calci complete assignments

CALCULUS I Assignment Problems Paul Dawkins

Calculus I

Table of Contents Preface ........................................................................................................................................... iii Outline ........................................................................................................................................... iii Review............................................................................................................................................. 2 Introduction .............................................................................................................................................. 2 Review : Functions ................................................................................................................................... 2 Review : Inverse Functions ...................................................................................................................... 7 Review : Trig Functions ........................................................................................................................... 8 Review : Solving Trig Equations .............................................................................................................. 9 Review : Solving Trig Equations with Calculators, Part I .....................................................................10 Review : Solving Trig Equations with Calculators, Part II ....................................................................12 Review : Exponential Functions .............................................................................................................13 Review : Logarithm Functions ................................................................................................................13 Review : Exponential and Logarithm Equations ...................................................................................14 Review : Common Graphs .......................................................................................................................17 Limits ............................................................................................................................................ 19 Introduction .............................................................................................................................................19 Rates of Change and Tangent Lines........................................................................................................20 The Limit ..................................................................................................................................................23 One-Sided Limits .....................................................................................................................................26 Limit Properties .......................................................................................................................................28 Computing Limits ....................................................................................................................................29 Infinite Limits ..........................................................................................................................................33 Limits At Infinity, Part I ...........................................................................................................................34 Limits At Infinity, Part II .........................................................................................................................36 Continuity .................................................................................................................................................37 The Definition of the Limit ......................................................................................................................42 Derivatives .................................................................................................................................... 42 Introduction .............................................................................................................................................42 The Definition of the Derivative .............................................................................................................43 Interpretations of the Derivative ...........................................................................................................44 Differentiation Formulas ........................................................................................................................48 Product and Quotient Rule .....................................................................................................................51 Derivatives of Trig Functions .................................................................................................................53 Derivatives of Exponential and Logarithm Functions ..........................................................................55 Derivatives of Inverse Trig Functions ....................................................................................................57 Derivatives of Hyperbolic Functions ......................................................................................................57 Chain Rule ................................................................................................................................................58 Implicit Differentiation ...........................................................................................................................62 Related Rates ...........................................................................................................................................64 Higher Order Derivatives ........................................................................................................................68 Logarithmic Differentiation ....................................................................................................................70 Applications of Derivatives ......................................................................................................... 71 Introduction .............................................................................................................................................71 Rates of Change........................................................................................................................................72 Critical Points ...........................................................................................................................................72 Minimum and Maximum Values .............................................................................................................76 Finding Absolute Extrema ......................................................................................................................80 The Shape of a Graph, Part I....................................................................................................................82 The Shape of a Graph, Part II ..................................................................................................................88 The Mean Value Theorem .......................................................................................................................93 Optimization ............................................................................................................................................95 More Optimization Problems .................................................................................................................97 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

Calculus I

Indeterminate Forms and LHospitals Rule ..........................................................................................99 Linear Approximations .........................................................................................................................101 Differentials ...........................................................................................................................................101 Newtons Method ...................................................................................................................................103 Business Applications ...........................................................................................................................104 Integrals ...................................................................................................................................... 106 Introduction ...........................................................................................................................................106 Indefinite Integrals ................................................................................................................................106 Computing Indefinite Integrals ............................................................................................................108 Substitution Rule for Indefinite Integrals ............................................................................................111 More Substitution Rule .........................................................................................................................114 Area Problem .........................................................................................................................................116 The Definition of the Definite Integral .................................................................................................117 Computing Definite Integrals ...............................................................................................................119 Substitution Rule for Definite Integrals ...............................................................................................122 Applications of Integrals ........................................................................................................... 124 Introduction ...........................................................................................................................................124 Average Function Value ........................................................................................................................124 Area Between Curves ............................................................................................................................125 Volumes of Solids of Revolution / Method of Rings ............................................................................127 Volumes of Solids of Revolution / Method of Cylinders .....................................................................129 More Volume Problems .........................................................................................................................131 Work .......................................................................................................................................................133

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

Calculus I

Preface Here are a set of problems for my Calculus I notes. These problems do not have any solutions available on this site. These are intended mostly for instructors who might want a set of problems to assign for turning in. I try to put up both practice problems (with solutions available) and these problems at the same time so that both will be available to anyone who wishes to use them.

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

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

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

Calculus I

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 LHospitals Rule and Indeterminate Forms Linear Approximations Differentials Newtons Method Business Applications

Integrals

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

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

Calculus I

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 problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. 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

Review : Functions For problems 1 6 the given functions perform the indicated function evaluations. 1. ( ) 10 3f x x= (a) ( )5f (b) ( )0f (c) ( )7f (d) ( )2 2f t + (e) ( )12f x (f) ( )f x h+ 2. ( ) 24 7 1h y y y= + (a) ( )0h (b) ( )3h (c) ( )5h


Calculus I

(d) ( )6h z (e) ( )1 3h y (f) ( )h y k+

3. ( ) 51tg t

t+

=

(a) ( )0g (b) ( )4g (c) ( )7g

(d) ( )2 5g x (e) ( )g t h+ (f) ( )4 9g t +

4. ( ) 4 5f z z= + (a) ( )0f (b) ( )1f (c) ( )2f (d) ( )5 12h y (e) ( )22 8f z + (f) ( )f z h+

5. ( )2 9

4 8xz xx+

=+

(a) ( )4z (b) ( )4z (c) ( )1z

(d) ( )2 7z x (e) ( )3 4z x + (f) ( )z x h+

6. ( ) 32 5

tY t tt

= +

(a) ( )0Y (b) ( )7Y (c) ( )4Y (d) ( )5Y t (e) ( )2 10Y t (f) ( )26Y t t The difference quotient of a function ( )f x is defined to be,

( ) ( )f x h f xh

+

For problems 7 13 compute the difference quotient of the given function. 7. ( ) 4 7Q t t= 8. ( ) 42g t = 9. ( ) 22 9H x x= + 10. ( ) 23 8z y y y=


Calculus I

11. ( ) 4 3g z z= +

12. ( ) 41 2

y xx

=

13. ( )2

7tf t

t=

+

For problems 14 21 determine all the roots of the given function. 14. ( ) 240 3y t t t= + 15. ( ) 4 3 26 5 4f x x x x= 16. ( ) 26 11Z p p p= 17. ( ) 6 5 44 10h y y y y= + + 18. ( ) 7 46 16g z z z z= +

19. ( )1 12 48 15f t t t=

20. ( ) 34 5 8

w wh ww w

= ++

21. ( ) 23 4 1

w wg ww w

+=

+

For problems 22 30 find the domain and range of the given function. 22. ( ) 2 8 3f x x x= + 23. ( ) 24 7z w w w= 24. ( ) 23 2 3g t t t= +


Calculus I

25. ( ) 5 2g x x=

26. ( ) 210 9 7B z x= + +

27. ( ) 1 6 7h y y= +

28. ( ) 12 5 2 9f x y= + 29. ( ) 6 5V t t=

30. ( ) 212 9 1y x x= + For problems 31 51 find the domain of the given function.

31. ( )24 12 8

16 9t tf t

t +

=+

32. ( )3 27

4 17yv y

y

=

33. ( ) 23 1

5 3 2xg x

x x+

=

34. ( )3 2

3 4 5

1 135 2t th t

t t t +

=+

35. ( )2

3 29 2z zf z

z z z+

= +

36. ( )4

2

34 10 2

pV pp p

=

+ +

37. ( ) 2 15g z z=

38. ( ) 236 9f t t=


Calculus I

39. ( ) 2 315 2A x x x x=

40. ( ) 3 24 4Q y y y y= +

41. ( )2

2

76tP t

t t+

=

42. ( )2

25 3th t

t t=

+

43. ( )2

67 3

h xx x

= +

44. ( )4 3 2

16 9zf z

z z z+

= +

45. ( ) 8 2S t t t= +

46. ( ) 5 8 2 9g x x x= +

47. ( ) 2494 12

yh y yy

=

48. ( ) 21 4 10 94

xA x x xx+

= + + +

49. ( ) 2 28 33 4 12 7 3

f tt t t t

= +

50. ( ) 5 24 23 6R x x x

x x= +

+

51. ( ) 43 6 2C z z z z= + For problems 52 55 compute ( )( )f g x and ( )( )g f x for each of the given pair of functions.


Calculus I

52. ( ) 5 2f x x= + , ( ) 8 23g x x=

53. ( ) 2f x x= , ( ) 22 9g x x= 54. ( ) 22 4f x x x= + , ( ) 27g x x x=

55. ( )3 2

xf xx

=+

, ( ) 8 5g 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. ( ) 11 8f x x= 2. ( ) 4 10g x x= 3. ( ) 72 9Z x x=

4. ( ) ( )37 2 1h x x= + +

5. ( ) 7 15 2W x x= +

6. ( ) 3 6 18h x x=

7. ( ) 2 146 1xR xx+

=+

8. ( ) 19 12

xg xx

=


Calculus I

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. 3tan4

2. 7sin6

3. 3sin4

4. 4cos3

5. 5cot4

6. 5sin6

7. sec6

8. 5cos4

9. 11cos

6

10. 11csc

6

11. 4cot3


Calculus I

12. cos4

13. 2csc3

14. 17sec

6

15. 23sin

3

16. 31tan

6

17. 15cos

4

18. 23sec

4

19. 11cot

4

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. ( )10cos 8 5t =

2. ( )10cos 8 5t = in ,4 4

3. 2sin 34z =


Calculus I

4. 2sin 34z =

in [ ]0,16

5. 22sin 2 03t + =

in [ ]0,5

6. ( )6 8 cos 3x= in 50,3

7. ( )10 7 tan 4 3x+ = in [ ],0

8. 0 2cos 22y =

in [ ]4 ,5

9. ( )3cos 5 1 7z = in [ ],

10. ( )7 3 7 cot 2 0w+ = in , 23

11. 2csc 8 03x + =

in [ ]0,2

12. ( )3 4sin 4 5t = in 3 ,2 2

13. 3sec 9 155y + =

in [ ]3 ,20

14. ( ) ( )12 cos 2 2sin 2 0z z = in 3 , 22

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.


Calculus I

1. 2 14sin 53t =

2. ( ) ( )4cos 4 8 10 cos 4x x+ = 3. ( )2 tan 3 3 25w + =

4. 3 7 12sin5 5 5x =

in [ ]0,15

5. 1 3 8cos2w = +

in [ ]20,5

6. 45sin 9 7sin 172 2x x = +

in [ ]10,20

7. ( )2 4 3sec 113

x= in [ ]0,1 8. ( ) ( )3sin 4 18cos 4 0v v+ = in [ ]2,5

9. 2 22 cos 3 7cos 67 7t t + = +

in [ ]10,30

10. 1 10 3csc2 3 7 14

y =

in [ ]0,32

11. 31 1 40cos8t = +

in [ ]50,60

12. ( ) ( )15csc 15 14 20 12csc 15x x+ = in [ ]1, 2

13. ( ) ( )1 1cos 6 3 1 cos 62 3

t t+ = + in [ ]0,5

14. 4 1 2sec 125z =

in [ ]0,15


Calculus I

15. 2 211 7sin 23 19sin13 13

x x =

in [ ]60,60

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. ( )22cos 8 10 0x + = 2. ( )10 tan 4 10 7 31x + =

3. ( )4 tan sin 2 tan 03 3w ww =

4. ( ) ( ) ( )3tan 4 sec 2 1 sec 2 1 0z z z + = 5. ( ) ( )22 sin 2 3sin 2y y = 6. ( ) ( )24cos 2 5 4cos 2 5 1t t+ + =

7. 26 5sin 7sin4 4x x =

8. ( ) ( )22 2 tan 8 3tan 8t t= + 9. ( ) ( )335csc 4 csc 4z z z= 10. ( )3 8 cos 5t t t= +

11. ( ) 65 1 sin 25 5 02

yx x + + + =

12. ( )2 2 45 20 8 2 sec 9ww w =


Calculus I

Review : Exponential Functions Sketch the graphs of each of the following functions.

1. ( )3

27t

g t

=

2. ( ) 4 13 5 xf x += 3. ( ) 2 16 3xh x = e

4. ( )3257 9t

f t

= + e

Review : Logarithm Functions Without using a calculator determine the exact value of each of the following. 1. 7log 343 2. 4log 1024

3. 38

27log512

4. 111log

121

5. 0.1log 0.0001 6. 16log 4 7. log10000

8. 5

1lne


Calculus I

Write each of the following in terms of simpler logarithms

9. ( )7 3 87log 10a b c

10. ( )32 2log 4z x +

11. 2 34

ln w tt w

+

Combine each of the following into a single logarithm with a coefficient of one. 12. 7 ln 6ln 5lnt s w +

13. ( )1 log 1 2log 4log 3log2

z x y z+

14. ( )3 312 log 6log3

x y x+ +

Use the change of base formula and a calculator to find the value of each of the following. 15. 7log 100

16. 57

1log8

Review : Exponential and Logarithm Equations For problems 1 14 find all the solutions to the given equation. If there is no solution to the equation clearly explain why. 1. 10 715 12 5 w= + e

2. 224 7 2x x+ =e

3. 4 98 3 1z+ =e


Calculus I

4. 2 2 24 3 0tt t =e

5. 3 57 16 0x xx x + =e

6. 7 8 53 12 0t t+ =e e

7. 2 1 52 7 0y yy y =e e

8. ( )16 4ln 2 7x+ + =

9. 3 11ln 13

zz

=

10. ( ) ( )2log log 3 7 1w w + = 11. ( ) ( )ln 3 1 ln 2x x+ = 12. ( ) ( )2log 6 1 3 log 6 1 0t t t t+ + =

13. ( ) ( )22 log log 4 1 0z z z + + = 14. ( ) ( )ln ln 2 3x x+ = 15. 9 111 5 3w = 16. 7 212 20 50t+ =

17. 2 21 3 5z + =

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,


Calculus I

r tA P= e dollars after t years.

18. We have $2,500 to invest and 80 months. How much money will we have if we put the money into an account that has an annual interest rate of 9% and interest is compounded (a) quarterly (b) monthly (c) continuously 19. We are starting with $60,000 and were going to put it into an account that earns an annual interest rate of 7.5%. How long will it take for the money in the account to reach $100,000 if the interest is compounded (a) quarterly (b) monthly (c) continuously 20. Suppose that we put some money in an account that has an annual interest rate of 10.25%. How long will it take to triple our money if the interest is compounded (a) twice a year (b) 8 times a year (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. 21. A population of bacteria initially has 90,000 present and in 2 weeks there will be 200,000 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 90,000 to a population of 150,000? 22. We initially have 2 kg grams of some radioactive element and in 7250 years there will be 1.5 kg 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 250 grams of the element left? 23. For a particular radioactive element the value of k in the exponential decay equation is given by 0.000825k = . (a) How long will it take for a quarter of the element to decay? (b) How long will it take for half of the element to decay? (c) How long will it take 90% of the element to decay?


Calculus I

Review : Common Graphs Without using a graphing calculator sketch the graph of each of the following. 1. 2 7y x= + 2. ( ) 4f x x= +

3. ( ) 5g x x=

4. ( ) tan3

g x x = +

5. ( ) ( )sec 2f x x= + 6. ( ) 2 4h x x= + 7. ( ) 3 6xQ x = +e

8. ( ) 6 3V x x= +

9. ( ) sin 16

g x x = +

10. ( ) ( )26 8h x x= +

11. ( ) ( )25 3W y y= + +

12. ( ) ( )29 2f y y=

13. ( ) ( )21 6f x x= + 14. ( ) ( )lnR x x= 15. ( ) ( )lng x x=


Calculus I

16. ( ) 2 8 1h x x x= + 17. ( ) 23 6 5Y x x x= + 18. ( ) 2 4 2f y y y= 19. ( ) 22 2 3h y y y= + 20. 2 26 8 24 0x x y y + + + = 21. 2 2 10 9x y y+ + =

22. ( ) ( )2 24 2 1

25 25x y+ +

+ =

23. 2 22 4 16 16 0x x y y + + =

24. ( ) ( )

226 16 5 1

4x

y+

+ =

25. ( ) ( )2 21 3 1

25 4y x

=

26. ( ) ( )2 24 9 7 1x y + =


Calculus I

Limits

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. 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 have been written for this section yet.


Calculus I

Rates of Change and Tangent Lines 1. For the function ( ) 3 23f x 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 ( ) 2 4xg x

x=

+ and the point P given by 0x = answer each of the following

questions.


(i) 1 (ii) 0.5 (iii) 0.1 (iv) 0.01 (v) 0.001 (vi) -1 (vii) -0.5 (viii) -0.1 (ix) -0.01 (x) -0.001

(b) Use the information from (a) to estimate the slope of the tangent line to ( )g x at 0x = and write down the equation of the tangent line.

3. For the function ( ) ( )22 2h x x= + and the point P given by 2x = answer each of the following questions.


(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 ( )h x at 2x = and write down the equation of the tangent line.


Calculus I

4. For the function ( ) 22 8xP x = e and the point P given by 0.5x = answer each of the following questions.


(i) 1 (ii) 0.51 (iii) 0.501 (iv) 0.5001 (v) 0.50001 (vi) 0 (vii) 0.49 (viii) 0.499 (ix) 0.4999 (x) 0.49999

(b) Use the information from (a) to estimate the slope of the tangent line to ( )h x at 0.5x = and write down the equation of the tangent line.

5. The amount of grain in a bin is given by ( ) 11 44

tV tt+

=+

answer each of the following

questions.

(a) Compute (accurate to at least 8 decimal places) the average rate of change of the amount of grain in the bin between 6t = and the following values of t.


(b) Use the information from (a) to estimate the instantaneous rate of change of the volume of air in the balloon at 6t = .

6. The population (in thousands) of insects is given by ( ) ( )12 cos 3 sin2tP 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 insects between 4t = 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 insects at 4t = .

7. The amount of water in a holding tank is given by ( ) 4 28 7V 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 amount of grain in the bin between 0.25t = and the following values of t.


Calculus I

(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 water in the tank at 0.25t = .

8. The position of an object is given by ( ) 2 721

s t xx

= ++

answer each of the following

questions.

(a) Compute (accurate to at least 8 decimal places) the average velocity of the object between 5t = and the following values of t. (i) 5.5 (ii) 5.1 (iii) 5.01 (iv) 5.001 (v) 5.0001 (vi) 4.5 (vii) 4.9 (viii) 4.99 (ix) 4.999 (x) 4.9999

(b) Use the information from (a) to estimate the instantaneous velocity of the object at 5t = 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).

9. The position of an object is given by ( ) ( ) ( )2cos 4 7sins t t t= . Note that a negative position here simply means that the position is to the left of the zero position and is perfectly acceptable. Answer each of the following questions.

(a) Compute (accurate to at least 8 decimal places) the average velocity of the object between 0t = 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 0t = 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).

10. The position of an object is given by ( ) 2 10 11s t t t= + . Note that a negative position here simply means that the position is to the left of the zero position and is perfectly acceptable. Answer each of the following questions.

(a) Determine the time(s) in which the position of the object is at 5s =


Calculus I

(b) Estimate the instantaneous velocity of the object at each of the time(s) found in part (a) using the method discussed in this section.

The Limit

1. For the function ( )2

2

6 93

x xg xx x+ +

=+

answer each of the following questions.

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

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

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

23

6 9lim3x

x xx x+ ++

.

2. For the function ( )2

2

10 91

z zf zz

=


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



21

10 9lim1z

z zz

.

3. For the function ( ) 2 4 2th tt

+= answer each of the following questions.

(a) Evaluate the function the following values of 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


2 4 2limt

tt

+.


Calculus I

4. For the function ( ) ( )cos 4 1

2 8g

=


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


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

0

cos 4 1lim

2 8

.

5. 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) 2a = (b) 1a = (c) 2a = (d) 3a =


( )limx a

f x


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


Calculus I


( )limx a

f x


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

8. Explain in your own words what the following equation means. ( )

12lim 6x

f x

=

9. Suppose we know that ( )

7lim 18x

f x

= . If possible, determine the value of ( )7f . If it is not possible to determine the value explain why not. 10. Is it possible to have ( )

1lim 23x

f x

= and ( )1 107f = ? Explain your answer.


Calculus I

One-Sided Limits

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

x af x

, ( )lim

x af x

+, and ( )lim

x af x


(a) 5a = (b) 2a = (c) 1a = (d) 4a =

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) 1a = (b) 1a = (c) 3a =

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


Calculus I

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

1 1lim 2 lim 3 1 6x x

f x f x f +

= = =

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

3 3lim 1 lim 1 3 4

x xf x f x f

+ = = =

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

( ) ( ) ( )

( ) ( )5 5

4

lim 1 lim 7 5 4

lim 6 4 does not existx x

x

f x f x f

f x f

+

= = =

=

7. Explain in your own words what each of the following equations mean. ( ) ( )

8 8lim 3 lim 1x x

f x f x +

= =

8. Suppose we know that ( )

7lim 18x

f x

= . If possible, determine the value of ( )7

limx

f x

and

the value of ( )7

limx

f x+

. If it is not possible to determine one or both of these values explain

why not. 9. Suppose we know that ( )6 53f = . If possible, determine the value of ( )

6limx

f x

and the

value of ( )6

limx

f x+

. If it is not possible to determine one or both of these values explain why

not.


Calculus I

Limit Properties 1. Given ( )

0lim 5x

f x

= , ( )0

lim 1x

g x

= and ( )0

lim 3x

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) ( )

0lim 11 7x

f x

+ (b) ( ) ( )0lim 6 4 10x g x h x

(c) ( ) ( ) ( )0

lim 4 12 3x

g x f x h x

+ (d) ( ) ( )( )0lim 1 2x g x f x + 2. Given ( )

12lim 2x

f x

= , ( )12

lim 6x

g x

= and ( )12

lim 9x

h x


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

(a) ( ) ( ) ( )( )121

limx

g xh x f x

g x +

+

(b) ( )( ) ( )( )12

lim 3 1 2x

f x g x

+

(c) ( )

( ) ( )121

lim3 2x

f xg x h x

+

(d) ( ) ( )

( ) ( )122

lim7xf x g x

h x f x

+

3. Given ( )

1lim 0x

f x

= , ( )1

lim 9x

g x

= and ( )1

lim 7x

h x


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

(a) ( )( ) ( )( )2 31

limx

g x h x

(b) ( ) ( )1

lim 3 6x

f x h x

+

(c) ( ) ( ) ( )1

limx

f x g x h x

(d) ( )( )

41

2lim

1 10xg xh 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. ( )24

lim 3 9 2x

x x

+


Calculus I

5. ( )( )221lim 3w w w +

6. ( )4 20

lim 4 12 8t

t t t

+

7. 2

2

10lim3 4z

zz

+

8. 278lim

14 49xx

x x +

9. 3

23

20 4lim8 1y

y yy y ++

10. 36

lim 8 7w

w

+

11. ( )21

lim 4 8 1t

t t

+

12. ( )48

lim 3 8 9 2x

x x

+ +

Computing Limits For problems 1 20 evaluate the limit, if it exists.

1. ( )39

lim 1 4x

x

2. ( )4 31

lim 6 7 12 25y

y y y

+ +

3. 2

20

6lim3t

tt+

4. 246lim

2 3zzz +

5. 222lim

6 16ww

w w+


Calculus I

6. 2

25

6 5lim2 15t

t tt t

+ ++

7. 2

23

5 16 3lim9x

x xx

+

8. 2

21

10 9lim3 4 7z

z zz z +

9. 3

22

8lim8 12x

xx x

++ +

10. ( )

28

5 24lim

8tt t

t t

11. ( )( )

2

4

16lim2 3 6ww

w w

+

12. ( )3

0

2 8limh

hh

+

13. ( )4

0

1 1limh

hh

+

14. 25

5lim25t

tt

15. 2

2lim2xx

x

16. 6

6lim3 2 4z

zz

17. 2

3 1 4lim2 4z

zz

+


Calculus I

18. 3

3lim1 5 11t

tt t

+

19. 7

1 17lim

7xx

x

20. 1

1 14 3lim

1yy y

y

++

+

21. Given the function

( )15 4

6 2 4x

f xx x

< =

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

7limx

f x

(b) ( )4

limx

f x


( )2 3 2

5 14 2t t t

g tt t

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

6limw

h w

(b) ( )2

limw

h w


( ) 25 24 3

3 41 2 4

x xg x x x

x x

+ < =

Calculus I

(a) ( )3

limx

g x

(b) ( )0

limx

g x

(c) ( )4

limx

g x

(d) ( )12

limx

g x

For problems 25 30 evaluate the limit, if it exists.

25. ( )10

lim 10 3z

t

+ +

26. ( )4

lim 9 8 2x

x

+

27. 0

limh

hh

28. 2

2lim2tt

t

29. 5

2 10lim

5www++

30. 244

lim16x

xx

31. Given that ( )3 2 1x f x x+ for all x determine the value of ( )

4limx

f x

.

32. Given that ( ) 172

xx f x + for all x determine the value of ( )9

limx

f x

.

33. Use the Squeeze Theorem to determine the value of 40

3lim cosx

xx

.

34. Use the Squeeze Theorem to determine the value of 0

1lim cosx

xx

.

35. Use the Squeeze Theorem to determine the value of ( )21

1lim 1 cos1x

xx

.


Calculus I

Infinite Limits For problems 1 8 evaluate the indicated limits, if they exist.

1. For ( )( )2

41

g xx

=

evaluate,

(a) ( )1

limx

g x

(b) ( )1

limx

g x+

(c) ( )1

limx

g x

2. For ( )( )3

174

h zz

=

evaluate,

(a) ( )4

limz

h z

(b) ( )4

limz

h z+

(c) ( )4

limz

h z

3. For ( )( )

2

74

3tg t

t=

+ evaluate,

(a) ( )3

limt

g t

(b) ( )3

limt

g t+

(c) ( )3

limt

g t

4. For ( ) 31

8xf x

x+

=+

evaluate,

(a) ( )2

limx

f x

(b) ( )2

limx

f x+

(c) ( )2

limx

f x

5. For ( )( )42

1

9

xf xx

=

evaluate,

(a) ( )3

limx

f x

(b) ( )3

limx

f x+

(c) ( )3

limx

f x

6. For ( ) ( )ln 8W t t= + evaluate, (a) ( )

8lim

wW t

(b) ( )

8lim

wW t

+ (c) ( )

8limw

W t

7. For ( ) lnh z z= evaluate, (a) ( )

0limz

h z

(b) ( )0

limz

h z+

(c) ( )0

limz

h z

8. For ( ) ( )cotR y y= evaluate, (a) ( )lim

yR y

(b) ( )lim

yR y

+ (c) ( )lim

yR y


Calculus I

For problems 9 12 find all the vertical asymptotes of the given function.

9. ( ) 69

h xx

=

10. ( )( )32

85 2xf x

x x+

=

11. ( ) ( )( )5

7 12tg t

x x x=

+

12. ( )( ) ( )

2

5 62

1

1 15

zg zz z

+=

+

Limits At Infinity, Part I 1. For ( ) 3 58 9 11f x x x x= + evaluate each of the following limits. (a) ( )lim

xf x

(b) ( )lim

xf x

2. For ( ) 2 410 6 2h t t t t= + + evaluate each of the following limits. (a) ( )lim

th t

(b) ( )lim

th t

3. For ( ) 3 47 8g z z z= + + evaluate each of the following limits. (a) ( )lim

zg z

(b) ( )lim

zg z

For problems 4 17 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.

4. ( )3

3

10 67 9

x xf xx

=+


Calculus I

5. ( ) 212

3 8 23xf x

x x+

= +

6. ( )8

3 5 8

5 910 3xf x

x x x

=+

7. ( )2

2

2 6 915 4

x xf xx x

=+

8. ( )4

2

5 74x xf x

x+

=

9. ( )3 2

3

4 3 2 110 5

x x xf xx x

+ =

+

10. ( )8

3

52 7 1

xf xx x

=

+

11. ( )3 21 4

9 10xf xx

+=

+

12. ( )2

25 75 2

xf xx+

=+

13. ( )28 11

9xf xx

+=

14. ( )4 2

2

9 2 35 2x xf x

x x+ +

=

15. ( )3

6

68 4

xf xx

+=

+

16. ( )3 32 84 7

xf xx

=

+


Calculus I

17. ( )44

15 2

xf xx

+=

+

Limits At Infinity, Part II

For problems 1 11 evaluate (a) ( )limx

f x

and (b) ( )limx

f x

.

1. ( ) 4 8x xf x += e

2. ( ) 2 52 4 2x x xf x + += e

3. ( )3

23 xx xf x

+= e

4. ( )5 97 3

xxf x

+= e

5. ( )6

45 2

8x

x xf x+

= e 6. ( ) 3 1012 2x x xf x = + e e e 7. ( ) 2 149 7x x xf x = e e e 8. ( ) 8 75 220 3x x x xf x = + e e e e

9. ( )15

15

4

46

11 6

x x

x xf x

+=

+e ee e

10. ( )3 10

7

9 42

x x x

x xf x

+

=

e e ee e

11. ( )14 18

20 9

32

x x

x x xf x

= e e

e e e

For problems 12 20 evaluate the given limit.


Calculus I

12. ( )2lim ln 5 12 6x

x x

+

13. ( )5lim ln 5 7y

y

14. 33lim ln

1 5xxx

+ +

15. 3

2

2 5lim ln4 3tt t

t +

16. 2

2

10 8lim ln1z

z zz

+

17. ( )1 3lim tan 7 4x

x x

+

18. ( )1 2 6lim tan 4w

w w

18. 3 2

1 4lim tan1 3tt t

t

+ +

19. 4

12 3

4lim tan3 5z

zz z

+ +

Continuity

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


Calculus I




Calculus I

For problems 4 13 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.

4. ( ) 6 27 14

xf xx+

=

(a) 3x = , (b) 0x = , (c) 2x = ?

5. ( ) 22

25yR y

y=

(a) 5y = , (b) 1y = , (c) 3y = ?

6. ( ) 25 20

12zg z

z z

=

(a) 1z = , (b) 0z = , (c) 4z = ?

7. ( ) 22

6 7xW x

x x+

=+

(a) 7x = , (b) 0x = , (c) 1x = ?

8. ( )22 1

4 6 1z z

h zz z

< =

+

(a) 6z = , (b) 1z = ?

9. ( ) 200

xx xg x

x x +

9.

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )2 0 0 0 3 0 6 0

0 on 0,3 , 6, 0 on , 2 , 2,0 , 3,6

g g g g

g x g x

= = = =

< >

10.

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )1 0 2 0 5 0

0 on , 1 , 1,2 0 on 2,5 , 5,

h h h

h x h x

= = =

< >

For problems 11 28 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.

11. ( ) 3 215 63 3f t t t t= + + 12. ( ) 2 3 420 8 4g x x x x= + + 13. ( ) 3 28 18 24 10Q w w w w= 14. ( ) 5 4 354 20 7f x x x x= + 15. ( ) 2 35 4 9 3P x x x x= 16. ( ) 5 4 36 5R z z z z= + + 17. ( ) 2 3 41 12 9 2h z z z z= 18. ( ) ( )7 sin 4Q t t t= + on 3 32 2,


Calculus I

19. ( ) ( )26 20cos zf z z= on [ ]0,22 20. ( ) ( )324cos 8 2xg x x= + + on [ ]30,25 21. ( ) ( )9 5sin 2h w w w= on [ ]5,0

22. ( ) ( )5 7h x x x= +

23. ( ) ( )( )2

2 310 2W z w w= +

24. ( ) ( ) 32 28 4f t t t=

25. ( )3 21

3 3x x xf x = e 26. ( ) ( )2 38 zh z z = e

27. ( ) ( )2ln 5 8A t t t= + +

28. ( ) ( )23 ln 1g x x x x= + + + 29. Answer each of the following questions.

(a) What is the minimum degree of a polynomial that has exactly one relative extrema? (b) What is the minimum degree of a polynomial that has exactly two relative extrema? (c) What is the minimum degree of a polynomial that has exactly three relative extrema? (d) What is the minimum degree of a polynomial that has exactly n relative extrema?

30. For some function, ( )f x , it is known that there is a relative minimum at 4x = . Answer each of the following questions about this function.

(a) What is the simplest form that 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.

(b) Using your answer from (a) determine the most general form that the function itself can take.

(c) Given that ( )4 6f = find a function that will have a relative minimum at 4x = . Note : There are many possible answers here so just give one of them.


Calculus I

31. For some function, ( )f x , it is known that there is a relative maximum at 1x = . Answer each of the following questions about this function.



(c) Given that ( )1 3f = find a function that will have a relative maximum at 1x = . Note : There are many possible answers here so just give one of them.

32. For some function, ( )f x , it is known that there is a critical point at 3x = that is neither a relative minimum or a relative maximum. Answer each of the following questions about this function.



(c) Given that ( )3 2f = find a function that will have a critical point at 3x = that is neither a relative minimum or a relative maximum. Note : There are many possible answers here so just give one of them.

33. For some function, ( )f x , it is known that there is a relative maximum at 1x = and a relative minimum at 4x = . Answer each of the following questions about this function.



(c) Given that ( )1 6f = and ( )4 2f = find a function that will have a relative maximum at 1x = and a relative minimum at 4x = . Note : There are many possible answers here so just give one of them.

34. Given that ( )f x and ( )g x are increasing functions will ( ) ( ) ( )h x f x g x= always be an increasing function? If so, prove that ( )h x will be an increasing function. If not, find increasing functions, ( )f x and ( )g x , so that ( )h x will be a decreasing function and find a different set of increasing functions so that ( )h x will be an increasing function.


Calculus I

35. Given that ( )f x is an increasing function. There are several possible conditions that we can impose on ( )g x so that ( ) ( ) ( )h x f x g x= will be an increasing function. Determine as many of these possible conditions as you can. 36. For a function ( )f x determine a set of conditions on ( )f x , different from those given in

#15 in the practice problems, for which ( ) ( ) 2h x f x= will be an increasing function.

37. For a function ( )f x determine a single condition on ( )f x for which ( ) ( ) 3h x f x= will be an increasing function. 38. Given that ( )f x and ( )g x are positive functions. Determine a set of conditions on them for which ( ) ( ) ( )h x f x g x= will be an increasing function. Note that there are several possible sets of conditions here, but try to determine the simplest set of conditions.

39. Repeat #38 for ( ) ( )( )f x

h xg x

= .

40. Given that ( )f x and ( )g x are increasing functions prove that ( ) ( )( )h x f g x= will also be an increasing function.

The Shape of a Graph, Part II For problems 1 & 2 the graph of a function is given. Determine the open intervals on which the function is concave up and concave down. 1.


Calculus I

2.

For problems 3 5 the graph of the 2nd derivative of a function is given. From this graph determine the open intervals in which the function is concave up and concave down. 3.


Calculus I

4.

5.

For problems 6 18 answer each of the following.

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

6. ( ) 3 29 24 6f x x x x= + + 7. ( ) 4 3 22 120 84 35Q t t t t t= + 8. ( ) 5 4 33 20 40h z z z z= +


Calculus I

9. ( ) 4 3 25 2 18 108 12g w w w w w= + 10. ( ) 4 5 610 360 20 3g x x x x x= + + + 11. ( ) ( )2 49 3 160sin xA x x x= on [ ]20,10 12. ( ) ( ) 23cos 2 14f x x x= on [ ]0,6 13. ( ) ( )21 2 sin 2h t t t= + on [ ]2,4

14. ( ) ( )138R v v v=

15. ( ) ( )( )251 3g x x x= +

16. ( ) 4 x xf x = e e 17. ( ) 2 wh w w = e

18. ( ) ( )2 2ln 1A w w w= + For problems 19 33 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. (d) Determine the open intervals on which the function is concave up and concave down. (e) Determine the inflection points of the function. (f) Use the information from steps (a) (e) to sketch the graph of the function.

19. ( ) 2 310 30 2f x x x= + 20. ( ) 3 314 4G t t t= + 21. ( ) 4 3 24 18 9h w w w w= + 22. ( ) 3 4 510 10 3g z z z z= + +


Calculus I

23. ( ) 6 5 49 20 10f z z z z= + + 24. ( ) ( )3 5sin 2Q t t t= on [ ]1,4 25. ( ) ( )1 12 3cosg x x x= + on [ ]25,0

26. ( ) ( )134h x x x=

27. ( ) 2 1f t t t= +

28. ( ) ( )45 27A z z z=

29. ( ) 4 6w wg w = e e

30. ( )21

413 tP t t = e

31. ( ) ( )31 xg x x = + e

32. ( ) ( )2ln 1h z z z= + +

33. ( ) ( )22 8ln 4f w w w= + 34. Answer each of the following questions.

(a) What is the minimum degree of a polynomial that has exactly two inflection points. (b) What is the minimum degree of a polynomial that has exactly three inflection points. (c) What is the minimum degree of a polynomial that has exactly n inflection points.

35. For some function, ( )f x , it is known that there is an inflection point at 3x = . Answer each of the following questions about this function.

(a) What is the simplest form that the 2nd derivative of this function? . (b) Using your answer from (a) determine the most general form that the function itself can

take. (c) Given that ( )0 6f = and ( )3 1f = find a function that will have an inflection point at

3x = .


Calculus I

For problems 36 39 ( )f x is a polynomial. Given the 2nd derivative of the function classify, if possible, each of the given critical points as relative minimums or relative maximum. If it is not possible to classify the critical point(s) clearly explain why they cannot be classified. 36. ( ) 23 4 15f x x x = . The critical points are : 3x = , 0x = and 5x = . 37. ( ) 3 24 21 24 68f x x x x = + . The critical points are : 2x = , 4x = and 7x = . 38. ( ) 2 323 18 9 4f x x x x = + . The critical points are : 4x = , 1x = and 3x = . 39. ( ) 2 3 4216 410 249 60 5f x x x x x = + + . The critical points are : 1x = , 4x = and

5x = .

40. Use ( ) ( ) ( )3 41 1f x x x= + for this problem. (a) Determine the critical points for the function. (b) Use the 2nd derivative test to classify the critical points as relative minimums or relative

maximums. If it is not possible to classify the critical point(s) clearly explain why they cannot be classified.

(c) Use the 1st derivative test to classify the critical points as relative minimums, relative maximums or neither.

41. Given that ( )f x and ( )g x are concave down functions. If we define ( ) ( ) ( )h x f x g x= + show that ( )h x is a concave down function.

42. Given that ( )f x is a concave up function. Determine a condition on ( )g x for which ( ) ( ) ( )h x f x g x= + will be a concave up function.

43. For a function ( )f x determine conditions on ( )f x for which ( ) ( ) 2h x f x= will be a concave up function. Note that there are several sets of conditions that can be used here. How many of them can you find?

The Mean Value Theorem For problems 1 4 determine all the number(s) c which satisfy the conclusion of Rolles Theorem for the given function and interval.


Calculus I

1. ( ) 3 24 3f x x x= + on [ ]0,4 2. ( ) 215 2Q z z z= + on [ ]2, 4

3. ( ) 2 91 th t = e on [ ]3,3 4. ( ) [ ]1 cosg w w= + on [ ]5,9 For problems 5 8 determine all the number(s) c which satisfy the conclusion of the Mean Value Theorem for the given function and interval. 5. ( ) 3 2 8f x x x x= + + on [ ]3,4 6. ( ) 3 22 7 1g t t t t= + + on [ ]1,6 7. ( ) 2 6 3tP t t= e 8. ( ) ( )29 8sin xh x x= on [ ]3, 1 9. Suppose we know that ( )f x is continuous and differentiable on the interval [ ]2,5 , that ( )5 14f = and that ( ) 10f x . What is the smallest possible value for ( )2f ?

10. Suppose we know that ( )f x is continuous and differentiable on the interval [ ]6, 1 , that ( )6 23f = and that ( ) 4f x . What is the smallest possible value for ( )1f ?

11. Suppose we know that ( )f x is continuous and differentiable on the interval [ ]3,4 , that ( )3 7f = and that ( ) 17f x . What is the largest possible value for ( )4f ?

12. Suppose we know that ( )f x is continuous and differentiable on the interval [ ]1,9 , that ( )9 0f = and that ( ) 8f x . What is the largest possible value for ( )1f ?

13. Show that ( ) 7 5 32 3 14 1f x x x x x= + + + + has exactly one real root. 14. Show that ( ) 3 26 2 4 3f x x x x= + has exactly one real root.


Calculus I

15. Show that ( ) 420 xf x x = e has exactly one real root.

Optimization 1. Find two positive numbers whose sum of six times one of them and the second is 250 and whose product is a maximum. 2. Find two positive numbers whose sum of twice the first and seven times the second is 600 and whose product is a maximum. 3. Let x and y be two positive numbers such the sum is 175 and ( )( )3 4x y+ + is a maximum. 4. Find two positive numbers such that the sum of one and the square of the other is 200 and whose product is a maximum. 5. Find two positive numbers whose product is 400 and such that the sum of twice the first and three times the second is a minimum. 6. Find two positive numbers whose product is 250 and such that the sum of the first and four times the second is a minimum. 7. Let x and y be two positive numbers such that ( )2 100y x + = and whose sum is a minimum. 8. Find a positive number such that the sum of the number and its reciprocal is a minimum. 9. We are going to fence in a rectangular field and have 200 feet of material to construct the fence. Determine the dimensions of the field that will enclose the maximum area. 10. We are going to fence in a rectangular field. The cost of material of each side is $6/ft, $9/ft, $12/ft and $14/ft respectively. If we have $1000 to buy fencing material determine the dimensions of the field that will maximize the enclosed area. 11. We are going to fence in a rectangular field that encloses 75 ft2. Determine the dimensions of the field that will require the least amount of fencing material to be used. 12. We are going to fence in a rectangular field that encloses 200 m2. If the cost of the material for of one pair of parallel sides is $3/ft and cost of the material for the other pair of parallel sides is $8/ft determine the dimensions of the field that will minimize the cost to build the fence around the field. 13. Show that a rectangle with a fixed area and minimum perimeter is a square.


Calculus I

14. Show that a rectangle with a fixed perimeter and a maximum area is a square. 15. We have 350 m2 of material to build a box whose base width is four times the base length. Determine the dimensions of the box that will maximize the enclosed volume. 16. We have $1000 to buy the materials to build a box whose base length is seven times the base width and has no top. If the material for the sides cost $10/cm2 and the material for the bottom cost $15/cm2 determine the dimensions of the box that will maximize the enclosed volume. 17. We want to build a box whose base length is twice the base width and the box will enclose 80 ft3. The cost of the material of the sides is $0.5/ft2 and the cost of the top/bottom is $3/ft2. Determine the dimensions of the box that will minimize the cost. 18. We want to build a box whose base is a square, has no top and will enclose 100 m3. Determine the dimensions of the box that will minimize the amount of material needed to construct the box. 19. We want to construct a cylindrical can with a bottom but no top that will have a volume of 65 in3. Determine the dimensions of the can that will minimize the amount of material needed to construct the can. 20. We want to construct a cylindrical can whose volume is 105 mm3. The material for the wall of the can costs $3/mm2, the material for the bottom of the can costs $7/mm2 and the material for the top of the can costs $2/mm2. Determine the dimensions of the can that will minimize the cost of the materials needed to construct the can. 21. We have a piece of cardboard that is 30 cm by 16 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. 22. We have a piece of cardboard that is 5 in by 20 in 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. 23. A printer needs to make a poster that will have a total of 500 cm2 that will have 3 cm margins on the sides and 2 cm margins on the top and bottom. What dimensions of the poser will give the largest printed area? 24. A printer needs to make a poster that will have a total of 125 in2 that will have inch margin on the bottom, 1 inch margin on the right, 2 inch margin on the left and 4 inch margin on the top. What dimensions of the poser will give the largest printed area?


Calculus I

More Optimization Problems 1. We want to construct a window whose bottom is a rectangle and the top of the window is an equilateral triangle. If we have 75 inches of framing material what are the dimensions of the window that will let in the most light? 2. We want to construct a window whose middle is a rectangle and the top and bottom of the window are equilateral triangles. If we have 4 feet of framing material what are the dimensions of the window that will let in the most light? 3. We want to construct a window whose middle is a rectangle, the top of the window is a semicircle and the bottom of the window is an equilateral triangle. If we have 1500 cm of framing material what are the dimensions of the window that will let in the most light? 4. Determine the area of the largest rectangle that can be inscribed in a circle of radius 5. 5. Determine the area of the largest rectangle whose base is on the x-axis and the top two corners lie on semicircle of radius 16. 6. Determine the area of the largest rectangle whose base is on the x-axis and the top two corners lie 24y x= .

7. Find the point(s) on 2 2

14 36x y

+ = that are closest to ( )0,1 .

8. Find the point(s) on 2 8x y= that are closest to ( )5,0 . 9. Find the point(s) on 22y x= that are closest to ( )0, 3 . 10. A 6 ft 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 twice the length of the other side. Determine where, if anywhere, the wire should be cut to minimize the area enclosed by the two figures. 11. A 250 cm piece of wire is cut into two pieces. One piece is bent into an equilateral triangle and the other will be bent into circle. Determine where, if anywhere, the wire should be cut to maximize the area enclosed by the two figures. 12. A 250 cm piece of wire is cut into two pieces. One piece is bent into an equilateral triangle and the other will be bent into circle. Determine where, if anywhere, the wire should be cut to minimize the area enclosed by the two figures.


Calculus I

13. A 4 m piece of wire is cut into two pieces. One piece is bent into a circle and the other will be bent into a rectangle with one side three 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. 14. A line through the point ( )4,1 forms a right triangle with the x-axis and y-axis in the 2nd quadrant. Determine the equation of the line that will minimize the area of this triangle. 15. A line through the point ( )3,3 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. 16. A piece of pipe is being carried down a hallway that is 14 feet wide. At the end of the hallway there is a right-angled turn and the hallway narrows down to 6 feet wide. What is the longest pipe (always keeping it horizontal) that can be carried around the turn in the hallway? 17. A piece of pipe is being carried down a hallway that is 9 feet wide. At the end of the hallway there is a right-angled turn and the hallway widens up to 21 feet wide. What is the longest pipe (always keeping it horizontal) that can be carried around the turn in the hallway? 18. Two poles, one 15 meters tall and one 10 meters tall, are 40 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? 19. Two poles, one 2 feet tall and one 5 feet tall, are 3 feet 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.? 20. Two poles, one 15 meters tall and one 10 meters tall, are 40 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 angle formed by the two pieces of wire at the stake is a maximum? 21. Two poles, one 34 inches tall and one 17 inches tall, are 3 feet 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 angle formed by the two pieces of wire at the stake is a maximum? 22. A trough for holding water is to be formed as shown in the figure below. Determine the angle that will maximize the amount of water that the trough can hold.


Calculus I

23. A trough for holding water is to be formed as shown in the figure below. Determine the angle that will maximize the amount of water that the trough can hold.

Indeterminate Forms and LHospitals Rule Use LHospitals Rule to evaluate each of the following limits.

1. 3 2

3 24

6 32lim5 4x

x xx x x

+ + +

2. 6

3lim 4w

w

w

+e

e

3. ( )( )0

sin 6lim

sin 11ttt

4. 2

3 21

8 9lim2 5 6x

x xx x x

+ +

5. 3 2

4 3 22

7 16 12lim4 4t

t t tt t t + +

6. 2

2

4 1lim3 7 4ww ww w

++

7. 2 6

2 7lim 4y

y

yyy+ee

8. ( )( )

2

20

2cos 4 4 2lim

sin 2 2xx x

x x x

9. 2 6 2

3 23

3 12lim6 9x

x xx x x

+

+ + +

e


Calculus I

10. ( )

( )6sin

limln 5z

zz

11.

2

0

0limx

x t dt

x e

12. 2lim ln 1

3ww

w

13. ( ) ( )0

lim ln sint

t t+

14. 2lim

z

zz

e

15. 7lim sin

xx

x

16. ( )220

lim lnz

z z+

17. 1

0limx

xx+

18. 1

0limt

t tt+ + e

19. 1

2lim 3x

x xx

e

20. Suppose that we know that ( )f x is a continuous function. Use LHospitals Rule to show that,

( ) ( ) ( )0

lim2h

f x h f x hf x

h+

=

21. Suppose that we know that ( )f x is a continuous function. Use LHospitals Rule to show that,

( ) ( ) ( ) ( )202

limh

f x h f x f x hf x

h+ +

=


Calculus I

Linear Approximations For problems 1 4 find a linear approximation to the function at the given point. 1. ( ) ( )cos 2f x x= at x =

2. ( ) ( )2ln 5h z z= + at 2z = 3. ( ) 2 32 9 3g x x x x= at 1x =

4. ( ) ( )sin tg t = e at 4t = 5. Find the linear approximation to ( ) ( )sin 1h y y= + at 0y = . Use the linear approximation to approximate the value of ( )sin 2 and ( )sin 15 . Compare the approximated values to the exact values.

6. Find the linear approximation to ( ) 5R t t= at 32t = . Use the linear approximation to approximate the value of 5 31 and 5 3 . Compare the approximated values to the exact values. 7. Find the linear approximation to ( ) 1 xh x = e at 1x = . Use the linear approximation to approximate the value of e and 4e . Compare the approximated values to the exact values. For problems 8 10 estimate the given value using a linear approximation and without using any kind of computational aid. 8. ( )ln 1.1

9. 8.9 10. ( )sec 0.1

Differentials For problems 1 5 compute the differential of the given function.


Calculus I

1. ( ) 6 3 23 8 9 4f x x x x x= + 2. ( )2 cos 2u t t=

3. ( )cos zy = e 4. ( ) ( ) ( )sin 3 cos 1g z z z=

5. ( ) 4 6 xR x x = + e 5. Compute dy and y for ( )siny x= as x changes from 6 radians to 6.05 radians.

6. Compute dy and y for ( )2ln 1y x= + as x changes from -2 to -2.1.

7. Compute dy and y for 12

yx

=

as x changes from 3 to 3.02.

8. Compute dy and y for 14 xy x= e as x changes from -10 to -9.99.

9. 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 surface area of the cube if we use this value of the length of the side to compute the surface area? 10. The radius of a circle is found to be 7 cm in length with a possible error of no more than 0.04 cm. What is the maximum possible error in the area of the circle if we use this value of the radius to compute the area? 11. The radius of a sphere is found to be 22 cm in length with a possible error of no more than 0.07 cm. What is the maximum possible error in the volume of the sphere if we use this value of the radius to compute the volume? 12. The radius of a sphere is found to be foot in length with a possible error of no more than 0.03 inches. What is the maximum possible error in the surface area of the sphere if we use this value of the radius to compute the surface area?


Calculus I

Newtons Method

For problems 1 3 use Newtons Method to determine 2x for the given function and given value

of 0x . 1. ( ) 37 8 4f x x x= + , 0 1x = 2. ( ) ( ) ( )cos 3 sinf x x x= , 0 0x = 3. ( ) 2 37 xf x = e , 0 5x = For problems 4 8 use Newtons Method to find the root of the given equation, accurate to six decimal places, that lies in the given interval. 4. 5 6x = in [ ]1,2 5. 3 22 9 17 20 0x x x + + = in [ ]1,1 6. 3 43 12 4 3 0x x x = in [ ]3, 1 7. ( )4cosx x=e in [ ]1,1

8. 22 2 xx = e in [ ]0,2

For problems 9 12 use Newtons Method to find all the roots of the given equation accurate to six decimal places. 9. 3 22 5 10 4 0x x x+ = 10. 4 3 24 54 92 105 0x x x x+ + =

11. ( )232 cosx x =e 12. ( ) ( )ln 2cosx x=


Calculus I

13. Suppose that we want to find the root to 3 27 8 3 0x x x + = . Is it possible to use 0 4x = as the initial point? What can you conclude about using Newtons Method to approximate roots from this example? 14. Use the function ( ) ( ) ( )2cos sinf x x x= for this problem.

(a) Plot the function on the interval [ ]0,9 . (b) Use 0 4x = to find one of the roots of this function to six decimal places. Did you get the

root you expected to? (c) Use 0 5x = to find one of the roots of this function to six decimal places. Did you get the

root you expected to? (d) Use 0 6x = to find one of the roots of this function to six decimal places. Did you get the

root you expected to? (e) What can you conclude about choosing values of 0x to find roots of equations using

Newtons Method. 15. Use 0 0x = to find one of the roots of

5 32 7 3 1 0x x x + = accurate to six decimal places.

Did we chose a good value of 0x for this problem?

Business Applications 1. A company can produce a maximum of 2500 widgets in a year. If they sell x widgets during the year then their profit, in dollars, is given by, ( ) 2 313500,000,000 1,540,000 1450P x x x x= + How many widgets should they try to sell in order to maximize their profit? 2. A company can produce a maximum of 25 widgets in a day. If they sell x widgets during the day then their profit, in dollars, is given by, ( ) 2 3133000 40 11P x x x x= + How many widgets should they try to sell in order to maximize their profit? 3. 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,


Calculus I

( ) 2 32736 211 15 50 15070,000C x x x x= + + The land they have purchased can hold a complex of at most 400 apartments. How many apartments should the complex have in order to minimize the maintenance costs? 4. The production costs of producing x widgets is given by,

( ) 90,0002000 4C x xx

= + +

If the company can produce at most 200 widgets how many should they produce to minimize the production costs? 5. The production costs, in dollars, per day of producing x widgets is given by, ( ) 2 3400 3 2 0.002C x x x x= + + What is the marginal cost when 20x = and 75x = ? What do your answers tell you about the production costs? 6. The production costs, in dollars, per month of producing x widgets is given by,

( ) 28,000,00010,000 14C x x

x= +

What is the marginal cost when 80x = and 150x = ? What do your answers tell you about the production costs? 7. The production costs, in dollars, per week of producing x widgets is given by, ( ) 2 365,000 4 0.2 0.00002C x x x x= + + and the demand function for the widgets is given by, ( ) 5000 0.5p x x= What is the marginal cost, marginal revenue and marginal profit when 2000x = and 4800x = ? What do these numbers tell you about the cost, revenue and profit? 8. The production costs, in dollars, per week of producing x widgets is given by,


Calculus I

( ) 2 56,000800 0.008C x xx

= + +

and the demand function for the widgets is given by, ( ) 2350 0.05 0.001p x x x= What is the marginal cost, marginal revenue and marginal profit when 175x = and 325x = ? What do these numbers tell you about the cost, revenue and profit?

Integrals

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. 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

Indefinite Integrals 1. Evaluate each of the following indefinite integrals.

(a) 9 310 12 5x x dx


Calculus I

(b) 9 310 12 5x x dx 2. Evaluate each of the following indefinite integrals.

(a) 7 233 8t t t dt+ + (b) 7 233 8t dt t t+ + 3. Evaluate each of the following indefinite integrals.

(a) 5 3 26 7 12 10x x x dx + (b) 5 3 26 7 12 10x x dx x + (c) 5 3 26 7 12 10x dx x x + 4. Evaluate each of the following indefinite integrals.

(a) 6 5 321 9x x x x dx (b) 6 5 321 9x x x dx x (c) 6 5 321 9x x dx x x For problems 5 9 evaluate the indefinite integral.

5. 5 28 15 1t t dt

6. 9 5 3120 24 4y y y dy

7. dw

8. 9 6 314 10 13x x x x dx+ +

9. 6 4 28 7 11 12x x x x dx + 10. Determine ( )f x given that ( ) 4 216 9f x x x x = . 11. Determine ( )g t given that ( ) 5 24 1

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