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CALCULUS I Practice Problems

Paul Dawkins

Calculus I

Table of Contents Preface ........................................................................................................................................... iii Outline ........................................................................................................................................... iii Review............................................................................................................................................. 2

Introduction .............................................................................................................................................. 2 Review : Functions ................................................................................................................................... 3 Review : Inverse Functions ...................................................................................................................... 6 Review : Trig Functions ........................................................................................................................... 6 Review : Solving Trig Equations .............................................................................................................. 7 Review : Solving Trig Equations with Calculators, Part I ...................................................................... 9 Review : Solving Trig Equations with Calculators, Part II ....................................................................10 Review : Exponential Functions .............................................................................................................10 Review : Logarithm Functions ................................................................................................................11 Review : Exponential and Logarithm Equations ...................................................................................12 Review : Common Graphs .......................................................................................................................14

Limits ............................................................................................................................................ 16 Introduction .............................................................................................................................................16 Rates of Change and Tangent Lines........................................................................................................17 The Limit ..................................................................................................................................................19 One-Sided Limits .....................................................................................................................................21 Limit Properties .......................................................................................................................................22 Computing Limits ....................................................................................................................................24 Infinite Limits ..........................................................................................................................................25 Limits At Infinity, Part I ...........................................................................................................................26 Limits At Infinity, Part II .........................................................................................................................27 Continuity .................................................................................................................................................28 The Definition of the Limit ......................................................................................................................31

Derivatives .................................................................................................................................... 31 Introduction .............................................................................................................................................31 The Definition of the Derivative .............................................................................................................32 Interpretations of the Derivative ...........................................................................................................33 Differentiation Formulas ........................................................................................................................35 Product and Quotient Rule .....................................................................................................................37 Derivatives of Trig Functions .................................................................................................................38 Derivatives of Exponential and Logarithm Functions ..........................................................................39 Derivatives of Inverse Trig Functions ....................................................................................................39 Derivatives of Hyperbolic Functions ......................................................................................................40 Chain Rule ................................................................................................................................................40 Implicit Differentiation ...........................................................................................................................42 Related Rates ...........................................................................................................................................43 Higher Order Derivatives ........................................................................................................................45 Logarithmic Differentiation ....................................................................................................................46

Applications of Derivatives ......................................................................................................... 47 Introduction .............................................................................................................................................47 Rates of Change........................................................................................................................................48 Critical Points ...........................................................................................................................................48 Minimum and Maximum Values .............................................................................................................49 Finding Absolute Extrema ......................................................................................................................52 The Shape of a Graph, Part I....................................................................................................................53 The Shape of a Graph, Part II ..................................................................................................................55 The Mean Value Theorem .......................................................................................................................57 Optimization ............................................................................................................................................58 More Optimization Problems .................................................................................................................58

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

Calculus I

Indeterminate Forms and L’Hospital’s Rule ..........................................................................................59 Linear Approximations ...........................................................................................................................60 Differentials .............................................................................................................................................61 Newton’s Method .....................................................................................................................................61 Business Applications .............................................................................................................................62

Integrals ........................................................................................................................................ 63 Introduction .............................................................................................................................................63 Indefinite Integrals ..................................................................................................................................64 Computing Indefinite Integrals ..............................................................................................................64 Substitution Rule for Indefinite Integrals ..............................................................................................66 More Substitution Rule ...........................................................................................................................68 Area Problem ...........................................................................................................................................69 The Definition of the Definite Integral ...................................................................................................69 Computing Definite Integrals .................................................................................................................70 Substitution Rule for Definite Integrals .................................................................................................72

Applications of Integrals ............................................................................................................. 73 Introduction .............................................................................................................................................73 Average Function Value ..........................................................................................................................74 Area Between Curves ..............................................................................................................................74 Volumes of Solids of Revolution / Method of Rings ..............................................................................75 Volumes of Solids of Revolution / Method of Cylinders .......................................................................76 More Volume Problems ...........................................................................................................................76 Work .........................................................................................................................................................78

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

Calculus I

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.

Outline Here is a list of sections for which problems have been written.

Review Review : Functions Review : Inverse Functions Review : Trig Functions Review : Solving Trig Equations Review : Solving Trig Equations with Calculators, Part I Review : Solving Trig Equations with Calculators, Part II Review : Exponential Functions Review : Logarithm Functions Review : Exponential and Logarithm Equations Review : Common Graphs

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

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

Calculus I

Limits Tangent Lines and Rates of Change The Limit One-Sided Limits Limit Properties Computing Limits Infinite Limits Limits At Infinity, Part I Limits At Infinity, Part II Continuity The Definition of the Limit - No problems written yet.

Derivatives

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

Applications of Derivatives

Rates of Change Critical Points Minimum and Maximum Values Finding Absolute Extrema The Shape of a Graph, Part I The Shape of a Graph, Part II The Mean Value Theorem Optimization Problems More Optimization Problems L’Hospital’s Rule and Indeterminate Forms Linear Approximations Differentials Newton’s Method Business Applications

Integrals

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

Calculus I

Indefinite Integrals Computing Indefinite Integrals Substitution Rule for Indefinite Integrals More Substitution Rule Area Problem Definition of the Definite Integral Computing Definite Integrals Substitution Rule for Definite Integrals

Applications of Integrals

Average Function Value Area Between Two Curves Volumes of Solids of Revolution / Method of Rings Volumes of Solids of Revolution / Method of Cylinders More Volume Problems Work

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

Calculus I

© 2007 Paul Dawkins 1 http://tutorial.math.lamar.edu/terms.aspx

Calculus I

Review

Introduction Here are a set of practice problems for the Review 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.




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. Review : Functions Review : Inverse Functions Review : Trig Functions Review : Solving Trig Equations Review : Solving Trig Equations with Calculators, Part I Review : Solving Trig Equations with Calculators, Part II Review : Exponential Functions Review : Logarithm Functions Review : Exponential and Logarithm Equations Review : Common Graphs



Calculus I

Review : Functions For problems 1 – 4 the given functions perform the indicated function evaluations. 1. ( ) 23 5 2f x x x= − −

(a) ( )4f (b) ( )0f (c) ( )3f −

(d) ( )6f t− (e) ( )7 4f x− (f) ( )f x h+

2. ( )2 6

tg tt

=+

(a) ( )0g (b) ( )3g − (c) ( )10g

(d) ( )2g x (e) ( )g t h+ (f) ( )2 3 1g t t− +

3. ( ) 21h z z= −

(a) ( )0h (b) ( )12h − (c) ( )1

2h

(d) ( )9h z (e) ( )2 2h z z− (f) ( )h z k+

4. ( ) 431

R x xx

= + −+

(a) ( )0R (b) ( )6R (c) ( )9R −

(d) ( )1R x + (e) ( )4 3R x − (f) ( )1 1xR −

The difference quotient of a function ( )f x is defined to be,

( ) ( )f x h f xh

+ −

For problems 5 – 9 compute the difference quotient of the given function. 5. ( ) 4 9f x x= −

6. ( ) 26g x x= −

7. ( ) 22 3 9f t t t= − +


Calculus I

8. ( ) 12

y zz

=+

9. ( ) 23

tA tt

=−

For problems 10 – 17 determine all the roots of the given function. 10. ( ) 5 4 34 32f x x x x= − −

11. ( ) 212 11 5R y y y= + −

12. ( ) 218 3 2h t t t= − −

13. ( ) 3 27g x x x x= + −

14. ( ) 4 26 27W x x x= + −

15. ( )5 43 37 8f t t t t= − −

16. ( ) 45 8

zh zz z

= −− −

17. ( ) 2 41 2 3

w wg ww w

−= +

+ −

For problems 18 – 22 find the domain and range of the given function. 18. ( ) 23 2 1Y t t t= − +

19. ( ) 2 4 7g z z z= − − +

20. ( ) 22 1f z z= + +

21. ( ) 3 14 3h y y= − +

22. ( ) 5 8M x x= − +


Calculus I

For problems 23 – 31 find the domain of the given function.

23. ( )3 3 112 7

w wf ww− +

=−

24. ( ) 3 2

510 9

R zz z z

=+ +

25. ( )3

2

67 4

t tg tt t−

=− −

26. ( ) 225g x x= −

27. ( ) 4 3 220h x x x x= − −

28. ( )3 2

5 18

tP tt t t

+=

− −

29. ( ) 1 6f z z z= − + +

30. ( ) 12 92

h y yy

= + −−

31. ( ) 24 369

A x xx

= − −−

32. ( ) 2 31 1Q y y y= + − −

For problems 33 – 36 compute ( )( )f g x and ( )( )g f x for each of the given pair of

functions.

33. ( ) 4 1f x x= − , ( ) 6 7g x x= +

34. ( ) 5 2f x x= + , ( ) 2 14g x x x= −

35. ( ) 2 2 1f x x x= − + , ( ) 28 3g x x= −


Calculus I

36. ( ) 2 3f x x= + , ( ) 25g x x= +

Review : Inverse Functions For each of the following functions find the inverse of the function. Verify your inverse by computing one or both of the composition as discussed in this section. 1. ( ) 6 15f x x= +

2. ( ) 3 29h x x= −

3. ( ) 3 6R x x= +

4. ( ) ( )54 3 21g x x= − +

5. ( ) 5 9 11W x x= −

6. ( ) 7 5 8f x x= +

7. ( ) 1 94

xh xx

+=

−

8. ( ) 6 108 7

xf xx−

=+

Review : Trig Functions Determine the exact value of each of the following without using a calculator. Note that the point of these problems is not really to learn how to find the value of trig functions but instead to get you comfortable with the unit circle since that is a very important skill that will be needed in solving trig equations.

1. 5cos6π


Calculus I

2. 4sin3π −

3. 7sin4π

4. 2cos3π −

5. 3tan4π

6. 11sec6π −

7. 8cos3π

8. tan3π −

9. 15tan4π

10. 11sin3π −

11. 29sec4π

Review : Solving Trig Equations Without using a calculator find the solution(s) to the following equations. If an interval is given then find only those solutions that are in the interval. If no interval is given then find all solutions to the equation. 1. ( )4sin 3 2t =


Calculus I

2. ( )4sin 3 2t = in 40,3π

3. 2cos 2 03x + =

4. 2cos 2 03x + =

in [ ]7 ,7π π−

5. ( )4cos 6 12z = in 0,2π

6. 32sin 3 02y + =

in 7 ,0

3π −

7. ( )8 tan 2 5 3x − = in 3,2 2π π −

8. ( )16 9sin 7 4x= − − in 92 ,4ππ −

9. 3 tan 5 44t + =

in [ ]0,4π

10. ( )3 csc 9 7 5z − = − in 4,3 9π π −

11. 21 14cos 65x − = −

in 405 ,

3ππ

12. 15 17 4cos7y = +

in [ ]10 ,15π π


Calculus I

Review : Solving Trig Equations with Calculators, Part I Find the solution(s) to the following equations. If an interval is given then find only those solutions that are in the interval. If no interval is given then find all solutions to the equation. These will require the use of a calculator so use at least 4 decimal places in your work. 1. ( )7cos 4 11 10x + =

2. 6 5cos 103x + =

in [ ]0,38

3. 3 6 11sin8t = −

4. ( ) 13 34sin 610 10

z + = − in [ ]0,2

5. 4 49cos 21sin 09 9z z + =

in [ ]10,10−

6. 3 tan 1 11 2 tan4 4w w − = −

in [ ]50,0−

7. 17 3sec 22z − =

in [ ]20,45

8. ( ) ( )12sin 7 11 3 4sin 7y y+ = + in 12,2

− −

9. ( )5 14 tan 8 30x− = in [ ]1,1−

10. 0 18 2csc3t = +

in [ ]0,5

11. 1 1 2cos2 8 4 3

x + =

in [ ]0,100

12. ( )4 1 3sec 23

t= + in [ ]4,6−


Calculus I

Review : Solving Trig Equations with Calculators, Part II Find all the solution(s) to the following equations. These will require the use of a calculator so use at least 4 decimal places in your work. 1. ( )3 14sin 12 7 13t− + =

2. ( )3sec 4 9 24 0z− − =

3. ( ) ( ) ( )4sin 2 15sin 2 tan 4 0x x x+ − + =

4. 3 33cos sin 14cos 07 2 7y y y + =

5. ( ) ( )27cos 3 cos 3 0x x− =

6. 2tan tan 124 4w w = +

7. ( ) ( )24csc 1 6 25csc 1t t− + = −

8. ( )4 sec 7 21y y y= −

9. ( ) ( )210 sin 3 2 7 sin 3 2x x x x+ = +

10. ( ) 62 3 tan 15 1011

tt t − = −

Review : Exponential Functions Sketch the graphs of each of the following functions. 1. ( ) 1 23 xf x +=

2. ( )3

42 7x

h x−

= −


Calculus I

3. ( ) 2 48 3 th t −= + e

4. ( ) 2 31104

zg z − −= − e

Review : Logarithm Functions Without using a calculator determine the exact value of each of the following. 1. 3log 81 2. 5log 125

3. 21log8

4. 1

4

log 16

5. 4ln e

6. 1log

100

Write each of the following in terms of simpler logarithms 7. ( )4 7log 3x y−

8. ( )2 2ln x y z+

9. 4 2 5

4log xy z

−

Combine each of the following into a single logarithm with a coefficient of one.


Calculus I

10. 4 4 412 log 5log log2

x y z+ −

11. ( ) ( )3ln 5 4ln 2ln 1t t s+ − − −

12. 1 log 6log 23

a b− +

Use the change of base formula and a calculator to find the value of each of the following. 13. 12log 35 14. 2

3

log 53

Review : Exponential and Logarithm Equations For problems 1 – 10 find all the solutions to the given equation. If there is no solution to the equation clearly explain why. 1. 7 312 4 7x+− =e

2. 2 21 10 3 z z−= − e

3. 6 12 0tt t −− =e

4. ( ) 2 24 1 12 3 xx x −+ = + e

5. 3 8 5 102 11 0y y+ −− =e e 6. 6 12 714 0x x− −+ =e e

7. 2 11 8ln 147

x − − =

8. ( ) ( )ln 1 1 ln 3 2y y− = + +


Calculus I

9. ( ) ( )log log 21 2w w+ − =

10. ( ) ( )2log log 7 1 0z z− − =

11. 216 17 11t−= + 12. 3 82 7 11w− − = Compound Interest. If we put P dollars into an account that earns interest at a rate of r (written as a decimal as opposed to the standard percent) for t years then,

a. if interest is compounded m times per year we will have,

1t mrA P

m = +

dollars after t years.

b. if interest is compounded continuously we will have, r tA P= e

dollars after t years. 13. We have $10,000 to invest for 44 months. How much money will we have if we put the money into an account that has an annual interest rate of 5.5% and interest is compounded (a) quarterly (b) monthly (c) continuously 14. We are starting with $5000 and we’re going to put it into an account that earns an annual interest rate of 12%. How long should we leave the money in the account in order to double our money if interest is compounded (a) quarterly (b) monthly (c) continuously Exponential Growth/Decay. Many quantities in the world can be modeled (at least for a short time) by the exponential growth/decay equation. 0

k tQ Q= e If k is positive then we will get exponential growth and if k is negative we will get exponential decay. 15. A population of bacteria initially has 250 present and in 5 days there will be 1600 bacteria present. (a) Determine the exponential growth equation for this population. (b) How long will it take for the population to grow from its initial population of 250 to


Calculus I

a population of 2000? 16. We initially have 100 grams of a radioactive element and in 1250 years there will be 80 grams left. (a) Determine the exponential decay equation for this element. (b) How long will it take for half of the element to decay? (c) How long will it take until there is only 1 gram of the element left?

Review : Common Graphs Without using a graphing calculator sketch the graph of each of the following.

1. 4 23

y x= −

2. ( ) 3f x x= −

3. ( ) ( )sin 6g x x= +

4. ( ) ( )ln 5f x x= −

5. ( ) cos2

h x x π = +

6. ( ) ( )23 4h x x= − +

7. ( ) 2 3xW x += −e

8. ( ) ( )21 2f y y= − +

9. ( )R x x= −

10. ( )g x x= −


Calculus I

11. ( ) 22 3 4h x x x= − +

12. ( ) 24 8 3f y y y= − + +

13. ( ) ( )2 21 5 9x y+ + − =

14. 2 24 6 87 0x x y y− + − − =

15. ( )2

225 2 14yx + + =

16. ( )2

2 61

9y

x−

+ =

17. 2 2

136 49x y

− =

18. ( ) ( )22 4

2 116

xy

++ − =


Calculus I

Limits

Introduction Here are a set of practice problems for the Limits 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.




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. Tangent Lines and Rates of Change The Limit One-Sided Limits Limit Properties Computing Limits Infinite Limits Limits At Infinity, Part I Limits At Infinity, Part II Continuity The Definition of the Limit – Problems for this section have not yet been written.



Calculus I

Rates of Change and Tangent Lines

1. For the function ( ) ( )23 2f x x= + and the point P given by 3x = − answer each of the

following questions.

(a) For the points Q given by the following values of x compute (accurate to at least 8 decimal places) the slope, PQm , of the secant line through points P and Q.

(i) -3.5 (ii) -3.1 (iii) -3.01 (iv) -3.001 (v) -3.0001 (vi) -2.5 (vii) -2.9 (viii) -2.99 (ix) -2.999 (x) -2.9999

(b) Use the information from (a) to estimate the slope of the tangent line to ( )f x at 3x = −

and write down the equation of the tangent line.

2. For the function ( ) 4 8g x x= + and the point P given by 2x = answer each of the



(i) 2.5 (ii) 2.1 (iii) 2.01 (iv) 2.001 (v) 2.0001 (vi) 1.5 (vii) 1.9 (viii) 1.99 (ix) 1.999 (x) 1.9999

(b) Use the information from (a) to estimate the slope of the tangent line to ( )g x at 2x =

and write down the equation of the tangent line. 3. For the function ( ) ( )4ln 1W x x= + and the point P given by 1x = answer each of the




(b) Use the information from (a) to estimate the slope of the tangent line to ( )W x at 1x =

and write down the equation of the tangent line.


Calculus I

4. The volume of air in a balloon is given by ( ) 64 1

V tt

=+

answer each of the following

questions.

(a) Compute (accurate to at least 8 decimal places) the average rate of change of the volume of air in the balloon between 0.25t = and the following values of t.

(i) 1 (ii) 0.5 (iii) 0.251 (iv) 0.2501 (v) 0.25001 (vi) 0 (vii) 0.1 (viii) 0.249 (ix) 0.2499 (x) 0.24999

(b) Use the information from (a) to estimate the instantaneous rate of change of the volume of air in the balloon at 0.25t = .

5. The population (in hundreds) of fish in a pond is given by ( ) ( )2 sin 2 10P t t t= + − answer

each of the following questions.

(a) Compute (accurate to at least 8 decimal places) the average rate of change of the population of fish between 5t = and the following values of t. Make sure your calculator is set to radians for the computations.


(b) Use the information from (a) to estimate the instantaneous rate of change of the population of the fish at 5t = .

6. The position of an object is given by ( ) 2 3 6cos2

xs t − =

answer each of the following

questions.

(a) Compute (accurate to at least 8 decimal places) the average velocity of the object between 2t = and the following values of t. Make sure your calculator is set to radians for the

computations. (i) 2.5 (ii) 2.1 (iii) 2.01 (iv) 2.001 (v) 2.0001 (vi) 1.5 (vii) 1.9 (viii) 1.99 (ix) 1.999 (x) 1.9999

(b) Use the information from (a) to estimate the instantaneous velocity of the object at 2t = and determine if the object is moving to the right (i.e. the instantaneous velocity is positive), moving to the left (i.e. the instantaneous velocity is negative), or not moving (i.e. the instantaneous velocity is zero).

7. The position of an object is given by ( ) ( )( )328 6s t x x= − + . Note that a negative position

here simply means that the position is to the left of the “zero position” and is perfectly acceptable.


Calculus I

Answer each of the following questions.

(a) Compute (accurate to at least 8 decimal places) the average velocity of the object between 10t = and the following values of t. (i) 10.5 (ii) 10.1 (iii) 10.01 (iv) 10.001 (v) 10.0001 (vi) 9.5 (vii) 9.9 (viii) 9.99 (ix) 9.999 (x) 9.9999

(b) Use the information from (a) to estimate the instantaneous velocity of the object at 10t = and determine if the object is moving to the right (i.e. the instantaneous velocity is positive), moving to the left (i.e. the instantaneous velocity is negative), or not moving (i.e. the instantaneous velocity is zero).

The Limit

1. For the function ( )3

2

84

xf xx−

=−

answer each of the following questions.

(a) Evaluate the function the following values of x compute (accurate to at least 8 decimal places).


(b) Use the information from (a) to estimate the value of 3

22

8lim4x

xx→

−−

.

2. For the function ( )22 31

tR tt

− +=

+ answer each of the following questions.

(a) Evaluate the function the following values of t compute (accurate to at least 8 decimal places).

(i) -0.5 (ii) -0.9 (iii) -0.99 (iv) -0.999 (v) -0.9999 (vi) -1.5 (vii) -1.1 (viii) -1.01 (ix) -1.001 (x) -1.0001

(b) Use the information from (a) to estimate the value of 2

1

2 3lim1t

tt→−

− ++

.

3. For the function ( ) ( )sin 7g

θθ

θ= answer each of the following questions.


Calculus I

(a) Evaluate the function the following values of θ compute (accurate to at least 8 decimal places). Make sure your calculator is set to radians for the computations.

(i) 0.5 (ii) 0.1 (iii) 0.01 (iv) 0.001 (v) 0.0001 (vi) -0.5 (vii) -0.1 (viii) -0.01 (ix) -0.001 (x) -0.0001

(b) Use the information from (a) to estimate the value of ( )

0

sin 7limθ

θθ→

.

4. Below is the graph of ( )f x . For each of the given points determine the value of ( )f a and

( )limx a

f x→

. If any of the quantities do not exist clearly explain why.

(a) 3a = − (b) 1a = − (c) 2a = (d) 4a =


( )limx a

f x→


(a) 8a = − (b) 2a = − (c) 6a = (d) 10a =


Calculus I


( )limx a

f x→


(a) 2a = − (b) 1a = − (c) 1a = (d) 3a =

One-Sided Limits

1. Below is the graph of ( )f x . For each of the given points determine the value of ( )f a ,

( )limx a

f x−→

, ( )limx a

f x+→

, and ( )limx a

f x→


(a) 4a = − (b) 1a = − (c) 2a = (d) 4a =


Calculus I

2. Below is the graph of ( )f x . For each of the given points determine the value of ( )f a ,

( )limx a

f x−→

, ( )limx a

f x+→

, and ( )limx a

f x→


(a) 2a = − (b) 1a = (c) 3a = (d) 5a =

3. Sketch a graph of a function that satisfies each of the following conditions. ( ) ( ) ( )

2 2lim 1 lim 4 2 1x x

f x f x f− +→ →

= = − =

4. Sketch a graph of a function that satisfies each of the following conditions.

( ) ( ) ( )( ) ( )

3 3

1

lim 0 lim 4 3 does not exist

lim 3 1 2x x

x

f x f x f

f x f− +→ →

→−

= =

= − − =

Limit Properties 1. Given ( )

8lim 9x

f x→

= − , ( )8

lim 2x

g x→

= and ( )8

lim 4x

h x→

= use the limit properties given in this

section to compute each of the following limits. If it is not possible to compute any of the limits clearly explain why not. (a) ( ) ( )

8lim 2 12x

f x h x→

− (b) ( )8

lim 3 6x

h x→

−

(c) ( ) ( ) ( )8

limx

g x h x f x→

− (d) ( ) ( ) ( )8

limx

f x g x h x→

− +


Calculus I

2. Given ( )4

lim 1x

f x→−

= , ( )4

lim 10x

g x→−

= and ( )4

lim 7x

h x→−

= − use the limit properties given in

this section to compute each of the following limits. If it is not possible to compute any of the limits clearly explain why not.

(a) ( )( )

( )( )4

limx

f x h xg x f x→−

−

(b) ( ) ( ) ( )

4limx

f x g x h x→−

(c) ( )

( )( ) ( )4

31limx

f xh x g x h x→−

−+ +

(d) ( ) ( ) ( )4

1lim 27x

h xh x f x→−

− +

3. Given ( )

0lim 6x

f x→

= , ( )0

lim 4x

g x→

= − and ( )0

lim 1x

h x→

= − use the limit properties given in this

section to compute each of the following limits. If it is not possible to compute any of the limits clearly explain why not.

(a) ( ) ( ) 3

0limx

f x h x→

+ (b) ( ) ( )0

limx

g x h x→

(c) ( ) 230

lim 11x

g x→

+ (d) ( )

( ) ( )0limx

f xh x g x→ −

For each of the following limits use the limit properties given in this section to compute the limit. At each step clearly indicate the property being used. If it is not possible to compute any of the limits clearly explain why not.

4. ( )3

2lim 14 6t

t t→−

− +

5. ( )2

6lim 3 7 16x

x x→

+ −

6. 2

3

8lim4 7w

w ww→

−−

7. 25

7lim3 10x

xx x→−

++ −

8. 2

0lim 6z

z→

+


Calculus I

9. ( )3

10lim 4 2x

x x→

+ −

Computing Limits For problems 1 – 9 evaluate the limit, if it exists.

1. ( )2

2lim 8 3 12x

x x→

− +

2. 23

6 4lim1t

tt→−

++

3. 2

25

25lim2 15x

xx x→−

−+ −

4. 2

8

2 17 8lim8z

z zz→

− +−

5. 2

27

4 21lim3 17 28y

y yy y→

− −− −

6. ( )2

0

6 36limh

hh→

+ −

7. 4

2lim4z

zz→

−−

8. 3

2 22 4lim3x

xx→−

+ −+

9. 0

lim3 9x

xx→ − +

10. Given the function

( ) 2

7 4 12 1x x

f xx x− <

= + ≥

Evaluate the following limits, if they exist.


Calculus I

(a) ( )6

limx

f x→−

(b) ( )1

limx

f x→

11. Given

( )6 4

1 9 4z z

h zz z

≤ −= − > −

Evaluate the following limits, if they exist. (a) ( )

7limz

h z→

(b) ( )4

limz

h z→−

For problems 12 & 13 evaluate the limit, if it exists.

12. ( )5

lim 10 5x

x→

+ −

13. 1

1lim1t

tt→−

++

14. Given that ( ) 27 3 2x f x x≤ ≤ + for all x determine the value of ( )

2limx

f x→

.

15. Use the Squeeze Theorem to determine the value of 4

0lim sinx

xxπ

→

.

Infinite Limits For problems 1 – 6 evaluate the indicated limits, if they exist.

1. For ( )( )5

93

f xx

=−

evaluate,

(a) ( )3

limx

f x−→

(b) ( )3

limx

f x+→

(c) ( )3

limx

f x→

2. For ( ) 26

th tt

=+

evaluate,

(a) ( )6

limt

h t−→−

(b) ( )6

limt

h t+→−

(c) ( )6

limt

h t→−

3. For ( )( )2

31

zg zz+

=+

evaluate,


Calculus I

(a) ( )1

limz

g z−→−

(b) ( )1

limz

g z+→−

(c) ( )1

limz

g z→−

4. For ( ) 2

74

xg xx+

=−

evaluate,

(a) ( )2

limx

g x−→

(b) ( )2

limx

g x+→

(c) ( )2

limx

g x→

5. For ( ) ( )lnh x x= − evaluate,

(a) ( )0

limx

h x−→

(b) ( )0

limx

h x+→

(c) ( )0

limx

h x→

6. For ( ) ( )tanR y y= evaluate,

(a) ( )32

limy

R yπ −

→ (b) ( )

32

limy

R yπ +

→ (c) ( )

32

limy

R yπ→

For problems 7 & 8 find all the vertical asymptotes of the given function.

7. ( )( )4

710 3

xf xx

=−

8. ( ) ( )( )8

5 9g x

x x−

=+ −

Limits At Infinity, Part I 1. For ( ) 7 34 18 9f x x x= − + evaluate each of the following limits.

(a) ( )limx

f x→−∞

(b) ( )limx

f x→∞

2. For ( ) 23 12 2h t t t t= + − evaluate each of the following limits.

(a) ( )limt

h t→−∞

(b) ( )limt

h t→∞

For problems 3 – 10 answer each of the following questions. (a) Evaluate ( )lim

xf x

→−∞.

(b) Evaluate ( )limx

f x→∞

.

(c) Write down the equation(s) of any horizontal asymptotes for the function.


Calculus I

3. ( )2

2

8 49 5

xf xx x−

=+

4. ( )7 2

2

3 4 15 10

x xf xx

− +=

−

5. ( )4 3

2 4

20 72 9 5

x xf xx x x

−=

+ +

6. ( )3

5

2 113 6

x xf xx

− +=

−

7. ( )6 4 2

6 3

17 4 10x x xf x

x x− + −

=+ +

8. ( )27 9

1 2xf xx

+=

−

9. ( )2

82 3xf xx+

=+

10. ( )2

2 4

8 46 7

x xf xx x

+ −=

+ +

Limits At Infinity, Part II

For problems 1 – 6 evaluate (a) ( )limx

f x→−∞

and (b) ( )limx

f x→∞

.

1. ( ) 38 2x xf x + −= e

2. ( )26

5 3x x

xf x+

+= e

3. ( ) 6 7 42 10x x xf x −= − −e e e


Calculus I

4. ( ) 5 103 8x x xf x − −= − −e e e

5. ( )3 8

8 3

29 7

x x

x xf x−

−

−=

−e ee e

6. ( )7 3

10 4

216 2

x x x

x x xf x−

− −

− −=

+ +e e e

e e e

For problems 7 – 12 evaluate the given limit. 7. ( )3lim ln 4 9

tt t

→−∞− −

8. 4

2

3 8lim ln2z

zz→−∞

− +

9. 3

11 8lim ln7x

xx x→∞

+ +

10. ( )1 5lim tan 7 3x

x x−

→−∞− +

11. 1 4 7lim tan2t

tt

−

→∞

+ −

12. 2 4

13

3 9lim tan4w

w ww w

−

→∞

− −

Continuity

1. The graph of ( )f x is given below. Based on this graph determine where the function is

discontinuous.


Calculus I

2. The graph of ( )f x is given below. Based on this graph determine where the function is

discontinuous.

For problems 3 – 7 using only Properties 1 – 9 from the Limit Properties section, one-sided limit properties (if needed) and the definition of continuity determine if the given function is continuous or discontinuous at the indicated points.

3. ( ) 4 59 3

xf xx

+=

−

(a) 1x = − , (b) 0x = , (c) 3x = ?

4. ( ) 2

63 10

g zz z

=− −

(a) 2z = − , (b) 0z = , (c) 5z = ?

5. ( )2 6

1 6x x

g xx x

<= − ≥


Calculus I

(a) 4x = , (b) 6x = ?

6. ( )2 26 2

t th t

t t < −

= + ≥ −

(a) 2t = − , (b) 10t = ?

7. ( ) 3

1 3 67 6

6 11 1

2 1

x xx

g x x xx

x x

− < − = −= − < < =

− >

(a) 6x = − , (b) 1x = ? For problems 8 – 12 determine where the given function is discontinuous.

8. ( )2

2

93 2 8

xf xx x

−=

+ −

9. ( ) 2

89 1tR t

t t=

− −

10. ( ) ( )1

2 4cos 3h z

z=

−

11. ( ) 2 37 xxy x +=

− e

12. ( ) ( )tan 2g x x=

For problems 13 – 15 use the Intermediate Value Theorem to show that the given equation has at least one solution in the indicated interval. Note that you are NOT asked to find the solution only show that at least one must exist in the indicated interval. 13. 2 325 8 0x x− − = on [ ]2,4−

14. ( )2 4 ln 5 2 0w w− + = on [ ]0,4

15. 24 10 0t tt + − =e e on [ ]1,3


Calculus I

The Definition of the Limit Problems for this section have not yet been written.

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.




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 Derivatives of Exponential and Logarithm Functions Derivatives of Inverse Trig Functions Derivatives of Hyperbolic Functions Chain Rule



Calculus I

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. ( ) ( ) ( )34 log 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

Applications of Derivatives

Introduction Here are a set of practice problems for the Applications of 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.




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. Rates of Change Critical Points Minimum and Maximum Values Finding Absolute Extrema The Shape of a Graph, Part I The Shape of a Graph, Part II The Mean Value Theorem Optimization Problems More Optimization Problems L’Hospital’s Rule and Indeterminate Forms Linear Approximations Differentials Newton’s Method Business Applications



Calculus I

Rates of Change As noted in the text for this section the purpose of this section is only to remind you of certain types of applications that were discussed in the previous chapter. As such there aren’t any problems written for this section. Instead here is a list of links (note that these will only be active links in the web version and not the pdf version) to problems from the relevant sections from the previous chapter. Each of the following sections has a selection of increasing/decreasing problems towards the bottom of the problem set.

Differentiation Formulas Product & Quotient Rules Derivatives of Trig Functions Derivatives of Exponential and Logarithm Functions Chain Rule

Related Rates problems are in the Related Rates section.

Critical Points Determine the critical points of each of the following functions. 1. ( ) 3 28 81 42 8f x x x x= + − −

2. ( ) 3 4 51 80 5 2R t t t t= + + −

3. ( ) 3 22 7 3 2g w w w w= − − −

4. ( ) 6 5 42 8g x x x x= − +

5. ( ) 3 24 3 9 12h z z z z= − + +

6. ( ) ( ) ( )34 22 8 9Q x x x= − −

7. ( ) 2

42 8

zf zz z

+=

+ +

8. ( ) 2

12 15

xR xx x

−=

+ −


Calculus I

9. ( ) 25 6r y y y= −

10. ( ) ( )1

2 315 3 8 7h t t t t = − − − +

11. ( ) ( )4coss z z z= −

12. ( ) ( )3 92sin y yf y = +

13. ( ) ( )2sin 3 1V t t= +

14. ( ) 9 25 xf x x −= e

15. ( ) 3 22 7w w wg w − −= e

16. ( ) ( )2ln 4 14R x x x= + +

17. ( ) ( )3 7 ln 8 2A t t t= − +

Minimum and Maximum Values 1. Below is the graph of some function, ( )f x . Identify all of the relative extrema and absolute

extrema of the function.


Calculus I

2. Below is the graph of some function, ( )f x . Identify all of the relative extrema and absolute

extrema of the function.

3. Sketch the graph of ( ) 2 4g x x x= − and identify all the relative extrema and absolute extrema

of the function on each of the following intervals. (a) ( ),−∞ ∞

(b) [ ]1,4−

(c) [ ]1,3

(d) [ ]3,5

(e) ( ]1,5−

4. Sketch the graph of ( ) ( )34h x x= − + and identify all the relative extrema and absolute

extrema of the function on each of the following intervals. (a) ( ),−∞ ∞

(b) [ ]5.5, 2− −

(c) [ )4, 3− − (d) [ ]4, 3− −

5. Sketch the graph of some function on the interval [ ]1,6 that has an absolute maximum at

6x = and an absolute minimum at 3x = . 6. Sketch the graph of some function on the interval [ ]4,3− that has an absolute maximum at

3x = − and an absolute minimum at 2x = .


Calculus I

7. Sketch the graph of some function that meets the following conditions :

(a) The function is continuous. (b) Has two relative minimums. (b) One of relative minimums is also an absolute minimum and the other relative

minimum is not an absolute minimum. (c) Has one relative maximum. (d) Has no absolute maximum.


Calculus I

Finding Absolute Extrema For each of the following problems determine the absolute extrema of the given function on the specified interval. 1. ( ) 3 28 81 42 8f x x x x= + − − on [ ]8,2−

2. ( ) 3 28 81 42 8f x x x x= + − − on [ ]4, 2−

3. ( ) 3 4 51 80 5 2R t t t t= + + − on [ ]4.5, 4−

4. ( ) 3 4 51 80 5 2R t t t t= + + − on [ ]0,7

5. ( ) 3 24 3 9 12h z z z z= − + + on [ ]2,1−

6. ( ) 4 3 23 26 60 11g x x x x= − + − on [ ]1,5

7. ( ) ( ) ( )34 22 8 9Q x x x= − − on [ ]3,3−

8. ( ) ( )532 2h w w w= + on 5 12 2,−

9. ( ) 2

42 8

zf zz z

+=

+ + on [ ]10,0−

10. ( ) ( )2

2 310A t t t= − on [ ]2, 10.5

11. ( ) ( )3 92sin y yf y = + on [ ]10,15−

12. ( ) 3 22 7w w wg w − −= e on 512 2,−

13. ( ) ( )2ln 4 14R x x x= + + on [ ]4, 2−


Calculus I

The Shape of a Graph, Part I For problems 1 & 2 the graph of a function is given. Determine the open intervals on which the function increases and decreases. 1.

2.

3. Below is the graph of the derivative of a function. From this graph determine the open intervals in which the function increases and decreases.


Calculus I

4. This problem is about some function. All we know about the function is that it exists everywhere and we also know the information given below about the derivative of the function. Answer each of the following questions about this function.

(a) Identify the critical points of the function. (b) Determine the open intervals on which the function increases and decreases. (c) Classify the critical points as relative maximums, relative minimums or neither.

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )5 0 2 0 4 0 8 0

0 on 5, 2 , 2, 4 , 8, 0 on , 5 , 4,8

f f f f

f x f x

′ ′ ′ ′− = − = = =

′ ′< − − − ∞ > −∞ −

For problems 5 – 12 answer each of the following.

(a) Identify the critical points of the function. (b) Determine the open intervals on which the function increases and decreases. (c) Classify the critical points as relative maximums, relative minimums or neither.

5. ( ) 3 22 9 60f x x x x= − −

6. ( ) 3 4 550 40 5 4h t t t t= + − −

7. 3 22 10 12 12y x x x= − + − 8. ( ) ( )cos 3 2p x x x= + on 3

2 , 2−

9. ( ) ( )22 5 14sin zR z z= − − on [ ]10,7−

10. ( ) 2 3 7h t t t= −


Calculus I

11. ( )21

22 wf w w −= e

12. ( ) ( )22 ln 1g x x x= − +

13. For some function, ( )f x , it is known that there is a relative maximum at 4x = . Answer

each of the following questions about this function. (a) What is the simplest form for the derivative of this function? Note : There really are many

possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative that you can come up with.

(b) Using your answer from (a) determine the most general form of the function. (c) Given that ( )4 1f = find a function that will have a relative maximum at 4x = . Note :

You should be able to use your answer from (b) to determine an answer to this part. 14. Given that ( )f x and ( )g x are increasing functions. If we define ( ) ( ) ( )h x f x g x= +

show that ( )h x is an increasing function.

15. Given that ( )f x is an increasing function and define ( ) ( ) 2h x f x= . Will ( )h x be an

increasing function? If yes, prove that ( )h x is an increasing function. If not, can you determine

any other conditions needed on the function ( )f x that will guarantee that ( )h x will also

increase?

The Shape of a Graph, Part II 1. The graph of a function is given below. Determine the open intervals on which the function is concave up and concave down.


Calculus I

2. Below is the graph the 2nd derivative of a function. From this graph determine the open intervals in which the function is concave up and concave down.


(a) Determine a list of possible inflection points for the function. (b) Determine the open intervals on which the function is concave up and concave down. (c) Determine the inflection points of the function.

3. ( ) 2 312 6f x x x= + −

4. ( ) 4 312 84 4g z z z z= − + +

5. ( ) 4 3 212 6 36 2h t t t t t= + + − +

6. ( ) ( )28 5 2 cos 3h w w w w= − + − on [ ]1,2−

7. ( ) ( )234R z z z= +

8. ( ) 24 xh x −= e


(a) Identify the critical points of the function. (b) Determine the open intervals on which the function increases and decreases. (c) Classify the critical points as relative maximums, relative minimums or neither. (d) Determine the open intervals on which the function is concave up and concave down.


Calculus I

(e) Determine the inflection points of the function. (f) Use the information from steps (a) – (e) to sketch the graph of the function.

9. ( ) 5 45 8g t t t= − +

10. ( ) 3 45 8f x x x= − −

11. ( ) 4 3 22 12h z z z z= − −

12. ( ) ( )23 8sin tQ t t= − on [ ]7,4−

13. ( ) ( )43 2f x x x= −

14. ( ) 4wP w w= e on [ ]1

42,−

15. Determine the minimum degree of a polynomial that has exactly one inflection point. 16. Suppose that we know that ( )f x is a polynomial with critical points 1x = − , 2x = and

6x = . If we also know that the 2nd derivative is ( ) 23 14 4f x x x′′ = − + − . If possible, classify

each of the critical points as relative minimums, relative maximums. If it is not possible to classify the critical points clearly explain why they cannot be classified.

The Mean Value Theorem For problems 1 & 2 determine all the number(s) c which satisfy the conclusion of Rolle’s Theorem for the given function and interval. 1. ( ) 2 2 8f x x x= − − on [ ]1,3−

2. ( ) 2 32g t t t t= − − on [ ]2,1−

For problems 3 & 4 determine all the number(s) c which satisfy the conclusion of the Mean Value Theorem for the given function and interval. 3. ( ) 3 24 8 7 2h z z z z= − + − on [ ]2,5


Calculus I

4. ( ) 38 tA t t −= + e on [ ]2,3−

5. Suppose we know that ( )f x is continuous and differentiable on the interval [ ]7,0− , that

( )7 3f − = − and that ( ) 2f x′ ≤ . What is the largest possible value for ( )0f ?

6. Show that ( ) 3 27 25 8f x x x x= − + + has exactly one real root.

Optimization 1. Find two positive numbers whose sum is 300 and whose product is a maximum. 2. Find two positive numbers whose product is 750 and for which the sum of one and 10 times the other is a minimum. 3. Let x and y be two positive numbers such that 2 50x y+ = and ( )( )1 2x y+ + is a maximum.

4. We are going to fence in a rectangular field. If we look at the field from above the cost of the vertical sides are $10/ft, the cost of the bottom is $2/ft and the cost of the top is $7/ft. If we have $700 determine the dimensions of the field that will maximize the enclosed area. 5. We have 45 m2 of material to build a box with a square base and no top. Determine the dimensions of the box that will maximize the enclosed volume. 6. We want to build a box whose base length is 6 times the base width and the box will enclose 20 in3. The cost of the material of the sides is $3/in2 and the cost of the top and bottom is $15/in2. Determine the dimensions of the box that will minimize the cost. 7. We want to construct a cylindrical can with a bottom but no top that will have a volume of 30 cm3. Determine the dimensions of the can that will minimize the amount of material needed to construct the can. 8. We have a piece of cardboard that is 50 cm by 20 cm and we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give a maximum volume.

More Optimization Problems


Calculus I

1. We want to construct a window whose middle is a rectangle and the top and bottom of the window are semi-circles. If we have 50 meters of framing material what are the dimensions of the window that will let in the most light? 2. Determine the area of the largest rectangle that can be inscribed in a circle of radius 1. 3. Find the point(s) on 23 2x y= − that are closest to ( )4,0− .

4. An 80 cm piece of wire is cut into two pieces. One piece is bent into an equilateral triangle and the other will be bent into a rectangle with one side 4 times the length of the other side. Determine where, if anywhere, the wire should be cut to maximize the area enclosed by the two figures. 5. A line through the point ( )2,5 forms a right triangle with the x-axis and y-axis in the 1st

quadrant. Determine the equation of the line that will minimize the area of this triangle. 6. A piece of pipe is being carried down a hallway that is 18 feet wide. At the end of the hallway there is a right-angled turn and the hallway narrows down to 12 feet wide. What is the longest pipe (always keeping it horizontal) that can be carried around the turn in the hallway? 7. Two 10 meter tall poles are 30 meters apart. A length of wire is attached to the top of each pole and it is staked to the ground somewhere between the two poles. Where should the wire be staked so that the minimum amount of wire is used?

Indeterminate Forms and L’Hospital’s Rule Use L’Hospital’s Rule to evaluate each of the following limits.

1. 3 2

22

7 10lim6x

x x xx x→

− ++ −

2. ( )

24

sinlim

16w

ww

π→− −

3. ( )

2

ln 3limt

tt→∞

4. ( )

( )

2

220

sin 2 7 2lim

1z

z z zz z→

+ −

+


Calculus I

5. 2

1limx x

x−→−∞ e

6. 2 4

lim2z

z

zz

z→∞

+−

ee

7. 3lim ln 1

tt

t→∞

+

8. ( )2 2

0lim ln 4w

w w+→

9. ( ) ( )21lim 1 tanx

x xπ+→ −

10. ( ) 21

0lim cos 2y

yy+→

11. 1

limx

x xx→∞

+ e

Linear Approximations For problems 1 & 2 find a linear approximation to the function at the given point. 1. ( ) 2 103 xf x x −= e at 5x =

2. ( ) 4 36 3 7h t t t t= − + − at 3t = −

3. Find the linear approximation to ( ) 4g z z= at 2z = . Use the linear approximation to

approximate the value of 4 3 and 4 10 . Compare the approximated values to the exact values. 4. Find the linear approximation to ( ) ( )cos 2f t t= at 1

2t = . Use the linear approximation to

approximate the value of ( )cos 1 and ( )cos 9 . Compare the approximated values to the exact

values. 5. Without using any kind of computational aid use a linear approximation to estimate the value of 0.1e .


Calculus I

Differentials For problems 1 – 3 compute the differential of the given function. 1. ( ) ( )2 secf x x x= −

2. 4 2 4x x xw − += e

3. ( ) ( ) ( )ln 2 sin 2h z z z=

4. Compute dy and y∆ for 2xy = e as x changes from 3 to 3.01.

5. Compute dy and y∆ for 5 32 7y x x x= − + as x changes from 6 to 5.9. 6. The sides of a cube are found to be 6 feet in length with a possible error of no more than 1.5 inches. What is the maximum possible error in the volume of the cube if we use this value of the length of the side to compute the volume?

Newton’s Method

For problems 1 & 2 use Newton’s Method to determine 2x for the given function and given value

of 0x . 1. ( ) 3 27 8 3f x x x x= − + − , 0 5x =

2. ( ) ( ) 2cosf x x x x= − , 0 1x =

For problems 3 & 4 use Newton’s Method to find the root of the given equation, accurate to six decimal places, that lies in the given interval. 3. 4 35 9 3 0x x x− + + = in [ ]4,6

4. 22 5 xx + = e in [ ]3,4


Calculus I

For problems 5 & 6 use Newton’s Method to find all the roots of the given equation accurate to six decimal places. 5. 3 2 15 1 0x x x− − + = 6. ( )22 sinx x− =

Business Applications 1. A company can produce a maximum of 1500 widgets in a year. If they sell x widgets during the year then their profit, in dollars, is given by, ( ) 2 31

330,000,000 360,000 750P x x x x= − + − How many widgets should they try to sell in order to maximize their profit? 2. A management company is going to build a new apartment complex. They know that if the complex contains x apartments the maintenance costs for the building, landscaping etc. will be, ( ) 24000 14 0.04C x x x= + − The land they have purchased can hold a complex of at most 500 apartments. How many apartments should the complex have in order to minimize the maintenance costs? 3. The production costs, in dollars, per day of producing x widgets is given by, ( ) 2 31750 6 0.04 0.0003C x x x x= + − + What is the marginal cost when 175x = and 300x = ? What do your answers tell you about the production costs? 4. The production costs, in dollars, per month of producing x widgets is given by,

( ) 10000200 0.5C x xx

= + +

What is the marginal cost when 200x = and 500x = ? What do your answers tell you about the production costs? 5. The production costs, in dollars, per week of producing x widgets is given by,


Calculus I

( ) 2 34000 32 0.08 0.00006C x x x x= − + + and the demand function for the widgets is given by, ( ) 2250 0.02 0.001p x x x= + − What is the marginal cost, marginal revenue and marginal profit when 200x = and 400x = ? What do these numbers tell you about the cost, revenue and profit?

Integrals

Introduction Here are a set of practice problems for the Integrals 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.




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. Indefinite Integrals Computing Indefinite Integrals Substitution Rule for Indefinite Integrals More Substitution Rule Area Problem



Calculus I

Definition of the Definite Integral Computing Definite Integrals Substitution Rule for Definite Integrals

Indefinite Integrals 1. Evaluate each of the following indefinite integrals. (a) 5 26 18 7x x dx− +∫

(b) 5 26 18 7x dx x− +∫

2. Evaluate each of the following indefinite integrals. (a) 3 240 12 9 14x x x dx+ − +∫

(b) 3 240 12 9 14x x x dx+ − +∫

(c) 3 240 12 9 14x x dx x+ − +∫

For problems 3 – 5 evaluate the indefinite integral. 3. 7 212 3t t t dt− − +∫

4. 4 310 9 7w w w dw+ +∫

5. 6 4 24z z z dz+ −∫

6. Determine ( )f x given that ( ) 8 4 26 20 9f x x x x′ = − + + .

7. Determine ( )h t given that ( ) 4 3 2 1h t t t t t′ = − + + − .

Computing Indefinite Integrals For problems 1 – 21 evaluate the given integral. 1. 6 34 2 7 4x x x dx− + −∫


Calculus I

2. 7 11 1648 5z z z dz− −∫

3. 3 9 310 12 4t t t dt− −+ +∫

4. 2 510 8w w dw− −+ −∫

5. 12 dy∫

6. 5 33 10w w dw+∫

7. 6 37 5 107 17x x x dx− +∫

8. 2 3

4 128

dxx x

+ −⌠⌡

9. 6 10 43

7 1 23

dyy y y

+ −⌠⌡

10. ( )( )2 1 4 3t t dt− +∫

11. 2 14

z z dzz

−

⌠⌡

12. 8 5 3

4

6 4 2z z z dzz

− + −⌠⌡

13. 4 3

6x x dx

x−⌠

⌡

14. ( ) ( )2sin 10cscx x dx+∫

15. ( ) ( ) ( )2cos sec tanw w w dw−∫

16. ( ) ( ) ( )12 csc sin csc dθ θ θ θ+ + ∫


Calculus I

17. 14 156

z dzz

+ −⌠⌡

e

18. 3 4t

tt dt−

−−

−⌠⌡

ee

19. 3

6 2 dww w

−⌠⌡

20. 2 2

1 121 1

dxx x

++ −

⌠⌡

21. ( )2

46cos1

z dzz

+−

∫

22. Determine ( )f x given that ( ) 212 4f x x x′ = − and ( )3 17f − = .

23. Determine ( )g z given that ( ) 3 732

zg z zz

′ = + − e and ( )1 15g = − e .

24. Determine ( )h t given that ( ) 224 48 2h t t t′′ = − + , ( )1 9h = − and ( )2 4h − = − .

Substitution Rule for Indefinite Integrals For problems 1 – 16 evaluate the given integral.

1. ( )( )428 12 4 12x x x dx− −∫

2. ( ) 74 33 2 4t t dt−− −+∫

3. ( )( )1023 4 4 6 7w w w dw− − +∫

4. ( ) 3 25 4 8z z z dz− −∫

5. ( )2 390 sin 2 6x x dx+∫


Calculus I

6. ( ) ( )sec 1 tan 1z z dz− −∫

7. ( ) ( )2 1 215 5 cos 6t t t t dt− −− +∫

8. ( ) 4 23 77 2 y yy y dy−−∫ e

9. 2

4 34 6 1

w dww w

++ −

⌠⌡

10. ( )( ) ( )( )52 3cos 3 sin 3t t t t dt− −∫

11. ( )14 cos lnz z z dzz

− − − +

⌠⌡

e e

12. ( ) ( )12 tansec vv dv+∫ e

13. ( ) ( ) ( )210sin 2 cos 2 cos 2 5x x x dx−∫

14. ( ) ( )

( )csc cot

2 cscx x

dxx−

⌠⌡

15. 2

67

dyy+

⌠⌡

16. 2

14 9

dww−

⌠⌡

17. Evaluate each of the following integrals.

(a) 2

31 9

x dxx+

⌠⌡

(b) ( )42

3

1 9

x dxx+

⌠⌡

(c) 2

31 9

dxx+

⌠⌡


Calculus I

More Substitution Rule Evaluate each of the following integrals.

1. ( )74 5 9 12 5 9t t dt+ + +∫

2. ( ) 43 4 3 27 cos 2 8 xx x x dx++ −∫ e

3. ( )

7 7

3 77

6 141 81 8

w w

wwdw+

−−

⌠⌡

e eee

4. ( )4 5 67 cos 2 3x x x dx− +∫

5. ( )( )

4sin 81 9cos 8

z zdz

z+

+⌠⌡

e

6. 2 8 2 8 3 320 1 7 6w w w w dw− −+ + −∫ e e

7. ( )3 244 7 9 5 3t t t dt+ − +∫

8. 2

23 2

6 3csc9 8 2

x x x dxx x

− − − + ⌠⌡

9. ( )( ) ( )327 3 2 4 3 sin 3 8y y y y dy+ + + +∫

10. ( ) ( ) ( )2 2sec 2 9 7 tan 2 tan 2t t t dt + − ∫

11. 2

84 9

w dww−+

⌠⌡

12. 2

7 21 25

x dxx

+

−

⌠⌡


Calculus I

13. ( )87 48 3z z dz+∫

Area Problem For problems 1 – 3 estimate the area of the region between the function and the x-axis on the given interval using 6n = and using, (a) the right end points of the subintervals for the height of the rectangles, (b) the left end points of the subintervals for the height of the rectangles and, (c) the midpoints of the subintervals for the height of the rectangles. 1. ( ) 3 22 4f x x x= − + on [ ]1,4

2. ( ) 24 2g x x= − + on [ ]1,3−

3. ( ) ( )3cos xh x x= − on [ ]0,3

4. Estimate the net area between ( ) 2 58 12f x x x= − − and the x-axis on [ ]2,2− using 8n =

and the midpoints of the subintervals for the height of the rectangles. Without looking at a graph of the function on the interval does it appear that more of the area is above or below the x-axis?

The Definition of the Definite Integral For problems 1 & 2 use the definition of the definite integral to evaluate the integral. Use the right end point of each interval for *

ix .

1. 4

12 3x dx+∫

2. ( )1

06 1x x dx−∫

3. Evaluate : ( )4 3 2

44

cos1

x xdx

x+

+⌠⌡

e


Calculus I

For problems 4 & 5 determine the value of the given integral given that ( )11

67f x dx = −∫ and

( )11

624g x dx =∫ .

4. ( )6

119 f x dx∫

5. ( ) ( )11

66 10g x f x dx−∫

6. Determine the value of ( )9

2f x dx∫ given that ( )

2

53f x dx =∫ and ( )

9

58f x dx =∫ .

7. Determine the value of ( )20

4f x dx

−∫ given that ( )0

42f x dx

−= −∫ , ( )

0

3119f x dx =∫ and

( )31

2021f x dx = −∫ .

For problems 8 & 9 sketch the graph of the integrand and use the area interpretation of the definite integral to determine the value of the integral.

8. 4

13 2x dx−∫

9. 5

04x dx−∫

For problems 10 – 12 differentiate each of the following integrals with respect to x.

10. ( )2 2

49cos 6 1

xt t dt− +∫

11. ( )sin 6 2

74

xt dt+∫

12. 2

1

3

1t

xdt

t

− −⌠⌡

e

Computing Definite Integrals 1. Evaluate each of the following integrals.


Calculus I

a. ( ) 5

3cos x dxx

−⌠⌡

b. ( )4

53

3cos x dxx−

−⌠⌡

c. ( )4

51

3cos x dxx

−⌠⌡

Evaluate each of the following integrals, if possible. If it is not possible clearly explain why it is not possible to evaluate the integral.

2. 6 3 2

112 9 2x x dx− +∫

3. 1 2

25 7 3z z dz

−− +∫

4. 0 4 2

315 13w w wdw− +∫

5. 4

3

1

8 12 t dtt−⌠

⌡

6. 2 3 2

31

1 17 4 2

z dzz z+ −⌠

⌡

7. 4

6 42

2

1x x dxx−

− +⌠⌡

8. ( )1 2

43 4x x dx

−

−−∫

9. 1 3 2

22

2 6y y dyy−⌠

⌡

10. ( ) ( )2

07sin 2cost t dt

π

−∫

11. ( ) ( )0

sec tan 1z z dzπ

−∫


Calculus I

12. ( ) ( ) ( )3

6

22sec 8csc cotw w w dwπ

π−∫

13. 2

20

11

x dxx

++

⌠⌡

e

14. 2

5

27 y dyy

−

−

+⌠⌡

e

15. ( )4

0f t dt∫ where ( ) 2

2 11 3 1

t tf t

t t>

= − ≤

16. ( )1

6g z dz

−∫ where ( )2 24 2z

z zg z

z− > −

= ≤ − e

17. 6

32 10x dx−∫

18. 0

14 3w dw

−+∫

Substitution Rule for Definite Integrals Evaluate each of the following integrals, if possible. If it is not possible clearly explain why it is not possible to evaluate the integral.

1. ( )( )1 64 2 5

03 4 10 2x x x x dx+ + −∫

2. ( )( )

4

0

8cos 29 5sin 2

tdt

t

π

−

⌠⌡

3. ( ) ( )0 3sin cosz z dzπ∫

4. 34 1

1

ww dw−∫ e


Calculus I

5. 1

3

4

75 25 2

y dyy

−

−

− +−

⌠⌡

6. 14

2 3

1

xx dx−

+∫ e

7. ( ) ( )32 6sin 2 7cosw w dwπ

π−∫

8. 5 3

4 2 21

21 4

x x x dxx x x

+−

+ + −⌠⌡

9. ( )

02

22

336 1

t t dtt−

+ +−

⌠⌡

10. ( ) ( ) ( )1 33

22 sin 3 2cosz z z dzπ π

−− + + ∫

Applications of Integrals

Introduction Here are a set of practice problems for the Integrals 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.






Calculus I

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. Average Function Value Area Between Two Curves Volumes of Solids of Revolution / Method of Rings Volumes of Solids of Revolution / Method of Cylinders More Volume Problems Work

Average Function Value

For problems 1 & 2 determine avgf for the function on the given interval.

1. ( ) 28 3 5 xf x x −= − + e on [ ]0,2

2. ( ) ( ) ( )2cos 2 sin xf x x= − on 2 ,π π−

3. Find avgf for ( ) 24 5f x x x= − + on [ ]2,3− and determine the value(s) of c in [ ]2,3− for

which ( ) avgf c f= .

Area Between Curves 1. Determine the area below ( ) 23 2f x x x= + − and above the x-axis.

2. Determine the area to the left of ( ) 23g y y= − and to the right of 1x = − .

For problems 3 – 11 determine the area of the region bounded by the given set of curves. 3. 2 2y x= + , ( )siny x= , 1x = − and 2x =

4. 8yx

= , 2y x= and 4x =


Calculus I

5. 23x y= + , 22x y= − , 1y = and 2y = − 6. 2 6x y y= − − and 2 4x y= +

7. 2 1y x x= + , 12 xy −= e , 3x = − and the y-axis

8. 4 3y x= + , 26 2y x x= − − , 4x = − and 2x =

9. 1

2y

x=

+, ( )22y x= + ,

32

x = − , 1x =

10. 2 1x y= + , 5x = , 3y = − and 3y = 11. 1 2 yx += e , 1 yx −= e , 2y = − and 1y =

Volumes of Solids of Revolution / Method of Rings For problems 1 – 8 use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by the given curves about the given axis. 1. Rotate the region bounded by y x= , 3y = and the y-axis about the y-axis. 2. Rotate the region bounded by 27y x= − , 2x = − , 2x = and the x-axis about the x-axis. 3. Rotate the region bounded by 2 6 10x y y= − + and 5x = about the y-axis. 4. Rotate the region bounded by 22y x= and 3y x= about the x-axis. 5. Rotate the region bounded by 26 xy −= e and 26 4 2y x x= + − between 0x = and 1x = about the line 2y = − . 6. Rotate the region bounded by 210 6y x x= − + , 210 6y x x= − + − , 1x = and 5x = about the line 8y = . 7. Rotate the region bounded by 2 4x y= − and 6 3x y= − about the line 24x = . 8. Rotate the region bounded by 2 1y x= + , 4x = and 3y = about the line 4x = − .


Calculus I

Volumes of Solids of Revolution / Method of Cylinders For problems 1 – 8 use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by the given curves about the given axis.

1. Rotate the region bounded by ( )22x y= − , the x-axis and the y-axis about the x-axis.

2. Rotate the region bounded by 1yx

= , 12

x = , 4x = and the x-axis about the y-axis.

3. Rotate the region bounded by 4y x= and 3y x= about the y-axis. For this problem assume that 0x ≥ . 4. Rotate the region bounded by 4y x= and 3y x= about the x-axis. For this problem assume that 0x ≥ . 5. Rotate the region bounded by 2 1y x= + , 3y = and 4x = about the line 10y = . 6. Rotate the region bounded by 2 4x y= − and 6 3x y= − about the line 8y = − . 7. Rotate the region bounded by 2 6 9y x x= − + and 2 6 1y x x= − + − about the line 8x = .

8. Rotate the region bounded by 12

2

x

yx

=+

e , 145y x= − , 1x = − and 6x = about the line

2x = − .

More Volume Problems 1. Find the volume of a pyramid of height h whose base is an equilateral triangle of length L. 2. Find the volume of the solid whose base is a disk of radius r and whose cross-sections are squares. See figure below to see a sketch of the cross-sections.


Calculus I

3. Find the volume of the solid whose base is the region bounded by 22x y= − and 2 2x y= − and whose cross-sections are isosceles triangles with the base perpendicular to the y-axis and the angle between the base and the two sides of equal length is 4

π . See figure below to see a sketch of the cross-sections.

4. Find the volume of a wedge cut out of a “cylinder” whose base is the region bounded by

4y x= − , 4x = − and the x-axis. The angle between the top and bottom of the wedge is 3π .

See the figure below for a sketch of the “cylinder” and the wedge (the positive x-axis and positive y-axis are shown in the sketch – they are just in a different orientation).


Calculus I

Work 1. A force of ( ) ( )2 cos 3 2F x x x= − + , x is in meters, acts on an object. What is the work

required to move the object from 3x = to 7x = ? 2. A spring has a natural length of 18 inches and a force of 20 lbs is required to stretch and hold the spring to a length of 24 inches. What is the work required to stretch the spring from a length of 21 inches to a length of 26 inches? 3. A cable that weighs ½ kg/meter is lifting a load of 150 kg that is initially at the bottom of a 50 meter shaft. How much work is required to lift the load ¼ of the way up the shaft? 4. A tank of water is 15 feet long and has a cross section in the shape of an equilateral triangle with sides 2 feet long (point of the triangle points directly down). The tank is filled with water to a depth of 9 inches. Determine the amount of work needed to pump all of the water to the top of the tank. Assume that the density of water is 62 lb/ft3.


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