BM306 Derivatives Assignments

5/28/2018 BM306 Derivatives Assignments

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2/31

Calculus I

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

Table of Contents

Preface ............................................................................................................................................. 1Derivatives ...................................................................................................................................... 1

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3/31

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

TheDefinitionoftheDerivativeUse 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 = - +
http://tutorial.math.lamar.edu/terms.aspxhttp://tutorial.math.lamar.edu/terms.aspx


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Calculus I


9. ( ) 21 10 7z z z= + -

10. ( ) 35f x x x= -

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

12. ( ) 3 22Z x x x x= - -

13. ( )2

3f t

t=

-

14. ( )2

1

xg x

+=

-

15. ( )2

2

tQ 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. ( )1

W tt

=

22. ( )4

1g x =

-

23. ( )f x x x= +


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Calculus I


24. ( )1

f x xx

= +

InterpretationsoftheDerivativeFor 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=


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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.


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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) ofzdoes 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 tis 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?


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Calculus I


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

of wdoes the function stop changing?

14. Suppose that the volume of air in a balloon for0 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) ofxdoes 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

2

tQ 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 tis 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 tdoes the function stop changing?


9/31

Calculus I


DifferentiationFormulasFor problems 1 20 find the derivative of the given function.

1. ( ) 3 88 4 2x 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 3x x x= - +

9. ( ) 9 34 7 427 2f x x x x= - +

10. ( ) 5629

76h y y y= + +

11. ( )2 5

4 1 1

7 2g z

z z z= + -

12.2 3

9 3

2 193 7 t tt t= + - -

13. ( ) 36 5 2

1 1W x x

x= - +

14. ( ) ( )( )25 1g w w w= - +


10/31

Calculus I


15. ( ) ( )31 9h x x x= -

16. ( ) ( )2

33 2f t t= -

17. ( ) ( ) ( )21 2 2t x x x= + - +

18.

24 8 2x xy

- +=

19. ( )4 2

3

2 7t t tY t

t

- +=

20. ( ) ( )2 52

3

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 2

9 10Q t t t t = - + -

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

25. ( ) 2 3 49 8 3x 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 1x

g x +

= at 2x= .

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


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Calculus I


30. The position of an object at any time tis given by ( ) 4 3 23 44 108 20s t t t 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 tis given by ( ) 3 4 51 150 45 2s t t t 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 3

3 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 line1 8

4 3y 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 1

9

xf x = + is parallel to the line

y x= .


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Calculus I


ProductandQuotientRuleFor 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

21

xg x

x=

+

5. ( )2

4

6

y yZ y

y

-=

-

6. ( )2

3

1 10

5 2

t tV t

t t

- +=

+

7. ( ) ( )( )1 4 2

3 9

w wf w

w

- +=

+

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. ( )3h

g

-


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Calculus I


10. ( )3f g

h

-

11. If ( ) ( )x f x h x= -

determine3x

dy

dx =-.

12. If( ) ( )

( )

1 g x h xy

f x

-=

+determine

3x

dy

dx =-.

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

4

2

xf x

x

-=

+at 1= .

15. Determine where ( )2

2

12

zg z

z

-=

+is increasing and decreasing.

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


2

7

1 2

t th t

t

-=

+is increasing and decreasing.


1f x

+=-

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 = + +

DerivativesofTrigFunctionsFor problems 1 6 evaluate the given limit.

1.( )0

3lim

sint

t

t

2.( )

0

sin 9lim

10w

w

w


14/31

Calculus I


3.( )( )0

sin 2lim

sin 17q

q

q

4. ( )0

sin 4lim3 12x

x

++

5.( )

0

cos 1lim

9x

x

-

6.( )

0

cos 8 1lim

2z

z

z

-

For problems 6 10 differentiate the given function.

6. ( ) ( ) ( )4 9sin 2 tanh x x x x= - +

7. ( ) ( ) ( ) ( )8sec cos 4csct 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 cos

1 sinY x x

+

= -

14. ( ) ( )

( )

sec3

1 9 tan

wf w w

w= -

+


15/31

Calculus I


15. ( ) ( )

2

cot

1

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 p= .

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

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 xdx

= .

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

xdx

= - .

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

x x xdx

= - .


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Calculus I


DerivativesofExponentialandLogarithmFunctionsFor problems 1 12 differentiate the given function.

1. ( ) 10 9zg 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. ( ) ( ) ( )64log lnU z z z z = -

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

9. ( )1

1 7

w

wf w

-=

+

e

e

10. ( ) ( )

3

1 4ln

5

tf t

t

+=

11. ( ) ( )2 7log

7rr r

g r+

=

12. ( )( )

4

ln

ttV t

t=

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 1= .

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


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Calculus I


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

(a) 2y= - (b) 0y= (c) 3y=

17. Determine if ( )( )

2

lny z

z= is increasing or decreasing at the following points.

(a)1

2z= (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=

DerivativesofInverseTrigFunctionsFor 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 4cosz 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

1

tan

zQ z

-

+=

8. ( ) ( )

( )

1

1

1 sin

1 cos

tA t

t

-

-

+=

-


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Calculus I


DerivativesofHyperbolicFunctionsFor each of the following problems differentiate the given function.

1. ( ) ( )2 3sinhh w w w= -

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

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

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

5. ( ) ( )coshxf x x= e

6. ( ) ( )sech 1

1

zf z

z

+=

-

7. ( ) ( )

( )

coth

sinh

wQ w

w w=

+

ChainRuleFor problems 1 46 differentiate the given function.

1. ( ) ( )11

3 8x x= -

2. ( ) 7 39z z=

3. ( ) ( )6

39 2h t t t = + -

4. 3 28y w w= +

5. ( ) ( )2

214 3R v v v-

= -

6. ( )( )

8

2

6 5H w

w=

-


19/31

Calculus I


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

8. ( ) ( )tan 1 2 xT x = - e

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

10. ( ) ( )2sech u u u= -

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

12. ( )21 tf t -= e

13. ( )612 z

J z -= e

14. ( ) ( )lnz z

f z +

= e

15. ( ) ( )cos

7 x

B x =

16.2 93x xz -=

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

18. ( ) ( )7 5 3lnh w w w w w= - + -

19. ( ) ( )( )ln 1 csct 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= -


20/31

Calculus I


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

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

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

27. ( ) ( )5

4 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 x

f xx

+

=+

e

36. ( ) ( )

( )

3

62

4 1xg x

x x

+=

-

37. ( ) ( )csc 1

1 tt

g t-

-=

+ e

38. ( ) ( )

( )

2

2

sin

1 cos

zV z

z=

+


21/31

Calculus I


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

40. ( ) ( ) ( )( )2tan 5 lnh t t t = -

41.2

3ln

2

xz

+ =

-

42. ( )7 2

v

g vv

=+

e

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

44. ( )( )5

6 cos 8u w= +

45. ( ) ( )2 42 5

7 z z

h z z z -

+= - + e

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

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

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

49. ( ) ( )3 6sin th t t -= e

50. ( ) ( ) ( )( )8

sinsin

r rB r = -e e

51.

( ) ( )( )

2 2cos 1 cosf z z= +

52. Find the tangent line to ( ) ( )5

22 4f x x= - at 0x= .

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


22/31

Calculus I


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

55. Is ( ) ( ) ( )4 5

2 1 2h x x x= + - increasing or decreasing more in the interval [ ]2,3- ?

56. Determine where ( ) 3cos 32

wU 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 2

1 w

H w w -= - e is increasing and decreasing.

60. What percentage of [ ]3,5- is the function ( )2 2

8 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 10hours of motion (assuming the motion starts at 0t= ) what percentage of the time is the objectmoving to the right?

62. For the function ( ) ( )21 ln 2 92

xf 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.

ImplicitDifferentiationFor problems 1 6 do each of the following.

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

1.2 9 2x y =


23/31

Calculus I


2.7

64

x

y=

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 by implicit differentiation.

7.2 3

12 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 4y y x- -- = +

11. ( ) ( ) 4sin cosx y+ = e

12. ( ) ( )ln secy y+ =

13. ( )2 2 74 9x y x- = +

14.2 3 26 4 0x x y x- - + =

15.4 3 38 2y x y x-+ =

16. ( ) ( )3 cos sin 7yx x y x- =

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

18.2 3 2

2x y y+ + =


24/31

Calculus I


19. ( ) 1tan 3 7 6 4x y x-+ = -

20.2 2 2 2

1x y x y+ = +e e

21.3 4sin 2

xy

y

+ = -

For problems 22 - 24 find the equation of the tangent line at the given point.

22.23 1y x+ = + at ( )4,3-

23.2 2 6y y x= - at ( )2,6

24. ( ) ( )2sin cos 1x y = at , 06

p

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

y-

= +e e at ( )1,0

27. ( ) ( )2sin cosx y x yp- + = at ,12

p

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 3z y- = -

29.4 2 3y y z=

30. ( )10

7 6 2 48yz y x z-= - +e

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


25/31

Calculus I


RelatedRates1. In the following assume that xandyare both functions of t. Given 3x= , 2y= and 7y=

determine for the following equation.3 4 2 7x y x y- = -

2. In the following assume that xandyare both functions of t. Given6

x p= , 4y= - and

12x= determine for the following equation.

( ) ( )2 2 16 6cos 2 1x y x y- - = +

3. In the following assume that x, yandzare all functions of t. Given 1x= - , 8y= , 2= ,

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

4 2 22 3

yx x z+ = -

4. In the following assume that x, yandzare all functions of t. Given 2x= - , 3y= , 4= ,

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 ft 3/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.


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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 12above 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


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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 cm 3/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 thewater 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


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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 0.75feet?

23. The angle of elevation (depression) is the angle formed by a horizontal line and a line joining

the observers 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

observers 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

observers 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. Assumingthat 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?

HigherOrderDerivativesFor 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 - - -= + -


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4. ( )3 5

3 1 3

4 2g x

x x x= - +

5. ( ) 9438 5h x x x x= - +

6. ( ) 2354

32 1

3h y y

y y= - +

7. ( ) ( ) ( )239sin 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 3x=

13. ( )csc 8w=

14. ( )2

4 9u uf u += e

15. ( ) ( )2ln 3h x x x= -

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

17. ( )4

1

6f x

x=

+

18. ( ) ( ) ( )3

3sin 8cos 2f x x x -

= +


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

20. ( ) ( )4tanA w w=

For problems 21 23 determine the third derivative of the given function.

21. ( ) ( )sec 3x x=

22.31 2t

y -= 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 2x x y- = -

26.34 1y x y+ = -e

27. ( ) 2cos 3 4x y= +

LogarithmicDifferentiationFor problems 1 6 use logarithmic differentiation to find the first derivative of the given

function.

1. ( ) ( )( )4

8 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

+=

+ +


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4. ( ) ( )

( )

1 sin 2

2 tang x

x

+=

-

5. ( ) ( )

( )

10

2

9 3

sin 7

t

h t t t

-

=

6.

( )

( )

( )4 7

2 2

cos 13 8

1 2 5

xxy

x x

-+=

+ +

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

6.( )ln xy x=

7. ( ) ( )6

sin 4 t

R t t=

8. ( ) ( )32 8

26

w wh w w

+ +

= -

9. ( ) [ ]212 3

zg z z z

-= +

BM306 Derivatives Assignments

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