BM306 Derivatives PracticeProblems

CALCULUS I Practice Problems

Derivatives

Paul Dawkins

Calculus I

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

Table of Contents Preface ............................................................................................................................................. 1 Derivatives ...................................................................................................................................... 2

Introduction .............................................................................................................................................. 2 The Definition of the Derivative .............................................................................................................. 3 Interpretations of the Derivative ............................................................................................................. 4 Differentiation Formulas .......................................................................................................................... 6 Product and Quotient Rule ....................................................................................................................... 8 Derivatives of Trig Functions .................................................................................................................. 9 Derivatives of Exponential and Logarithm Functions ......................................................................... 10 Derivatives of Inverse Trig Functions ................................................................................................... 10 Derivatives of Hyperbolic Functions ..................................................................................................... 11 Chain Rule ............................................................................................................................................... 11 Implicit Differentiation........................................................................................................................... 13 Related Rates .......................................................................................................................................... 14 Higher Order Derivatives ....................................................................................................................... 16 Logarithmic Differentiation ................................................................................................................... 17

Preface Here are a set of practice problems for my Calculus I notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. Solutions can be found in a number of places on the site.

1. If you’d like a pdf document containing the solutions go to the note page for the section you’d like solutions for and select the download solutions link from there. Or,

2. Go to the download page for the site http://tutorial.math.lamar.edu/download.aspx and select the section you’d like solutions for and a link will be provided there.

3. If you’d like to view the solutions on the web or solutions to an individual problem you can go to the problem set web page, select the problem you want the solution for. At this point I do not provide pdf versions of individual solutions, but for a particular problem you can select “Printable View” from the “Solution Pane Options” to get a printable version.

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. I add problems to this document as I get a chance and so that means that sometimes some sections are apparently given more attention than others. Eventually I’ll have a full complement of problems for all the sections, however it takes time to write the problems, write the solutions for

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Calculus I

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

each problem and format them for presentation on the web. If there aren’t practice problems for a given section (or there aren’t that many problems in a section) then know that I will eventually get around to writing problems for the section (or more problems for the section as the case may be). Please do not email me asking when I’m going to get problems for a particular section written. As I’ve already said, I write problems as I have time to do so. Please understand that I have my own classes to teach and other responsibilities to my department that I have to also take care of. This web site and all of its information has been written and maintained in my spare time and that often means that I don’t have as much time to work on it as I’d like to.

Derivatives

Introduction Here are a set of practice problems for the Derivatives chapter of my Calculus I notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. Solutions can be found in a number of places on the site.

4. If you’d like a pdf document containing the solutions go to the note page for the section you’d like solutions for and select the download solutions link from there. Or,

5. Go to the download page for the site http://tutorial.math.lamar.edu/download.aspx and select the section you’d like solutions for and a link will be provided there.

6. If you’d like to view the solutions on the web or solutions to an individual problem you can go to the problem set web page, select the problem you want the solution for. At this point I do not provide pdf versions of individual solutions, but for a particular problem you can select “Printable View” from the “Solution Pane Options” to get a printable version.

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 practice problems written for them. The Definition of the Derivative Interpretation of the Derivative Differentiation Formulas Product and Quotient Rule Derivatives of Trig Functions


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Calculus I


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. ( ) 6f x =

2. ( ) 3 14V t t= −

3. ( ) 2g x x=

4. ( ) 210 5Q t t t= + −

5. ( ) 24 9W z z z= −

6. ( ) 32 1f x x= −

7. ( ) 3 22 1g x x x x= − + −

8. ( ) 5R zz

=

9. ( ) 14

tV tt

+=

+

10. ( ) 3 4Z t t= −

11. ( ) 1 9f x x= −


Calculus I


Interpretations of the Derivative

For problems 1 and 2 use the graph of the function, ( )f x , estimate the value of ( )f a′ for the

given values of a. 1. (a) 2a = − (b) 3a =

2. (a) 1a = (b) 4a =

For problems 3 and 4 sketch the graph of a function that satisfies the given conditions. 3. ( )1 3f = , ( )1 1f ′ = , ( )4 5f = , ( )4 2f ′ = −

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


Calculus I


For problems 5 and 6 the graph of a function, ( )f x , is given. Use this to sketch the graph of the

derivative, ( )f x′ .

5.

6.

7. Answer the following questions about the function ( ) 24 9W z z z= − .

(a) Is the function increasing or decreasing at 1z = − ? (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? 8. What is the equation of the tangent line to ( ) 3 14f x x= − at 8x = .


Calculus I


9. The position of an object at any time t is given by ( ) 14

ts tt

+=

+.

(a) Determine the velocity of the object at any time t. (b) Does the object ever stop moving? If yes, at what time(s) does the object stop moving?

10. What is the equation of the tangent line to ( ) 5f xx

= at 12

x = ?

11. Determine where, if anywhere, the function ( ) 3 22 1g x x x x= − + − stops changing.

12. Determine if the function ( ) 3 4Z t t= − increasing or decreasing at the given points.

(a) 5t = (b) 10t = (c) 300t = 13. Suppose that the volume of water in a tank for 0 6t≤ ≤ is given by ( ) 210 5Q t t t= + − .

(a) Is the volume of water increasing or decreasing at 0t = ? (b) Is the volume of water increasing or decreasing at 6t = ? (c) Does the volume of water ever stop changing? If yes, at what times(s) does the volume stop changing?

Differentiation Formulas For problems 1 – 12 find the derivative of the given function. 1. ( ) 36 9 4f x x x= − +

2. 4 22 10 13y t t t= − + 3. ( ) 7 74 3 9g z z z z−= − +

4. ( ) 4 3 29 8 12h y y y y− − −= − + +

5. 3 48 2y x x x= + −


Calculus I


6. ( ) 5 33 7 810 6 3f x x x x= − + −

7. ( ) 3 5

4 1 86

f tt t t

= − +

8. ( ) 4 103

6 1 18 3

R zz zz

= + −

9. ( )23 9z x x= −

10. ( ) ( ) ( )24 2g y y y y= − +

11. ( )34 7 8x xh x

x− +

=

12. ( )5 3

3

5 2y y yf yy

− +=

13. Determine where, if anywhere, the function ( ) 3 29 48 2f x x x x= + − + is not changing.

14. Determine where, if anywhere, the function 4 3 22 3y z z z= − − is not changing.

15. Find the tangent line to ( ) 16 4g x xx

= − at 4x = .

16. Find the tangent line to ( ) 4 67 8 2f x x x x−= + + at 1x = − .

17. The position of an object at any time t is given by ( ) 4 3 23 40 126 9s 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? 18. Determine where the function ( ) 3 4 56 40 5 4h z z z z= + − − is increasing and decreasing.

19. Determine where the function ( ) ( ) ( )21 2R x x x= + − is increasing and decreasing.


Calculus I


20. Determine where, if anywhere, the tangent line to ( ) 3 25f x x x x= − + is parallel to the line

4 23y x= + .

Product and Quotient Rule For problems 1 – 6 use the Product Rule or the Quotient Rule to find the derivative of the given function.

1. ( ) ( )( )2 3 24 8 12f t t t t t= − − +

2. ( )( )3 3 31 2y x x x−= + −

3. ( ) ( )( )2 2 31 2 3 5 8h z z z z z z= + + + −

4. ( )26

2xg x

x=

−

5. ( )4

2

32 1w wR ww

+=

+

6. ( ) 2

27 4

x xf xx x

+=

−

7. If ( )2 8f = − , ( )2 3f ′ = , ( )2 17g = and ( )2 4g′ = − determine the value of ( ) ( )2f g ′ .

8. If ( ) ( )3f x x g x= , ( )7 2g − = , ( )7 9g′ − = − determine the value of ( )7f ′ − .

9. Find the equation of the tangent line to ( ) ( ) ( )21 12 4f x x x= + − at 9x = .

10. Determine where ( )2

21 8x xf x

x−

=+

is increasing and decreasing.

11. Determine where ( ) ( )( )2 24 1 5V t t t= − + is increasing and decreasing.


Calculus I


Derivatives of Trig Functions For problems 1 – 3 evaluate the given limit.

1. ( )

0

sin 10limz

zz→

2. ( )( )0

sin 12lim

sin 5α

αα→

3. ( )

0

cos 4 1limx

xx→

−

For problems 4 – 10 differentiate the given function. 4. ( ) ( ) ( )2cos 6sec 3f x x x= − +

5. ( ) ( ) ( )10 tan 2cotg z z z= −

6. ( ) ( ) ( )tan secf w w w=

7. ( ) ( )3 2 sinh t t t t= −

8. ( )6 4 cscy x x= +

9. ( ) ( ) ( )1

2sin 4cosR t

t t=

−

10. ( ) ( )( )

tan1 cscv v

Z vv

+=

+

11. Find the tangent line to ( ) ( ) ( )tan 9cosf x x x= + at x π= .

12. The position of an object is given by ( ) ( )2 7coss t t= + determine all the points where the

object is not moving. 13. Where in the range [ ]2,7− is the function ( ) ( )4cosf x x x= − is increasing and

decreasing.


Calculus I


Derivatives of Exponential and Logarithm Functions For problems 1 – 6 differentiate the given function. 1. ( ) 2 8xxf x = −e

2. ( ) ( ) ( )34log lng t t t= −

3. ( ) ( )3 logwR w w=

4. ( )5 lnzy z z= −e

5. ( )1 y

yh y =− e

6. ( ) ( )1 5ln

tf tt

+=

7. Find the tangent line to ( ) 7 4x xf x = + e at 0x = .

8. Find the tangent line to ( ) ( ) ( )2ln logf x x x= at 2x = .

9. Determine if ( ) ttV t =e

is increasing or decreasing at the following points.

(a) 4t = − (b) 0t = (c) 10t = 10. Determine if ( ) ( ) ( )6 lnG z z z= − is increasing or decreasing at the following points.

(a) 1z = (b) 5z = (c) 20z =

Derivatives of Inverse Trig Functions For each of the following problems differentiate the given function. 1. ( ) ( ) ( )12cos 6cosT z z z−= +


Calculus I


2. ( ) ( ) ( )1 1csc 4cotg t t t− −= −

3. ( )6 15 secy x x−= −

4. ( ) ( ) ( )2 1sin tanf w w w w−= +

5. ( ) ( )1sin1

xh x

x

−

=+

Derivatives of Hyperbolic Functions For each of the following problems differentiate the given function. 1. ( ) ( ) ( ) ( )sinh 2cosh sechf x x x x= + −

2. ( ) ( ) ( )2tan cschR t t t t= +

3. ( ) ( )1

tanhzg z

z+

=

Chain Rule

For problems 1 – 26 differentiate the given function.

1. ( ) ( )426 7f x x x= +

2. ( ) ( ) 224 3 2g t t t−

= − +

3. 3 1 8y z= − 4. ( ) ( )csc 7R w w=

5. ( ) ( )( )2sin 3 tanG x x x= +


Calculus I


6. ( ) ( )tan 4 10h u u= +

7. ( ) 745 t tf t += + e

8. ( ) ( )1 cos xg x −= e

9. ( ) 1 62 zH z −=

10. ( ) ( )1tan 3 1u t t−= −

11. ( ) ( )2 3ln 1 5F y y y= − +

12. ( ) ( ) ( )( )ln sin cotV x x x= −

13. ( ) ( ) ( )6 6sin sinh z z z= +

14. ( ) 7 wS w w −= + e

15. ( ) ( )7 23 sin 6g z z z= − +

16. ( ) ( )( ) ( )104ln sin 3f x x x x= − −

17. ( ) 6 25h t t t t= −

18. ( ) ( )2 5lnq t t t=

19. ( ) ( ) ( )cos 3 sec 1g w w w= −

20. ( )

2

sin 31

ty

t=

+

21. ( ) ( )21

tan 12

xK x

x x

−+=

+e


Calculus I


22. ( ) ( )2cos xf x x= e

23. ( )5 tan 4z x x= +

24. ( ) ( )( )36 sin 2tf t t−= + −e

25. ( ) ( ) ( )( )102 1ln 1 tan 6g x x x−= + −

26. ( ) ( )4 2tan 1h z z= +

27. ( ) ( )( ) 123 12 sin 3f x x x−

= +

28. Find the tangent line to ( ) 24 2 6 xf x x −= − e at 2x = .

29. Determine where ( ) ( )34 2 8V z z z= − is increasing and decreasing.

30. The position of an object is given by ( ) ( )sin 3 2 4s t t t= − + . Determine where in the

interval [ ]0,3 the object is moving to the right and moving to the left.

31. Determine where ( ) 2 5 tA t t −= e is increasing and decreasing.

32. Determine where in the interval [ ]1, 20− the function ( ) ( )4 3ln 20 100f x x x= − − is

increasing and decreasing.

Implicit Differentiation For problems 1 – 3 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. 3 1xy

=


Calculus I


2. 2 3 4x y+ = 3. 2 2 2x y+ = For problems 4 – 9 find y′ by implicit differentiation. 4. 3 2 62 4y x y x+ − = 5. ( )2 47 sin 3 12y x y+ = −

6. ( )sinx y x− =e

7. 2 7 5 34 2 4x y x x y− = +

8. ( ) 22cos 2 1yx y x+ + =e

9. ( )2 4 2tan 3x y x y= +

For problems 10 & 11 find the equation of the tangent line at the given point.

10. 4 2 3x y+ = at ( )1, 2− .

11. 2 2 23xy y x= +e at ( )0,3 .

For problems 12 & 13 assume that ( )x x t= , ( )y y t= and ( )z z t= and differentiate the given

equation with respect to t. 12. 2 3 4 1x y z− + =

13. ( ) ( )2 3cos sin 4x y y z= +

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


Calculus I


2 2 3 4 46 2 yy x x −+ = − e 2. In the following assume that x, y and z are all functions of t. Given 4x = , 2y = − , 1z = ,

9x′ = and 3y′ = − determine z′ for the following equation.

( ) 3 2 2 21 5 3x y z y z x− + = + − 3. For a certain rectangle the length of one side is always three times the length of the other side.

(a) If the shorter side is decreasing at a rate of 2 inches/minute at what rate is the longer side decreasing?

(b) At what rate is the enclosed area decreasing when the shorter side is 6 inches long and is decreasing at a rate of 2 inches/minute?

4. A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the area of the sheet is decreasing at a rate of 0.5 m2/sec at what rate is the radius decreasing when the area of the sheet is 12 m2 ? 5. A person is standing 350 feet away from a model rocket that is fired straight up into the air at a rate of 15 ft/sec. At what rate is the distance between the person and the rocket increasing (a) 20 seconds after liftoff? (b) 1 minute after liftoff? 6. A plane is 750 meters in the air flying parallel to the ground at a speed of 100 m/s and is initially 2.5 kilometers away from a radar station. At what rate is the distance between the plane and the radar station changing (a) initially and (b) 30 seconds after it passes over the radar station? See the (probably bad) sketch below to help visualize the problem.

7. Two people are at an elevator. At the same time one person starts to walk away from the elevator at a rate of 2 ft/sec and the other person starts going up in the elevator at a rate of 7 ft/sec. What rate is the distance between the two people changing 15 seconds later? 8. Two people on bikes are at the same place. One of the bikers starts riding directly north at a rate of 8 m/sec. Five seconds after the first biker started riding north the second starts to ride directly east at a rate of 5 m/sec. At what rate is the distance between the two riders increasing 20 seconds after the second person started riding?


Calculus I


9. A light is mounted on a wall 5 meters above the ground. A 2 meter tall person is initially 10 meters from the wall and is moving towards the wall at a rate of 0.5 m/sec. After 4 seconds of moving is the tip of the shadow moving (a) towards or away from the person and (b) towards or away from the wall? 10. A tank of water in the shape of a cone is being filled with water at a rate of 12 m3/sec. The base radius of the tank is 26 meters and the height of the tank is 8 meters. At what rate is the depth of the water in the tank changing with the radius of the top of the water is 10 meters? 11. 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 500 feet way from the launch point of a hot air balloon. The hot air balloon is starting to come back down at a rate of 15 ft/sec. At what rate is the angle of elevation, θ , changing when the hot air balloon is 200 feet above the ground. See the (probably bad) sketch below to help visualize the angle of elevation if you are having trouble seeing it.

Higher Order Derivatives For problems 1 – 5 determine the fourth derivative of the given function. 1. ( ) 7 4 33 6 8 12 18h t t t t t= − + − +

2. ( ) 3 2 1V x x x x= − + −

3. ( ) 5 32

148

f x x xx

= − −

4. ( ) ( ) ( )37sin cos 1 2wf w w= + −

5. ( )5 48ln 2zy z−= +e

For problems 6 – 9 determine the second derivative of the given function.


Calculus I


6. ( ) ( )3sin 2 9g x x x= −

7. ( )3ln 7z x= −

8. ( )( )42

2

6 2Q v

v v=

+ −

9. ( ) ( )2cos 7H t t=

For problems 10 & 11 determine the second derivative of the given function. 10. 3 22 1 4x y y+ = − 11. 26 1y xy− =

Logarithmic Differentiation For problems 1 – 3 use logarithmic differentiation to find the first derivative of the given function.

1. ( ) ( )72 25 3 6 8 12f x x x x= − + −

2. ( )

( )

2

34

sin 3

6

z zy

z

+=

−

3. ( ) ( )3

24

5 8 1 9cos 4

10

t th t

t t

+ −=

+

For problems 4 & 5 find the first derivative of the given function.

4. ( ) ( )43 7 wg w w= −

5. ( ) ( ) ( )sin 282xxf x x= − e


Calculus I



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