6 3f ′ =

6. ( )1 9f − = − , ( )1 0f ′ − = , ( )2 1f = − , ( )2 3f ′ = , ( )5 4f = , ( )5 1f ′ = −

For problems 7 – 9 the graph of a function, ( )f x , is given. Use this to sketch the graph of the

derivative, ( )f x′ .

7.

8.


Calculus I


9.

10. Answer the following questions about the function ( ) 21 10 7g z z z= + − .

(a) Is the function increasing or decreasing at 0z = ? (b) Is the function increasing or decreasing at 2z = ? (c) Does the function ever stop changing? If yes, at what value(s) of z does the function stop changing? 11. What is the equation of the tangent line to ( ) 35f x x x= − at 1x = .

12. The position of an object at any time t is given by ( ) 22 8 10s t t t= − + .

(a) Determine the velocity of the object at any time t. (b) Is the object moving to the right or left at 1t = ? (c) Is the object moving to the right or left at 4t = ? (d) Does the object ever stop moving? If so, at what time(s) does the object stop moving?


Calculus I


13. Does the function ( ) 2 8 20R w w w= − + ever stop changing? If yes, at what value(s)

of w does the function stop changing? 14. Suppose that the volume of air in a balloon for 0 6t≤ ≤ is given by ( ) 26V t t t= − .

(a) Is the volume of air increasing or decreasing at 2t = ? (b) Is the volume of air increasing or decreasing at 5t = ? (c) Does the volume of air ever stop changing? If yes, at what times(s) does the volume stop changing? 15. What is the equation of the tangent line to ( ) 5 7f x x= + at 4x = − ?

16. Answer the following questions about the function ( ) 3 22Z x x x x= − − .

(a) Is the function increasing or decreasing at 1x = − ? (b) Is the function increasing or decreasing at 2x = ? (c) Does the function ever stop changing? If yes, at what value(s) of x does the function stop changing?

17. Determine if the function ( ) 14 3V t t= + increasing or decreasing at the given points.

(a) 0t = (b) 5t = (c) 100t =

18. Suppose that the volume of water in a tank for 0t ≥ is given by ( )2

2tQ t

t=

+ .

(a) Is the volume of water increasing or decreasing at 0t = ? (b) Is the volume of water increasing or decreasing at 3t = ? (c) Does the volume of water ever stop changing? If so, at what times(s) does the volume stop changing? 19. What is the equation of the tangent line to ( ) 10g x = at 16x = ?

20. The position of an object at any time t is given by ( ) 1 4Q t t= + .

(a) Determine the velocity of the object at any time t. (b) Does the object ever stop moving? If so, at what time(s) does the object stop moving? 21. Does the function ( ) 32 9 5Y t t t= + + ever stop changing? If yes, at what value(s)

of t does the function stop changing?


Calculus I


Differentiation Formulas For problems 1 – 20 find the derivative of the given function. 1. ( ) 3 88 4 2g x x x= − +

2. ( ) 10 5 3 27 2f z z z z z= − + −

3. 4 38 10 9 4y x x x= − − + 4. ( ) 4 43 3f x x x x−= + −

5. ( ) 10 109 8 12R t t t−= + +

6. ( ) 6 3 13 8 9h y y y y− − −= − +

7. ( ) 7 3 2 42 6 8 1g t t t t t− − −= + − + −

8. 6 47 3z x x x= − +

9. ( ) 9 34 7 427 2f x x x x= − +

10. ( ) 5629

76h y y yy

= + +

11. ( ) 2 5

4 1 17 2

g zz z z

= + −

12. 2 39 3

2 1 93 7

y t tt t

= + − −

13. ( ) 36 5 2

1 1W x xx x

= − +

14. ( ) ( )( )25 1g w w w= − +


Calculus I


15. ( ) ( )31 9h x x x= −

16. ( ) ( )233 2f t t= −

17. ( ) ( )( )21 2 2g t x x x= + − +

18. 24 8 2x xy

x− +

=

19. ( )4 2

3

2 7t t tY tt

− +=

20. ( ) ( )2 523

w w wS w

w− +

=

For problems 21 – 26 determine where, if anywhere, the function is not changing. 21. ( ) 3 22 9 108 14f x x x x= − − +

22. ( ) 2 3 445 300 20 3u t t t t= + + −

23. ( ) 3 29 10Q t t t t= − + −

24. ( ) 3 22 3 4 5h w w w w= + + +

25. ( ) 2 3 49 8 3g x x x x= + + −

26. ( ) ( )22 1G z z z= −

27. Find the tangent line to ( ) 5 23 4 9 12f x x x x= − + − at 1x = − .

28. Find the tangent line to ( )2 1xg xx+

= at 2x = .

29. Find the tangent line to ( ) 42 8h x x x= − at 16x = .


Calculus I


30. The position of an object at any time t is given by ( ) 4 3 23 44 108 20s t t t t= − + + .

(a) Determine the velocity of the object at any time t. (b) Does the object ever stop changing? (c) When is the object moving to the right and when is the object moving to the left? 31. The position of an object at any time t is given by ( ) 3 4 51 150 45 2s t t t t= − + − .

(a) Determine the velocity of the object at any time t. (b) Does the object ever stop changing? (c) When is the object moving to the right and when is the object moving to the left? 32. Determine where the function ( ) 3 24 18 336 27f x x x x= − − + is increasing and decreasing.

33. Determine where the function ( ) 4 3 22 15 9g w w w w= + − − is increasing and decreasing.

34. Determine where the function ( ) 3 224 192 50V t t t t= − + − is increasing and decreasing.

35. Determine the percentage of the interval [ ]6,4− on which ( ) 3 4 57 10 5 2f x x x x= + − − is

increasing. 36. Determine the percentage of the interval [ ]5,2− on which ( ) 4 3 23 8 144f x x x x= − − is

decreasing. 37. Is ( ) 2 33 2h z x x x= − + + increasing or decreasing more on the interval [ ]1,1− ?

38. Determine where, if anywhere, the tangent line to ( ) 212 9 3f x x x= − + is parallel to the

line 1 7y x= − . 39. Determine where, if anywhere, the tangent line to ( ) 2 38 4 2f x x x x= + + − is perpendicular

to the line 1 84 3

y x= − + .

40. Determine where, if anywhere, the tangent line to ( ) 3 8f x x x= − is perpendicular to the

line 2 11y x= − .

41. Determine where, if anywhere, the tangent line to ( ) 13 19xf x

x= + is parallel to the line

y x= .


Calculus I


Product and Quotient Rule For problems 1 – 7 use the Product Rule or the Quotient Rule to find the derivative of the given function.

1. ( ) ( )( )3 22 3 8h z z z= − +

2. ( ) ( )32 7 2f x x xx

= − −

3. ( )( )2 35 1 12 2y x x x x= − + + −

4. ( )3

21xg xx

=+

5. ( )24

6y yZ y

y−

=−

6. ( )2

3

1 105 2

t tV tt t

− +=

+

7. ( ) ( )( )1 4 23 9w w

f ww

− +=

+

For problems 8 – 12 use the fact that ( )3 12f − = , ( )3 9f ′ − = , ( )3 4g − = − , ( )3 7g′ − = ,

( )3 2h − = − and ( )3 5h′ − = determine the value of the indicated derivative.

8. ( ) ( )3f g ′ −

9. ( )3hg

′ −


Calculus I


10. ( )3f gh

′ −

11. If ( ) ( )y x f x h x= − determine 3x

dydx =−

.

12. If ( ) ( )

( )1 g x h x

yx f x

−=

+ determine

3x

dydx =−

.

13. Find the equation of the tangent line to ( ) ( )( )2 28 1f x x x x= − + + at 2x = − .

14. Find the equation of the tangent line to ( )3

2

42xf x

x x−

=+

at 1x = .

15. Determine where ( ) 2

212

zg zz

−=

+ is increasing and decreasing.

16. Determine where ( ) ( )( )23 1 2R x x x x= − − + is increasing and decreasing.

17. Determine where ( )2

2

71 2

t th tt

−=

+ is increasing and decreasing.

18. Determine where ( ) 11

xf xx

+=

− is increasing and decreasing.

19. Using the Product Rule for two functions prove the Product Rule for three functions.

( )f g h f g h f g h f g h′ ′ ′ ′= + +

Derivatives of Trig Functions For problems 1 – 6 evaluate the given limit.

1. ( )0

3limsint

tt→

2. ( )

0

sin 9lim

10w

ww→


Calculus I


3. ( )( )0

sin 2lim

sin 17θ

θθ→

4. ( )

0

sin 4lim

3 12x

xx→

++

5. ( )

0

cos 1lim

9x

xx→

−

6. ( )

0

cos 8 1lim

2z

zz→

−

For problems 6 – 10 differentiate the given function. 6. ( ) ( ) ( )4 9sin 2 tanh x x x x= − +

7. ( ) ( ) ( ) ( )8sec cos 4cscg t t t t= + −

8. ( ) ( )6cot 8cos 9y w w= − +

9. ( ) ( ) ( )8sec cscf x x x=

10. ( ) ( )98 tanh t t t= −

11. ( ) ( )5 26 8 sinR x x x x= +

12. ( ) ( )3

cos3

zh z z

z= −

13. ( ) ( )( )

1 cos1 sin

xY x

x+

=−

14. ( ) ( )( )

sec3

1 9 tanw

f w ww

= −+


Calculus I


15. ( ) ( )2

cot1

t tg t

t=

+

16. Find the tangent line to ( ) ( )2 tan 4f x x x= − at 0x = .

17. Find the tangent line to ( ) ( )secf x x x= at 2x π= .

18. Find the tangent line to ( ) ( ) ( )cos secf x x x= + at x π= .

19. The position of an object is given by ( ) ( ) ( )9sin 2cos 7s t t t= + − determine all the points

where the object is not changing. 20. The position of an object is given by ( ) ( )8 10sins t t t= + determine where in the interval

[ ]0,12 the object is moving to the right and moving to the left.

21. Where in the range [ ]6,6− is the function ( ) ( )3 8cosf z z z= − is increasing and

decreasing. 22. Where in the range [ ]3,5− is the function ( ) ( ) ( )7cos sin 3R w w w= − + is increasing and

decreasing. 23. Where in the range [ ]0,10 is the function ( ) ( )9 15sinh t t= + is increasing and decreasing.

24. Using the definition of the derivative prove that ( )( ) ( )cos sind x xdx

= − .

25. Prove that ( )( ) ( ) ( )sec sec tand x x xdx

= .

26. Prove that ( )( ) ( )2cot cscd x xdx

= − .

27. Prove that ( )( ) ( ) ( )csc csc cotd x x xdx

= − .


Calculus I


Derivatives of Exponential and Logarithm Functions For problems 1 – 12 differentiate the given function. 1. ( ) 10 9z zg z = −

2. ( ) ( ) ( )4 119log 12logf x x x= +

3. ( ) 6 4t th t = − e

4. ( ) ( ) ( )12320ln logR x x x= +

5. ( ) ( )2 6 3 tQ t t t= − + e

6. 8 9v vy v= + 7. ( ) ( ) ( )6

4log lnU z z z z= −

8. ( ) ( ) ( )3log logh x x x=

9. ( ) 11 7

w

wf w −=

+ee

10. ( ) ( )3

1 4ln5

tf t

t+

=

11. ( ) ( )27log

7rr r

g r+

=

12. ( ) ( )4

ln

ttV tt

=e

13. Find the tangent line to ( ) ( )1 8 xf x x= − e at 1x = − .

14. Find the tangent line to ( ) ( )23 lnf x x x= at 1x = .

15. Find the tangent line to ( ) ( )3 8lnxf x x= +e at 2x = .


Calculus I


16. Determine if ( ) 4 3y yU y = − e is increasing or decreasing at the following points.

(a) 2y = − (b) 0y = (c) 3y =

17. Determine if ( ) ( )2

lnzy z

z= is increasing or decreasing at the following points.

(a) 12

z = (b) 2z = (c) 6z =

18. Determine if ( ) 2 xh x x= e is increasing or decreasing at the following points.

(a) 1x = − (b) 0x = (c) 2x =

Derivatives of Inverse Trig Functions For each of the following problems differentiate the given function. 1. ( ) ( ) ( )1sin 9sinf x x x−= +

2. ( ) ( ) ( )1 15sin cosC t t t− −= −

3. ( ) ( ) ( )1 1tan 4cosg z z z− −= +

4. ( ) ( ) ( )1 3 1sec cosh t t t t− −= −

5. ( ) ( ) ( )2 1sinf w w w w−= −

6. ( )( ) ( )( )1 1cot 1 cscy x x x− −= − +

7. ( ) ( )1

1tan

zQ zz−

+=

8. ( ) ( )( )

1

1

1 sin1 cos

tA t

t

−

−

+=

−


Calculus I


Derivatives of Hyperbolic Functions For each of the following problems differentiate the given function. 1. ( ) ( )2 3sinhh w w w= −

2. ( ) ( ) ( )cos coshg x x x= +

3. ( ) ( ) ( )3csch 7sinhH t t t= +

4. ( ) ( ) ( )tan tanhA r r r=

5. ( ) ( )coshxf x x= e

6. ( ) ( )sech 11

zf z

z+

=−

7. ( ) ( )( )

cothsinh

wQ w

w w=

+

Chain Rule

For problems 1 – 46 differentiate the given function.

1. ( ) ( )113 8g x x= −

2. ( ) 7 39g z z=

3. ( ) ( )639 2h t t t= + −

4. 3 28y w w= +

5. ( ) ( ) 2214 3R v v v−

= −

6. ( )( )8

26 5

H ww

=−


Calculus I


7. ( ) ( )4sin 4 7f x x x= +

8. ( ) ( )tan 1 2 xT x = − e

9. ( ) ( )( )2cos sing z z z= +

10. ( ) ( )2sech u u u= −

11. ( )( )cot 1 coty x= +

12. ( ) 21 tf t −= e

13. ( ) 612z zJ z −= e

14. ( ) ( )lnz zf z += e

15. ( ) ( )cos7 xB x =

16. 2 93x xz −=

17. ( ) ( )ln 6 zR z z= + e

18. ( ) ( )7 5 3lnh w w w w w= − + −

19. ( ) ( )( )ln 1 cscg t t= −

20. ( ) ( )1tan 3 2f v v−= −

21. ( ) ( )1sin 9h t t−=

22. ( ) ( ) ( )6cos 1 sinA t t t= − −

23. ( ) ( ) ( )ln 6 4secH z z z= −


Calculus I


24. ( ) ( ) ( )4 4tan tanf x x x= +

25. ( ) 24 86 7 uu u uf u − −= − +e e e

26. ( ) ( ) ( )8 8sec secg z z z= +

27. ( ) ( )54 1 2 9k w w w= − + +

28. ( ) ( ) 73 2 5 1 9 4h x x x x −= − + + +

29. ( ) ( ) ( )5 432 1 5 3T x x x= − −

30. ( ) ( )2 4 sin 1 2w z z z= + −

31. ( ) ( )8 4cosY t t t=

32. ( ) ( )46 ln 10 3f x x x= − +

33. ( ) ( ) ( )2sec 4 tanA z z z=

34. ( ) ( )4 6 95 ln vh v v v += + e

35. ( )2 8

4 7

x xf x

x

+

=+

e

36. ( ) ( )( )

3

62

4 1xg x

x x

+=

−

37. ( ) ( )csc 11 t

tg t −

−=

+ e

38. ( ) ( )( )

2

2

sin1 cos

zV z

z=

+


Calculus I


39. ( ) ( )( )ln coswU w w= e

40. ( ) ( ) ( )( )2tan 5 lnh t t t= −

41. 2

3ln2

xzx

+ = −

42. ( )7 2

vg v

v=

+e

43. ( ) 2 1 4f x x x= + +

44. ( )( )56 cos 8u w= +

45. ( ) ( )2 42 57 z zh z z z

−+= − + e

46. ( ) ( )( )3 2ln 7 sinA y y y= +

47. ( ) ( )6csc 8g x x=

48. ( ) ( ) ( )24 cos 9 ln 6 5V w w w= − + +

49. ( ) ( )3 6sin th t t −= e

50. ( ) ( ) ( )( )8sin sinr rB r = −e e

51. ( ) ( )( )2 2cos 1 cosf z z= +

52. Find the tangent line to ( ) ( )522 4f x x= − at 0x = .

53. Find the tangent line to ( ) ( )2 4 28ln 3xf x x+= − −e at 2x = − .


Calculus I


54. Determine where ( ) ( )43 9A t t t= − is increasing and decreasing.

55. Is ( ) ( ) ( )4 52 1 2h x x x= + − increasing or decreasing more in the interval [ ]2,3− ?

56. Determine where ( ) 3cos 32wU w w = + −

is increasing and decreasing in the interval

[ ]10,10− .

57. If the position of an object is given by ( ) ( )4sin 3 10 7s t t= − + . Determine where, if

anywhere, the object is not moving in the interval [ ]0,4 .

58. Determine where ( ) ( ) ( )6sin 2 7cos 3 3f x x x= − − is increasing and decreasing in the

interval [ ]3,2− .

59. Determine where ( ) ( ) 22 21 wH w w −= − e is increasing and decreasing.

60. What percentage of [ ]3,5− is the function ( ) 2 28 1 23z zg z − −= +e e decreasing?

61. The position of an object is given by ( ) ( )3 2ln 2 21 36 200s t t t t= − + + . During the first 10

hours of motion (assuming the motion starts at 0t = ) what percentage of the time is the object moving to the right?

62. For the function ( ) ( )21 ln 2 92xf x x x= − − + − determine each of the following.

(a) The interval on which the function is defined. (b) Where the function is increasing and decreasing.

Implicit Differentiation For problems 1 – 6 do each of the following.

(a) Find y′ by solving the equation for y and differentiating directly. (b) Find y′ by implicit differentiation. (c) Check that the derivatives in (a) and (b) are the same.

1. 2 9 2x y =


Calculus I


2. 7

6 4xy

=

3. 4 31 5x y= + 4. 28 3x y− = 5. 2 24 6x y xy− = 6. ( )ln x y x=

For problems 7 – 21 find y′ by implicit differentiation. 7. 2 312 8y x y− = 8. 7 10 2 33 6 2y x y x−+ = − + 9. 3 1 14 8y x y− − −+ = 10. 4 6 3 310 7 4x y y x− −− = + 11. ( ) ( ) 4sin cos yx y+ = e

12. ( ) ( )ln secx y y+ =

13. ( )2 2 74 9y x y x− = +

14. 2 3 26 4 0x x y x− − + = 15. 4 3 38 2xy x y x−+ = 16. ( ) ( )3 cos sin 7yx x y x− =

17. ( ) ( )cos sin 9x y xy+ =e

18. 2 3 22x x y y+ + =


Calculus I


19. ( ) 1tan 3 7 6 4x y x−+ = −

20. 2 2 2 2

1x y x y+ = +e e

21. 3 4sin 2x x yy

+ = −

For problems 22 - 24 find the equation of the tangent line at the given point. 22. 23 1x y x+ = + at ( )4,3−

23. 2 2 6x y y x= − at ( )2,6

24. ( ) ( )2sin cos 1x y = at , 06π

For problems 25 – 27 determine if the function is increasing, decreasing or not changing at the given point. 25. 2 3 4 9x y y− = + at ( )2, 1−

26. 21 3x y x y− = +e e at ( )1,0

27. ( ) ( )2sin cosx y x yπ − + = at ,12π

For problems 28 - 31 assume that ( )x x t= , ( )y y t= and ( )z z t= and differentiate the given

equation with respect to t. 28. 4 26 3x z y− = − 29. 4 2 3x y y z=

30. ( )107 6 2 48yz y x z−= − +e

31. ( )2 3 2 2cos 0z x y x+ + =


Calculus I


Related Rates 1. In the following assume that x and y are both functions of t. Given 3x = , 2y = and 7y′ = determine x′ for the following equation. 3 4 2 7x y x y− = − 2. In the following assume that x and y are both functions of t. Given 6x π= , 4y = − and

12x′ = determine x′ for the following equation.

( ) ( )2 2 16 6 cos 2 1x y x y− − = + 3. In the following assume that x, y and z are all functions of t. Given 1x = − , 8y = , 2z = ,

4x′ = − and 7y′ = determine z′ for the following equation.

4 2 22 3yx x zz

+ = −

4. In the following assume that x, y and z are all functions of t. Given 2x = − , 3y = , 4z = ,

6y′ = and 0z′ = determine x′ for the following equation.

2 2 3 4 8x y z x z y= − − 5. The sides of a square are increasing at a rate of 10 cm/sec. How fast is the area enclosed by the square increasing when the area is 150 cm2. 6. The sides of an equilateral triangle are decreasing at a rate of 3 in/hr. How fast is the area enclosed by the triangle decreasing when the sides are 2 feet long? 7. A spherical balloon is being filled in such a way that the surface area is increasing at a rate of 20 cm2/sec when the radius is 2 meters. At what rate is air being pumped in the balloon when the radius is 2 meters? 8. A cylindrical tank of radius 2.5 feet is being drained of water at a rate of 0.25 ft3/sec. How fast is the height of the water decreasing? 9. A hot air balloon is attached to a spool of rope that is 125 feet away from the balloon when it is on the ground. The hot air balloon rises straight up in such a way that the length of rope increases at a rate of 15 ft/sec. How fast is the hot air balloon rising 20 seconds after is lifts off? See the (probably bad) sketch below to help visualize the problem.


Calculus I


10. A rock is dropped straight off a bridge that is 50 meters above the ground. Another person is 7 meters away on the same bridge. At what rate is the distance between the rock and the second person increasing just as the rock hits the ground? 11. A person is 8 meters away from a road and there is a car that is initially 800 meters away approaching the person at a speed of 45 m/sec. At what rate is the distance between the person and the car changing (a) 5 seconds after the start, (b) when the car is directly in front of the person and (c) 10 seconds after the car has passed the person. See the (probably bad) sketch below to help visualize the problem.

12. Two cars are initially 1200 miles apart. At the same time Car A starts driving at 35 mph to the east while Car B starts driving at 55 mph to the north (see sketch below for this initial setup). At what rate is the distance between the two cars changing after (a) 2 hours of travel, (b) 20 hours of travel and (c) 40 hours of travel?

13. Repeat problem 12 above except for this problem assume that Car A starts traveling 4 hours after Car B starts traveling. For parts (a), (b) and (c) assume that these are travel times for Car B. 14. Two people are on a city block. See the sketch below for placement and distances. Person A is on the northeast corner and Person B is on the southwest corner. Person A starts walking towards the southeast corner at a rate of 3 ft/sec. Four seconds later Person B starts walking


Calculus I


towards the southeast corner at a rate of 2 ft/sec. At what rate is the distance between them changing (a) 3 seconds after Person A starts walking and (b) after Person A has covered half the distance?

15. A person is standing 75 meters away from a kite and has a spool of string attached to the kite. The kite starts to rise straight up in the air at a rate of 2 m/sec and at the same time the person starts to move towards the kites launch point at a rate of 0.75 m/sec. Is the length string increasing or decreasing after (a) 4 seconds and (b) 20 seconds. 16. A person lights the fuse on a model rocket and starts to move away from the rocket at a rate of 3 ft/sec. Five seconds after lighting the fuse the rocket launches straight up into the air at a rate of 10 ft/sec. Is the distance between the person and the rocket increasing or decreasing (a) 5 seconds after launch and (b) 10 seconds after launch? 17. A light is suspended above the ground and is being lowered towards the ground at a rate of 9 in/sec. A 6 foot tall person is on the ground and 8 feet away from the light. At what rate is the persons shadow increasing then the light is 15 feet above the ground? 18. A light is fixed on a wall 10 meters above the floor. Twelve meters away from the wall a pole is being raised straight up at a rate of 45 cm/sec. When the pole is 6 meters tall at what rate is the tip of the shadow moving (a) away from the pole and (b) away from the wall? 19. A light is on the top of a 15 foot tall pole. A 5 foot tall person moves away from the pole at a rate of 2.5 ft/sec. After moving for 8 seconds at what rate is the tip of the shadow moving (a) away from the person and (b) away from the pole? 20. A tank of water is in the shape of a cone and is leaking water at a rate of 35 cm3/sec. The base radius of the tank is 1 meter and the height of the tank is 2.5 meters. When the depth of the water is 1.25 meters at what rate is the (a) depth changing and (b) the radius of the top of the water changing? 21. A trough of water is 20 meters long and its ends are in the shape of an isosceles triangle whose width is 7 meters and height is 10 meters. Assume that the two equal length sides of the triangle are the sides of the water tank and the other side of the triangle is the top of the tank and is parallel to the ground. Water is being pumped into the tank at a rate of 2 m3/min. When the


Calculus I


water is 6 meters deep at what rate is (a) depth changing and (b) the width of the top of the water changing? 22. A trough of water is 9 feet long and its ends are in the shape of an equilateral triangle whose sides are 1.5 feet long. Assume that the top of the tank is parallel to the ground. If water is being pumped out of the tank at what rate is the depth of the water changing when the depth is 0.75 feet? 23. The angle of elevation (depression) is the angle formed by a horizontal line and a line joining the observer’s eye to an object above (below) the horizontal line. Two people are on the roof of buildings separated by at 25 foot wide road. Person A is 100 feet above Person B and drops a rock off the roof of their building and it falls at a rate of 3 ft/sec. (a) At what rate is the angle of elevation changing as Person B watches the rock fall when the rock is 25 feet above Person B? (b) At what rate is the angle of depression changing as Person B watches the rock fall when the rock is 65 feet below Person B? 24. The angle of elevation is the angle formed by a horizontal line and a line joining the observer’s eye to an object above the horizontal line. A person is standing 15 meters away from a building and watching an outside elevator move down the face of the building. When the angle of elevation is 1 radians it is changing at a rate of 0.15 radians/sec. At this point in time what is the speed of the elevator? 25. The angle of elevation is the angle formed by a horizontal line and a line joining the observer’s eye to an object above the horizontal line. A person is 24 feet away from a building and watching an outside elevator move up the face of the building. The elevator is moving up at a rate of 4 ft/sec and the person is moving towards the building at a rate of 0.75 ft/sec. Assuming that the elevator started moving from the ground at the same time that the person started walking is the angle of elevation increasing or decreasing after 10 seconds?

Higher Order Derivatives For problems 1 – 9 determine the fourth derivative of the given function. 1. ( ) 8 6 4 22 7 20 3f z z z z z= + − + −

2. 4 3 26 5 4 3 2y t t t t= − + − + 3. ( ) 2 3 46 7V t t t t− − −= + −


Calculus I


4. ( ) 3 5

3 1 34 2

g xx x x

= − +

5. ( ) 9438 5h x x x x= − +

6. ( ) 2354

32 13

h y yy y

= − +

7. ( ) ( ) ( )2

39sin sin 4 7cos zy z z= − +

8. ( ) ( )1 82 3 9 ln 6x xR x x− += − +e e

9. ( ) ( ) ( ) ( )6 7ln cos 4 9sin 2 tf t t t t= − + + e

For problems 10 – 20 determine the second derivative of the given function.

10. ( ) ( )2cos 2 7Q w w= −

11. ( ) ( )2sin 1 xf z = + e

12. ( )tan 3y x=

13. ( )csc 8z w=

14. ( ) 24 9u uf u += e

15. ( ) ( )2ln 3h x x x= −

16. ( ) ( )( )ln 3 cosg z z= +

17. ( )4

16

f xx x

=+

18. ( ) ( ) ( ) 33sin 8cos 2f x x x

−= +


Calculus I


19. ( ) ( )3sin 2f t t=

20. ( ) ( )4tanA w w=

For problems 21 – 23 determine the third derivative of the given function. 21. ( ) ( )sec 3g x x=

22. 31 2ty −= e

23. ( ) ( )2cosh w w w= −

For problems 24 - 27 determine the second derivative of the given function. 24. 2 46 3 9y y x x− = + 25. 3 2 24 11 2y x x y− = − 26. 34 1y x y+ = −e 27. ( ) 2cos 3 4y x y= +

Logarithmic Differentiation For problems 1 – 6 use logarithmic differentiation to find the first derivative of the given function.

1. ( ) ( )( )48 2cos 3 6 3h x x x x= +

2. ( ) 52 3 54 2 9 7 2f w w w w w w= + − + +

3. ( ) ( )( )

32

42

1 7

2 3 4

zh z

z z

+=

+ +


Calculus I


4. ( ) ( )( )

1 sin 22 tan

xg x

x x+

=−

5. ( ) ( )( )

10

2

9 3sin 7

th t

t t−

=

6. ( )

( )( )4 72 2

cos 13 8

1 2 5

xxyx x x

−+=

+ +

For problems 6 – 9 find the first derivative of the given function.

6. ( )ln xy x=

7. ( ) ( ) 6sin 4

tR t t=

8. ( ) ( )32 826

w wh w w

+ += −

9. ( ) [ ]212 3 zg z z z −= +


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