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SALAS / HILLE / ETGEN

CALCULUSOne and Several

Variables

Eighth Edition

PREPARED BY

Deborah Betthauser Britt

John Wiley & SonsNew York Chichester Weinheim Brisbane Singapore Toronto

C O N T E N T S

Chapter 1 Introduction............................................................................. 1

Chapter 2 Limits and Continuity.............................................................. 11

Chapter 3 Differentiation......................................................................... 27

Chapter 4 The Mean-Value Theorem and Applications .......................... 41

Chapter 5 Integration............................................................................... 57

Chapter 6 Some Applications of the Integral ........................................... 71

Chapter 7 The Transcendental Functions................................................. 81

Chapter 8 Techniques of Integration........................................................ 97

Chapter 9 Conic Sections; Polar Coordinates; Parametric Equations ....... 109

Chapter 10 Sequences; Indeterminate Forms; Improper Integrals.............. 123

Chapter 11 Infinite Series .......................................................................... 133

Chapter 12 Vectors.................................................................................... 149

Chapter 13 Vector Calculus....................................................................... 157

Chapter 14 Functions of Several Variables................................................ 171

Chapter 15 Gradients; Extreme Values; Differentials ................................ 177

Chapter 16 Double and Triple Integrals..................................................... 191

Chapter 17 Line Integrals and Surface Integrals ........................................ 203

Chapter 18 Elementary Differential Equations .......................................... 219

TO THE INSTRUCTOR

The Test Bank has been prepared for the eighth edition of Salas, Hille, and Etgen’s Calculus:One and Several Variables. Each chapter is divided into sections that correspond to the sectionsin the textbook. In all, the Test Bank consists of 2,223 problems, together with their solutionsthat appear immediately after the end of each chapter. Numerous charts and illustrations havealso been drawn where appropriate to strengthen the presentation. Far more items are offeredwithin each chapter than any user of the Test Bank would ordinarily need. Due to differences inrounding, some of the answers requiring calculations by students may differ slightly from theones given in the Test Bank.

The entire Test Bank has been incorporated into MICROTEST, a computerized test preparationsystem, by Delta Software. This software has been designed to retrieve the questions from theTest Bank and print your tests and answer keys using a PC running on Windows 3.1 or higher.The questions on your tests printed by the software will be exact duplicates of those shown here.The software includes a host of powerful features and flexibility for selecting questions andproducing multiple versions of tests without your having to retype the items or draw the pictures.To obtain MICROTEST software for this textbook, please contact your John Wiley salesrepresentative.

I wish to thank Edwin and Shirley Hackleman of Delta Software for their composition andeditorial assistance in preparing the hard copy manuscript and Donald Newell of Delta Softwarefor developing and checking the computerized version of the Test Bank. We have endeavored asa team to produce test items that we trust instructors will find a useful supplement to Calculus:One and Several Variables.

DEBORAH BETTHAUSER BRITT

1

CHAPTER 1

Introduction

1.2 Notions and Formulas from Elementary Mathematics

1. Is the number13 122 2− rational or irrational?

2. Is the number 5.121122111222 . . . rational or irrational?

3. Write 6.27272727 . . . in rational form p / q.

4. Find, if any, upper and lower bounds for the set S = x : x3 > 1.

5. Find, if any, upper and lower bounds for the set S =

=−

3,... 2, 1, :12

nn

n.

6. Rewrite 27 – 8x3 in factored form.

7. Rewrite x4 – 18x2 + 81 in factored form.

8. Evaluate 46

!!.

9. Evaluate 7

4 3!

! !.

10. What is the ratio of the surface area of a cube of side x to the surface area of a sphere of diameter x?

1.3 Inequalities

11. Solve x + 3 < 2x – 8.

12. Solve − ≥ −2

3

1

6

3

4x x .

13. Solve 8 – x2 > 7x.

14. Solve x2 – 5x + 5 ≥ 1.

15. Solve 3 112

32 2x x− ≥ +( ) .

16. Solve 2 34 1

1x

x

+−

< .

17. Solve1 1

1x x<

+.

18. Solve 12

3 2 223

5 3( ) ( )x x+ < − − .

Calculus: One and Several Variables2

19. Solve 3

22

1x x+<

−.

20. Solve 2 3

21

x

x

−+

≥ .

21. Solve x(2x – 1)(3x + 2) < 0.

22. Solve x

x

2

2 90

−< .

23. Solve x2(x – 1)(x + 2)2 > 0.

24. Solve x

x x

++

>2

30

2( ).

25. Solve |x| > 2.

26. Solve x + ≥113

.

27. Solve 21

1 ≤−x .

28. Solve 01

41< − <x .

29. Solve |2x – 1| < 5.

30. Solve |3x – 5|3

2≥ .

31. Find the inequality of the form x c− < δ whose solution is the open interval (–1, 5).

32. Find the inequality of the form x c− < δ whose solution is the open interval (–2, 3).

33. Determine all values of A > 0 for which the following statement is true. If |2x – 5| < 1, then |4x – 10| < A.

34. Determine all values of A > 0 for which the following statement is true. If |2x – 3| < A then |6x – 9| < 4.

1.4 Coordinate Plane; Analytic Geometry

35. Find the distance between the points P0(2, –4) and P1(1, 5).

36. Find the midpoint of the line segment from P0(a, 2b) to P1(3a, 5b).

37. Find the slope of the line through P0(–4, 2) and P1(3, 5).

38. Find the slope of the line through P0(–2, –4) and P1(3, 5).

39. Find the slope and the y-intercept for the line 2x + y – 10 = 0.

40. Find the slope and the y-intercept for the line 8x + 3y = 6.

Introduction 3

41. Write an equation for the line with the slope –3 and y-intercept –4.

42. Write an equation for the horizontal line 4 units below the x-axis.

43. Write an equation for the vertical line 2 units to the left of the y-axis.

44. Find an equation for the line that passes through the point P(2, –1) and is parallel to the line 3y + 5x – 6 = 0.

45. Find an equation for the line that passes through the point P(1, –1) and is perpendicular to the line2x – 3y – 8 = 0.

46. Find an equation for the line that passes through the point P(2, 2) and is parallel to the line 2x + 3y = 18.

47. Find an equation for the line that passes through the point P(3, 5) and is perpendicular to the line6x – 7y + 17 = 0.

48. Determine the point(s) where the line y = 2x intersects the circle x2 + y2 = 4.

49. Find the point where the lines l1 and l2 intersect. l1: x + y – 2 = 0; l2: 3x + y = 0

50. Find the area of the triangle with vertices (–1, 1), (4, –2), (3, 6).

51. Find the area of the triangle with vertices (–1, –1), (–4, 3), (3, 3).

52. Find an equation for the line tangent to the circle x2 + y2 – 4x – 2y = 0 at the point P(4, 2).

1.5 Functions

53. If f xx

x( ) =

++2

42, calculate (a) f ( )0 , (b) f ( )1 , (c) f ( )−2 , (d) f ( / )−5 2 .

54. If f xx

x( ) =

+

3

2 2, calculate (a) f x( )− , (b) f x( / )1 , (c) f a b( )+ .

55. f x x( ) = −1 2 Find the number(s), if any, where ƒ takes on the value 1.

56. f x x( ) cos= −1 Find the number(s), if any, where ƒ takes on the value 1.

57. Find the exact value(s) of x in the interval [0, 2π) which satisfy cos 23

2x = − .

58. Find the domain and range for f x x x( ) = − −2 2 .

59. Find the domain and range for f x x( ) = +3 4 .

60. Find the domain and range for 24)( xxh −−= .

61. Find the domain and range for f xx

( ) =−

1

1.


( )( )

=+1

3 2.



( ) =+

1

32.

64. Find the domain and range for f x x( ) cos= − 1

2.

65. Sketch the graph of f x x( ) = −3 4 .

66. Sketch the graph of f x x x( ) = − −6 2 .

67. Sketch the graph of f x xx

( ) = − 4.

68. Sketch the graph of g x x( ) sin= +2 .

69. Sketch the graph of

>−=<

=1 if12

1 if3

1 if2

)(

xx

x

x

xg and give its domain and range.


≥<

=1 if2

1 if)(

2

xx

xxxf and give its domain and range.

71. Is an ellipse the graph of a function?

72. Determine whether f x x x( ) = − +4 2 1 is odd, even, or neither.

73. Determine whether f x x x x( ) = + −5 3 3 is odd, even, or neither.

74. Determine whether f xx x x

x( ) = − + +3 22 5 1

is odd, even, or neither.

75. Determine whether f x x( ) cos( / )= + π 6 is odd, even, or neither.

76. Determine whether f x x x( ) sin= −3 2 is odd, even, or neither.

77. Determine whether f x x x( ) cos sec= + is odd, even, or neither.

78. A given rectangle is twice as long as it is wide. Express the area of the rectangle as a function of the (a)width, (b) length, (c) diagonal.

1.6 The Elementary Functions

79. Find all real numbers x for which 2

2 523)(

xx

xxxR

−++= is undefined.

80. Find all real numbers x for which 2

34)(

2

23

−++−=

xx

xxxxR is zero.

81. Find the inclination of the line 323 +− yx = 0.

Introduction 5

82. Write an equation for the line with inclination 45° and y-intercept –2.

83. Find the distance between the line 4x + 3y + 4 = 0 and (a) the origin (b) the point P(1, 3).

84. Find the distance between the line 2x – 5y – 10 = 0 and (a) the origin (b) the point P(–2, –1).

85. In the triangle with vertices (0, 0), (2, 6), (7, 0), which vertex is farthest from the centroid?

1.7 Combinations of Functions

86. Given that f xx x

x( ) = − −2 6

and g x x( ) = − 3, find (a) f g+ (b) f g− (c) f

g

87. Sketch the graphs of the following functions with ƒ and g as shown in the figure.

(a) 12

f (b) –g (c) g f−

88. xxgxxf =+= )( ,5)( .

(a) Form the composition of f go . (b) Form the composition of g fo .

89.x

xgxxf2

)( ,1)( 2 =+= .


90. 23)( ,11

)( 2 +=+= xxgx

xf .


91.x

xgxxf2

)( ,4)( 2 =+= .


92. 1)( ,)( 3 +== xxgxxf .


93. Form the composition of f g ho o if 12)( ,4

1)( −== xxgxxf , and h x x( ) = 3 2 .


94. Form the composition of f g ho o if 12)( ,)( 2 +== xxgxxf , and h x x( ) = 2 2 .

95. Form the composition of f g ho o if 23

2)( ,

2)(

+==

xxg

xxf , and h x x( ) = 2 .

96. Find ƒ such that f g Fo = given that g(x) = 2x2 and F(x) = x + 2x2 + 3.

97. Find ƒ such that f g Fo = given that g x x( ) = + 1 and F x x x( ) = + 2 .

98. Find g such that f g Fo = given that f x x( ) = −2 1 for all real x and F x x( ) = −3 1 for x ≥ 0.

99. Find g such that f g Fo = given that f x x( ) = 2 and F x x( ) ( )= +2 5 2 .

100. Form f go and g fo given that f x x( ) = +4 1 and g x x( ) = 4 2 .

101. Form f go and g fo given that

≥−

<=

0 if12

0 if1

)(xx

xxxf and that

≥<

=1 if

1 if2)(

2 xx

xxxg .

102. Decide whether f x x( ) = +4 3 and g x x( ) = −14

3 are inverses of each other.

103. Decide whether f x x( ) ( )= − +1 15 and g x x( ) ( ) /= − +1 11 5 are inverses of each other.

1.8 A Note on Mathematical Proof; Mathematical Induction

104. Show that 3n ≤ 3n for all positive integers n.

105. Show that n(n + 1)(n + 2)(n + 3) is divisible by 8 for all positive integers n.

106. Show that 1 + 5 + 9 + . . . + (4n – 3) = 2n2 – n for all positive integers n.

Introduction 7

Answers to Chapter 1 Questions

1. rational

2. irrational

3. 69/11

4. lower bound 1, no upper bound

5. lower bound 1, upper bound 2

6. (3 – 2x)(9 + 6x + 4x2)

7. (x – 3) 2(x + 3)2

8. 1/30

9. 35

10. 6/π

11. (11, ∞)

12. [2, ∞)

13. (–8, 1)

14. (–∞, 1] ∪ [4, ∞)

15. (–∞, 1] ∪ [1, ∞)

16. (–∞, 1/4] ∪ (2, ∞)

17. (–1, 0)

18. (14/3, ∞)

19. (–∞, –2) ∪ (1, 7)

20. (–∞, –2) ∪ [5, ∞)

21. (–∞, –2/3] ∪ (0, 1/2)

22. (–3, 0) ∪ (0, 3)

23. (1, ∞)

24. (–∞, –3) ∪ (–2, 0) ∪ (0, ∞)

25. (–∞, –2) ∪ (2, ∞)

26. (–∞, –4/3] ∪ [–2/3, ∞)

27. [1/2, 3/2]

28. (–3/4, 1/4) ∪ (1/4, 5/4)

29. (–1, 4)

30. (–∞, 13/9) ∪ (17/9, ∞)

31. |x – 2| < 3

32. |x – 1/2| < 5/2

33. A ≥ 2

34. 0 ≤ A ≤ 4/3

35. 82

36. 27

2a b,

37. m = 3/7

38. m = 9/5

39. m = –2; y-intercept: 10

40. m = –8/3; y-intercept: 2

41. y = –3x – 4

42. y = –4

43. x = –2

44. y x= − +5

3

7

3

45. y x= − +32

12

46. y x= − +23

103

47. y x= − +7

6

17

2

48.2 5

5

4 5

5

2 5

5

4 5

5, , ,

− −

49. The point of intersection is P(–1, 3).

50. 37/2

51. 14

52. y = –2x + 10


53. (a) 1/2 (b) 3/5 (c) 0 (d) 2/41

54. (a) −

+x

x

3

2 2(b)

1

2 3x x+(c)

( )

( )

a b

a b

++ +

3

2 2

55. x = 0, 1

56. ( )2 12

n + π, n = integer

57. 5π/12, 7π/12, 17π/12, 19π/12

58. domain: (ƒ) = (–∞, ∞)range: (ƒ) = (–∞, ∞)

59. domain: (ƒ) = [–4/3, ∞)range: (ƒ) = [0, ∞)

60. domain: (ƒ) = [–2, 2]range: (ƒ) = [–2, 0]

61. domain: (ƒ) = [0, 1) ∪ (1, ∞)range: (ƒ) = [–1, 0) ∪ (0, ∞)

62. domain: (ƒ) = (–∞, –3) ∪ (–3, ∞)range: (ƒ) = (0, ∞)

63. domain: (ƒ) = (–∞, ∞)range: (ƒ) = (0, 1/3]

64. domain: (ƒ) = (–∞, ∞)range: (ƒ) = [0, 3/2]

65. y = 3 – 4x

66. 26 xxy −−=

67.

68. y = 2 + sin x

Introduction 9

69. domain: (ƒ) = (–∞, +∞)range: (ƒ) = [2, +∞)

70. domain: (ƒ) = (–∞, +∞)range: (ƒ) = [0, +∞)

71. no

72. even

73. odd

74. neither

75. neither

76. odd

77. even

78. (a) A = 2w2 (b) Al=2

2(c) A d= 2

52

79. 0, 1

80. 0, 3

81. θ = π/6

82. x – y – 2 = 0

83. (a) 4/5 (b) 17/5

84. (a) 10

29(b)

9

29

85. (7, 0)

86. (a) 2 4 62x x

x

− −(c)

x

xx

+ ≠263,

(b) 2 6x

x

−

87.

88. (a) x + 5 (b) x + 54

89. (a) 4

1 0x

x+ >, (b) 2

12x +

90. (a) 1

3 21

2x ++ (b) 3

11 2

2

x+

+

91. (a) 442

+x

(b) 2

4 2+ x

92. (a) |x3 + 1| (b) |x|3 + 1

93.14

6 12( x − )

94. (4x2 + 1)2

95. 3x2 + 2

96. ƒ(x) = x + 3

97. ƒ(x) = x2 – 1


98. g x x( ) = 3

99. g(x) = 2x + 5

100. ( )( )f g x xo = +16 12

( )( ) ( )g f x xo = +4 4 1 2

101.

≥−

<≤−

<

=

1 if12

10 if14

0 if2

1

))((2 xx

xx

xx

xgf o

≥−

<≤−<

=

1 if)12(

10 if)12(2

0 if/2

))((2 xx

xx

xx

xfg o

102. not inverses

103. inverses

104. True for n = 1: 3 ≤ 3. Assume true for n. Then3(n + 1) = 3n + 3 ≤ 3n + 3 ≤ 3n + 3n = 2(3n) < 3(3n)= 3n + 1, so the inequality is true for n + 1.Therefore, by induction, it is true for n ≥ 1.

105. True for n = 1: 1 • 2 • 3 • 4 = 3 • 8. Assume truefor n. Then (n + 1)(n + 2)(n + 3)(n + 4) = n(n +1)(n + 2)(n + 3) + 4(n + 1)(n + 2)(n + 3).The firstterm is divisible by 8 by the inductionhypothesis, and the second term is divisible by 8since at least one of (n + 1), (n + 2), (n + 3) iseven. Hence the result is true for n + 1.Therefore, by induction, it is true for all n ≥ 1.

106. True for n = 1: 1 = 2(1)2 – 1. Assume true for n.Then 1 + 5 + 9 + . . . + [4(n + 1) – 3] = 1 + 5 + 9 +. . . + (4n – 3) + [4(n + 1) – 3] = 2n2 – n + (4n + 1)= 2(n + 1)2 – (n + 1), so the result is true for n + 1.Therefore, by induction, it is true for all n ≥ 1.

11

CHAPTER 2

Limits and Continuity

2.1 The Idea of Limit

1. For the function ƒ graphed below, c = 0. Use the graph of ƒ to find(a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x) (d) ƒ(c)

x→c- x→c+ x→c


x→c- x→c+ x→c

3. For the function ƒ graphed below, c = −3. Use the graph of ƒ to find(a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x) (d) ƒ(c)

x→c- x→c+ x→c



x→c- x→c+ x→c

5. For the function g graphed below, c = −2. Use the graph of g to find(a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x) (d) ƒ(c)

x→c- x→c+ x→c


x→c- x→c+ x→c

Limits and Continuity 13


x→c- x→c+ x→c


x→c- x→c+ x→c


x→c- x→c+ x→c


10. Consider the function ƒ graphed below. State the values of c for which lim ƒ(x) does not exist.x→c





14. Evaluate lim (2x – 5), if it exists.x→2

15. Evaluate lim π2, if it exists.x→0

16. Evaluate lim x3, if it exists.x→–3

17. Evaluatex x→ − −2

5

2lim , if it exists.

18. Evaluate lim (x3 + 6x2 – 16), if it exists.x→–2

19. Evaluatex

x

x→

+

−4

2

2

9


20. Evaluatex

x

x→

−−2

3 6

2 4lim , if it exists.

21. Evaluatex

x x

x→

−0


22. Evaluate2

8lim

3

2 −−

→ x

xx

, if it exists.

23. Evaluate1

3lim 2

2

1 +−

−→ x

x

x, if it exists.

24. Evaluateax

axax −

−→

22

lim , if it exists.

25. Evaluate20

16lim 2

2

4 −+−

→ xx

xx

, if it exists.


26. Evaluate22

2

0 2

2lim

xx

xxx −

+→

, if it exists.

27. Evaluate )(lim2

xfx→

, if it exists.

=≠

=2 ,2

2 ,3)(

x

xxf

28. Evaluate )(lim1

xfx −→

, if it exists.

−≥−<−

=1 ,3

1 ,12)(

xx

xxxf

29. Evaluate )(lim1

xfx→

, if it exists.

>−

<+=

1 ,21

1 ,2

1)(

xx

xxxf

30. Evaluate1

54lim

2

1 −−+

→ x

xx

, if it exists.

2.2 Definition of Limit

31. Evaluate13

2lim

1 +→ x

xx

, if it exists.

32. Evaluatex

x x

x→

+0

3

2

4 1

5lim

( ), if it exists.

33. Evaluatex

x

x→ +2

9


34. Evaluatex

x

x x→

−+ −1

2

2

1


35. Evaluatex

x x

x x→

+ −

− +1

2

2

2


36. Evaluatex

x x x

x→

− +−1

3 23 2


37. Evaluateh

h

h→

− −

1


38. Evaluatex

x

x→ −

−−2

2


39. Evaluatex

x

x→ +

−−1

1


40. Evaluatex

f x→ −3lim ( ) , if it exists, if

>

<−

−=

3,

3,3

3)(

xx

xx

xxf


41. Evaluate the right hand limit at x = 1, if it exists, for

>−=<+

=0 ,1

1 ,6

1,1

)(

xx

x

xx

xf

42. Evaluate the right hand limit at x = 0, if it exists, for

≥−

<+=

0,1

0,1)(

3 xx

xxxf

43. Evaluatex

f x→1

lim ( ) , if it exists, if

>−

=<

=

1 ,12

1 ,5

1,3

)(2

2

xx

x

xx

xf

44. Evaluate the largest δ that “works” for a given arbitrary ε. x

x→

=15 5lim

45. Evaluate the largest δ that “works” for a given arbitrary ε. x

x→

=10

3

3

52lim

46. Give an ε, σ proof forx

x→

− =2

3 2 4lim ( ) .


x→

− =1

5 2 3lim ( ) .


x→

− =5

3 2lim .

49. Give the four equivalent limit statements displayed in (2.2.5), taking f xx

c( ) ,=+

=4

2 12 .


x→

=3

2 9lim .

51. Evaluatex

f x→ 3

lim ( ) , if it exists, if

>−

<−=

3 ,)1(

3 ,1)(

3

2

xx

xxxf

2.3 Some Limit Theorems

52. Given that lim f (x) = 0, lim g(x) = 3, lim h(x) = −4, evaluate the limits that exist.x→c x→c x→c

(a) [ ]x c

f x g x→

−lim ( ) ( ) (e) x c

g x

f x→lim

( )

( )

(b) [ ]x c

h x→

lim ( )2

(f) x c g x h x→ −lim

( ) ( )

1

(c) x c

g x

h x→lim

( )

( )(g) [ ]

x cf x g x h x

→− −lim ( ) ( ) ( )3 2

(d) x c

f x

g x→lim

( )

( )

53. Evaluate lim 5, if it exists.x→1


54. Evaluate lim (2 − 3x)2, if it exists.x→2

55. Evaluate lim (2x3 − 3x2 + 2), if it exists.x→2

56. Evaluate lim 22x – 1, if it exists.x→-1

57. Evaluatex

x

x→ −1

2


58. Evaluatex

xx→

−02


59. Evaluateh

h h

h h→

−−2

3

3 2

4


60. Evaluatex

x xx→

+

0

23

lim , if it exists.

61. Evaluatex

x

x→

−

−2 3

2


62. Evaluatex

x

x→

−−2

3 8


63. Evaluatex

x x

x x→

−

− −1

2

2

2 3

2 1 4lim

( )( ), if it exists.

64. Evaluatex

x x

x→ −

+ −+5

2 22 15

5lim

( ), if it exists.

65. Evaluatex

x x

x→

+ −

+3

2

2

2 15

5lim

( ), if it exists.

66. Evaluatex

x x

x→ −

+ −

+5

2

2

2 15

5lim

( ), if it exists.

67. Evaluatex

x

x x→ −

+

+ −3

2

2

5

2 15lim

( ), if it exists.

68. Evaluatex a

x ax a→

−

−lim

1 1

, if it exists.

69. Evaluatehh

h

21

21

lim0

−+

→, if it exists.


70. Evaluateax

axax +

+−→

33

lim , if it exists.

71. Evaluate

x

xx 2

1

41

lim2

2 −

−

→, if it exists.

72. Evaluatex

xx

xx

→

+

−0

2


73. Evaluatex

x

x x→ −+

−

3

2

3

5


74. Evaluatex

x

x x→ −+

−−

3

2

3

6


75. Evaluate the following limits that exist.

(a) x x→

−

2

3 1

2lim (c)

x xx

→−

−

2

3 1

22lim ( )

(b) x x x→

−

−

2

3 1

2

1

2lim (d)

x x x→−

−

2

23 1

2

1

6lim

76. Given that ƒ(x) = x2 – 3x, evaluate the limits that exist.

(a) x

f x f

x→

−−3

3

3lim

( ) ( )(d)

x

f x f

x→

−−1

1

1lim

( ) ( )

(b) x

f x f

x→

−−5

5

5lim

( ) ( )(e)

2)1()(

lim2 −

−→ x

fxfx

(c) 3

)5()(lim

3 +−

−→ x

fxfx

(f) x

f x f

x→ −

− −+2

2

2lim

( ) ( )

2.4 Continuity

77. Determine whether or not ƒ(x) = 2x2 – 3x – 5 is continuous at x = 1. If not, determine whether thediscontinuity is a removable discontinuity, a jump discontinuity, or neither.

78. Determine whether or not f x x( ) ( )= − +2 22 is continuous at x = 2. If not, determine whether thediscontinuity is a removable discontinuity, a jump discontinuity, or neither.

79. Determine whether or not ƒ(x) = |5 – 2x2| is continuous at x = 3. If not, determine whether the discontinuity isremovable discontinuity, a jump discontinuity, or neither.

80. Determine whether or not f xx

x( ) =

−−

9 4

3 2

2

is continuous at x = 2/3. If not, determine whether the

discontinuity is a removable discontinuity, a jump discontinuity, or neither.


81. Determine whether or not

−>−=−<−

=1 ,2

1 ,1

1 ,3

)(

x

x

x

xf is continuous at x = –1. If not, determine whether the



>+=<

=1 ,12

1 ,3

1 ,

)(

2

xx

x

xx

xf is continuous at x = 1. If not, determine whether the



>=

<

=2 ,2

2 ,1

2 ,2

1

)(

3

xx

x

xx

xf is continuous at x = 2. If not, determine whether the



=

≠−=

4 ,1

1 ,4

1)(

x

xxxf is continuous at x = 4. If not, determine whether the



x x( )

( )=

−+

1

1 is continuous at x = 0. If not, determine whether the



( )( )

=−1

1 3 is continuous at x = 1. If not, determine whether the


87. Sketch the graph of f xx

x( ) =

−+

2 9

3 and classify the discontinuities, if any.

88. Sketch the graph of f xx

x( ) =

+

−

2

42 and classify the discontinuities, if any.

89. Sketch the graph of ƒ(x) = | x – 3 | and classify the discontinuities, if any.


≥<<−+

−≤+

=1 ,5

12 ,1

2 ,1

)(

x

xx

xx

xf and classify the discontinuities, if any.


>−=<+

=1 ,12

1 ,1

1 ,12

)(

xx

x

xx




>=<

=1 ,2

1 ,0

1 ,

)(

2

xx

x

xx


93. Define f xx

x( ) =

++

3 1

1 at x = –1 so that it becomes continuous at x = –1.

94. Define 3

9)(

2

+−

=x

xxf at x = –3 so that it becomes continuous at x = –3.

95. Define f xx x

x( ) =

+ −−

2 6

2 at x = 2 so that it becomes continuous at 2.

96. Let

−<

−≥+

−−=

1 ,

1 ,1

2)(

2

xA

xx

xxxf . Find A given that ƒ is continuous at –1.

97. Prove that if ƒ(x) has a removable discontinuity at c, then lim (x – c) ƒ(x) = 0.x→c

2.5 The Pinching Theorem; Trigonometric Limits

98. Evaluatex

x

x→ 0

7lim

sin, if it exists.

99. Evaluatex

xx→ 0

2

2

limsin

, if it exists.

100. Evaluatex

x

x→ −0

24

1 3lim

cos, if it exists.

101. Evaluatex

x

x→

−0

2

2

1 3

5lim

cos, if it exists.

102. Evaluateθ

θθ→ 0

limtan

, if it exists.

103. Evaluateθ

θθ→ 0

2lim

sin

tan, if it exists.

104. Evaluateα

α α

α→

−0 3lim

sin tan

sin, if it exists.

105. Evaluateθ

θ θ→ 0

4lim cot , if it exists.

106. Evaluateθ

θ

θ→ 0

2lim

sin, if it exists.


107. Evaluateθ

θ

θ→ 0

2

23lim

sin, if it exists.

108. Evaluateθ θ θ→ 0

3lim

csc, if it exists.

109. Evaluateθ

θθ→ 0

3

2lim

sin

sin, if it exists.

110. Evaluateα

αα→ 0

limcos

, if it exists.

111. Evaluatet

t

t→ −0

2

21lim

cos, if it exists.

112. Evaluateθ

θθ→ 0

3

2lim

cos, if it exists.

113. Evaluateθ

θθ→ 0

2

limsin

tan, if it exists.

114. For ƒ(x) = sin x and a = π/3, find h

f a h f a

h→

+ −0

lim( ) ( )

and give an equation for the tangent line to the

graph of ƒ at (a, ƒ (a)).

115. Use the pinching theorem to find x

xx→ 0 2

1lim cos .

2.6 Some Basic Properties of Continuous Functions

116. Sketch the graph of a function ƒ that is defined on [0, 1] and meets the following conditions (if possible): ƒ is continuous on [0, 1], minimum value ½, maximum value 1.

117. Sketch the graph of a function ƒ that is defined on [0, 1] and meets the following conditions (if possible): ƒ is continuous on (0, 1], no minimum value, maximum value ½.

118. Sketch the graph of a function ƒ that is defined on [0, 1] and meets the following conditions (if possible): ƒ is continuous on (0, 1), takes on the values ½ and 1 but does not take on the value 0.

119. Sketch the graph of a function ƒ that is defined on [0, 1] and meets the following conditions (if possible): ƒ is continuous on [0, 1], does not take on the value 0, minimum value –1, maximum value ½.

120. Sketch the graph of a function ƒ that is defined on [0, 1] and meets the following conditions (if possible): ƒ is discontinuous at x = ¾ but takes on both a minimum value and a maximum value.

121. Show the equation x3 – cos2 x = 0 has a root in [0, 2].

122. Given that ƒ (x) = x4 – x2 + 5x + 2, show that there exist at least two real numbers x such that ƒ (x) = 3.



1. (a) 4 (c) does not exist(b) does not exist (−∞) (d) 0

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

3. (a) −2 (c) does not exist(b) −4 (d) −2

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

5. (a) −1 (c) does not exist(b) 1 (d) −1

6. (a) does not exist (+∞) (c) does not exist(b) does not exist (−∞) (d) does not exist

7. (a) does not exist (c) does not exist(b) 0 (d) 0

8. (a) does not exist (+∞) (c) does not exist (+∞)(b) does not exist (+∞) (d) does not exist

9. (a) 1 (c) does not exist(b) 2 (d) 0

10. c = 0

11. c = −3 and c = 0

12. c = 2

13. c = −2 and c = 1

14. −1

15. π2

16. −27

17. −5/4

18. 0

19. 5/3

20. 3/2

21. 7

22. 12

23. −1

24. 2a

25. 8/9

26. 2

27. 3

28. −3

29. does not exist

30. 5 5/

31. ½

32. 0

33.3 10

5

34. −2/7

35. −3/2

36. 2

37. −1

38. −1

39. 1

40. −1

41. 0

42. 1

43. does not exist

44.1

5ε

45.53

ε

46. Since |(3x – 2) – 4| = |3x – 6| = 3|x – 2|, we can

take δ = 13

ε : if 0 < |x – 2| < 13

ε , then

|(3x – 2) – 4| = 3|x – 2| < ε .


47. Since |(5x – 2) – 3| = |5x – 5| = 5|x – 1|, we can

take σ = 15

ε : if 0 < |x – 1| < 15

ε , then

|(5x – 2) – 3| = 5|x – 1| < ε .

48. Since |(x – 3) – 2| = |x – 5|, we can take δ = ε :if 0 < |x – 5| < ε, then |(x – 3) – 2| = |x – 5| < ε .

49. (i)x x→ +

=2

4

2 1

4

5lim

(ii)h h→ + +

=0

4

2 1

4

5lim

(2 )

(iii)x x→ +

−

=2

4

2 1

4

50lim

(iv)x x→ +

− =2

4

2 1

4

50lim

50. If |x – 3| < 1, then –1 < x – 3 < 1, 2 < x < 4,5 < x + 3 < 7, and |x + 3| < 7.Take δ = minimum of 1 and ε/7.If 0 < |x – 3| < σ, then2 < x < 4 and|x – 3| < ε/7. Therefore,|x2 – 9| = |x + 3||x – 3| < 7|x – 3| < 7(ε/7) = ε.

51. 8

52. (a) −3 (e) does not exist(b) 16 (f) 1/7(c) −3/4 (g) −2(d) 0

53. 5

54. 16

55. 6

56. 6

57. 1

58. does not exist

59. 2

60. 3

61. 1/12

62. 12

63. 1/3

64. 0

65. 0

66. does not exist

67. does not exist

68.−1

2a

69.−14

70. 3a2

71. 2

72.−23

73. does not exist

74. 2

75. (a) 1 (c) 0(b) does not exist (d) −1/4

76. (a) 3 (d) −1/4(b) 7 (e) 1(c) does not exist (f) −7

77. continuous

78. continuous

79. continuous

80. removable discontinuity at x = 2/3.

81. jump discontinuity

82. jump discontinuity

83. removable discontinuity

84. discontinuity of neither type



87. removable discontinuity at x = −3.


88. nonremovable, nonjump discontinuity at x = 2.

89. no discontinuities

90. jump discontinuity at x = −2 and x = 1.

91. jump discontinuity at x = 1.

92. jump discontinuity at x = 1.

93. ƒ(−1) = 3

94. ƒ(−3) = −6

95. ƒ(2) = 5

96. −3

97. Since ƒ has a removable discontinuity at c, forLxf

cx=

→)(lim some real number L. Then

.00)(lim .)()(lim =•=−=−→→

LcxLxfcxcxcx

98. 7

99. 2 2

100. 8/9

101. 0

102. 1

103. 2

104. −½

105. 1/4

106. 2

107. 1/3

108. 3

109. 3/2

110. 0

111. 1


112. 0

113. 0

114. limit: ½; tangent line: y x= −

+1

2 3

3

2

π

115. 0

116.

117. impossible

118.

119. impossible

120.

121. If ƒ (x) = x3 – cos2 x, then ƒ is continuous, andƒ (0) = –1 < 0, ƒ (2) = 8 – cos2 (2) ≥ 7 > 0, soby the intermediate-value theorem ƒ (c) = 0 forsome c in [0, 2].

122. ƒ is continuous, andƒ (0) = 2, ƒ (–2) = 4, ƒ (1)= 7, so by the intermediate-value theoremƒ (x) = 3 for some x in [–2, 0] and for some x in[0, 1].

27

CHAPTER 3

Differentiation

3.1 The Derivative

1. Differentiate ƒ(x) = 7 by forming a difference quotient f x h f x

h

( ) ( )+ −and taking the limit as h tends to 0.

2. Differentiate ƒ(x) = 3 – 4x by forming a difference quotient f x h f x

h

( ) ( )+ −and taking the limit as h tends

to 0.

3. Differentiate ƒ(x) = 2x2 + x by forming a difference quotient f x h f x

h


to 0.

4. Differentiate ƒ(x) = x3 by forming a difference quotient f x h f x

h

( ) ( )+ −and taking the limit as h tends to 0.

5. Differentiate ƒ(x) = x + 2 by forming a difference quotient f x h f x

h

( ) ( )+ −and taking the limit as h

tends to 0.

6. Differentiate ƒ(x) = 1

1x + by forming a difference quotient

f x h f x

h


to 0.

7. Differentiate ƒ(x) = 1

2 2x by forming a difference quotient

f x h f x

h


to 0.

8. Find ƒ ′ (2) for ƒ(x) = (2x + 3)2 by forming a difference quotient f h f

h

( ) ( )2 2+ −and taking the limit as

h → 0.

9. Find ƒ ′ (2) for ƒ(x) = x3 – 2x by forming a difference quotient f h f

h


h → 0.

10. Find ƒ ′ (2) for ƒ(x) = 2 2x x+ + by forming a difference quotient f h f

h


h → 0.

11. Find ƒ ′ (2) for ƒ(x) = 3

2 1x + by forming a difference quotient

f h f

h


h → 0.

12. Find equations for the tangent and normal to the graph of ƒ(x) = 2x3 + 1 at the point (1, ƒ(1)).

13. Find equations for the tangent and normal to the graph of ƒ(x) = x3 – 3x at the point (2, ƒ(2)).


14. Find equations for the tangent and normal to the graph of ƒ(x) = 2x at the point (2, ƒ(2)).

15. Find equations for the tangent and normal to the graph of ƒ(x) = 2 1x + at the point (3/2, ƒ(3/2)).

16. Draw a graph of ƒ(x) = |2x – 1| and indicate where it is not differentiable.

17. Draw a graph of

>≤+

=1 ,2

1 ,1)(

2

x

xxxf and indicate where it is not differentiable.

18. Find ƒ ′ (c) if it exists.

>+

≤+=

1 ,3

1 ,22)(

4

2

xx

xxxf c = 1.

19. Find ƒ ′ (c) if it exists.

−>+

−≤+=

1 ,)1(2

1 ,13)(

2 xx

xxxf c = –2.

20. Sketch the graph of the derivative of the function with the graph shown below.

21. Sketch the graph of the derivative of the function with the graph shown below.

22.h

h

h→

+ −0

2 38 4lim

( ) /

represents the derivative of a function ƒ at a point c. Determine ƒ and c.

23.h

h

h→0lim

sinrepresents the derivative of a function ƒ at a point c. Determine ƒ and c.

Differentiation 29

3.2 Some Differentiation Formulas

24. Differentiate F(x) = 1 – 3x.

25. Differentiate F(x) = 4x5 – 8x2 + 9x.

26. Differentiate F(x) = 24x

.

27. Differentiate F(x) = (2x2 – 1)(3x + 1).

28. Differentiate F(x) = ( )2 52

4

x

x

−.

29. Differentiate F(x) = 3 5

1

4x

x

+−

.

30. Differentiate F(x) = 12

122

+

+

x x

.

31. Find ƒ ' (0) and ƒ ' (1) for ƒ (x) = x3(2x + 3).

32. Find ƒ ' (0) and ƒ ' (1) for ƒ (x) = 1214

−+

x

x.

33. Given that h(0) = 4 and h ' (0) = 3, find ƒ ' (0) for f(x) = 2x2h(x) – 3x.

34. Find an equation for the tangent to the graph of ƒ(x) = 252xx

− at the point (–1, ƒ(–1)).

35. Find the points where the tangent to the curve is horizontal for ƒ(x) = (x + 1)(x2 – 3x – 8).

36. Find the points where the tangent to the curve for ƒ(x) = –x3 + 2x is parallel to the line y = 2x + 5.

37. Find the points where the tangent to the curve for ƒ(x) = 3x + x2 is perpendicular to the line 3x + 2y + 1 = 0.

38. Find the area of the triangle formed by the x-axis and the lines tangent and normal to the curveƒ(x) = 2x + 3x2 at the point (–1, 3).

3.3 The d/dx Notation; Derivatives of Higher Order

39. Finddy

dxfor y

x y

x=

+ +3 5 23 2

2.

40. Finddy

dxfor y x

x= +5

13 .

41. Finddy

dxfor y

x x

x=

+−

2 3

7 2.

42. Findd

dxx x x[ ( )( )]− − +2 5 32 7 .


43. Findd

dx

x

x

2

2

5

3 1

−−

.

44. Evaluate dy

dxat x = 2 for y = (x2 + 1)(x3 – x).

45. Find the second derivative for f xx

x( ) =−

+8 1

525 .

46. Find the second derivative for ƒ(x) = (x2 – 2)(x3 + 5x).

47. Findd y

dx

3

3for y x

x= −3 1

.

48. Findd y

dx

4

4for y

xx=

−− −1

5 2 .

49. Findd

dxx

d

dxx x[ ( )]2

2

22 53 − .

50. Determine the values of x for which (a) ƒ ′ ′ (x) = 0, (b) ƒ ′ ′ (x) > 0, and (c) ƒ ′ ′ (x) < 0 for ƒ(x) = 2x4 + 2x3 – x.

3.4 The Derivative as a Rate of Change

51. Find the rate of change of the area of a circle with respect to the radius r when r = 3.

52. Find the rate of change of the volume of a cube with respect to the length s of a side when s = 2.

53. Find the rate of change of the area of a square with respect to the length z of a diagonal when z = 5.

54. Find the rate of change of the volume of a ball with respect to the radius r when r = 4.

55. Find the rate of change of y = 6 – x − x2 with respect to x at x = −1.

56. Find the rate of change of the volume V of a cube with respect to the length w of a diagonal on one of thefaces when w = 2.

57. The volume of a cylinder is given by the formula V = π r2 h where r is the base radius and h is the height.(a) Find the rate of change of V with respect to h if r remains constant.(b) Find the rate of change of V with respect to r if h remains constant.(c) Find the rate of change of h with respect to r if V remains constant.

58. An object moves along a coordinate line, its position at each time t ≥ 0 given by x(t) = 3t2 – 7t + 4. Find theposition, velocity, acceleration, and speed at time t0 = 4.

59. An object moves along a coordinate line, its position at each time t ≥ 0 given by x(t) = t3 – 6t2 – 15t.Determine when, if ever, the object changes direction.

60. An object moves along the x-axis, its position at each time t ≥ 0 given by x(t) = t4 – 12t3 + 28t2. Determine thetime interval(s), if any, during which the object moves left.

61. An object moves along the x-axis, its position at each time t ≥ 0 given by x(t) = 5t4 – t5. Determine the timeinterval(s), if any, during which the object is speeding up to the right.

Differentiation 31

62. An object is dropped and hits the ground 5 seconds later. From what height was it dropped? Neglect airresistance.

63. A stone is thrown upward from ground level. The initial speed is 24 feet per second. (a) In how manyseconds will the stone hit the ground? (b) How high will it go? (c) With what minimum speed should thestone be thrown to reach a height of 40 feet?

64. An object is projected vertically upward from ground level with a velocity of 32 feet per second. What is theheight attained by the object? (Take g as 32 ft/sec2).

65. If C x xx

( ) = + +700 5100

is the cost function for a certain commodity, find the marginal cost at a production

level of 400 units, and find the actual cost of producing the 401st unit .

66. If C(x) = 25,000 + 30x + (0.003)x2 is the cost function for a certain commodity and R(x) = 60x – (0.002)x2 isthe revenue function, find:(a) the profit function(b) the marginal profit(c) the production level(s) at which the marginal profit is zero.

3.5 The Chain Rule

67. Differentiate ƒ(x) = (x3 + 1)3 : (a) by expanding before differentiation, (b) by using the chain rule. Thenreconcile the results.

68. Differentiate ƒ(x) = (x – x3)3.

69. Differentiate f xx

x( ) =

+−

1

1

2

.

70. Differentiate f xx x

( ) = +

1 12

4

.

71. Differentiate4

2

2

7

7)(

−+=

x

xxf .

72. Differentiate ƒ(x) = (x + 4)4(3x + 2)3.

73. Finddy

dxat x = 0 for y

u=

+1

1and u = (3x + 1)3.

74. Finddy

dtat t = 1 for y = u3 – u2, u

x

x=

−+

1

1and x = 2t – 5.

75. Finddy

dxat x = 1 for y =

+−

1 5

1 5,and s t= + 1 , and t

x=

3

4

2

.

76. Given that ƒ(1) = 2, g(1) = 1, ƒ ' (1) = 3, g ' (1) = 2, and ƒ ' (2) = 0, evaluate (f • g) ' (1).

77. Findd

dxf x[ ( )]3 1− .

78. Given that ƒ(x) = (1 + 2x2)–2, determine the values of x for which (a) ƒ ' (x) = 0, (b) ƒ ' (x) > 0 and (c) ƒ ' (x) < 0.


79. An object moves along a coordinate line, its position at each time t ≥ 0 given by x(t) = (t2 – 3)3(t2 + 1)2.Determine when the object changes direction.

80. Differentiate ƒ(x) = [(x3 – x–3)2 – x2]3.

81. Find ƒ ′ ′ (x) for ƒ(x) = (x2 + 2x)17.

82. The edge of a cube is decreasing at the rate of 3 centimeters per second. How is the volume of the cubechanging when the edge is 5 centimeters long?

83. The diameter of a sphere is increasing at the rate of 3 centimeters per second. How is the volume of thesphere changing when the diameter is 6 centimeters?

3.6 Differentiating the Trigonometric Functions

84. Differentiate y = x tan x.

85. Differentiate y = sin x tan x.

86. Differentiate yx

x=

−sin

cos1.

87. Differentiate yx

x=

sin2

.

88. Differentiate y = sec x tan x.

89. Find the second derivative for y = x sin x.

90. Find the second derivative for y x xx

= + +5 72

2

cos sin

91. Find d

dxx

3

3(sin )

92. Find an equation for the tangent to the curve y – sin x at x = π/6

93. Determine the numbers x between 0 and 2π on y = sin x, where the tangent to the curve is parallel to the liney = 0.

94. An object moves along the y-axis, its position at each time t given by x(t) = sin 2t. Determine those timesfrom t = 0 to t = π when the object is moving to the right with increasing speed.

95. Finddy

dtfor y u= +

1

21

3

( ) , u = sin x, and x = 2π t.

96. A rocket is launched 2 miles away from one observer on the ground. How fast is the rocket going when theangle of elevation of the observer’s line of sight to the rocket is 50° (from the horizontal) and is increasing at5 °/sec?

97. An airplane at a height of 2000 meters is flying horizontally, directly toward an observer on the ground, witha speed of 300 meters per second. How fast is the angle of elevation of the plane changing when this angle is45°?

Differentiation 33

3.7 Implicit Differentiation: Rational Powers

98. Use implicit differentiation to obtain dy

dx in terms of x and y for x2 – 4xy + 2y2 = 5.


dx in terms of x and y for x2y + y2 = 6.


dx in terms of x and y for xy xy2 2+ = .


dx in terms of x and y for y = sin (x + y) + cos x.

102. Express d y

dx

2

2in terms of x and y for x2 + 3y2 = 10.

103. Express d y

dx

2

2in terms of x and y for x2 + 2xy – y2 + 8 = 0.

104. Express dy

dxat the point P(–1, –1) for 3x2 + xy = y2 + 3.

105. Express dy

dxand

d y

dx

2

2 at the point P(2, –1) for x2 – xy + y2 = 7.

106. Find the equations for the tangent and normal at the point P(–1, –1) for 2x2 – 3xy + 3y2 = 2.

107. Find dy

dxfor y = (x4 + x3)3/2.

108. Find dy

dxfor y x= +3 234 .

109. Find dy

dxfor y = (x2 + 1)1/4(x2 + 2)1/2.

110. Compute d

dxx

x4

4

1+

.

111. Compute d

dx

x

x

4 3

2 5

+−

.

112. Find the second derivative for y x= +9 3 .

113. Find the second derivative for y x= −4 4 3 .

114. Compute [ ]d

dxf x( )− 1 .


115. In economics, the elasticity of demand is given by the formula ε =P

Q

dQ

dP where P is price and Q quantity.

The demand is said to be

>=<

1 whereelastic

1 ereunitary wh

1 whereinelastic

εεε

. Describe the elasticity of Q = (400 – P)3/5.

3.8 Rates of Change Per Unit Time

116. A shark, looking for dinner, is swimming parallel to a straight beach and is 90 feet offshore. The shark isswimming at a constant speed of 30 feet per second. At time t = 0, the shark is directly opposite a lifeguardstation. How fast is the shark moving away from the lifeguard station when the distance between them is 150feet?

117. A boat sails parallel to a straight beach at a constant speed of 12 miles per hour, staying 4 miles offshore.How fast is it approaching a lighthouse on the shoreline at the instant it is exactly 5 miles from thelighthouse?

118. A ladder 13 feet long is leaning against a wall. If the base of the ladder is moving away from the wall at therate of ½ foot per second, at what rate will the top of the ladder be moving when the base of the ladder is 5feet from the wall?

119. A spherical balloon is inflated so that its volume is increasing at the rate of 3 cubic feet per minute. How fast

is the radius of the balloon increasing at the instant the radius is ½ foot? V r=43

3π

120. Sand is falling into a conical pile so that the radius of the base of the pile is always equal to one-half of itsaltitude. If the sand is falling at a rate of 10 cubic feet per minute, how fast is the altitude of the pile

increasing when the pile is 5 feet deep? V r h=1

32π

121. A spherical balloon is inflated so that its volume is increasing at the rate of 20 cubic feet per minute. How

fast is the surface area of the balloon increasing at the instant the radius is 4 feet? V r S r= =4

343 2π π,

122. Two ships leave port at noon. One ship sails north at 6 miles per hour, and the other sails east at 8 miles perhour. At what rate are the two ships separating 2 hours later?

123. A conical funnel is 14 inches in diameter and 12 inches deep. A liquid is flowing out at the rate of 40 cubic

inches per second. How fast is the depth of the liquid falling when the level is 6 inches deep? V r h=1

32π

124. A baseball diamond is a square 90 feet on each side. A player is running from home to first base at the rate of25 feet per second. At what rate is his distance from second base changing when he has run half way to firstbase?

125. A ship, proceeding southward on a straight course at a rate of 12 miles/hr. is, at noon, 40 miles due north of asecond ship, which is sailing west at 15 miles/hr.(a) How fast are the ships approaching each other 1 hour later?(b) Are the ships approaching each other or are they receding from each other at 2 o’clock and at what rate?

126. An angler has a fish at the end of his line, which is being reeled in at the rate of 2 feet per second from abridge 30 feet above water. At what speed is the fish moving through the water towards the bridge when theamount of line out is 50 feet? (Assume the fish is at the surface of the water and that there is no sag in theline.)

Differentiation 35

127. A kite is 150 feet high and is moving horizontally away from a boy at the rate of 20 feet per second. How fastis the string being payed out when the kite is 250 feet from him?

128. An ice cube is melting so that its edge length x is decreasing at the rate of 0.1 meters per second. How fast isthe volume decreasing when x = 2 meters?

129. Consider a rectangle where the sides are changing but the area is always 100 square inches. One side changesat the rate of 3 inches per second. When that side is 20 inches long, how fast is the other side changing?

130. The sides of an equilateral triangle are increasing at the rate of 5 centimeters per hour. At what rate is thearea increasing when the side is 10 centimeters?

131. A circular cylinder has a radius r and a height h feet. If the height and radius both increase at the constant rateof 10 feet per minute, at what rate is the lateral surface area increasing? S = 2πrh

132. The edges of a cube of side x are contracting. At a certain instant, the rate of change of the surface area isequal to 6 times the rate of change of its edge. Find the length of the edge.

133. A particle is moving along the parabola y = x2. If the x-coordinate of its position P is increasing at the rate of10 m/sec, what is the rate of change of the angle of inclination of the line OP when x = 3 m?

3.9 Differentials; Newton-Raphon Approximations

134. Estimate 144 by differentials.


136. Estimate 5 30 by differentials.


138. Use differentials to estimate cos 59°.

139. Use differentials to estimate sin 31°.

140. Use differentials to estimate tan 43°.

141. Estimate ƒ (3.2), given that ƒ (3) = 2 and ƒ ' (x) = (x3 + 5)1/5.

142. How accurately must we measure the edge of a cube to determine the volume within 1%?

143. Use differentials to estimate the volume of gold needed to cover a sphere of radius 10 cm with a layer of gold0.05 cm thick.



1. 0

2. −4

3. 4x + 1

4. 3x2

5.1

2 2x +

6.−+

1

1 2( )x

7. −1/x3

8. 28

9. 10

10. 9/4

11. −6/25

12. tangent: y – 3 = 6(x – 1)

normal: y x− =−

−31

61( )

13. tangent: y – 2 = 9(x – 2)

normal: y x− =−

−21

92( )

14. tangent: y x− = −212

2( )

normal: y – 2 = –2(x – 2)

15. tangent: y x− = −

21

2

3

2

normal: y x− = − −

2 23

2

16. no derivative at x = −1/2

17. no derivative at x = 1

18. ƒ ′ (1) = 4

19. ƒ ′ (−2) = 3

20.

21.

22. ƒ(x) = x2/3; c = 8

23. ƒ(x) = sin x; c = 0

24. −3

25. 20x4 – 16x + 9

26. −8/x5

27. 18x2 + 4x – 3

28.53

204

xx+

−

29.9 12 5

1

4 3

2

x x

x

− −

−( )

Differentiation 37

30.−

− −2 4 122 3 4x x x

31. ƒ ′ (x) = 8x3 + 9x2

ƒ ′ (0) = 0ƒ ′ (1) = 17

32. ƒ ′ (x) = −−6

2 1 2( )x

ƒ ′ (0) = −6ƒ ′ (1) = −6

33. ƒ ′ (x) = 4xh(x)+ 2x2h ′ (x) − 3ƒ ′ (0) = −3

34. y – 7 = (1)(x + 1)y – 7 = x + 1y = x + 8

35. ƒ ′ (x) = 0 at x =±2 37

3

36. at x = 0

37. at x = −9/4

38. 153/8

39. 3 – 4/x3

40. 151

22x

x x−

41.21 14 2

7 2

2

2

+ −−

x x

x( )

42. –18x8 + 80x7 – 12x + 30

43.28

3 12 2

x

x( )−

44. 79

45.−

+48

44

3

xx

46. 20x3 + 18x

47. 6 + 6/x4

48. − − =−

−− −24 60024 6005 65 6x x

x x

49. 12x – 100x4

50. (a) x = 0 or x = –½(b) x < –½ or x > 0(c) –½ < x < 0

51. 6π

52. 12

53. 5

54. 64π

55. 1

56. 3 2

57. (a) dV

dhr= π 2

(b) dV

drrh= 2π

(c) dh

dr

V

r=

−23π

58. x(4) = 24; v(4) = 17; a(4) = 6Speed = | v(4) | = 17

59. The object chanhes direction (from left to right)at t = 5.

60. v(t) < 0 when 2 < t < 7.

61. 0 < t < 3

62. 400 ft

63. (a) 1½ sec(b) 9 ft

(c) 16 10 ft/sec

64. 16 ft

65. $4.99; $4.99

66. (a) P(x) = 30x – (0.005)x2 – 25,000(b) P ′ (x) = 30 – (0.010)x(c) x = 3000

67. ƒ(x) = x9+ 3x6 + 3x3 + 1ƒ ′ (x) = 9x8 + 18x5 + 9x2

ƒ(x) = (x3+ 1)3

ƒ ′ (x) = 9x2(x3 + 1)2 = 9x8 + 18x5 + 9x2

68. 3(x – x3)2(1 – 3x2)


69.− +

−4 1

1 3

( )

( )

x

x

70. − +

+

41 1 1 2

2

3

2 3x x x x

71.52

32

)7(

)7(112

−+−

x

xx

72. (x + 4)3(3x + 2)2(21x + 44)

73. –9/4

74. –16

75.2)72(7

12

−or

112 44 7

21

+

76. 6

77. 3x2ƒ ′ (x3 – 1)

78. (a) x = 0 (b) x < 0 (c) x > 0

79. The object changes direction (from left to right)

at t =15

5and at t = 3 (from right to left).

80. 3[(x3 – x–3)2 – x2]2 [2(x3 – x–3)(3x2 + 3x–4) – 2x]

81. 17(x2 + 2x)15(66x2 + 132x + 64)

82. decreasing at a rate of 225 cm3/sec

83. decreasing at a rate of 54π cm3/sec

84. tan x + x sec2 x

85. cos x tan x + sin x sec2 x = sin x + sin x sec2 x= sin x (1 + sec2 x)

86.1

1cos x −

87.x x x

x

cos sin− 23

88. 2sec3 x – sec x

89. 2cos x – x sin x

90. –5cos x – 7sin x + 1

91. –cos x

92. y x− = −

1

2

3

2 6

π

93. x = π/2, 3π/2

94. 3π/4 < t < π

95. tt πππ

2cos)2sin1(4

3 2+

96.π18

50 15212sec ° ≅ mph

97. 3/40 radian/sec

98.dy

dx

y x

y x=

−−

2

2 2

99.dy

dx

xy

x y=

−+2

22

100.dy

dx

y y xy

x xy x=

− −

+

2

4

2

101.dy

dx

x y x

x y=

+ −− +

cos( ) sin

cos( )1

102.d y

dx y

2

2 3

10

9=

−

103.16

3( )y x−

104. 7

105. at (2, –1), dy

dx

d y

dx= =

5

4

21

32

2

2,

106. tangent: y x+ =−

+11

31( )

normal: y + 1 = 3(x + 1)

107.3

24 3

24 3 1 2x

x x x( )( ) /+ +

108.9

43 2

23 3 4x

x( ) /+ −

109.3 4

2 1 2

3

2 3 4 2 1 2

x x

x x

+

+ +( ) ( )/ /

Differentiation 39

110.1

4

1

4

1

4

1

43 4 5 4

34 4x x

x x x− −− = −/ /

111.−

− + −

13

2 5 4 3 2 5( ) ( )( )x x x

112.3 108

4 9 9

4

3 3

x x

x x

+

+ +( )

113.15 96

4 4 4

5 2

3 3

x x

x x

−

− −( )

114.1

21

xf x′ −( )

115. with 0 < p < 400, ∈ < 1 for P < 250∈ = 1 for P = 250∈ > 1 for P > 250

116. 24 ft/sec

117.365

mi/hr

118.−5

24 ft/sec

119.3π

ft/min

120.8

5π ft/min

121. 10 ft2/min

122. 10 mi/hr

123.−16049π

in/sec

124. − 5 5 ft/sec

125. (a) 111

1009≈ 3.498 mi/hr

(b) 12917

mi/hr (receding from each other)

126. 5/2 ft/sec

127. 16 ft/sec

128. –1.2 m3/sec

129.−34

in/sec

130. 25 3 cm2/hr

131. 20π (r + h) ft2/min

132. x = ½

133. 1 radian/sec (increasing)

134. 31/16

135. 25/12

136. 79/40

137. 13/6

138. ≈ + ≈1

2

3

3600515

π.

139. 515.01802

3

2

1≈

+≈

π

140. ≈ − ≈145

0 930π

.

141. 2.4

142. within 13

%

143. approximately 20π cm3

41

CHAPTER 4

The Mean-Value Theorem and Applications

4.1 The Mean-Value Theorem

1. Determine whether the function ƒ(x) = x3 – 3x + 2 satisfies the conditions of the mean-value theorem on theinterval [–2, 3]. If so, find the admissible values of c.

2. Determine whether the function ƒ(x) = x2 + 2x – 1 satisfies the conditions of the mean-value theorem on theinterval [0, 1]. If so, find the admissible values of c.

3. Determine whether the function ƒ(x) = 1/x2 satisfies the conditions of the mean-value theorem on the interval[–1, 1]. If so, find the admissible values of c.

4. Determine whether the function ƒ(x) = x2 + 4 satisfies the conditions of the mean-value theorem on theinterval [0, 2]. If so, find the admissible values of c.

5. Determine whether the function ƒ(x) = x3 – 3x + 1 satisfies the conditions of the mean-value theorem on theinterval [–2, 2]. If so, find the admissible values of c.

6. Determine whether the function ƒ(x) = x3 – 2x + 4 satisfies the conditions of the mean-value theorem on theinterval [1, 2]. If so, find the admissible values of c.

7. Determine whether the function ƒ(x) = x3 – 3x2 – 3x + 1 satisfies the conditions of the mean-value theorem onthe interval [0, 2]. If so, find the admissible values of c.

8. Determine whether the function f x x( ) = satisfies the conditions of the mean-value theorem on theinterval [0, 4]. If so, find the admissible values of c.

9. Determine whether the function f x x( ) = 3 satisfies the conditions of the mean-value theorem on theinterval [–1, 1]. If so, find the admissible values of c.

10. Determine whether the function ƒ(x) = x3 – x satisfies the conditions of the mean-value theorem on theinterval [–1, 1]. If so, find the admissible values of c.

11. Determine whether the function ƒ(x) = x3 – 4x satisfies the conditions of Rolle’s theorem on the interval[–2, 2]. If so, find the admissible values of c.

12. Determine whether the function f x x( ) = 3 satisfies the conditions of the mean-value theorem on theinterval [0, 1]. If so, find the admissible values of c.

4.2 Increasing and Decreasing Functions

13. Find the intervals on which ƒ(x) = x4 – 24x2 increases and the intervals on which ƒ decreases.

14. Find the intervals on which ƒ(x) = x4 – 4x3 increases and the intervals on which ƒ decreases.

15. Find the intervals on which ƒ(x) = x4 – 6x2 + 2 increases and the intervals on which ƒ decreases.

16. Find the intervals on which ƒ(x) = 5x4 – x5 increases and the intervals on which ƒ decreases.


17. Find the intervals on which ƒ(x) = 4x3 – 15x2 – 18x + 10 increases and the intervals on which ƒ decreases.

18. Find the intervals on which ƒ(x) = x(x – 6)2 increases and the intervals on which ƒ decreases.

19. Find the intervals on which ƒ(x) = xx

2 2+ increases and the intervals on which ƒ decreases.

20. Find the intervals on which ƒ(x) = x(x – 4)4 + 4 increases and the intervals on which ƒ decreases.

21. Find the intervals on which ƒ(x) = sin 2x, 0 ≤ x ≤ π , increases and the intervals on which ƒ decreases.

22. Find ƒ given that ƒ ′ (x) = 3x2 – 10x + 3 for all real x and ƒ(0) = 1.

23. Find ƒ given that ƒ ′ (x) = 12x3 – 12x2 for all real x and ƒ(0) = 1.

24. Find ƒ given that ƒ ′ (x) = 3(x – 2)2 for all real x and ƒ(0) = 1.

25. Find the intervals on which ƒ increases and the intervals on which ƒ decreases given that

≤−

<≤−<+

=

xx

xx

xx

xf

3 ,)1(

30 ,27

0 ,2

)(2

.

26. Given the graph of ƒ ′ (x) below, and given that ƒ(0) = 0, sketch the graph of ƒ.

27. Sketch the graph of a differentiable function ƒ that satisfies ƒ(2) = 1, ƒ(–1) = 0, and ƒ ′ (x) > 0, for all x, ifpossible.

28. Sketch the graph of a differentiable function ƒ that satisfies ƒ(0) = 0, ƒ ′ (x) < 0 for x < 0, and ƒ ′ (x) > 0 forx > 0, if possible.

4.3 Local Extreme Values

29. Find the critical numbers and the local extreme values of ƒ(x) = 3x5 – 5x4.

30. Find the critical numbers and the local extreme values of ƒ(x) = 12x2/3 – 16x.

31. Find the critical numbers and the local extreme values of ƒ(x) = x2/3(5 – x).

32. Find the critical numbers and the local extreme values of f x x x( ) / /= −13

43

4 3 1 3 .

33. Find the critical numbers and the local extreme values of f xx

x( ) = − +4

2

42 1.

The Mean-Value Theorem and Applications 43

34. Find the critical numbers and the local extreme values of ƒ(x) = (x + 1)(x – 1)3.

35. Find the critical numbers and the local extreme values of ƒ(x) = 2x + 2x2/3.

36. Find the critical numbers and the local extreme values of 33

11)(

xxxf −= .

37. Find the critical numbers and the local extreme values of ƒ(x) = x4/3 – 4x–1/3.

38. Find the critical numbers and the local extreme values of ƒ(x) = 6x2 – 9x + 5.

39. Find the critical numbers and the local extreme values of ƒ(x) = x4 – 6x2 + 17.

40. Find the critical numbers and the local extreme values of x

xxf1

)( −= .

41. Find the critical numbers and the local extreme values of ƒ(x) = (x + 1)2/3.

42. Find the critical numbers and the local extreme values of ƒ(x) = x – sin 2x, 0 < x < π.

43. Show that ƒ(x) = x3 – 4x2 + 2x – 5 has exactly one critical number in (0, 1).

4.4 Endpoint and Absolute Extreme Values

44. Find the critical numbers and classify the extreme values for f xx

x( ) , [ ].= + ∈2

2 0, 100

45. Find the critical numbers and classify the extreme values for f x x x x x( ) , [ ].= − − + ∈ −2 3 12 8 23 2 , 2


x x x( ) , [ ].= − − + ∈ −3

2

33 1 1, 2

47. Find the critical numbers and classify the extreme values for f x x x x( ) , [ ].= − + ∈ −3 26 5 1, 5

48. Find the critical numbers and classify the extreme values for f x x x x x( ) , [ ].= − − + ∈2 3 12 5 03 2 , 4

49. Find the critical numbers and classify the extreme values for f x x x x x( ) , [ ].= − − ∈ −4 6 9 13 2 , 2

50. Find the critical numbers and classify the extreme values for f x x x x( ) , [ ].= − + ∈3 3 6 0, 3 / 2

51. Find the critical numbers and classify the extreme values for f x x x( ) , [ ]./= − ∈ −1 12 3 , 1

52. Find the critical numbers and classify the extreme values for f x x x x( ) , [ ].= − + ∈ −3 12 8 4, 3

53. Find the critical numbers and classify the extreme values for ].8 ,1[,3)( 3/13/4 −∈−= xxxxf


xx( ) , [ ].=

+∈ ∞

2 30,


xx( ) , [ ].=

+∈

2 10 2,


56. Find the critical numbers and classify the extreme values for f xx x

x( ) , [ ].=−

∈1

0 12

,

57. Find the critical numbers and classify the extreme values for

≥

<=

0 ,

0 ,)(

3

2

xx

xxxf .

58. Find the critical numbers and classify the extreme values for ].2 ,2[ ,

1 ,1

11 ,1

1 ,1

)( 2 −∈

>−≤≤−−

−<−−

= x

xx

xx

xx

xf

59. Find the critical numbers and classify the extreme values for ].1 ,2[ ,1 ,1

0 ,1)(

3

2

−∈

≥−

<−−= x

xx

xxxf

4.5 Some Max-Min Problems

60. Find the dimensions of the rectangle of greatest area that can be inscribed in a circle of radius a.

61. Find the dimensions of the rectangle of greatest area that can be inscribed in a semicircle of radius 1.

62. An open field is to be surrounded with a fence that also divides the enclosure into three equal areas as shownin the figure below. The fence is 4000 feet long. For what value of x will the total area be a maximum?

y

x

63. Find the dimension of the rectangle of maximum area that may be embedded in a right triangle with sides oflength 12, 16, and 20 feet as shown in the figure below.

64. The infield of a 440-yard track consists of a rectangle and two semicircles as shown below. To whatdimensions should the track be built in order to maximize the area of the rectangle?


65. Find the dimensions of the largest circular cylinder that can be inscribed in a hemisphere of radius 1.

66. A long strip of copper 8 inches wide is to be made into a rain gutter by turning up the sides to form a troughwith a rectangular cross section. Find the dimensions of the cross section if the carrying capacity of thetrough is to be a maximum.

67. An isosceles triangle is drawn with its vertex at the origin and its base parallel to the x-axis. The vertices ofthe base are on the curve 5y = 25 – x2. Find the area of the largest such triangle.

68. The strength of a beam with a rectangular cross section varies directly as x and as the square of y. What arethe dimensions of the strongest beam that can be sawed out of a round log whose diameter is d? See thefigure below.

69. Find the area of the largest possible isosceles triangle with 2 sides equal to 6.

70. A lighthouse is 8 miles off a straight coast and a town is located 18 miles down the seacoast. Supplies are tobe moved from the town to the lighthouse on a regular basis and at a minimum time. If the supplies can bemoved at the rate of 7 miles/hour on water and 25 miles/hour over land, how far from the town should a dockbe constructed for shipment of supplies?

71. Find the circular cylinder of largest lateral area that can be inscribed in a sphere of radius 4 feet. [Surfacearea of a cylinder, S = 2πrh, where r = radius, h = height].

72. If three sides of a trapezoid are 10 inches long, how long should the fourth side be if the area is to be amaximum? [Area of a trapezoid = (a + b)h/2 where a and b are the lengths of the parallel sides and h =height].

73. The stiffness of a beam of rectangular cross section is proportional to the product xy3. Find the stiffest beamthat can be cut from a round log two feet in diameter. See the figure below.

74. Find the dimensions of the maximum rectangular area that can be laid out within a triangle of base 12 andaltitude 4 if one side of the rectangle lies on the base of the triangle.

75. Find the dimensions of the rectangle of greatest area with its base on the x-axis and its other two verticesabove the x-axis and on 4y = 16 – x2.

76. Find the dimensions of the trapezoid of greatest area with its longer base on the x-axis and its other twovertices above the x-axis on 4y = 16 – x2. [Area of a trapezoid = (a + b)h/2 where a and b are the lengths ofthe parallel sides and h = height].


77. A poster is to contain 50 in2 of printed matter with margins of 4 in. each at top and bottom and 2 in. at eachside. Find the overall dimensions if the total area of the poster is to be a minimum.

78. A rancher is going to build a 3-sided cattle enclosure with a divider down the middle as shown below.

Back Wall

The cost per foot of the three side walls will be $6/foot, while the back wall, being taller, will be $10/foot. Ifthe rancher wishes to enclose an area of 180 ft2, what dimensions of the enclosure will minimize his cost?

79. A can containing 16 in3 of tuna and water is to be made in the form of a circular cylinder. What dimensionsof the can will require the least amount of material? (V = πr2h, S = 2πrh, A = πr2)

80. Find the maximum sum of two numbers given that the first plus the square of the second is equal to 30.

81. An open-top shipping crate with a square bottom and rectangular sides is to hold 32 in3 and requires aminimum amount of cardboard. Find the most economical dimensions.

82. Find the minimum distance from the point (3, 0) to y x= .

83. The product of two positive numbers is 48. Find the numbers, if the sum of one number and the cube of theother is to be minimized.

84. Find the values for x and y such that their product is a minimum, if y = 2x – 10.

85. A container with a square base, vertical sides, and an open top is to be made from 192 ft2 of material. Findthe dimensions of the container with greatest volume.

86. The cost of fuel used in propelling a dirigible varies as the square of its speed and costs $200/hour when thespeed is 100 miles/hour. Other expenses amount to $300/hour. Find the most economical speed for a voyageof 1000 miles.

87. A rectangular garden is to be laid out with one side adjoining a neighbor’s lot and is to contain 675 ft2. If theneighbor agrees to pay for half of the dividing fence, what should the dimensions of the garden be to ensure aminimum cost of enclosure?.

88. A rectangle is to have an area of 32 in2. What should be its dimensions if the distance from one corner to themidpoint of the nonadjacent side is to be a minimum?

89. A slice of pizza, in the form of a sector of a circle, is to have a perimeter of 24 inches. What should be theradius of the pan to make the slice of pizza the largest? (Hint: the area of a sector of circle is A = r2θ /2 whereθ is the central angle in radians and the arc length along a circle is C = rθ with θ in radians).

90. Find the minimum value for the slope of the tangent to the curve of ƒ(x) = x5 + x3 – 2x.

91. A line is drawn through the point P(3, 4) so that it intersects the y-axis at A(0, y) and the x-axis at B(x, 0).Find the triangle formed if x and y are positive.

92. An open cylindrical trashcan is to hold 6 ft3 of material. What should be its dimension if the cost of materialused is to be a minimum? [Surface Area, S = 2πrh where r = radius and h = height].


93. Two fences, 16 feet apart, are to be constructed so that the first fence is 2 feet high and the second fence ishigher than the first. What is the length of the shortest pole that has one end on the ground, passes over thefirst fence and reaches the second fence. See the figure below.

94. A line is drawn through the point (3, 4) so that it intersects the y-axis at A(0, y) and the x-axis at B(x, 0). Findthe equation of the line through AB if the triangle is to have a minimum area and both x and y are positive.

4.6 Concavity and Points of Inflection

95. Describe the concavity of the graph of ƒ(x) = x4 – 24x2 and find the points of inflection, if any.

96. Describe the concavity of the graph of ƒ(x) = x4 – 4x3 and find the points of inflection, if any.

97. Describe the concavity of the graph of ƒ(x) = x4 – 6x2 + 2 and find the points of inflection, if any.

98. Describe the concavity of the graph of ƒ(x) = 5x4 – x5 and find the points of inflection, if any.

99. Describe the concavity of the graph of ƒ(x) = 4x3 – 15x2 – 18x + 10 and find the points of inflection, if any.

100. Describe the concavity of the graph of ƒ(x) = x(x – 6)2 and find the points of inflection, if any.

101. Describe the concavity of the graph of ƒ(x) = x3 – 5x2 + 3x + 1 and find the points of inflection, if any.

102. Describe the concavity of the graph of ƒ(x) = 3x4 – 4x3 + 1 and find the points of inflection, if any.

103. Describe the concavity of the graph of ƒ(x) = x2 + 2/x and find the points of inflection, if any.

104. Describe the concavity of the graph of ƒ(x) = (x – 2)3 + 1 and find the points of inflection, if any.

105. Describe the concavity of the graph of ƒ(x) = (x – 4)4 + 4 and find the points of inflection, if any.

106. Describe the concavity of the graph of ƒ(x) = sin 2x, x ∈ (0, π) and find the points of inflection, if any.

4.7 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps

107. Find the vertical and horizontal asymptotes for f xx

x( ) =

−−

3

1

2

.


x( ) =

+

2

2 1.

109. Find the vertical and horizontal asymptotes for f xx x

x( )

( )=

−+

2

21.



x( ) =

−

3

4

2

2.


( ) =−

8

4 2.


x( ) =

−

2

2 9.


x( ) =

−−

12

.

114. Determine whether the graph of ƒ(x) = 1 + (x – 2)1/3 has a vertical tangent or a vertical cusp at c = 2.

115. Determine whether the graph of ƒ(x) = (x + 1)1/3(x – 4) has a vertical tangent or a vertical cusp at c = –1.

116. Determine whether the graph of ƒ(x) = (x + 1)2/3 has a vertical tangent or a vertical cusp at c = –1.

117. Determine whether the graph of ƒ(x) = (x – 2)2/3 – 1 has a vertical tangent or a vertical cusp at c = 2.

4.8 Curve Sketching

When graphing the following functions, you need not indicate the extrema or inflection points, but show allasymptotes (vertical, horizontal, or oblique).

118. Sketch the graph of ƒ(x) = 5 – 2x – x2.

119. Sketch the graph of ƒ(x) = x3 – 9x2 + 24x – 7.

120. Sketch the graph of ƒ(x) = x3 + 6x2.

121. Sketch the graph of ƒ(x) = x3 – 5x2 + 8x – 4.

122. Sketch the graph of ƒ(x) = x3 – 12x2 + 6.

123. Sketch the graph of ƒ(x) = x3 – 6x2 + 9x + 6.

124. Sketch the graph of ƒ(x) = 3x4 – 4x3 + 1.

125. Sketch the graph of ƒ(x) = x2(9 – x2).

126. Sketch the graph of ƒ(x) = x4 – 2x2 + 7.

127. Sketch the graph of ƒ(x) = x x x3 232

6 12+ − + .

128. Sketch the graph of ƒ(x) = x1/3 (x + 4).

129. Sketch the graph of ƒ(x) = x2/3 (x + 5).

130. Sketch the graph of ƒ(x) = x (x – 3)2/3 .

131. Sketch the graph of ƒ(x) = 1 − x .


132. Sketch the graph of ƒ(x) = 1 2− x .

133. Sketch the graph of ƒ(x) = 4 2− x .

134. Sketch the graph of ƒ(x) = x

x4 −.


x

−+

32

.


x x

3

2

1

3 3 6

−

− −.



1. c = ± = ±7

3

21

3

2. c = ½

3. Since f is not differentiable at x = 0, which is in(−1, 1), the function f(x) does not satisfy theconditions of the mean-value theorem.

4. c = 1

5. c = ±2 3

3

6. c = =7

3

21

3

7. c =±3 3

3

8. c = 1

9. Since f is not differentiable at x = 0, which is in(−1, 1), the function f(x) does not satisfy theconditions of the mean-value theorem.

10. c = ±3

3

11. c = ±2 3

3

12. c =3

9

13. f increases on [ , ] and [ , ]− ∞2 3 0 2 3

f decreases on [ , ] and [ , ]−∞ −2 3 0 2 3

14. f increases on [3, ∞)f decreases on (−∞, 0) and [0, 3]

15. f increases on [ , ] and [ , ]− ∞3 0 3

f decreases on [ , ] and [ , ]−∞ − 3 0 3

16. f increases on [0, 4]f decreases on (−∞, 0] and [4, ∞)

17. f increases on (−∞, −1/2] and [3, ∞)f decreases on [−1/2, 3]

18. f increases on (−∞, 2] and [6, ∞)f decreases on [2, 6]

19. f increases on [1, ∞)f decreases on (−∞, 0] and (0, 1]

20. f increases on [4, ∞)f decreases on (−∞, 4]

21. f increases on [0, π/4] and [3π/4, π]f decreases on [π/4, 3π/4]

22. f(x) = x3 – 5x2 + 3x + 1

23. f(x) = 3x4 – 4x3 + 1

24. f(x) = (x – 2)3 + 9

25. f increases on (−∞, 0) and [3, ∞)f decreases on [0, 3)

26.

27. impossible

28.

29. critical numbers x = 0, 4/3;local maximum f (0) = 0;local minimum f (4/3) = −256/81


30. critical numbers x = 0, 1/8;local maximum f (1/8) = 1;local minimum f (0) = 0

31. critical numbers x = 0, 2;local maximum f (2) = 3(2)2/3;local minimum f (0) = 0

32. critical numbers x = 0, 1;local minimum f (1) = –1;no local extreme at x = 0

33. critical numbers x = –2, 0, 2;local minimum f (–2) = –3;local maximum f (0) = 1;local minimum f (2) = –3

34. critical numbers x = –1/2, 1;local minimum f (–1/2) = –27/16;no local extreme at x = 1

35. critical numbers x = –8/27, 1;local maximum f (–8/27) = 8/27;local minimum f (0) = 0

36. critical numbers x = –1, 1;local minimum f (–1) = –2/3;local maximum f (1) = 2/3

37. critical number x = –1;local minimum f (–1) = 5

38. critical number x = 3/4;local minimum f (3/4) = 13/8

39. critical numbers x = − 3 , 0, 3 ;

local minimum f ( )− 3 = 8;local maximum f (0) = 17;

local minimum f ( )3 = 8

40. no critical numbers; no local extreme values

41. critical number x = –1;local maximum f (–1) = 0

42. critical numbers x = π/6, 5π/6;

local minimum f (π/6) = π/6 – 3 2/ ;

local maximum f (5π/6) = 5π/6 + 3 2/

43. ƒ ′ (x) = 3x2 – 8x + 2, ƒ ′ (0) = 2, ƒ ′ (1) = –3, soƒ ′ has at least one zero in (0, 1). ƒ ′ ′ (x) = 6x – 8< 0 on (0, 1) so ƒ ′ is decreasing on (0, 1), andtherefore, it has exactly one zero in (0, 1).Hence ƒ has exactly one critical number in(0, 1).

44. no critical numbers; f (0) = 2 endpointminimum and absolute minimum; f (100) = 52endpoint maximum and absolute maximum

45. critical numbers x = –1, 2; f (–2) = 4 endpointminimum; f (–1) = 15 local and absolutemaximum; f (2) = –12 endpoint and absoluteminimum

46. critical numbers x = –1, 3 but x = 3 is outsidethe interval; f (–1) = 8/3 endpoint and absolutemaximum; f (2) = –19/3 endpoint and absoluteminimum

47. critical numbers x = 0, 4; f (–1) = –2 endpointminimum; f (0) = 5 local and absolutemaximum; f (4) = –27 endpoint and absoluteminimum; f (5) = –20 endpoint maximum

48. critical numbers x = –1, 2 but x = –1 is outsidethe interval; f (0) = 5 endpoint maximum; f (2)= –15 local and absolute minimum; f (4) = 37local and absolute maximum

49. critical numbers x = –1/2, 3/2; f (–1) = –1endpoint minimum; f (–1/2) = 5/2 local andabsolute maximum; f (3/2) = –27/2 local andabsolute minimum; f (2) = –10 endpointmaximum

50. critical numbers x = –1, 1 but x = –1 is outsidethe interval; f (0) = 6 endpoint maximum; f (1)= 4 local and absolute minimum; f (3/2) = 39/8endpoint minimum

51. critical number x = 0; f (–1) = 0 endpointminimum; f (0) = 1 absolute maximum; f (1) =0 endpoint minimum

52. critical numbers x = –2, 2; f (–4) = –8 endpointand absolute minimum; f (–2) = 24 local andabsolute maximum; f (2) = –8 local andabsolute minimum; f (3) = –1 endpointmaximum

53. critical numbers x = 3/4, 0; f (–1) = 4 endpointmaximum; f (3/4) = 9/4(3/4)1/3 ≈ –2.04 localand absolute minimum; f (8) = 10 endpoint andabsolute maximum

54. critical numbers x = –1, 0, 1 but x = –1, 0 areoutside the interval; f (1) = 1/4 local andabsolute maximum; no minimum

55. critical numbers x = –1, 1 but x = –1 is outsidethe interval; f (0) = endpoint and absoluteminimum; f (1) = ½ local and absolutemaximum; f (2) = 2/5 endpoint minimum


56. critical number x = ½; f (½) = 4 local andabsolute minimum; no maximum

57. critical number x = 0; f (x) = 0 local andabsolute minimum; no maximum

58. critical numbers x = –1, 0, 1; f (–2) = 1endpoint maximum; f (–1) = 0 local andabsolute minimum; f (0) = 1 local and absolutemaximum; f (1) = 0 local and absoluteminimum; f (2) = 1 endpoint maximum

59. critical number x = 0; f (–2) = –5 endpointminimum; f (1) = 0 endpoint maximum

60. a 2 by a 2

61. 2 by 2 2/

62. 1000 ft

63. x = 6, y = 8

64. x = 110, y = 220/π

65. 3 3/ by 6 3/

66. 2 by 4

67. 50 3 9/

68. x d y d= =3

3

6

3,

69. 18

70. 15 2/3 mi

71. largest lateral area = 32π22,24 == rh

72. 20

73. x = 1, y = 3

74. 6 by 2

75. x y= =4 3

3

8

3, ;

8 3

3

8

3 by

76. x = 4/3, y = 32/9; Vertices of the trapezoid areat (–4, 0), (–4/3, 32/9), (4/3, 32/9), (4, 0);Lengths of the bases are 8 8/3. Lengths of thesides are 40/9 and 40/9.

77. 5 in. by 10 in.

78. 18 ft by 10 ft

79. r h= =2 4

3 3π πin. , in.

80. Numbers are ½ and 119/4.

81. 4 in. by 4 in. by 2 in.

82. Minimum distance is 26

2, when

x = =5

2

5

2 and y

83. Numbers are 24 and 2.

84. x = 5/2 and y = –5

85. 8 ft by 8 ft by 4 ft

86. 50 6 miles/hr

87. 30 ft by 45/2 ft

88. 4 in. by 8 in.

89. r = 6 in

90. Minimum slope of the tangent is −2 when x = 0.

91. x = 6 and y = 8

92. 336

and ft 6

ππ== hr ft

93. 10 5 ft

94. 3y – 4x – 48 = 0

95. concave up on (−∞, −2); concave down on(−2, 2); concave up on (2, ∞); points ofinflection (−2, −80) and (2, 80)

96. concave up on (−∞, 0); concave down on(0, 2); concave up on (2, ∞); points of inflection(0, 0) and (2, −16)

97. concave up on (−∞, −1); concave down on(−1, 1); concave up on (1, ∞); points ofinflection (−1, −3) and (1, −3)


98. concave up on (−∞, 0); concave up on (0, 3);concave down on (3, ∞); point of inflection(3, 162)

99. concave down on (−∞, 5/4); concave up on(5/4, ∞); point of inflection (5/4, −28 1/4)

100. concave down on (−∞, 4); concave up on(4, ∞); point of inflection (4, 16)

101. concave down on (−∞, 5/3); concave up on(5/3, ∞); point of inflection (5/3, −88/27)

102. concave up on (−∞, 0); concave down on(0, 2/3); concave up on (2/3, ∞); points ofinflection (0, 1) and (2/3, 11/27)

103. concave up on (−∞, − 23 ); concave down on

( − 23 , 0); concave up on (0, ∞); point of

inflection ( − 23 , 0)

104. concave down on (−∞, 2); concave up on(2, ∞); point of inflection (2, 1)

105. concave up on (−∞, 4); concave up on (4, ∞);no points of inflection

106. concave down on (0, π/2); concave up on(π/2, π); point of inflection (π/2, 0)

107. vertical: x = 1; horizontal: y = 1

108. vertical: none; horizontal: y = 1

109. vertical: x = −1; horizontal: y = 1

110. vertical: x = ±2; horizontal: y = 3



113. vertical: x = 2; horizontal: y = 1

114. vertical tangent at (2, 1)

115. vertical tangent at (−1, 0)

116. cusp at (−1, 0)

117. cusp at (2, −1)

118. ƒ(x) = 5 – 2x – x2

119. ƒ(x) = x3 – 9x2 + 24x – 7

120. ƒ(x) = x3 + 6x2

121. ƒ(x) = x3 – 5x2 + 8x – 4


122. ƒ(x) = x3 – 12x + 6

123. ƒ(x) = x3 – 6x2 + 9x + 6

124. ƒ(x) = 3x4 – 4x3 + 1

125. ƒ(x) = x2(9 – x2)

126. ƒ(x) = x4 – 2x2 + 7

127. ƒ(x) = x3 + 32

x2 – 6x + 12

128. ƒ(x) = x1/3(x + 4)

129. ƒ(x) = x2/3(x + 5)


130. ƒ(x) = x(x – 3)2/3

131. f x x( ) = +1

132. f x x( ) = −1 2

133. f x x( ) = −4 2

134. f xx

x( ) =

−4

135.

136.

57

CHAPTER 5

Integration

5.1 The Definite Integral of a Continuous Function

1. Find Lf(P) and Uf(P) for ƒ(x) = 3x, x ∈ [–1, 0];

−−−

−= 0 ,4

1,

2

1,

4

3,1P .

2. Find Lf(P) and Uf(P) for ƒ(x) = 1 + x, x ∈ [0, 1];

= 1 ,

2

1,

4

1,0P .

3. Find Lf(P) and Uf(P) for ƒ(x) = 1 + x2, x ∈ [0, 2];

= 2 ,

2

3,1 ,

2

1,0P .

4. Find Lf(P) and Uf(P) for f x x( ) = − 1 , x ∈ [1, 2];

= 2 ,

4

7,

2

3,

4

5,1P .

5. Find Lf(P) and Uf(P) for ƒ(x) = |x|, x ∈ [0, 1];

= 1 ,

4

3,

4

1,0P .

6. Find Lf(P) and Uf(P) for ƒ(x) = x2, x ∈ [0, 1];

= 1 ,

4

3,

3

2,

2

1,

3

1,0P .

5.2 The Function F(x) f(t)dtax= ∫

7. Given that ∫ ∫ ∫ ===1

0

3

0

7

31 )( ,3 )( ,5 )( dxxfdxxfdxxf find each of the following:

(a) f x dx( ) 0

7∫ (b) f x dx( )

1

3∫ (c) f x dx( )

1

7∫ (d) f x dx( )

3

0∫

.

8. Given that ∫ ∫ ∫ ===5

1

5

3

7

112 )( ,3 )( ,7 )( dxxfdxxfdxxf find each of the following:

(a) f x dx( ) 5

7∫ (b) f x dx( )

1

3∫ (c) f x dx( )

3

7∫ (d) f x dx( )

5

3∫

.

9. For x > –1, set F x t dtx

( ) = +∫ 2 20

(a) Find F(0) (b) Find F ′ (x) (c) Find F ′ (1)

10. For x > 0, set F xdt

t

x( ) =

+∫ 11

(a) Find F(1) (b) Find F ′ (x) (c) Find F ′ (1)

11. For the function F xdt

t

x( ) =

+∫ 20 4(a) Find F ′ (–1) (b) Find F ′ (0) (c) Find F ′ (1) (d) Find F ′ ′ (x)

12. For the function F x t t dtx

( ) = +∫ 20

9

(a) Find F ′ (–1) (b) Find F ′ (0) (c) Find F ′ (1) (d) Find F ′ ′ (x)


13. For the function F x t t dtx

( ) = +∫ 20

4


14. For the function F x t dtx

( ) ( )= +∫ 2 20


5.3 The Fundamental Theorem of Integral Calculus

15. Evaluate ( )3 20

1x dx−∫ .

16. Evaluate 3 52

1x dx

−

−∫ .

17. Evaluate ( )x x dx20

22 5+ +∫ .

18. Evaluate ( )x x dx31

15− +

−∫ .

19. Evaluate xx

dx+

∫

11

2 .

20. Evaluate 7

1

2

xdx ∫ .

21. Evaluate x

xdx

4

31

2 1+∫ .

22. Evaluate x xx

dx221

28

3+ +

∫ .

23. Evaluate ( )3 10

1x dx+∫ .

24. Evaluate xx

dx241

2 3−

∫ .

25. Evaluate 741

2

tdt∫ .

26. Evaluate ( )x dx+∫ 1 110

1 .

27. Evaluate x x dx30

12 1−∫ .

28. Evaluate ( )x dx2 20

11+∫ .

29. Evaluate cos/

/x dx

π

π

3

3 2∫ .

Integration 59

30. Evaluate sin/

x dx 0

4π∫ .

31. Evaluate sec/ 2

0

4x dx

π∫ .

32. Evaluate sec tan/

x x dx 0

6π∫ .

33. Evaluate f x dx( ) 1

3∫ , where

≤<−

≤≤+=

32 ,3

21 ,)1()(

2

2

xx

xxxf .

34. Evaluate f x dx( ) 0

π∫ , where

≤<≤≤

=ππ

πxx

xxxf

3/ ,sin

3/0 ,)( .

35. Find the area between the graph of ƒ(x) = x2 and the x-axis for x ∈ [0, 2].

36. Find the area between the graph of f xx

( ) =12

and the x-axis for x ∈ [1, 2].

37. Find the area between the graph of ƒ(x) = x3 + 2 and the x-axis for x ∈ [1, 4].

38. Find the area between the graph of ƒ(x) = x2 – x and the x-axis for x ∈ [3, 8].

39. Find the area between the graph of ƒ(x) = x2 – x – 6 and the x-axis for x ∈ [0, 2].

40. Find the area between the graph of f xx

( ) =12

and the x-axis for x ∈ [1, 4].

41. Find the area between the graph of ƒ(x) = (2x + 1)–2 and the x-axis for x ∈ [0, 2].

42. Find the area between the graph of ƒ(x) = sin x and the x-axis for x ∈ [π/6, 2π/3].

43. Find the area between the graph of f x x( ) = + 3 and the x-axis for x ∈ [1, 6].

44. Find the area between the graph of ƒ(x) = 2(x + 5)–1/2 and the x-axis for x ∈ [–1, 4].

45. Sketch the region bounded by the curves y x=1

22 and y = x + 4, and find its area.

46. Sketch the region bounded by the curves y = x2 and 2x – y + 3 = 0, and find its area.

47. Sketch the region bounded by the curves x2 = 8y and x = 2y – 8, and find its area.

48. Sketch the region bounded by the curves y = x2 – 4x + 4 and y = x, and find its area.

49. Sketch the region bounded by the curves y = x + 5 and y = x2 – 1 and find its area.

50. Sketch the region bounded by the curves y = x3, x = –1, x = 2, and y = 0 and find its area.

51. Sketch the region bounded by the curves y = 2 – x2 and y = –x and find its area.

52. Sketch the region bounded by the curves y = x3 + 1, x = –1, x = 2, and the x-axis, and find its area.


5.4 Indefinite Integrals

53. Calculatedx

x5∫ .

54. Calculate ( )3 2 2x dx+∫ .

55. Calculate ( )2 52x dx+∫ .

56. Calculate2

4 5

dx

x +∫ .

57. Calculatex

xdx

5

7

2+∫ .

58. Calculate3 23

2

x x

xdx

+∫ .

59. Calculate( )

/

1 2

1 2

+∫

x

xdx .

60. Calculate ( )x dx3 22+∫ .

61. Calculatex

xdx

2

23

4−∫ .

62. Calculate ( )x x dx+∫ 1 .

63. Calculate ( )x dx+∫ 2 2 .

64. Find ƒ given that ƒ ′ (x) = 3x + 1 and ƒ(2) = 3.

65. Find ƒ given that ƒ ′ (x) = 2x2 + 3x + 1 and ƒ(0) = 2.

66. Find ƒ given that ƒ ′ (x) = sin x and ƒ(π/2) = 2.

67. Find ƒ given that ƒ ′ ′ (x) = 4x – 1, ƒ ′ (1) = 3, and ƒ(0) = 1.

68. Find ƒ given that ƒ ′ ′ (x) = x2 + 2x, ƒ ′ (0) = 3, and ƒ(2) = 3.

69. Find ƒ given that ƒ ′ ′ (x) = cos x, ƒ ′ (π) = 2, and ƒ(π) = 1.

70. An object moves along a coordinate line with velocity v(t) = 2t2 – 6t – 8 units per second. Its initial position(position at time t = 0) is 3 units to the left of the origin.(a) Find the position of the object 2 seconds later.(b) Find the total distance traveled by the object during those 2 seconds.

71. An object moves along a coordinate line with acceleration a t t( ) ( )= +12

1 3 units per second per second.

(a) Find the velocity function given that the initial velocity is 4 units per second.(b) Find the position function given that the initial velocity is 4 units per second and the initial position is the

origin.

Integration 61

72. An object moves along a coordinate line with acceleration a(t) = (2t + 1)–1/2 units per second per second.(a) Find the velocity function given that the initial velocity is 2 units per second.(b) Find the position function given that the initial velocity is 2 units per second and the initial position is the

origin.

73. An object moves along a coordinate line with velocity v(t) = 3 – 2t2 units per second. Its initial position is 3units to the left of the origin.(a) Find the position of the object 5 seconds later.(b) Find the total distance traveled by the object during those 5 seconds.

74. A ball is rolled across a level floor with an initial velocity of 28 feet per second. How far will the ball roll ifthe speed diminishes by 4 feet/sec2 due to friction?

75. A particle, initially moving at 16 cm/sec, is slowing down at the rate of 0.8 m/sec2. How far will the particletravel before coming to rest.

76. A jet plane moves with constant acceleration a from rest to a velocity of 300 ft/sec in a distance of 450 ft.Find a.

77. A particle which starts at the origin moves along the x-axis from time t = 0 to time t = 3 with velocity v(t) =t2 – t – 2. Determine the final position of the particle and the total distance traveled.

78. A particle which starts at the origin moves along the x-axis from time t = 0 to time t = 6 with velocity v(t) =4 – t. Determine the final position of the particle and the total distance traveled.

79. A particle which starts at the origin moves along the x-axis from time t = 0 to time t = 5 with velocity v(t) =8 – 2t. Determine the final position of the particle and the total distance traveled.

80. A particle which starts at the origin moves along the x-axis from time t = 0 to time t = 3 with velocity v(t) =t2 – 3t + 2. Determine the final position of the particle and the total distance traveled.

81. A particle which starts at the origin moves along the x-axis from time t = 0 to time t = 4 with velocity v(t) =t2 – 4t + 3. Determine the final position of the particle and the total distance traveled.

82. A particle which starts at the origin moves along the x-axis from time t = 0 to time t = 2 with velocity v(t) =t2 + t – 2. Determine the final position of the particle and the total distance traveled.

83. A particle which starts at the origin moves along the x-axis from time t = 1 to time t = 3 with velocity v(t) =t – 8/t2. Determine the final position of the particle and the total distance traveled.

84. A rapid transit trolley moves with a constant acceleration and covers the distance between two points 300feet apart in 8 seconds. Its velocity as it passes the second point is 50 ft/sec.(a) Find its acceleration.(b) Find the velocity of the trolley as it passes the first point.

5.5 The u-Substitution; Change of Variables

85. Calculate 3 1 2 2x x dx−∫ .

86. Calculate t t dt2 3 32 3( )−∫ .

87. Calculate4

8 23

x

xdx

−∫ .

88. Calculate dxxx∫ − 185 43 .


89. Calculate x x dx−∫ 5 .

90. Calculatedx

x( )+∫1 2 .

91. Calculate ( )( )x x x dx2 3 101 3+ +∫ .

92. Calculate x x dx3 2−∫ .

93. Calculatex

xdx

2

1+∫ .

94. Calculatex

x xdx

−− +∫

2

4 42 2( ).

95. Evaluatedx

x +∫

10

1.

96. Evaluate ( )x x x dx2 31

21 2 6+ +∫ .

97. Evaluate x x dx4 20

32+ +∫ 1 .


1+∫ .

99. Evaluate x

xdx

( )1 2 20

1

+∫ .


3−∫ .

101. Evaluatex

xdx

9 20

4

+∫ .

102. Evaluatex

xdx

3

41

2

3 1+∫ .


21−∫ .


5−∫ .

105. Calculate sin( )5 3x dx+∫ .

106. Calculate cos ( )2 2 1x dx+∫ .

107. Calculate cos sin2 2 2x x dx ∫ .

Integration 63

108. Calculate x x dx1 2 3 2/ /sin ∫ .

109. Calculate ( cos ) sin/2 3 33 2+∫ x x dx .

110. Calculatecos

( sin )

2

1 2 2

x

xdx

+∫ .

111. Calculatedx

xcos2 ∫ .

112. Calculate ( sec sin )x x x dx− +∫ 2 2 3 2 .

113. Calculatedx

xsin2 3 ∫ .

114. Calculate dttt /∫ + 3cos)3sin2( 21 .

115. Calculate csc cot2 2t t dt ∫ .

116. Calculate tan sec3 25 5x x dx ∫ .

117. Calculate

cos x

3

sin x dx∫ .

118. Calculate x x dxsec2 2 ∫ .

119. Calculate x x dx3 4sin( + 2) ∫ .

120. Evaluate ( t t t) dtcos csc cot/

/−∫

π

π

6

3.

121. Evaluate sec/ 2

0

4t dt

π∫ .

122. Evaluate ( csc cot )/

/t t t dt−∫

π

π

3

2.

123. Evaluatesin(1 / t)

t

2dt

4

2

/

/

π

π∫ .

124. Evaluate cos sin/ 2

0

43 3t t dt

π∫ .

125. Evaluatet

sin (t

2 2 / )/

/

23

2dt

π

π∫ .

126. Evaluate ( sec tan )/

2 2 20

8x x x dx+∫

π.

127. Find the area bounded by y = cos πx, y = sin πx, x = ¼, and x = ½.


128. Find the area bounded by y = sin x, y = 2x/π, and x = 0.

129. Find the area bounded by y = ½ cos2 πx, y = –sin2 πx, x = 0, and x = ½.

5.6 Additional Properties of the Definite Integral

130. Calculated

dxt dt

x( )3

11+

∫ .

131. Calculated

dxt dt

x( ) /+

∫ 1 1 2

0 .

132. Calculated

dxt dt

x( ) /2 2 3

2

34

2

−

∫ .

133. Calculated

dx tdt

x

x 1

1 3 2

2

−

∫ .

134. Calculated

dx

dt

tx

x

2 51

1

+

−

+∫ .

135. Calculated

dxt dt

x

xsin

/

/ 21

1 ∫

.

136. Find H ′ (2) given that H xt

dtx

x x( ) =

+

+∫

3

2 2

2

.

137. Find H ′ (2) given that H xx

t H t dtx

( ) [ ( )= + ′∫1

22

] .

138. Suppose ƒ is continuous and f x dxa

b( ) =∫ 0 .

(a) Does it necessarily follow that f x dxa

b( ) =∫ 0 ?

(b) What can you conclude about ƒ(x) on [a, b]?

139. Let Ω be the region below the graph of ƒ(x) = 2x + 3, x ∈ [0, 2]. Draw a figure showing the Riemann sum

S*(P) as an estimate for this area. Take

= 2 ,

2

3,1 ,

4

3,

4

1,0P and let the xi

* be the midpoints of the

subintervals. Evaluate the Riemann sum.

140. Set ƒ(x) = 3x + 1, x ∈ [0, 1]. Take

= 1 ,

2

1,

8

3,

4

1,

8

1,0P and set x x x x1 2 3 4

1

16

3

16

3

8

5

8* * * *, , , ,= = = =

x53

4* = . Calculate the following:

(a) ∆x1, ∆x2, ∆x3, ∆x4, ∆x5

(b) ||P||(c) m1, m2, m3, m4, m5

(d) f x f x f x f x f x( ), ( ), ( ), ( ), ( )* * * * *1 2 3 4 5

(e) M1, M2, M3, M4, M5

(f) Lf(P)

Integration 65

(g) S*(P)(h) Uf(P)

(i) f x dx( ) 0

1∫

5.7 Mean-Value Theorems for the Integrals; Average Values

141. Determine the average value of f x x( ) = +4 1 on the interval [0, 2] and find a point c in this interval atwhich the function takes on this average value.

142. Determine the average value of f x x x( ) = +3 43 1 on the interval [–1, 2].

143. Determine the average value of ƒ(x) = x3 + 1 on the interval [0, 2] and find a point c in this interval at whichthe function takes on this average value.

144. Determine the average value of ƒ(x) = x cos x2 on the interval [0, π/2].

145. Determine the average value of ƒ(x) = cos x on the interval [0, π/2] and find a point c in this interval at whichthe function takes on this average value.

146. Find the average distance of the parabolic arc y x x= + ∈2 1 0 22( ) , [ , ] from (a) the x-axis; and (b) the y-axis.

147. A rod lies on the x-axis from x = 1 to x = L > 1. If the density at any point x on the rod is 3/x3, find the centerof mass of the rod.



1. Lf(P) = −15/8; Uf(P) = −9/8

2. Lf(P) = 21/16; Uf(P) = 27/16

3. Lf(P) = 15/4; Uf(P) = 23/4

4.L P

U P

f

f

( ) .

( ) .

=+ +

≈

=+ +

≈

1 2 38

05183

3 2 38

07683

5. Lf(P) = 13/8; Uf(P) = 19/8

6.L P

U P

f

f

( ) .

( ) .

= ≈

= ≈

137576

0 2378

259576

0 4497

7. (a) 4 (c) −1(b) −2 (d) −3

8. (a) 5 (c) 8(b) 4 (d) −3

9. (a) 0 (c) 2

(b) x x2 2+

10. (a) 0 (c) ½

(b) 1

1x +

11. (a) 1/5 (c) 1/5

(b) 1/4 (d) −

+2

42 2

x

x( )

12. (a) 10 (c) 10

(b) 3 (d) x

x 2 9+

13. (a) − 5 (c) 5

(b) 0 (d) 2 4

4

2

2

x

x

+

+

14. (a) 1 (c) 9(b) 4 (d) 2(x + 2)

15. −½

16. −63/2

17. 50/3

18. 10

19.13

10 2 8( )−

20. 14( 2 1− )

21. 15/8

22. 95/6

23. 3

24. 35/24

25. 49/24

26. 1024/3

27. 5/32

28. 28/15

29. −2

30.12

2 2( )−

31. 1

32.2 3

3

2 3 33

−=

−

33. 3

34.π 2 27

18+

35. 8/3

36. ½

37. 279/4

38. 805/6

39. 34/3

40. ¾

41. 2/5

42.1 3

2+

Integration 67

43. 38/3

44. 4

45. 18

46. 32/3

47. 36

48. 9/2

49. 125/6

50. 17/4


51. 9/2

52. 27/4

53. − +1

4 4xC

54.19

3 2 3( )x C+ +

55.23

53x x C+ +

56. 4 5x C+ +

57. − −1 1

3 6x xC

58.32

22x x C+ +ln

59. 243

25

3 2 5 2x x x C+ + +/ /

60.x

x x C7

4

74+ + +

61.37

127 3 1 3x x C/ /− +

62.25

23

5 3 3 2x x C/ /+ +

63.x

x x C2

3 2

283

4+ + +/

64.32

52x x+ −

65.23

32

23 2x x x+ + +

66. −cos x + 2

67.23 2

2 132

xx

x+ + +

68.x x

x4 3

12 33 7+ + −

69. −cos x + 2x − 2π

70. (a) –77/3 (b) 68/3

71. (a) 18

1318

4( )t + +

(b) 140

1318

140

5( )t t+ + −

72. (a) 2 1 1t + +

(b) 13

2 113

3 2( ) /t t+ + −

73. (a) –71 1/3 units to the left of the origin

(b) 205 3 6

370 78

+≈ . units

74. 98 ft

75. 160 cm

76. 100 ft/sec2

77. –3/2; 31/6

78. 6; 10

79. 15; 17

80. 3/2; 11/6

81. 4/3; 4

82. 2//3; 3

Integration 69

83. −4/3; 11/3

84. (a) 25/8 ft/sec2 (b) 25 ft/sec

85. − − +12

1 2 2 3 2( ) /x C

86. − − +136

2 3 3 4( )t C

87. − − +3 8 2 2 3( ) /x C

88.130

5 184 3 2( ) /x C− +

89.25

5103

55 2 3 2( ) ( )/ /x x C− + − +

90. −+

+1

1xC

91.133

33 11( )x x C+ +

92.

29

2127

2245

2

163

2

9 2 7 2 5 2

3 2

( ) ( ) ( )

( )

/ / /

/

x x x

x C

− + − + −

+ − +

93.25

143

1 2 15 2 3 2 1 2( ) ( ) ( )/ / /x x x C+ + + + + +

94. Cxx

++−

−)44(2

12

95. 2 2 1( )−

96.19

28 8 13953 2 3 2( ) ./ /− ≈

97. 12

98. 61/27

99. ¼

100. 9

101. 2

102. 5/6

103. 184/105

104. 428/15

105. − + +15

5 3cos( )x C

106.x

x C2

18

2(2 1+ + +sin )

107. − +16

23cos x C

108. − +23

3 2cos /x C

109. − + +215

3 5 2(2 cos ) /x C

110. −+

+1

2 1 2( sin )xC

111. tan x + C

112. − + − +1 3

22

xx x Ctan cos

113. − +13

3cot x C

114.29

2 3 3 2( sin )+ +t C/

115. − +12

2csc t C

116.120

54tan x C+

117.12

2sec x C+

118. Cx +2tan21

119. − + +14

24cos( )x C

120.7 3 15

60479

−≈ − .

121. 1

122. 1572

2

305307

2

+ − ≈ −π

.


123.− 2

2

124. 7/72

125. 3 1−

126. 3613.021

22

64

2

−≈−+π

127.1

2 1 01318π

( ) .− ≈

128. 1 − π/4

129. 3/8

130. x3 + 1

131. (x + 1)1/2

132. 6x(9x4 – 4)2/3

133.2

1 3

1

1 34 2

x

x x−−

−

134.1

2 7

1

2 7x x++

− +

135.sin( / ) sin( / )

/

1 1

2

2

2 3 2

x

x

x

x−

136.15

2 2 3

3

4 2 4 21 4+−

+ ( )/

137. 4

138. (a) yes(b) f(x) = 0 for all x ∈ [a, b]

139. 10

140. (a) ∆x1 = 1/8, ∆x2 = 1/8, ∆x3 = 1/8, ∆x4 = 1/8,∆x5 = ½

(b) ||P|| = ½

(c) m1 = 1, m2 = 11/8, m3 = 7/4, m4 = 17/8,m5 = 5/2

(d) f(x*1) = 19/16, f(x*

2) = 25/16, f(x*3) = 17/8,

f(x*4) = 23/8, f(x*

5) = 13/4

(e) M1 = 11/8, M2 = 7/4, M3 = 17/8, M4 = 5/2,M5 = 4

(f) Lf(P) = 65/32

(g) S*(P) = 83/32

(h) Uf(P) = 95/32

(i) f x dx( ) / /= =∫ 5 2 80 320

1

141. 13/6; c = 133/144

142. 335/54

143. 3; c = ≈2 1263 .

144.1

2π

145. 2/π; c ≈ 0.8807 rad

146. (a) 10 6 2

361618

+≈ . (b) 2/3

147. 2L(L + 1)

71

CHAPTER 6

Some Applications of the Interval

6.1 More on Area

1. Sketch the region bounded by y = x2 – 4x + 5 and y = 2x – 3. Represent the area of the region by one or moreintegrals (a) in terms of x; (b) in terms of y.

2. Sketch the region bounded by x = y2 – 4y + 2 and x = y – 2. Represent the area of the region by one or moreintegrals (a) in terms of x; (b) in terms of y.

3. Sketch the region bounded by y = 2x – x2 and y = –3. Represent the area of the region by one or moreintegrals (a) in terms of x; (b) in terms of y.

4. Sketch the region bounded by y = x + 4/x2, the x-axis, x = 2, and x = 4.(a) Represent the area of the region by one or more integrals. (b) Find the area.

5. Sketch the region bounded by y = 4x – x2 and y = 3. Represent the area of the region by one or more integrals(a) in terms of x; (b) in terms of y. (c) Find the area.

6. Sketch the region bounded by x = y2 – 4y and x = y and find its area.

7. Sketch the region bounded by y = 3 – x2, y = –x + 1, x = 0 and x = 2, and find its area.

8. Sketch the region bounded by x = 3y – y2 and x + y = 3, and find its area.

6.2 Volume by Parallel Cross Sections; Discs and Washers

9. Sketch the region Ω bounded by x + y = 4, y = 0, and x = 0, and find the volume of the solid generated byrevolving the region about the x-axis.

10. Sketch the region Ω bounded by y2 = 4x, y = 2, and x = 4, and find the volume of the solid generated byrevolving the region about the x-axis.

11. Sketch the region Ω bounded by y = 4 – x2 and y = x + 2, and find the volume of the solid generated byrevolving the region about the x-axis.

12. Sketch the region Ω bounded by y = x2, y = 4, and x = 0, and find the volume of the solid generated byrevolving the region about the x-axis.

13. Sketch the region Ω bounded by y2 = x3, x = 1, and y = 0, and find the volume of the solid generated byrevolving the region about the x-axis.

14. Sketch the region Ω bounded by y2 = x2, x = 0, and y = 4, and find the volume of the solid generated byrevolving the region about the y-axis.

15. Sketch the region Ω bounded by y = x , y = 0, and x = 9, and find the volume of the solid generated byrevolving the region about the y-axis.

16. Sketch the region Ω bounded by y2 = 4x, x = 4, and y = 0, and find the volume of the solid generated byrevolving the region about the y-axis.


17. Sketch the region Ω bounded by y2 = x3, x = 1, and y = 0, and find the volume of the solid generated byrevolving the region about the y-axis.

18. Sketch the region Ω bounded by y = x3, x = 2, and y = 0, and find the volume of the solid generated byrevolving the region about the line x = 2.

19. Sketch the region Ω bounded by y = x2, y = 0, and x = 2, and find the volume of the solid generated byrevolving the region about the line x = 2.

20. Sketch the region Ω bounded by y = x3, x = 1, and y = –1, and find the volume of the solid generated byrevolving the region about the line y = –1.

21. Sketch the region Ω bounded by y = x3/2, x = 2, and y = 0, and find the volume of the solid generated byrevolving the region about the line y = 4.

22. The base of a solid is a circle or radius 2. All sections that are perpendicular to the diameter are squares. Findthe volume of the solid.

23. The steeple of a church is constructed in the form of a pyramid 45 feet high. The cross sections are allsquares, and the base is a square of side 15 feet. Find the volume of the steeple.

6.3 Volume by the Shell Method

24. Sketch the region Ω bounded by y = x , y = 0, and x = 9, and use the shell method to find the volume ofthe solid generated by revolving Ω about the y-axis.

25. Sketch the region Ω bounded by y2 = 4x, x = 4, and y = 0, and use the shell method to find the volume of thesolid generated by revolving Ω about the y-axis.

26. Sketch the region Ω bounded by y2 = 4x, y = 2, and x = 4, and use the shell method to find the volume of thesolid generated by revolving Ω about the y-axis.

27. Sketch the region Ω bounded by y = 2x + 3, x = 1, and x = 4, and y = 0, and use the shell method to find thevolume of the solid generated by revolving Ω about the y-axis.

28. Sketch the region Ω bounded by y = x + 1 , x = 0, y = 0, and x = 3, and use the shell method to find thevolume of the solid generated by revolving Ω about the y-axis.

29. Sketch the region Ω bounded by y = x2, y = 4, and x = 0, and use the shell method to find the volume of thesolid generated by revolving Ω about the x-axis.

30. Sketch the region Ω bounded by y = x2 and x = y2, and use the shell method to find the volume of the solidgenerated by revolving Ω about the x-axis.

31. Sketch the region Ω bounded by y = x3 and y = x, and use the shell method to find the volume of the solidgenerated by revolving Ω about the x-axis.

32. Sketch the region Ω bounded by the first quadrant of the circle x2 + y2 = r2 and use the shell method to findthe volume of the solid generated by revolving Ω about the x-axis.

33. Sketch the region Ω bounded by x = 2y – y2 and x = 0, and use the shell method to find the volume of thesolid generated by revolving Ω about the x-axis.

34. Sketch the region Ω bounded by y = 2x, x = 0, and y = 2, use the shell method to find the volume of the solidgenerated by revolving Ω about x = 1.

Some Applications of the Integral 73

35. Sketch the region Ω bounded by y2 = 4x, y = x, and use the shell method to find the volume of the solidgenerated by revolving Ω about x = 4.

36. Sketch the region Ω bounded by y = x2, y = 0, and x = 2, and use the shell method to find the volume of thesolid generated by revolving Ω about x = 2.

37. Sketch the region Ω bounded by y = x3 and x = 2, and use the shell method to find the volume of the solidgenerated by revolving Ω about x = 2.

6.4 The Centroid of a Region; Pappus’s Theorem on Volumes

38. Sketch the region bounded by y = x2, y = x . Determine the centroid of the region and the volumegenerated by revolving the region about each of the coordinate axes.

39. Sketch the region bounded by y2 = x3, x = 1, and y = 0. Determine the centroid of the region and the volumegenerated by revolving the region about each of the coordinate axes.

40. Sketch the region bounded by y2 = 8x, y = 0, and x = 2. Determine the centroid of the region and the volumegenerated by revolving the region about each of the coordinate axes.

41. Sketch the region bounded by y = 2x, y = 4, y = 0, and x = 2. Determine the centroid of the region and thevolume generated by revolving the region about each of the coordinate axes.

42. Sketch the region bounded by x2 = 2y and 2x – y = 0. Determine the centroid of the region and the volumegenerated by revolving the region about each of the coordinate axes.

43. Find the centroid of the bounded region determined by y = 2x2 and x – 2y + 3 = 0.

44. Find the centroid of the bounded region determined by y + 1 = 0 and x2 + y = 0.

45. Find the centroid of the bounded region determined by y + x2 + 2x and y = 2x + 1.

46. Locate the centroid of a solid cone of base radius 2 cm and height 4 cm.

47. Locate the centroid of a solid generated by revolving the region below the graph of f(x) = 2 – x2, x ∈ [0, 1].(a) about the x-axis.(b) about the y-axis.

6.5 The Notion of Work

48. Find the work done by the force F x x x( ) ( ) /= + 1 pounds in moving an object from x = 1 foot to x = 4feet along the x-axis.

49. Find the work done by the force F x x x( ) = + 3 Newtons in moving an object from x = 1 meter to x = 6meters along the x-axis.

50. A spring exerts a force of 2 pounds when stretched 6 inches. How much work is required in stretching thespring from a length of 1 foot to a length of 2 feet?

51. A spring whose natural length is 10 feet exerts a force of 400 pounds when stretched 0.4 feet. How muchwork is required to stretch the spring from its natural length to 12 feet?

52. A spring exerts a force of 1 ton when stretched 10 feet beyond its natural length. How much work is requiredto stretch the spring 8 feet beyond its natural length?


53. A spring whose natural length is 18 inches exerts a force of 10 pounds when stretched 16 inches. How muchwork is required to stretch the spring from 4 inches beyond its natural length?

54. A dredger scoops a shovel full of mud weighing 2000 pounds from the bottom of a river at a constant rate.Water leaks uniformly at such a rate that half the weight of the contents is lost when the scoop has been lifted25 feet. How much work is done by the dredger in lifting the mud this distance?

55. A 60-foot length of steel chain weighing 10 pounds per foot is hanging from the top of a building. How muchwork is required to pull half of it to the top?

56. A 50-foot chain weighing 10 pounds per foot supports a steel beam weighing 1000 pounds. How much workis done in winding 40 feet of the chain onto a drum?

57. A bucket weighing 1000 pounds is to be lifted from the bottom of a shaft 20 feet deep. The weight of thecable used to hoist it is 10 pounds per foot. How much work is done lifting the bucket to the top of the shaft?

58. A cylindrical tank 8 feet in diameter and 10 feet high is filled with water weighing 62.4 lbs/ft3. How muchwork is required to pump the water over the top of the tank?

59. A cylindrical tank is to be filled with gasoline weighing 50 lbs/ft3. If the tank is 20 feet high and 10 feet indiameter, how much work is done by the pump in filling the tank through a hole in the bottom of the tank?

60. A cylindrical tank 5 feet in diameter and 10 feet high is filled with oil whose density is 48 lbs/ft3. How muchwork is required to pump the water over the top of the tank?

61. A conical tank (vertex down) has a diameter of 9 feet and is 12 feet deep. If the tank is filled with water ofdensity 62.4 lbs/ft3, how much work is required to pump the water over the top?

62. A conical tank (vertex down) has a diameter of 8 feet and is 10 feet deep. If the tank is filled to a depth of 6feet with water of density 62.4 lbs/ft3, how much work is required to pump the water over the top?

6.6 Fluid Pressure and Fluid Forces

63. A flat rectangular plate, 6 feet long and 3 feet wide, is submerged vertically in water (density 62.4 lbs/ft3)with the 3-foot edge parallel to and 2 feet below the surface. Find the force against the surface of the plate.

64. A flat rectangular plate, 6 feet long and 3 feet wide, is submerged vertically in water (density 62.4 lbs/ft3)with the 6-foot edge parallel to and 2 feet below the surface. Find the force against the surface of the plate.

65. A flat triangular plate whose dimensions are 5, 5, and 6 feet is submerged vertically in water (density 62.4lbs/ft3) so that its longer side is at the surface and parallel to it. Find the force against the surface of the plate.

66. A flat triangular plate whose dimensions are 5, 5, and 6 feet is submerged vertically in water (density 62.4lbs/ft3) so that its longer side is at the bottom and parallel to the surface, and its vertex is 2 feet below thesurface. Find the force against the surface of the plate.

67. A flat plate, shaped in the form of a semicircle 6 feet in diameter, is submerged in water (density 62.4 lbs/ft3)as shown. Find the force against the surface of the plate.



1. (a) [( ) ( )]2 3 4 522

4x x x dx− − − +∫

(b) ( )12

1 2 11

5+ − −∫ y y dy

2. (a) ( )2 2 21

2

2

1x dx x x dx+ + + −

−−

−∫∫

(b) [( ) ( )]y y y dy− − − +∫ 2 4 221

4

3. (a) ∫−−−−

3

1

2 )]3()2[( dxxx

(b) 2 13

1−

−∫ y dy

4. (a) xx

dx+

∫

422

4

(b) 7

5. (a) ( )4 321

3x x dx− −∫

(b) ∫ −4

3 42 dyy

(c) 4/3

6. 125/6

7. 10/3


8. 4/3

9. 64π/3

10. 18π

11. 108π/5

12. 128π/5

13. π/4

14. 8π

15. 972π/5


16. 256π/5

17. 4π/7

18. 16π/5

19. 8π/3

20. 16π/7

21. 80π/7

22. 128/3

23. 3375 ft3

24. 972π/5


25. 256π/5

26. 98π/5

27. 129π

28. 232π/15

29. 128π/5

30. 3π/10

31. 4π/21

32. 2πr3/3


33. 8π/3

34. 4π/3

35. 64π/5

36. 8π/3

37. 16π/5

38. ( , )x y V Vx y=

= =920

310

,920

; π

39. ( , ) ,x y V Vx y=

= =57 4

47

,5

16;

π π

40. ( , ) ,x y V Vx y=

= =65

1664

5,

32

; ππ


41. ( , )x y V Vx y=

= =

43

323

,43

; π

42.3

64,

15512

; 5

16,2),(

ππ==

= yx VVyx

43. ( , ) ,x y =

18

1920

44. ( , ) ,x y =−

03

5

45. ( , ) ,x y =

035

46. ( )( , , ) , ,x y z = 0 0 1

47. (a) ( , , ) , ,x y z =

920

0 0

(b) ( , , ) , ,x y z =

04350

0

48. 20/3 ft-lbs

49. 232/5 joules

50. 6 ft-lbs

51. 2000 ft-lbs

52. 6400 ft-lbs

53. 5/12 ft-lbs

54. 37,500 ft-lbs

55. 13,500 ft-lbs

56. 52,000 ft-lbs

57. 22,000 ft-lbs

58. 49,920π ft-lbs

59. 250,000π ft-lbs

60. 15,000π ft-lbs

61. 15,163.2π ft-lbs

62. 3953.7π ft-lbs

63. 2995.2 lbs

64. 3931.2 lbs

65. 998.4 lbs

66. 3494.4 lbs

67. 1123.2 lbs

81

CHAPTER 7

The Transcendental Functions

7.1 One-to-One Functions; Inverses

1. Determine whether or not f( x) = 4x + 3 is one-to-one, and, if so, find its inverse.

2. Determine whether or not f( x) = 5x – 7 is one-to-one, and, if so, find its inverse.

3. Determine whether or not f( x) = 2x2 – 1 is one-to-one, and, if so, find its inverse.

4. Determine whether or not f( x) = 2x3 + 1 is one-to-one, and, if so, find its inverse.

5. Determine whether or not f( x) = (1 – 3x)3 is one-to-one, and, if so, find its inverse.

6. Determine whether or not f( x) = (x + 1)4 is one-to-one, and, if so, find its inverse.

7. Determine whether or not f( x) = (x – 1)5 + 1 is one-to-one, and, if so, find its inverse.

8. Determine whether or not f( x) = (4x + 5)3 is one-to-one, and, if so, find its inverse.

9. Determine whether or not f( x) = 2 + (x + 1)5/3 is one-to-one, and, if so, find its inverse.


( ) =−1

2is one-to-one, and, if so, find its inverse.

11. Determine whether or not f xxx

( ) =−+

2 33 1

is one-to-one, and, if so, find its inverse.


( ) =−

+1

12 is one-to-one, and, if so, find its inverse.


( ) =−

2

13is one-to-one, and, if so, find its inverse.

14. Sketch the graph of f –1 given the graph of f shown below.


15. Sketch the graph of f –1 given the graph of f shown below.

16. Given that f is one-to-one and f(2) = 1, f ′ (2) = 3, f(3) = 2, f ′ (3) = 4, deduce, if possible, (f –1) ′ (2).

17. Show that f(x) = x3 + 2x – 5 has an inverse and find (f –1) ′ (7).

18. Show that f(x) = sin 2x – 4x has an inverse and find (f –1) ′ (0).

19. Find a formula for (f –1) ′ (x) given that f is one-to-one and satisfies )(

1)(

xfxf =′ .

20. Show that f x t dtx

( ) ( sin )/

= +∫ 1 22π

has an inverse and find (f –1) ′ (0).

7.2 The Logarithm Function; Part I

21. Estimate ln 25 on the basis of the following table.

n ln n n ln n12345

0.000.691.101.391.61

6789

10

1.791.952.082.202.30

22. Estimate ln 2.2 on the basis of the following table.

n ln n n ln n12345

0.000.691.101.391.61

6789

10

1.791.952.082.202.30


n ln n n ln n12345

0.000.691.101.391.61

6789

10

1.791.952.082.202.30

The Transcendental Functions 83


n ln n n ln n12345

0.000.691.101.391.61

6789

10

1.791.952.082.202.30

25. Estimate ∫=5.3

15.3ln

tdt

using the approximation [ ]12

L P U Pf f( ) ( )+ with

P = = =

144

54

64

74

84

94

104

52

, , , , , ,

26. Taking ln 4 ≈ 1.39, use differentials to estimate(a) ln 4.25 ; (b) ln 3.75.

27. Solve ln x + ln (x + 2) = 0 for x.

28. Solve ln (x + 1) – ln (x – 2) = 1 for x.

29. Solve 2 ln x = ln (x + 2) for x.

30. Solve (5 – ln x)(2 ln x) = 0 for x.

7.3 The Logarithm Function; Part II

31. Determine the domain and find the derivative of f(x) = ln(x3 – 1).

32. Determine the domain and find the derivative of f(x) = ln(1 – x2).

33. Determine the domain and find the derivative of f x x x( ) ln= +2 2 .

34. Determine the domain and find the derivative of f(x) = x ln(2x + x2).

35. Determine the domain and find the derivative of f x x x( ) ln= −3 3 2 .

36. Determine the domain and find the derivative of f(x) = ln(sec x).

37. Calculate dx

x5 3−∫ .

38. Calculate ( )4 2

2

x dx

x x

−

−∫ .

39. Calculate x

xdx

2

33 5−∫ .

40. Calculate cot 3x dx ∫ .

41. Calculate sin cos

sinx x

xdx

+∫ .


42. Calculatex

xdx

6 120

1

+∫ .

43. Calculate x

xdx

1 3 21 3

1 2

−∫ /

/ .

44. Calculate sin

cos/ x

xdx

20

2

−∫π

.

45. Calculate g ′ (x) by logarithmic differentiation if g xx x

x( )

( ) /=

++

2

2 3

1

1.

46. Calculate g ′ (x) by logarithmic differentiation if g xx x

x( )

( ) cos

( )=

+−

3 5 2

8

2 4

3 23 .

47. Calculate g ′ (x) by logarithmic differentiation if g xx

x( )

sin

( )=

+1 5 35 .

48. The region bounded by the graph of f xx

( ) =−

1

5 2 and the x-axis for 0 ≤ x ≤ 2 is revolved around the

y-axis. Find the volume of the solid that is generated.

49. Calculate tan 5x dx∫ .

50. Calculate sec2

3πx

dx∫ .

51. Calculate csc 22

π −

∫

xdx .

52. Calculate cot( )2 3πx dx−∫ .

53. Calculate e e dxx x2 2sin∫ .

54. Calculate ∫ + xdxx2tan2 2sec2

.

55. Calculate 3

2 3

sec tan

( sec )

x x dx

x

+∫ .

56. Evaluate e 0

2 23 x x/e dxcsc

ln π∫ .

57. Evaluate tan( )25 6

x dx/

−∫ ππ

π

7 /12 .

58. Evaluate 3

22 csc cot

cscx x dx

x/

/ 4 +∫π

π.


7.4 The Exponential Function

59. Differentiate y = e–3x/2.

60. Differentiate xxey −=22 .

61. Differentiate y e x= +3 1 .

62. Differentiate y e xx=3 3ln .

63. Differentiate y e x x= ++( )3

1 2 .

64. Differentiate ye e

e e

x x

x x=

−

+

3 2

3 2.

65. Differentiate y e x x= sin .

66. Differentiate y e xx= −2 3sin .

67. Differentiate y e x= sin 2 .

68. Differentiate y e x= 1/ .

69. Differentiate ye

x

x

=2

2.

70. Calculate e

edx

x

x1 −∫ .

71. Calculate e e

e edx

x x

x x

2 2

2 2

+

−

−

−∫ .

72. Calculate e dxx5 2/∫ .

73. Calculate e

xdx

x3

23∫ .

74. Calculate 12

2 22

e e dxx x−

−∫ .

75. Calculate e

edx

x

x

3

33 2−∫ .

76. Calculate 2

2

3

3

2

2

xe

edx

x

x +∫ .


77. Evaluate e x / 20

1∫ dx .

78. Evaluate e -3x0

ln / 2π∫ dx .

79. Evaluate e

x

0

1

xdx∫ .

80. Evaluate e

2x

0

1

2 2+∫

edx

x.

81. Sketch the region bounded by y = e3x, y = ex, and x = 2, and find its area.

82. Sketch the graph of f(x) = xe–2x.

7.5 Arbitrary Powers; Other Bases

83. Find log10 0.001

84. Find log9 243

85. Find the derivative of f x x( ) cos= 3 .

86. Find the derivative of f xx

( ) = 52 .

87. Calculate 42x dx∫ .

88. Calculate 3 2−∫ x dx/ .

89. Calculate 3 52

x dxx−∫ .

90. Calculate log3 2x

xdx∫ .

91. Calculate log4 2 2x

xdx

+∫ .

92. Find f ′ (e) if f(x) = ex + xe.

93. Find f ′ (e) if f(x) = ln(ln(ln x2)) .

94. Find [ ]ddx

x x(sin ) by logarithmic differentiation.

95. Find ddx

x x( )4 4 by logarithmic differentiation.

96. Find [ ]ddx

x x(tan ) by logarithmic differentiation.


97. Evaluate 3 1

2 -x dx∫ .

98. Evaluate 2 0

1 3x dx∫ .

99. Evaluate 0

5 4

2 1

2 1a

xdx

x-

−∫ .

100. Evaluate 9(ln 5) / (ln 3) .

7.6 Exponential Growth and Decay

101. Find the amount of interest earned by $700 compounded continuously for 10 years (a) at 8% (b) at 9% (c) at10%.

102. The population of a certain city increases at a rate proportional to the number of its inhabitants at any time. Ifthe population of the city was originally 10,000 and it doubled in 15 years, in how many years will it triple?

103. A certain radioactive substance has a half-life of 1300 years. Assume an amount y0 was initially present. (a)Find a formula for the amount of substance present at any time. (b) In how many years will only 1/10 of theoriginal amount remain?

104. A tank initially contains 100 gal. of pure water. At time t = 0, a solution containing 4 lb. of dissolved salt pergal. flows into the tank at 3 gal./min. The well-stirred mixture is pumped out of the tank at the same rate. (a)How much salt is present at the end of 30 min.? (b) How much salt is present after a very long time?

105. A tank initially contains 150 gal. of brine in which there is dissolved 30 lb. of salt. At t = 0, a brine solutioncontaining 3 lb. of dissolved salt per gallon flows into the tank at 4 gal./min. The well-stirred mixture flowsout of the tank at the same rate. How much salt is in the tank at the end of 10 min.?

106. An object of unknown temperature is put in a room held at 30° F. After 10 minutes, the temperature of theobject is 0° F; 10 minutes later it is 15° F. What was the object’s initial temperature?

107. Show that dydx

e xy− =− sec2 0 is a separable equation and then find its solution.

7.7 The Inverse Trigonometric Functions

108. Determine the exact value for tan ( / )−1 3 3 .

109. Determine the exact value for [ ]cot sin ( / )− −1 1 4 .

110. Determine the exact value for ( )[ ]tan sin /− −1 1 4 .

111. Determine the exact value for [ ]tan sec (3 / )−1 2 .

112. Determine the exact value for [ ]sin cot( / )−1 4π .

113. Determine the exact value for [ ]sec sin (3 / )−1 4 .


114. Determine the exact value for [ ]cos sin ( / )− −1 3 4 .

115. Determine the exact value for [ ]sin tan ( / )2 1 31− .

116. Determine the exact value for [ ]sin tan ( / )− −1 2 3 .

117. Determine the exact value for [ ]cos sin ( / )− −1 3 4 .

118. Taking 0 < x < 1, calculate sin (cos–1 x).

119. Taking 0 < x < ½, calculate tan (cos–1 2x).

120. Differentiate f(x) = x tan–1 3x.

121. Differentiate f(x) = x sin–1 2x.

122. Differentiate f(x) = sec(tan–1 x).

123. Differentiate y = cos–1 (cos x).

124. Differentiate f(x) = e3x sin–1 2x.

125. Differentiate f xxx

( ) tan=−+

−1 11

.

126. Differentiate f x x x( ) sin= + −−1 21 .

127. Differentiate 2

1

1

1sin

xy

+= − .

128. Differentiate f x x xx

( ) ln( ) tan= + − −2 142

.

129. Evaluate

dx

x16 9 20

1

+∫ .

130. Evaluate

dx

x1 4 20

1 4

−∫

/.

131. Evaluate

dx

x1 4 21 2

0

+−∫ /

.

132. Evaluate

dx

x x21

3

6 13− +∫ .

133. Calculate x

xdx

1 4+∫ .

134. Calculate dx

x1 9 2+∫ .


135. Calculate dx

x x1 2−∫

(ln ).

7.8 The Hyperbolic Sine and Cosine Functions

136. Differentiate y = sinh x3.

137. Differentiate y = cosh x2.

138. Differentiate y = e2ln sinh 3x.

139. Differentiate y = cosh2 3x.

140. Differentiate y x x= sinh .

141. Differentiate y = (sinh 2x)3x.

142. Differentiate y = cosh (e–2x).

143. Differentiate y = 2cosh 3x.

144. Differentiate y = sinh 2x cosh2 2x.

145. Calculate sinh

coshx

xdx

1 +∫ .

146. Calculate sinh cosh2 2x x dx ∫ .

147. Calculate sinh cosh5 π πx x dx ∫ .

7.9 The Other Hyperbolic Functions

148. Differentiate y = sech x tanh x.

149. Differentiate y = e3x tanh 2x.

150. Differentiate y x= +coth( )2 12 .

151. Differentiate y = csch (tan e3x).

152. Differentiate y = tanh2 x.

153. Differentiate y = tanh3 (3x + 2).

154. Differentiate y = x sech x.

155. Differentiate y x x= + +1 2 tanh .

156. Differentiate y = (tanh 3x)2.

157. Differentiate y = tanh (sin 2x).


158. Calculate tanh 5x dx ∫ .

159. Calculate x x x dxtanh sech2 2 2 ∫ .



1. one-to-one; f xx− =

−1 34

( )


+1 75

( )

3. not one-to-one


−1 31

2( )


−131

3( )

6. not one-to-one

7. one-to-one; f −1(x) = (x – 1)1/5 + 1


−13 5

4( )

9. one-to-one; f −1(x) = (x – 2)3/5 – 1

10. one-to-one; f xx

x− =

+1 1 2( )


x− =

+−

1 32 3

( )

12. one-to-one; f xxx

− =−−

1 12

( )


x− =

+1 32

( )

14.

15.

16. 1/4

17. f ′ (x) = 3x2 + 2 > 0, so f is increasing andtherefore, one-to-one. (f −1) ′ (7) = 1/14.

18. f ′ (x) = 2cos 2x − 4 < 0, so f is decreasing andtherefore, one-to-one. (f −1) ′ (0) = −½.

19. (f −1) ′ (x) = x

20. f ′ (x) = 1 + sin2 x > 0, so f is increasing andtherefore, one-to-one. (f −1) ′ (0) = ½.

21. 3.22

22. 0.79

23. 6.44

24. 3.03

25. 1.3

26. (a) 1.45 (b) 1.33

27. x = − +1 2 ( − −1 2 is not a solution

because h( )− −1 2 is undefined.)

28. xe

e=

+−

2 11

29. x = 2 (−1 is not a solution because ln (−1) isundefined.)

30. x = 1, e5

31. dom(f) = (1, ∞); f xx

x' ( ) =

−

3

1

2

3


32. dom(f) = (−1, 1); f xx

x' ( ) =

−

2

12

33. dom(f) = (0, ∞);

f xx x

xx x

' ( )( ) ( )

= ++

=+

+1 1

2 24 3

2 2

34. dom(f) = (−∞, −2) ∪ (0, ∞);

f x x xx

x x' ( ) ln( )= + +

+

+2

2 2

22

2

35. dom(f) = (0, 3 ); f xx

x x' ( )

( )=

−−

3 2

3

2

2

36. dom(f) = (2πk − π/2, 2πk + π/2),k = 0, ±1, ±2, … ; f ′ (x) = tan x

37. Cx +−− |35|ln31

38. 2 ln | x2 – x | + C

39. Cx +− |53|ln91 3

40. Cx +|3sin|ln31

41. x + ln | sin x | + C

42.1

127ln

43.16

83

ln

44. ln 2

45.

+

−+

++

+)1(3

21

1)1(

123/2

2

xxx

xxxx

46.13

3 5 2

8

2

58 2

6

8

2 4

3 23

2

2

3

( ) cos

( )tan

x x

x

x

xx

x

x

+− +

− −−

47.sin

( )cot

( )

x

xx

x

x1

15

3

15 35

4

5+−

+

48. V = π ln 5

49. Cx +|5sec|ln51

50. Cxx

++3

2tan

32

secln23 πππ

51. − −

− −

+2 22

22

ln csc cotπ πx x

C

52. Cx +−−|)32sin(|ln

21 ππ

53. Ce x +− 2cos21

54. Cx ++ |2tan2|ln21

55.−

++

32

1

2 2( sec )xC

56.12

9 91 1

2 2

lncsc cot

csc cot

π π−

−

57.14

3ln

58. 32 2

3ln

+

59. − −32

3 2e x /

60. ( )4 1 2 2x e x x− −

61.13

2 3 13x e x− +/

62. e x xx

x3 2 3 2 32

− +

/ ln

63. 2 1 13 32( )(3 )e x ex x x x+ ++ +

64.2 5

3 2 2

e

e e

x

x x( )+

65. ex sin x (x cos x + sin x)

66. e−2x (3 cos 3x − 2 sin 3x)

67. 2esin 2x cos 2x


68.− e

x

x1

2

/

69.12

2 122

ex

xx −

70. − − +2 1 e Cx

71.12

2 2ln( )e e Cx x+ +−

72.25

5 2e Cx / +

73. 33

e Cx +

74.1

1614

4 4e x e Cx x− − +−

75.19

3 23ln e Cx − +

76. Ce x ++ )2ln(31 23

77. 2(e1/2 – 1)

78.13

12 3

−

π

79. 2(e – 1)

80.12

23

2

ln+ e

81.13

23

6 2e e− +

82.

83. −3

84. 5/2

85.− 3 3

2

cos sin (ln )x x

x

86. 5 2 5 22x x (ln )(ln )

87.1

2 442

lnx C+

88.−

+−23

3 2

ln/x C

89.−

+−32 5

52

lnx C

90.1

2 32 2

ln(ln )x C+

91.1

4 42 22

ln[ln( )] lnx x C+ +

92. ee

e 11

+

93.1

2e ln

94. (sin x)x [ln sin x + x cot x]

95. xx

x4 44

4+

ln

96. (tan x)x [ln tan x + x(cot x + tan x)]

97.2

9 3ln

98.7

3 2ln

99.4 3

ln( )

aa a−

100. 25


101. (a) $857.88 (b) $1021.72 (c) $1202.80

102. 23.8 yrs

103. (a) y(t) = y0e−0.0005332t

(b) approximately 4319 yrs

104. (a) 237.4 lbs (b) 400 lbs

105. approximately 128.3 lbs

106. −30° F

107. The equation can be rewriten as eydy = sec2 x dx;y = ln (tan x + C)

108. π/6

109. − 15

110.−

=−1

15

1515

111.5

2

112. π/2

113.4

7

4 77

=

114.7

4

115. 3/5

116.−

=−2

13

2 1313

117.7

4

118. 1 2− x

119.1 4

2

2− xx

120.3

1 93

21x

xx

++ −tan

121.2

1 42

2

1x

xx

−+ −sin

122.x

x 2 1+

123. 1

124.2

1 43 2

3

2

3 1e

xe x

xx

−+ −sin

125.1

12x +

126.11

−+

xx

127. −+

1

1 2x

128. − −tan 1

2x

129.1

1234

1tan−

130. π/12

131. π/8

132. π/8

133.12

1 2tan− +x C

134.13

31tan− +x C

135. sin–1 (ln x) + C

136. 3x2 cosh x2

137. 2x sinh x2

138. 3 sinh 6x

139. 6 cosh 3x sinh 3x

140.x

x x2

cosh sinh+

141. (sinh 2x)3x (6x coth 2x + 3 ln sinh 2x)

142. −2e−2x sinh (e−2x)


143. 2cosh 3x (3 ln 2 sinh 3x)

144. 4 sinh2 2x cosh 2x + 2 cosh3 2x

145. ln (1 + cosh x) + C

146.13

2 3 2(sinh ) /x C+

147.1

66

ππsinh x C+

148. sech2 x − sech x tanh2 x

149. 2e3x sech2 2x + 3e3x tanh 2x

150.−

++

1

2 12 1

2

2 2

xxcsch ( )

151. −3e3x csch (tan e3x) coth (tan e3x ) sec2 e3x

152. 2 tanh x sech2 x

153. 9 tanh2 (3x + 2) sech2 (3x + 2)

154. sech x − x sech x tanh x = sech x(1 – x tanh x)

155.x

xx

1 2

2

++ sech

156. −6 tanh 3x sech2 3x

157. 2 sech2 (sin 2x) cos 2x

158.15

5ln(cosh )x C+

159.14

2 2tanh x C+

97

CHAPTER 8

Techniques of Integration

8.2 Integration by Parts

1. Calculate xe dxx−∫ 2 .

2. Calculate x x dxsin 2 ∫ .

3. Calculate ln( )1 2+∫ x dx .

4. Calculate x x dx1 +∫ .

5. Calculate e x dxx−∫ cos 2 .

6. Calculate ∫ dxx ln 2 .

7. Calculate x

x +dx

3

2 1 ∫ .

8. Calculate x x dx2 2 ∫ .

9. Calculate x x dxln 2 ∫ .

10. Calculate e x dxx5 2sin ∫ .


2ln ∫ .


22sin

/

π∫ .

13. Evaluate x x dxsec−∫ 11

2 .

14. Find the centroid of the region under the graph of ƒ(x) = e2x, x ∈ [0, 1].

15. Find the volume generated by revolving the region under the graph of ƒ(x) = e2x, x ∈ [0, 1] about the y-axis.

8.3 Powers and Products of Trigonometric Functions

16. Calculate ∫ dxxx )2(sin)(2cos 23 .

17. Calculate sin cos3 x x dx5∫ .


18. Calculate sin cos4 θ θ

θ2 2

3 d∫ .

19. Calculate sin3 3θ θd∫ .

20. Evaluate dx

xcos/

/

24

3

π

π∫ .

21. Calculate cos4 2x dx ∫ .

22. Calculate x x x dxsin cos2 2 2 2∫ .

23. Evaluate ∫2/

0 2

2

csccot

π

θθθ

d .

24. Calculate ∫ dxxx )3(sin)3(cos 22 .

25. Calculate ∫ dxxx

2

cos2

sin2 .

26. Calculate sin cos2 t t

dt2 2

5∫ .

27. Calculate sin

cos

x

xdx

5 ∫ .

28. Calculate sin4 2x dx ∫ .

29. Evaluate sin cos/ 2 2

0

22 2θ θ θ

π d∫ .

30. Evaluate sin cos/

/ 4 33

2 3θ θ θ

π

π d∫ .

31. Evaluate (tan sec )/

/ 2 2 46

3x x dx−∫

π

π.

32. Calculate ∫ dxxx )4(cot)4(csc 33 .

33. Calculate ∫ dxxx )2(sec)2(tan 63 .

34. Calculate sec tan3

2 2x x

dx ∫ .

35. Calculate tan5 x dx ∫ .

36. Calculate tan3 3θ θ d∫ .

37. Calculate sec tan6 2

3 3x x

dx ∫ .

Techniques of Integration 99

38. Calculate 1

2 2sec tanx xdx ∫ .

39. Calculate tan sec3 4

2 2x x

dx ∫ .

40. Evaluate (tan sec )/

x x dx+∫ 20

4

π.

41. Calculate cot 4 2θ θ d∫ .

42. Calculate tan sec5 4t t dt ∫ .

43. Calculate cot2 2x dx ∫ .

8.4 Trigonometric Substitutions

44. Calculate x

xdx

3

225 4−∫ .

45. Calculate 1

42 3 2( ) /xdx

+∫ .

46. Calculate 1

4 92 3 2( ) /xdx

−∫ .

47. Calculate 1

252 2x xdx

+∫ .

48. Calculate 1

2 4 2+∫

xdx .

49. Calculate x

xdx

2 4−∫ .

50. Evaluate 16 20

4−∫ x dx .

51. Calculate x dx3 21 + x ∫ .

52. Calculate x

xdx

2

2 3−∫ .

53. Calculate x

xdx

2

23 −∫ .

54. Calculate 1

42 2x xdx

−∫ .


55. Evaluate 1

x x

dx2 22

4

4+∫ .

56. Calculate x

xdx

3

24 −∫ .

57. Calculate 1

92 2x xdx

−∫ .

58. Calculate 1

42x xdx

−∫ .

59. Calculate x dx3 2x −∫ 1 .

60. Evaluate dx

x 22

3

1−∫ .

61. Calculate 1

4 92x xdx

+∫ .

8.5 Partial Fractions

62. Calculate x

x xdx

2

2

6

1

−

−∫

( ) .

63. Calculate x

x x xdx

+− − +∫

3

1 4 42( )( ) .

64. Calculate x

x xdx

+

−∫

23

.

65. Calculate x

x xdx

2

2 2 1− +∫ .

66. Calculate 4 3

2 1

2

2

x x

x xdx

−− +∫

( )( ) .

67. Calculate 2 3

3 23 2

x

x x xdx

−

− +∫ .

68. Calculate 2 1

2 23 2

x

x x xdx

+

+ + +∫ .

69. Calculate x

xdx

( )+∫1 2

.


70. Calculate x

x xdx

+

+ −∫

1

2 32 .

71. Calculate x

x x xdx

+

+ −∫

4

3 103 2 .

72. Calculate x

x xdx

+−∫1

12 ( ) .

73. Calculate 1

1 12( )( )x xdx

+ +∫ .

74. Calculate ∫ −+θ

θθθ

d 5sin4sin

cos2 .

75. Calculate ∫ +−−dx

xxxx

1

423

.

76. Calculate x

x xdx

+

+∫

43

.

77. Calculate x x

xdx

2

3

3 1

1

+ −

−∫ .

8.6 Some Rationalizing Substitutions

78. Calculate dx

x2 3+∫ .

79. Calculate ( )

x dx

x

3 1 +∫ .

80. Calculate 3 2+∫ e dxx .

81. Calculate ( )∫ −12 xx

dx.

82. Calculate ( )x x dx+ −∫ 2 3 .

83. Calculate 2

2 1

x

xdx

+∫ .

84. Calculate x x dx3 1+∫ .

85. Calculate dx

e x2 2−∫ .

86. Evaluate x

xdx

( ) /4 3 3 20

2

+∫ .


87. Evaluate ∫ −

10

7

53dx

x

x.

88. Calculate dx

x1 −∫ sin.

89. Calculate 3

2 1cos xdx

+∫ .

90. Calculate dx

x xtan sin−∫ .

91. Calculate cot

sinx

xdx

1 +∫ .

92. Evaluate cos

cos/ x

xdx

20

2

−∫ π

.

8.7 Numerical Integration

93. Estimate 4 30

2+∫ x dx using

(a) the left-endpoint estimate, n = 4.(b) the right-endpoint estimate, n = 4.(c) the midpoint estimate, n = 4.(d) the trapezoidal rule, n = 4.

94. Estimate 4 40

4+∫ x dx using

(a) the left-endpoint estimate, n = 4.(b) the right-endpoint estimate, n = 4.

95. Estimate 4

1

1

1

0 2

π=

+∫ dxx

using

(a) the trapezoidal rule, n = 8.(b) Simpson’s rule, n = 8.

96. Estimate 1

1 30

2

+∫

xdx using

(a) the trapezoidal rule, n = 4.(b) Simpson’s rule, n = 4.

97. Estimate ( ) /1 2 3 20

3+∫ x dx using

(a) the left-endpoint estimate, n = 6.(b) the right-endpoint estimate, n = 6.(c) the trapezoidal rule, n = 6.(d) Simpson’s rule, n = 6.

98. Determine the values of n for which a theoretical error less than 0.001 can be guaranteed if the integral isestimated using (a) the trapezoidal rule; (b) Simpson’s rule.


8.8 Differential Equations; First-Order Linear Equations

99. Find the general solution of y′ – 3y = 6.

100. Find the general solution of y′ – 2xy = x.

101. Find the general solution of 44xy

xy' =+ .

102. Find the general solution of 4210

2=

++ y

xy' .

103. Find the general solution of y′ – y = –ex .

104. Find the general solution of x ln x y′ + y = ln x.

105. Find the particular solution of y′ + 10y = 20 determined by the side condition y(0) = 2.

106. Find the particular solution of y′ – y = –ex determined by the side condition y(0) = 3.

107. Find the particular solution of xy′ – 2y = x3 cos 4x determined by the side condition y(π) = 1.

108. A 100-gallon mixing tank is full of brine containing 0.8 pounds of salt per gallon. Find the amount of saltpresent t minutes later if pure water is poured into the tank at the rate of 4 gallons per minute and the mixtureis drawn off at the same rate.

109. Determine the velocity of time t and the terminal velocity of a 2-kg object dropped with a velocity 3 m/s, ifthe force due to air resistance is –50v Newtons.

110. Use a suitable transformation to solve the Bernoulli equation y′ + xy = xy2 .

111. Use a suitable transformation to solve the Bernoulli equation xy′ + y = x3y6 .

8.9 Separable Equations

112. Find the general solution of y′ = y2x3.


2

37yy

xy'

−+

= .

114. Find the general solution of y′ = y2 + 1.

115. Find the general solution of x(y2 + 1)y′ + y3 – 2y = 0.

116. Find the particular solution of exdx – ydy = 0 determined by the side condition y(0) = 1.

117. Find the particular solution of y′ = y(x – 2) determined by the side condition y(2) = 5.

118. Verify that the equation x

xyy'

+= is homogeneous, then solve it.

119. Verify that the equation 3

442xy

xyy'

+= is homogeneous, then solve it.


120. Verify that the equation xyx

yy'

++ is homogeneous, then solve it.

121. Verify that the equation [2x sinh (y/x) + 3y cosh (y/x)]dx – 3x cosh (y/x)dy = 0 is homogeneous, then solve it.

122. Find the orthogonal trajectories for the family of curves x2 + y2 = C.

123. Find the orthogonal trajectories for the family of curves x2 + y2 = Cx.



1.−

− +− −xe e Cx x

214

2 2

2.−

+ +x

x x C2

214

2cos sin

3. x ln (1 + x2) – 2x + 2 tan-1 x + C

4.23

14

1513 2 5 2x

x x C( ) ( )/ /+ − + +

5.e

x x Cx−

− +5

2 2 2( sin cos )

6. x2 ln x – 2x ln x + 2x + C

7. Cxxx ++−+ 2/322/122 )1(32

)1(

8.2

22

22

22

2

xx

xC

ln ln (ln )− +

+

9. x x C3 2 223

89

/ ln −

+

10.1

295 2 2 25e x x Cx ( sin cos )− +

11. 4 ln 2 – 15/16

12. π2/8 – ½

13.23

32

π−

14. ( , )( )

,x ye

e

e=

+−

+

2

2

21

2 1

14

15.π2

12( )e +

16. Cxx ++ 2sin101

2sin61 53

17. Cxx ++− 86 sin81

cos61

18.25 2

27 2

5 7sin sinθ θ

+ + C

19.−

+ +1

33

19

33cos cosθ θ C

20. 3 1−

21.38

18

41

648

xx x C+ + +sin sin

22.x

x C2

2

161

644− +sin

23. π/4

24.x

x C8

196

12− +sin

25.23 2

3sinx

C+

26.23 2

45 2

27 2

2 5 7sin sin sint t t

C− + +

27.14

4sec x C+

28.38

18

41

648

xx x C− + +sin sin

29. π/16

30. 0

31. π/6

32.120

41

1245 3csc cscx x C+ +

33. Cxxx +++ 2tan81

2tan61

2tan161 468

34.23 2

3secx

C+

35.14

12

4 2tan tan ln cosx x x C− − +

36.16

313

32tan ln cosθ θ+ + C

37.37 3

65 3 3

7 5 3tan tan tanx x x

C+ + +


38.12

2 212

2ln csc cot cosx x x C− + +

39.13 2

12 2

6 4tan tanx x

C+ +

40. 2 2 4− π /

41.−

+ + +1

62

12

23cos cosθ θ θ C

42.18

16

8 6tan tant t C+ +

43. − − +12

2cot x x C

44.−

− − − +25

1625 4

148

25 42 1 2 2 3 2( ) ( )/ /x x C

45.x

xC

4 42 ++

46.−

−+

x

xC

9 4 92

47.25

25

2++

xx

C

48.12

1 2 22ln + + +x x C

49. xx

C2 14 22

− − +−sec

50. 4π

51.15

113

12 5 2 2 3 2( ) ( )/ /+ − + +x x C

52.x

x x x C2

332

32 2− + + − +ln

53.32 3 2

31 2sin− − − +x x

x C

54.− −

+44

2xx

C

55.2 2 5

8−

56. − − − − +x x x C2 2 2 3 2423

4( ) /

57.− −

+99

2xx

C

58.12 2

1sec− +x

C

59.x

x x C2

2 3 2 2 5 2

31

215

1( ) ( )/ /− − − +

60. ln3 8

2 3

+

+

61.13

4 92

32

2

lnx

x xC

+− +

62. − + − +−

+6 7 15

1ln lnx x

xC

63. 412

52

ln( )

xx x

C−−

−−

+

64. 232

112

1ln ln lnx x x C− − − + +

65. x xx

C+ − −−

+2 11

1ln

66. 2 2 12 1ln ln tanx x x C− + + + +−

67. Cxxx +−+−+−

2ln21

1lnln23

68. − + + + + +−13

116

25

3 2 22 1ln ln( ) tanx x

xC

69. ln xx

C+ ++

+11

1

70.12

2 32ln x x C+ − +

71. − + − − + +25

37

2135

5ln ln lnx x x C

72. 21 1

lnx

x xC

−+ +


73.12

114

112

2 1ln ln( ) tanx x x C+ − + + +−

74.16

15

lnsinsin

θθ

−+

+ C

75. ln lnxx

x C− −−

− + +12

11

76. ( ) Cxxx +++− −12 tan1ln2ln4

77. Cx

x ++

+− −

3

12tan

3

41ln 1

78.2

3

43

2 3x

x C− + +ln

79.x

x x C3

23

23

1− + + +ln

80. 33

23 3

3 3

22

2+ +

+ −

+ ++e

e

eCx

x

xln

81.1

21 1

x x

x

xC+ +

−

+ln

82.25

3103

35 2 3 2( ) ( )/ /x x C− + − +

83. x x x C− + + +2 2 1ln

84.

29

167

165

1

23

1

9 2 7 2 5 2

3 2

( ) ( ) ( )

( )

/ / /

/

x x x

x C

+ − + + +

− + +

85.1

2 2

2 2

2 2

2

2ln

− −

− ++

e

eC

x

x

86.14

7

113−

87. 152/27

88.2

1 2−+

tan( / )xC

89. 32 3

2 3ln

tan( / )

tan( / )

x

xC

+

−+

90. −

− +−1

4 212 2

2

tan ln tanx x

C

91. lntan( / )

[ tan( / )]

x

xC

2

1 2 2++

92. π4

3 3

12

−

93. (a) 4.4914 (c) 4.8030(b) 5.2234 (d) 4.8574

94. (a) 15.5927 (b) 30.6239

95. (a) 0.7847 (b) 0.7854

96. (a) 1.0865 (b) 1.0968

97. (a) 20.8933 (c) 28.0490(b) 36.2047 (d) 27.9586

98. (a) n ≥ 19 (b) n ≥ 2

99. y = Ce–3x

100.212

−= xCey

101. 54 9

1x

x

Cy +=

102.x

Cxxy

210440 2

+++

=

103. y = (C – x)ex

104.x

Cxy

ln2ln2 +

=

105. y = 2 (identically)

106. y = (3 – x)ex

107.2

2 4sin41

+=

πx

xxy

108. 80e–0.04t pounds

109. v = 0.392 + 2.608e–25t;terminal velocity 0.392 m/s

110.2/2

1

1xCe

y+

=

111.5/1

53

25

−

+= Cxxy


112.Cx

y+

−=

4

4

113. Cxxyy =+−− 731

53

101 3510

114. y = tan(x + C)

115. (y2 – 2)3x4 = Cy2

116.21

ln,12 >−= xey x

117. 2/)2( 25 −= xey

118. y = x ln |Cx|

119. x8 = C(y4 + x4)

120. Cyyx =+− ln/2

121. x2 = C sinh3 (y/x)

122. y = Kx

123. x2 + y2 = Ky

109

CHAPTER 9

Conic Sections; Polar Coordinates; Parametric Equations

9.1 Translations; The Parabola

1. Sketch and give an equation for the parabola with vertex (0, 0) and directrix x = 5/2.

2. Sketch and give an equation for the parabola with vertex (1, 2) and focus (1, 4).

3. Sketch and give an equation for the parabola with focus (6, –2) and directrix y = 2.

4. Sketch and give an equation for the parabola with vertex (–1, 2) and focus (2, 2).

5. Find the vertex, focus, axis, and directrix for the parabola y2 + 6y + 6x = 0.

6. Find the vertex, focus, axis, and directrix for the parabola x2 – 4x – 2y – 8 = 0.

7. Find the vertex, focus, axis, and directrix for the parabola 2x2 – 10x + 5y = 0.

8. Find an equation for the parabola with directrix x = –2 and vertex (1, 3). Where is the focus?

9. Find an equation for the parabola with directrix y = 3 and vertex (–2, 2). Where is the focus?

10. Find an equation for the parabola with directrix x = 5 and focus (–1, 0). Where is the vertex?

11. Find an equation for the parabola that has vertex (2, 1), passes through (5, –2), and has axis of symmetryparallel to the x-axis.

12. Find the length of the latus rectum for the parabola (y – 2) = 3(x + 1)2.

9.2 The Ellipse and Hyperbola

13. For the ellipse x y2 2

16 91+ = , find

(a) the center(b) the foci(c) the length of the major axis(d) the length of the minor axisThen sketch the figure.

14. For the ellipse 5x2 + 3y2 = 15, find(a) the center(b) the foci(c) the length of the major axis(d) the length of the minor axisThen sketch the figure.

15. For the ellipse 36(x – 1)2 + 4y2 = 144, find(a) the center(b) the foci(c) the length of the major axis(d) the length of the minor axisThen sketch the figure.


16. For the ellipse 9x2 + 16y2 – 36x + 96y + 36 = 0, find(a) the center(b) the foci(c) the length of the major axis(d) the length of the minor axisThen sketch the figure.

17. For the ellipse 9x2 + 5y2 + 36x – 30y + 36 = 0, find(a) the center(b) the foci(c) the length of the major axis(d) the length of the minor axisThen sketch the figure.

18. Find an equation for the ellipse with foci at (0, 3), (0, –3) and major axis 10.

19. Find an equation for the ellipse with foci at (1, 0), vertices at (–1, 0) (3, 0) and foci at ( , )1 3 0− and

( , )1 3 0+ .

20. Find an equation for the ellipse with focus (2, 2), center at (2, 1), and major axis 10.

21. Find an equation for the ellipse with foci at (1, –1), (7, –1) and minor axis 6.

22. Find the equation of the parabola that has vertex at the origin and passes through the ends of the minor axisof the ellipse y2 – 10y + 25x2 = 0.

23. Determine the eccentricity of the ellipse x y2 2

251

91+

−=

( ).

24. Write an equation for the ellipse with major axis from (–2, 0) to (2, 0), eccentricity ½.

25. Find an equation for the hyperbola with foci at (3, 0), (–3, 0) and transverse axis 4.

26. Find an equation for the hyperbola with asymptotes y x= ±43

and foci at (10, 0), (–10, 0).

27. Find an equation for the hyperbola with center at (2, 2), a vertex at (2, 10), and a focus at (2, 11).

28. Find an equation for the hyperbola with vertices at (7, –1), (–5, –1) and a focus at (9, –1).

29. For the hyperbola xy2

2

41− = , find the center, the vertices, the foci, the asymptotes, and the length of the

transverse axis. Then sketch the figure.

30. For the hyperbola ( )x y−

− =1

4 161

2 2

, find the center, the vertices, the foci, the asymptotes, and the length

of the transverse axis. Then sketch the figure.

31. For the hyperbola 9x2 – 16y2 = 144, find the center, the vertices, the foci, the asymptotes, and the length ofthe transverse axis. Then sketch the figure.

32. For the hyperbola 16x2 – 9y2 – 160x – 72y + 112 = 0, find the center, the vertices, the foci, the asymptotes,and the length of the transverse axis. Then sketch the figure.

Conic Sections; Polar Coordinates; Parametric Equations 111

33. For the hyperbola ( ) ( )x y−

++

=3

94

161

2 2

, find the center, the vertices, the foci, the asymptotes, and the

length of the transverse axis. Then sketch the figure.

34. For the hyperbola 4y2 – 9x2 – 36x – 8y – 68 = 0, find the center, the vertices, the foci, the asymptotes, and thelength of the transverse axis. Then sketch the figure.

35. Determine the eccentricity of the hyperbola x

y2

2

491− = .

36. Determine the eccentricity of the hyperbola y x2 2

16 121− = .

9.3 Polar Coordinates

37. Find the rectangular coordinates of the point with polar coordinates [4, 2π/3].

38. Find the rectangular coordinates of the point with polar coordinates [3, –π/4].

39. Find the rectangular coordinates of the point with polar coordinates [–2, –π/3].

40. Find all possible polar coordinates for the point with rectangular coordinates ( , ).−4 4 3

41. Find all possible polar coordinates for the point with rectangular coordinates (2, –2).

42. Find the point [r, θ] symmetric to the point [2, π/6] about(a) the x-axis(b) the y-axis(c) the originExpress your answer with r > 0 and θ ∈ [0, 2π).

43. Find the point [r, θ] symmetric to the point [2/3, 5π/4] about(a) the x-axis(b) the y-axis(c) the originExpress your answer with r > 0 and θ ∈ [0, 2π).

44. Test the curve r = 3 + 2 cos θ for symmetry about the coordinate axes and the origin.

45. Test the curve r sin 2θ = 1 for symmetry about the coordinate axes and the origin.

46. Write the equation x + y2 = x – y in polar coordinates.

47. Write the equation x2 + y2 – 6y = 0 in polar coordinates.

48. Write the equation x2 + y2 + 8y = 0 in polar coordinates.

49. Write the equation x4 + x2y2 = y2 in polar coordinates.

50. Write the equation y3 + x2y = x in polar coordinates.

51. Write the equation y6 + x2y4 = x4 in polar coordinates.

52. Write the equation (x2 + y2)2 = 4xy in polar coordinates.


53. Write the equation x(x2 + y2) = 2(3x2 – y2) in polar coordinates.

54. Write the equation (x2 + y2)3/2 = x2 – y2 – 2xy in polar coordinates.

55. Identify the curve given by r sin θ = 2 and write the equation in rectangular coordinates.

56. Identify the curve given by r = 4 cos θ and write the equation in rectangular coordinates.

57. Identify the curve given by θ π2 249

= and write the equation in rectangular coordinates.

58. Identify the curve given by r = 4 sin θ – 6 cos θ and write the equation in rectangular coordinates.

59. Identify the curve given by r = 3 cos θ – sin θ and write the equation in rectangular coordinates.

60. Identify the curve given by r =−

11cosθ

and write the equation in rectangular coordinates.

61. Identify the curve given by r =+

102 cosθ


62. Identify the curve given by r =−

11 cosθ


63. The parabola r =+

11 cosθ

has focus at the pole and directrix x = 2. Without resorting to xy-coordinates,

(a) locate the vertex of the parabola(b) find the width of the latus rectum(c) sketch the parabola.

64. The ellipse r =+

108 5cosθ

has right focus at the pole, major axis horizontal. Without resorting to xy-

coordinates,(a) find the eccentricity of the ellipse(b) locate the ends of the major axis(c) locate the center of the ellipse(d) locate the second focus(e) determine the length of the minor axis(f) determine the width of the ellipse at the foci(g) sketch the ellipse

65. The hyperbola r =+

92 6 cosθ

has left focus at the pole, transverse axis horizontal. Without resorting to xy-

coordinates,(a) find the eccentricity of the hyperbola(b) locate the ends of the transverse axes(c) locate the center of the hyperbola(d) locate the second focus(e) determine the width of the hyperbola at the foci, and sketch the hyperbola.

66. Find the points at which the curves r = 2 cos θ and r = –1 intersect. Express your answers in rectangularcoordinates.

67. Find the points at which the curves r = 1 + cos θ and r = 2 cos θ intersect. Express your answers inrectangular coordinates.


68. Find the points at which the curves r = sin 3θ and r = 2 sin θ intersect. Express your answers in rectangularcoordinates.

69. Find the points at which the curves r = ½ + cos θ and θ = π/4 intersect. Express your answers inrectangular coordinates.

70. Find the points at which the curves r =+

11 cosθ

and r sin θ = 2 intersect. Express your answers in

rectangular coordinates.

9.4 Graphing in Polar Coordinates

71. Sketch and identify the polar curve r2 = 9 sin 2θ .

72. Sketch and identify the polar curve r = 1 + cos θ .

73. Sketch and identify the polar curve r = 2 cos θ .

74. Sketch and identify the polar curve r = sin 3θ .

75. Sketch and identify the polar curve r = 4 + 4 cos θ .

76. Sketch and identify the polar curve r = 2 .

77. Sketch and identify the polar curve r2 = 4 cos 2θ .

78. Sketch and identify the polar curve r = 2 – 4 sin θ .

79. Sketch and identify the polar curve r = cos 3θ .

80. Sketch and identify the polar curve r = 2 sin 2θ .

81. Sketch and identify the polar curve r = 2 + 4 sin θ .

82. Sketch and identify the polar curve r = 4 + 2 sin θ .

83. Sketch and identify the polar curve r = 3 sin θ .

84. Sketch and identify the polar curve r = 1 – 2 cos θ .



87. Sketch and identify the polar curve r = 4(1 – cos θ ).

88. Sketch and identify the polar curve r = 4(1 – sin θ ).

9.5 Area in Polar Coordinates

89. Find the area of the region enclosed by r = 4 sin 3θ.

90. Find the area of the region enclosed by r = 2 + sin θ.


91. Find the area of the region that is inside r = 5 sin θ but outside r = 2 + sin θ.

92. Find the area of the region that is outside r = 1 + sin θ but inside r = 3 + sin θ.

93. Find the area of the region that is common to r = 3 cos θ and r = 1 + cos θ.

94. Find the area of the region that is common to r = 1 + sin θ and r = 1.

95. Find the area of the region that is inside r = 1 but outside r = 1 – cos θ.

96. Find the area of the region enclosed by r = 2 + cos θ.

97. Find the area of the region that is inside r = 2 cos θ but outside r = sin θ.

98. Find the area of the region enclosed by r = 1 – sin θ.

99. Find the area of the region that is common to r = 3a cos θ and r = a(1 + cos θ).

100. Find the area of the region that is inside r = 2 but outside r = 1 + cos θ.

101. Find the area of the region enclosed by r = 2 cos 3θ.

102. Find the area of the region that is inside r = 3(1 + sin θ) but outside r = 3 sin θ.

103. Find the area of the region that is common to r = a cos 3θ and r = a/2. Take a > 0.

104. Find the area of the region enclosed by r2 = cos 2θ.

105. Find the area of the region enclosed by the inner loop of r = 1 – 2 sin θ.

106. Find the centroid of the region enclosed by r = 2 cos θ.

9.6 Curves Given Parametrically

107. Express the curve by an equation in x and y: x(t) = 2t sin t, y(t) = 3 – cos t.

108. Express the curve by an equation in x and y: x(t) = et – 1, y(t) = 3 + e2t.

109. Express the curve by an equation in x and y: x(t) = 2 cos t, y(t) = 3 sin t.

110. Express the curve by an equation in x and y: x(t) = 3t cosh t, y(t) = 2t sinh t.

111. Express the curve by an equation in x and y: x(t) = 3 + cos t, y(t) = 3 – 2 sin t.

112. Express the curve by an equation in x and y: x(t) = 2 t2 + t – 3 , y(t) = t –1 .

113. Express the curve by an equation in x and y: x(t) = cos 2t, y(t) = sin t.

114. Express the curve by an equation in x and y: x(t) = –1 + 3 cos t, y(t) = sin t.

115. Find the parametrization x = x(t), y = y(t), t ∈ [0, 1], for the line segment from (2, 5) to (5, 8).

116. Find the parametrization x = x(t), y = y(t), t ∈ [0, 1], for the curve y = x2 from (1, 1) to (3, 9).


9.7 Tangents to Curves Given Parametrically

117. Find an equation in x and y for the line tangent to the curve x(t) = 3t, y(t) = t2 – 1 at t = 1.

118. Find an equation in x and y for the line tangent to the curve x(t) = 2t2, y(t) = (1 – t)2 at t = 1.

119. Find an equation in x and y for the line tangent to the curve x(t) = 2et, y t e t( ) = −12

at t = 0.

120. Find an equation in x and y for the line tangent to the curve x(t) = 2/t, y(t) = 2t2 + 3 at t = 1.

121. Find an equation in x and y for the line tangent to the polar curve r = 3 + 2 sin θ at θ = π/2.

122. Find an equation in x and y for the line tangent to the polar curve r = 3 sin 3θ at θ = π/6.

123. Find an equation in x and y for the line tangent to the polar curve r =+

32 cosθ

at θ = π/2.

124. Find the points (x, y) at which the curve x(t) = 2t – t3, y(t) = t – 1 has (a) a horizontal tangent; (b) a verticaltangent.

125. Find the points (x, y) at which the curve x(t) = 4 + 3 sin t, y(t) = 3 + 4 cos t has (a) a horizontal tangent; (b) avertical tangent.

126. Find the tangent(s) to the curve x(t) = t2 – 2t, y(t) = 1 – t at the point (–1, 0).

127. Calculate d y

dx

2

2 at the point t = 1 without eliminating the parameter if x(t) = et – 1 and y(t) = 3 + e2t.

128. Calculate d y

dx

2

2 at the point t = 3π/4 without eliminating the parameter if x(t) = 5 – 2 cos t and

y(t) = 3 + sin t.

9.8 Arc Length and Speed

129. Find the arc length of the curve f(x) = 2x3/2, x ∈ [0, 8/9] and compare it to the straight-line distancebetween the endpoints.

130. Find the arc length of the curve f xx

x( ) = +

3

61

2, x ∈ [1, 3] and compare it to the straight-line distance

between the endpoints.

131. Find the arc length of the curve f x x( ) ( ) /= +23

1 3 2 , x ∈ [1, 2] and compare it to the straight-line distance

between the endpoints.

132. Find the arc length of the curve f(x) = x2/3, x ∈ [0, 8] and compare it to the straight-line distance betweenthe endpoints.

133. Find the arc length of the curve f x x x( ) / /= −23

12

3 2 1 2 , x ∈ [1, 4] and compare it to the straight-line

distance between the endpoints.


134. Find the arc length of the curve f y y x( ) / /= −35

34

5 3 1 3 , y ∈ [1, 8] and compare it to the straight-line


135. Find the arc length of the curve f(x) = 2x3/2, x ∈ [0, 3] and compare it to the straight-line distance betweenthe endpoints.

136. Find the arc length of the curve f x x( ) ( ) /= +13

22 3 2 , x ∈ [0, 3] and compare it to the straight-line


137. The equations x(t) = 2 + sin t, y(t) = 3 – cos t give the position of a particle at time t from t = 0 to t = π/2.Find the initial speed of the particle, the terminal speed, and the distance traveled.

138. The equations x(t) = et sin t, y(t) = et cos t give the position of a particle at time t from t = 0 to t = π. Find theinitial speed of the particle, the terminal speed, and the distance traveled.

139. The equations x(t) = t2 + 2, y(t) = t3 – 3 give the position of a particle at time t from t = 0 to t = 1. Find theinitial speed of the particle, the terminal speed, and the distance traveled.

140. The equations x(t) = 3(t – 1)2, y(t) = 8t3/2 give the position of a particle at time t from t = 0 to t = 1. Find theinitial speed of the particle, the terminal speed, and the distance traveled.

141. Find the length of the polar curve r = 2e3θ from θ = 0 to θ = π.

142. Find the length of the polar curve r = 2 cos 2θ from θ = 0 to θ = 2π.

9.9 The Area of a Surface of Revolution; The Centroid of a Curve; Pappus’s Theorem on SurfaceArea

142. Find the length of the curve, locate its centroid, and determine the area of the surface generated by revolvingf(x) = 3x, x ∈ [0, 1] about the x-axis.

144. Find the length of the curve, locate its centroid, and determine the area of the surface generated by revolving

y x=12

, x ∈ [0, 2] about the x-axis.

145. Find the area of the surface generated by revolving f x x( ) = +2 1 , x ∈ [–1, 1] about the x-axis.

146. Find the area of the surface generated by revolving y = sin x, x ∈ [0, π/2] about the x-axis.



1. y2 = −10x

2. 8(y – 2) = (x – 1)2

3. 8y = −(x – 6)2

4. 12(x + 1) = (y – 2)2

5. V: (3/2, −3) F: (−3/2, −3)axis: y = −3 directrix: x = 9/2

6. V: (2, −6) F: (2, −11/2)axis: x = 2 directrix: y = −13/2

7. V: (5/2, 5/2) F: (5/2, 15/8)axis: x = 5/2 directrix: y = 25/8

8. (y – 3)2 = 12(x – 1); F: (4, 3)

9. (x + 2)2 = –4(y – 2); F: (–2, 1)

10. y2 = −12(x – 2); V: (0, 2)

11. (y − 1)2 = 3(x – 2)

12. 3

13. (a) (0, 0) (c) 8

(b) )0 ,7( ),0 ,7( − (d) 6

14. (a) (0, 0) (c) 52

(b) )2 ,0( ),2 ,0( − (d) 32

15. (a) (1, 0) (c) 12

(b) )24 ,1( ),24 ,1( − (d) 4

16. (a) (2, −3) (c) 8

(b) )3 ,72( ),3 ,72( −−−+ (d) 6

17. (a) (−2, 3) (c) 6

(b) (−2, 1), (−2, 5) (d) 52

18. 13425

22

=+yx

19. 14

)1( 22

=+−

yx

20. 1)1(25

)2( 22

=−+−

yx

21. 19

)1(18

)4( 22

=+

+− yx

22. yx512 =

23. 4/5


24. 134

22

=+yx

25. 154

22

=−yx

26. 16436

22

=−yx

27. 117

)2(64

)2( 22

=−

−− xy

28. 128

)1(36

)1( 22

=+

−− yx

29. center: (0, 0)vertices: (–1, 0), (1, 0)

foci: )0 ,5( ),0 ,5(−asymptotes: y = ±2xlength of transverse axis: 2

30. center: (1, 0)vertices: (–1, 0), (3, 0)

foci: )0 ,521( ),0 ,521( +−asymptotes: y = ±2(x – 1)length of transverse axis: 4

31. center: (0, 0)vertices: (–4, 0), (4, 0)foci: (–5, 0), (5, 0)

asymptotes: xy43

±=

length of transverse axis: 8

32. center: (5, –4)vertices: (2, –4), (8, –4)foci: (0, –4), (10, –4)

asymptotes: )5(34

4 −±−= xy


33. center: (3, –4)vertices: (0, –4), (6, –4)foci: (–2, –4), (8, –4)

asymptotes: )3(34

4 −±−= xy


34. center: (–2, 1)vertices: (–2, –2), (–2, 4)

foci: )131 ,2( ),131 ,2( +−−−

asymptotes: )2(23

1 +±= xy


35.7

25

36.27

37. )32 ,2(−

38. )2 ,2( −

39. )3 ,1(−

40.

+−

+ π

ππ

πnn 2

35

,8 ,23

2 ,8

41.

+−

+ π

ππ

πnn 2

43

,22 ,24

7 ,22

42. (a)

611 ,2

π(b)

65

,2π

(c)

67

,2π

43. (a)

43

,32 π

(b)

47

,32 π

(c)

4 ,

32 π

44. x-axis only

45. symmetric with respect to x-axis, y-axis, origin

46.θθ

θθcossin21

sincos+

−=r

47. r = 6 cos θ

48. r = –8 sin θ

49. r = tan θ

50. r2 = cot θ

51. r = cot2 θ

52. r2 = 2 sin 2θ

53. r = 2(3 cos θ – sin θ tan θ )

54. r = cos θ – sin 2θ

55. y = 2; line

56. circle; center (2, 0)x2 + y2 = 4x or (x – 2)2 + y2 = 4


57. half lines; 3

2πθ ±=

)0( 3

)0( 3

≤=

≤−=

xxy

xxy

58. (x + 3)2 + (y – 2)2 = 13; circle

59. (x – 3/2)2 + (y + 1/2)2 = 5/2; circle

60. y2 = –2(x – ½); parabola

61. 112/4009/400

)3/10( 22

=++ yx

; ellipse

62. y2 = 2(x + ½); parabola

63. (a) (1, 0) (b) 4

64. (a) 5/8 (d) (−100/39, 0)

(b) (−10/3, 0), (10/13, 0) (e) 39/20(c) (−50/39, 0) (f) 5/2

65. (a) 3 (d) (27/8, 0)(b) (9/8, 0), (9/4, 0) (e) 9(c) (27/16, 0)

66.

−

23

,21

, 23

,21

67. (2, 0), (0, 0)

68.

−21

,23

, 21

,2

2, (0, 0)

69. (0, 0),

−−

++4

22,

422

, 4

22,

422

70. (−3/2, 2)

71.

72.

73.

74.

75.


76.

77.

78.

79.

80.

81.

82.

83.


84.

85.

86.

87.

88.

89. 4π

90. 9π/2

91. 8π/3 + 3

92. 8π

93. 5π/4

94. 5π/4 – 2

95. 2 – π/4

96. 9π/2

97.21

2tan43

21 ++ −π

98. 3π/2

99. 5πa2/4

100. 5π/2

101. π

102. 45π/4

103.

−

83

62 π

a

104. 1

105.2

33−π

106. (1, 0)

107. (x – 2)2 + (y – 3)2 = 1; circle

108. y = x2 + 2x + 4 = (x + 1)2 + 3; parabola


109. x2/4 + y2/9 = 1; ellipse

110. (x – 3)2 – (y – 2)2 = 1; hyperbola

111. 14

)3()3(

22 =

−+−

yx ; ellipse

112. x = 2y2 + 5y; parabola

113. x = 1 – 2y2; parabola

114. 119

)1( 22

=++ yx

; ellipse

115. x = 2 + 3t; y = 5 + 3t

116. x = 2 + 3t; y = 1 + 4t + 4t2

117. 2x – 3y = 6

118. y = 0

119. x + 4y = 4

120. 2x + y = 9

121. y = 5

122. 63 =+ yx

123. x + 2y = 3

124. (a) None

(b)

−

−−

− 1

3

2,

33

24;1

3

2,

33

24

125. (a) (4, 7), (4, –11)(b) (7, 3), (1, 3)

126. vertical tangent

127. 2

128.2

2−

129. 52/27

130. 14/3

131. )338(32

−

132. )11010(278

−

133. 31/6

134. 387/20

135. )1756(272

−

136. 12

137. initial speed = 1; terminal speed = 1distance = π/2

138. initial speed = 2 ; terminal speed = 2πe

distance = ( )12 −πe

139. initial speed = 0; terminal speed = 13

distance = )81313(271

−

140. initial speed = 0; terminal speed = 12distance = 9

141. )1(3102 3 −πe

142. 4π

143. length: 10 ; centroid:

23

,21

area: 103π

144. length: 5 ; centroid:

21

,1

area: 5π

145.

−

31

38π

146. )]21ln(2[ ++π

123

CHAPTER 10

Sequences; Intermediate Forms; Improper Integrals

10.1 The Least Upper Bound Axiom

1. Find the least upper bound (if it exists) and the greatest lower bound (if it exists) for the interval (0, 4).

2. Find the least upper bound (if it exists) and the greatest lower bound (if it exists) for the set x: x2 < 5.

3. Find the least upper bound (if it exists) and the greatest lower bound (if it exists) for the set x: x2 > 9.

4. Find the least upper bound (if it exists) and the greatest lower bound (if it exists) for the set x: |x – 2| > 1.

5. Find the least upper bound (if it exists) and the greatest lower bound (if it exists) for the set x: |x – 3| < 1.

6. Find the least upper bound (if it exists) and the greatest lower bound (if it exists) for the setx: x2 + 3x + 2 ≥ 0.

10.2 Sequences of Real Numbers

7. Give an explicit formula for the nth term of the sequence 1, 4/5, 6/8, 8/11, 10/14, 12/17, . . . .

8. Give an explicit formula for the nth term of the sequence 1/2, –1/4, 1/8, –1/16, . . . .

9. Give an explicit formula for the nth term of the sequence 1, 4, 1/9, 16, 1/25, 36, . . . .

10. Determine the boundedness and monotonicity of the sequence 1

2+nn

.


2−nn

.


+−

nn

.

13. Determine the boundedness and monotonicity of the sequence n2

1− .


3

+n

n

.


131

+−

nn.

16. Determine the boundedness and monotonicity of the sequence 2)2(

4+n

n

.

17. Determine the boundedness and monotonicity of the sequence n

n

e

3.


18. Determine the boundedness and monotonicity of the sequence )!1(

3+n

n

.

19. Determine the boundedness and monotonicity of the sequence ne

n 2+.

20. Determine the boundedness and monotonicity of the sequence n

109

.


sinπn

.

22. Write the first six terms of the sequence given by a1 = 1; nn ann

a12

1 ++

=+ , and then give the general term.

23. Write the first six terms of the sequence given by a1 = 1; an + 1 = 2an – 1, and then give the general term.

24. Write the first six terms of the sequence given by a1 = 1; an + 1 = 2an – n(n + 1), and then give the generalterm.

25. Write the first six terms of the sequence given by a1 = 1; an + 1 = 1 – 2an, and then give the general term.

26. Write the first six terms of the sequence given by a1 = 1; a2 = 4; an + 1 = 2an – an – 1, and then give the generalterm.

27. Write the first six terms of the sequence given by a1 = 2; a2 = 2; an + 1 = 2an – an – 1, and then give the generalterm.

28. Write the first six terms of the sequence given by a1 = 3; a2 = 5; an + 1 = 3an – 2an – 1 – 2, and then give thegeneral term.

10.3 Limit of a Sequence

29. State whether or not the sequence nnπ

sin1

converges as n → ∞, and, if it does, find the limit.

30. State whether or not the sequence nnln


31. State whether or not the sequence 2

)1( 1

+− +

nnn converges as n → ∞, and, if it does, find the limit.

32. State whether or not the sequence (1 + n)1/n converges as n → ∞, and, if it does, find the limit.

33. State whether or not the sequence 2+n

nconverges as n → ∞, and, if it does, find the limit.

34. State whether or not the sequence 1 + (–1) n converges as n → ∞, and, if it does, find the limit.


611623

23

+++++

nnnnn


Sequences; Intermediate Forms; Improper Integrals 125

36. State whether or not the sequence nn

lnconverges as n → ∞, and, if it does, find the limit.


2

321

nn

+−


38. State whether or not the sequence n

n1

sin converges as n → ∞, and, if it does, find the limit.


22 −n


40. State whether or not the sequence 12 +n



nsinconverges as n → ∞, and, if it does, find the limit.


6+

+n



n

4

)1( 2+converges as n → ∞, and, if it does, find the limit.

44. State whether or not the sequence !

3n

n



+−

nn



en



2

321

nn

+−


48. State whether or not the sequence

+12

lnn



+142

lnn



+1

ln2

nn


51. State whether or not the sequence defined recursively by a1 = 1, an + 1 = na31

− converges as n → ∞, and, if it

does, find the limit.

52. State whether or not the sequence defined recursively by a1 = 1, an + 1 = (–1)nan converges as n → ∞, and, ifit does, find the limit.


53. State whether or not the sequence defined recursively by a1 = 1, an + 1 = na21

1− converges as n → ∞, and, if

it does, find the limit.

10.4 The Indeterminate Form (0 / 0)

54. Find

−→ 1ln)ln(ln

limx

xex

.

55. Find

x

xx 1

sinlim

2/3−

∞→.

56. Find x

xx sin1lim

0 +→.

57. Find xxxx

x −−

→ tansin

lim0

.

58. Find x

xx

tanlim

0→.

59. Find xx exx

xx

222

sinlim

20 −++−

→.

60. Find x

xx

x

35lim

0

−→

.

61. Find

+−

→ 20

)1ln(lim

xxx

x.

62. Find xx

x 2costan1

lim4/

−→π

.

63. Find xx

e x

x sin

1lim

2

2

0 −−

→.

64. Find x

x

x e

xe3

3

0 1lim

−→.

65. Find 20

16lim

2

2

4 −+−

→ xxx

x.

66. Find 1

lnlim

1 −→ xx

x.

67. Find 43

32lim

2

2

1 −+−+

→ xx

xxx

.

68. Find 1cosh

sinhlim

0 −→ xx

x.


69. Find x

x

x exe−→ 1

lim0

.

70. Find xxx

x cos1tan

lim0 −

−→

.

71. Find axax

ax −

−

→

11

lim .

10.5 The Indeterminate Form (∞∞/∞∞); Other Indeterminate Forms

72. Find 24

124lim

3

3

++−

∞→ xxx

x.

73. Find 3

limxe x

x ∞→.

74. Find )tan(seclim

2

xxx

−−

→π

.

75. Find )/1(csclim0

xxx

−→

.

76. Find 2

tan)1(lim1

xx

x

π−

−→.

77. Find xxx

lnsinlim0+→

.

78. Find xxx

lnlim0+→

.

79. Find xx

xex /2)(lim +

∞→.

80. Find x

x

x cos

2

)(tanlim+

→π

.

81. Find

−

→ xx

121

lim0

.

82. Find x

xx /1

0)3(coslim

→.

83. Find xxx

sinlnlim0+→

.

84. Find xx

xxe /1

0)3(lim +

→.

85. Find

2

2

11lim

x

x x

+

∞→.


86. Find x

xx)(sinlim

0→.

87. Find x

xx /1

0)12(sinlim +

→.

88. Find xx

xxe /32 )2(lim +

∞→.

89. Find x

xx /4

0)(coshlim

→.

10.7 Improper Integrals

90. Evaluate dxx

x

1

1

0 2∫ −.

91. Evaluate dxx

x

tan

sec4/

0

2

∫π

.

92. Evaluate ∫∞

1 3xdx

.

93. Evaluate dxx

11

0∫ .

94. Evaluate dxx

3

14

1 3∫ −.

95. Evaluate dxx

x

)1(

3

0 3/22∫ −.

96. Evaluate dxx

)4(

14

3 3∫ −.

97. Evaluate dxx

18

0 3/1∫ .

98. Evaluate dxxe x 0

2

∫∞

− .

99. Evaluate dxx

1

11

0 2∫ −.

100. Evaluate dxx

12

1 2∫−.

101. Evaluate dxx

2

10

2∫− +.

102. Evaluate dxx

)1(

14

0 3∫ −.


103. Evaluate dxx

1

1∫∞

.

104. Evaluate dxxx

ln12

1∫ .

105. Evaluate dxx

1

0 3/1∫∞

.

106. Evaluate dxx

)1(

12 3∫∞

−.

107. Evaluate dxexex

1)(∫∞

− .

Sequences: Indeterminate Forms; Improper Integrals 133


1. lub: 4 ; glb: 0

2. lub: 5 ; glb: 5−

3. lub: none; glb: none


5. lub: 4 ; glb: 2


7.13

2−nn

8. n

n

2)1( 1+−

9.n

n )1(2 −

10. Bounded: below by 1, above by 2;monotone increasing

11. Bounded: below by 1, above by 2;monotone decreasing

12. Bounded: below by –3/5, above by 2/3;monotone increasing

13. Bounded: below by –1, above by 1;monotone increasing

14. Bounded: below by 3/5, no upper bound;monotone increasing

15. Bounded: below by 0, above by 1/12;monotone decreasing

16. Bounded: below by 4/9, no upper bound;monotone increasing

17. Bounded: below by 3/e, no upper bound;monotone increasing


19. Bounded: below by 0, above by 3/e;monotone decreasing


21. Bounded: below by 2

3−, above by

23

;

not monotone

22. 1, 3/2, 2, 5/2, 3, 7/2; 2

1+=

nan

23. 1, 1, 1, 1, 1, 1; an = 1

24. 1, 0, –6, –24, –68, –166;

43)2(27 2 +++−= nna n

n

25. 1, –1, 3, –5, 11, –21; ])2(1[31 n

na −−=

26. 1, 4, 7, 10, 13, 16; an = 3n – 2

27. 2, 2, 2, 2, 2, 2; an = 2

28. 3, 5, 7, 9, 11, 13; an = 2n + 1

29. converges; 0

30. converges; 0

31. diverges

32. converges; 1

33. converges; 1

34. diverges

35. converges; ½

36. diverges

37. converges; –1/3

38. converges; 1

39. converges; 2

40. converges; ½

41. converges; 0

42. converges; ½

43. converges; 0

44. converges; 0

45. converges; 2/3


46. diverges

47. converges; –1/3

48. converges; ln 2

49. converges; 0

50. diverges

51. converges; 0

52. diverges

53. converges; 2/3

54. 1

55. 0

56. 0

57. –½

58. 1

59. –½

60. ln 5/3

61. ½

62. 1

63. –2

64. –1/3

65. 8/9

66. 1

67. 4/5

68. +∞

69. –1

70. 0

71.2

1

a

−

72. 1

73. +∞

74. 0

75. 0

76. −2/π

77. 0

78. 0

79. e2

80. 1

81. +∞

82. 1

83. 0

84. e4

85. e

86. 1

87. e2

88. e3

89. 1

90. 1

91. 2

92. ½

93. 2

94. )41(23 3−

95. 9/2

96. divergent

97. 6

98. ½

99. π/2

100. divergent

101. divergent

102. divergent

103. divergent

Sequences: Indeterminate Forms; Improper Integrals 135

104. divergent

105. divergent

106. ½

107.ee

11−

133

CHAPTER 11

Infinite Series

11.1 Infinite Series

1. Evaluate ∑=

+3

0

)24(k

k .

2. Evaluate ∑=

−4

1

)25(k

k .

3. Evaluate ∑=

+3

0

)13(k

k .

4. Evaluate ∑=

−3

0

3)1(k

kk .

5. Evaluate ∑=

++−4

1

113)1(k

kk .

6. Evaluate ∑=

5

3

2

31

k

k

.

7. Evaluate ∑=

4

0

2

kke

.

8. Evaluate ∑=

+

3

012

3

kk

k

.

9. Express 20 + 21 + … + 210 in sigma notation.

10. Express 2x2 – 22 x3 + … –26 x7 in sigma notation.

11. Express 723

2

2

3

31

. . .33

eee+++ in sigma notation.

12. Express )10(9

1 . . .

)5(41

)4(31

+++ in sigma notation.

13. Determine whether ∑∞

= +1 )1(21

k kk

converges or diverges. If it converges, find the sum.


=

−

2 5)1(

k

k

k



=+

+

01

2

43

k

k

k




=0 103

k

k converges or diverges. If it converges, find the sum.


=1

3

kke



= ++1 )4)(3(1

k kk



=1 52

k

k


20. Determine whether the series given by ...125

8254

52

10

+−+−=∑∞

=kku converges or diverges. If it converges,

find the sum.


=

+

1

1k

4(-1)

k

kconverges or diverges. If it converges, find the sum.


=

3 21

k

k



= +−1 )12)(12(2

k kk



=

+

−

1

1

73

k

k



=

−

1

14k

k converges or diverges. If it converges, find the sum.


=

+

−

1

1

32

k

k



= ++1 )2)(1(1

k kk


28. Write the decimal fraction 0.21212121 … as an infinite series and express the sum as the quotient of twointegers.

29. Write the decimal fraction 0.251251251251 … as an infinite series and express the sum as the quotient oftwo integers.

30. Write the decimal fraction 0.315315315315 … as an infinite series and express the sum as the quotient oftwo integers.

31. Find a series expansion for 1,1 3

<−

xx

x.

Infinite Series 135

32. Find a series expansion for 3 ,3

2<

+x

x.

11.2 The Integral Test; Comparison Tests


= +1 233

k k

converges or diverges. Justify your answer.


=1

1

k kk converges or diverges. Justify your answer.


= +1 431

k k



= +1

2

12

k kk



= +1 2 12

k k

k converges or diverges. Justify your answer.


=1

3

kke



= +1 141

k k



=23)(ln

1



= +13)32(

1

k k converges or diverges. Justify your answer.


= ++

1 )2(1

k kk


43. Find the sum of ∑∞

=

−

0 1006

105

k

kk.


= +1 3)1(

1



= +141

2

k kk



= −1 2 1

1




=1

k

kek



=12cosh

1



=1

ln

k kk


50. Which of the following statements about series is true?(a) If 0lim =

∞→ kk

u , then ∑ uk converges.

(b) If 0lim ≠∞→ k

ku , then ∑ uk diverges.

(c) If ∑ uk diverges, then 0lim ≠∞→ k

ku .

(d) ∑ uk converges if and only if 0lim =∞→ k

ku .

(e) None of the preceding.


= +12 1

k k



= +122

2

)12(

k kk



= ++1 )4)(3(1

k kk



= −+1 )4)(3(1

k kk



= +1 231

k k



= +1 231

k



=1

ln

k kk



= ++1

2

)4)(2(

k kkk



= ++

13 1

1

k kk



= +1 1

1



= ++

15 72

27

k kk


Infinite Series 137


= −1 231

k

kconverges or diverges. Justify your answer.


= −−−

13 75

34

k kkk



= −12 11

k k



= −2 ln1

k kk



= +12/3 131

k k



= +++

1

2

)2)(1(3

k kkk


68. Which of the following statements about ∑∞

=2 ln1

k kk

is true?

(a) converges because 0ln1

lim =∞→ kkk

.

(b) converges because kkk1

ln1

< .

(c) converges by ratio test.(d) diverges by ratio test.(e) diverges by integral test.

11.3 The Root Test; The Ratio Test


=1

2

k

kek

converges or diverges. Justify your answer by citing a relevant test.


=1 2

kk

k converges or diverges. Justify your answer by citing a relevant test.


=1410!

k

k



=−+1 32

1

kk



=

+1 1002

k

k

kk



=

+1 123

k

k

kk




=

−+

5 9223

k

k

kk



=1 2

kk

k! converges or diverges. Justify your answer by citing a relevant test.


=1

2

2

kk



= +1 921

k k



=1 !

k

k

ke



=1 !10

k

k



=1

3

3

kk



=12

!

k kk



=

1

ln

k

k

kk



=1 !

k

k

kk



=1

2

)!2(3

k

k


11.4 Absolute and Conditional Convergence; Alternating Series


= +

−

1

)1(

k

k

kk converges absolutely, converges conditionally, or diverges. Justify your

answer by citing a relevant test.


=

+

++−

1

1

)1(2)1(

k

k

kkk

converges absolutely, converges conditionally, or diverges. Justify your



=

+

+−

1

1

132)1(

k

k

kk



Infinite Series 139


=

−

2

ln)1(

k

k

kk




=

+−

12

1 2)1(

k

kk

k converges absolutely, converges conditionally, or diverges. Justify your



=

−

1

)1(

k

k




=

+−

1

1

3)1(

k

k

k




= +−

122

2

)12()1(

k

k

kk




=

−

1

3

3)1(

k

k

k k converges absolutely, converges conditionally, or diverges. Justify your



=

−

1

)1(

k

k

kk converges absolutely, converges conditionally, or diverges. Justify your



= +−

1 2)1(

k

k

kk




=

−

1

2)1(

kk

k

ek




=

+

+−−

13

1

1)12()1(

k

k

kk

converges absolutely, converges conditionally, or diverges. Justify

your answer by citing a relevant test.


=

+

+−

1

1

43)1(

k

k




=

+

+−

1

21

12)1(

k

k

kk





= +−

1 )!32(!)1(

k

k

kk




= +−

12 1

2)1(

k

k

kk




=

−

1

2)1(

kk

k

ek



104. Estimate the error if the partial sum S20 is used to approximate the sum of the series ∑∞

= +

−

0 13

)1(

k

k

k.

105. Find the smallest integer N such that SN will approximate the sum of the series ∑∞

= +−

02 2

)1(

k

k

k within 0.01.

11.5 Taylor Polynomials in x; Taylor Series in x

106. Find the Taylor polynomial P4(x) for ln (1 + x).

107. Find the Taylor polynomial P4(x) for cos x sin x.

108. Find the Taylor polynomial P5(x) for ln (1 + x)–2 .

109. Find the Taylor polynomial P5(x) for sin–1 x.

110. Find the Taylor polynomial P5(x) for cosh x.

111. Determine P0(x), P1(x), P2(x), P3(x) for 2x3 + x2 – 2x + 5.

112. Determine P0(x), P1(x), P2(x), P3(x) for (x + 2)3.

113. Use Taylor polynomials to estimate sin 1.3 within 0.01.

114. Use Taylor polynomials to estimate ln 2.4 within 0.01.

115. Use Taylor polynomials to estimate e0.2 within 0.01.

116. Find the Lagrange form of the remainder R3 for the function xxf −= 2)( .

117. Find the Lagrange form of the remainder Rn+1 for the function f(x) = e–3x.

11.6 Taylor Polynomials and Taylor Series in x - a

118. Expand g(x) = 2x4 – x3 + 3x2 – x + 1 in powers of x – 2 and specify the values of x for which the expansion isvalid.

119. Expand g(x) = 1/x in powers of x – 3 and specify the values of x for which the expansion is valid.

120. Expand g(x) = ln (x – 1) in powers of x – 2 and specify the values of x for which the expansion is valid.

Infinite Series 141

121. Expand g(x) = e2x in powers of x – 3 and specify the values of x for which the expansion is valid.

122. Expand g(x) = cos x in powers of x – π/3 and specify the values of x for which the expansion is valid.

123. Expand g(x) = ex sin πx in powers of x – 1 and specify the values of x for which the expansion is valid.

124. Expand g(x) = tan x in powers of x – π/3.

125. Expand g(x) = ln x in powers of x – 2.

126. Expand 2

)( xexg = in powers of x – 1.

11.7 Power Series

127. Find the interval of convergence for ∑∞

=

−1

)1(2

k

kk

xk

.


=

−−

12

)2(2)1(

k

kkk

xk

.


= +−−

1 1)2()1(

k

kk

kx

.


=

−1

)1( k

kx

kk

.


=1

2

k

kk

k

x.


=1 32

k

k

kk x.


= +−

1 12)1(

k

k

kx.


= +−

1 1)3(

k

k

kx

.


= +−

1 )!1()3(

k

k

kx

.


=1 )!2(!

k

kxkk

.

137. Find the interval of convergence for kk

k

xk )2(3 1

−∑∞

=

.


=

−

1

)1(

kk

kk

ex

.


139. Find the interval of convergence for k

k

x∑∞

=

−

0 31

.


=

+

+1

1

)1(2

kk

k

kx

.


=

−

12

)3(2

k

kk

kx

. For which values of x is the convergence absolute?


=

−−

133

)1()1(

kk

kk

k

x.


=

+

1

2

k

kxkk

.


=

+

123)1(

k

k

k

kx

.


=0 2!

k

k

kxk.

11.8 Differentiation and Integration of Power Series

146. Expand 4)1(

1

x− in powers of x, basing your calculation on the geometric series

......1)1(

1 2 +++++=−

nxxxx

147. Expand ln (1 – 2x2) in powers of x, basing your calculation on the geometric series

......1)1(

1 2 +++++=−

nxxxx

148. Find f 7 (0) for f(x) = x sin x2 .

149. Expand ex sin x in powers of x.

150. Expand e–x cos x in powers of x.

151. Expand x

e x

−1 in powers of x.

152. Expand x

x

+1

cos in powers of x.

153. Expand coth x in powers of x.

154. Expand sec2 x in powers of x.

155. Estimate ∫1

0

2cos dxx within 0.001.

Infinite Series 143

156. Estimate ∫1

0

sindx

xx

within 0.001.

157. Estimate ∫2/1

0

3cos dxx within 0.001.

158. Use a series to show 1sin

lim0

=→ x

xx

.

159. Use a series to show 1tan

lim0

=→ x

xx

by first obtaining a series for tan x.

160. Use a series to show 1sin

1lim

0=

−→ x

e x

x.

11.9 The Binomial Series

161. Expand 31 x+ in powers of x up to x6.

162. Expand 31 x− in powers of x up to x6.

163. Expand x−1

1 in powers of x up to x4.

164. Expand 3 1

1

x− in powers of x up to x4.

165. Expand 5 1 x− in powers of x up to x4.

166. Expand 5 1

1

x+ in powers of x up to x4.

167. Estimate 102 by using the first three terms of a binomial expansion, rounding off your answer to fourdecimal places.

168. Estimate 5 28 by using the first three terms of a binomial expansion, rounding off your answer to fourdecimal places.



1. 32

2. 42

3. 44

4. –20

5. –180

6. 91/310

7. 1426.3)1(2 234

4≈++++ eeee

e

8. 65/16

9. ∑=

10

0

2k

k

10. ∑=

++−6

1

112)1(k

kkk x

11. ∑=

+−−5

0

)2(33k

kk e

12. ∑= +

9

3 )1(

1

k kk

13. 1/2

14. 1/30

15. 9

16. 10/3

17.1

3

−e

18. 1/4

19. diverges

20. 5/7

21. 1/5

22. 1/4

23. 1

24. 9/70

25. diverges

26. 4/15

27. 1/2

28.33

7

10

21

10

21

10

21642

=+++ L

29.999

251

10

251

10

251

10

251963

=+++ L

30.111

35

10

315

10

315

10

315963

=+++ L

31. ∑∞

=

+

0

13

k

kx

32. ∑∞

=+

−0

13

)1(2

kk

kk x

33. diverges

34. converges

35. diverges

36. diverges

37. diverges

38. converges

39. diverges

40. converges

41. converges

42. diverges

43.99

50−

44. converges

45. converges

46. converges

47. converges

48. converges

Infinite Series 145

49. diverges

50. b

51. converges

52. converges

53. converges

54. converges

55. diverges

56. converges

57. diverges

58. diverges

59. converges

60. diverges

61. converges

62. converges

63. converges

64. converges

65. diverges

66. converges

67. diverges

68. e

69. converges by ratio test


71. diverges by ratio test

72. diverges since 0lim ≠∞→ k

ku

73. converges by root test

74. diverges by root test

75. diverges by root test



78. diverges by integral test





83. converges by root test



86. converges conditionally by limit comparison

test with ∑∞

=1

1

k k and by alternating series test

87. converges conditionally by limit comparison

test with ∑∞

=1

1

k k and by alternating series test

88. converges absolutely by comparison with

geometric series ∑∞

=

1 3

2

k

k

89. converges conditionally: converges by

alternating series test but ∑∞

=1

ln

k k

k diverges by

integral test


91. converges conditionally: converges by

alternating series test but ∑∞

=1

1

k k divergent p

series

92. converges absolutely by geometric series with

13

1 <=r

93. converges absolutely by comparison with p

series ∑∞

=12

141

k k

94. converges absolutely by ratio test

95. converges absolutely: convergent p series


96. divergent since 012

lim ≠=+∞→ k

kk

97. converges absolutely by geometric series with

11 <=e

r

98. converges absolutely by limit comparison test

with convergent p series, ∑∞

=12

1

k k

99. converges conditionally: converges byalternating series test but diverges by integraltest

100. diverges by integral test


102. converges absolutely by limit comparison test

with converging p series ∑∞

=12/3

12

k k


104. 1/8

105. N = 9

106.432

432 xxxx ++−

107. 3

3

2xx −

108. 1 – 2x + 3x2 – 4x3 + 5x4 – 6x5

109. 53

40

3

6x

xx +−

110.!4!2

42 xxx ++

111. P0(x) = 5; P1(x) = 5 – 2x; P2(x) = 5 – 2x + x2;P3(x) = 5 – 2x + x2 + 2x3

112. P0(x) = 8; P1(x) = 8 + 12x; P2(x) = 8 + 12x + 6x2;P3(x) = 8 + 12x + 6x2 + x3

113. 0.96

114. 0.88

115. 1.22

116.2/5

3

3)2(16

)(c

xxR

−−=

117. 131

1 )!1()3(

)( +−+

+ +−= n

cn

n xn

exR

118. 35 + 63(x – 2) + 45(x – 2)2 + 15(x – 2)3 + 2(x – 2)4

valid for –∞ < x < ∞

119. ∑∞

=+ −−

01

)3(3

1)1(

k

kk

k x ; valid for 0 < x < 6

120. ∑∞

=

+ −−1

1 )2(1

)1(k

kk xk

; valid for 1 < x < 3

121. ∑∞

=

−0

6 )3(!

2

k

kk

xk

e ; valid for –∞ < x < ∞

122. ∑ ∑∞

=

∞

=

+

+

−

−−

−

−0 0

122

)!12(3

)1(2

3

!23

)1(2

1

k k

k

k

k

k

k

x

k

xππ


123.

−−−−−−−+−− L!4

)1)(1(4

!3

)1)(3()1(

!2

2)1(

42322 xx

xxeπππ


124.

L+

−+

−+

−+

−+

4

32

33

3

44

33

40

334

343

π

πππ

x

xxx

125. ∑∞

=

+ −−+1

1 )2(!2

1)1(2ln

k

kk

k xk

126.

+−+−+−+−+ L432 )1(

6

19)1(

3

10)1(3)1(21 xxxxe

127. (–1, 3 )

128. [3/2, 5/2)

129. (1, 3]

130. (–1, 1]

Infinite Series 147

131. [–½, ½)

132. (–3/2, 3/2)

133. [0, 2)

134. [2, 4)

135. (–∞, +∞)

136. (–∞, +∞)

137. (5/3, 7/3)

138. (–e, e)

139. (–2, 4)

140. [–2, 2)

141. [5/2, 7/2]

142. (–2, 4]

143. (–∞, –1) ∪ (1, +∞)

144. [–4, 2]

145. x = 0

146. ∑∞

=

−++1

1)2)(1(6

1

k

kxkkk

147. ∑∞

=

−1

2 )2(

k

k

k

x

148. –840

149. L+−++303

532 xx

xx

150. L+−+−63

143 xx

x

151. L+++++ 432

2465

38

25

21 xxxx

152. L++−−− 432

38449

161

81

21

1 xxxx

153. L++−+ 53

9452

451

311

xxxx

154. L+++++ 8642

31562

4517

32

1 xxxx

155. 0.905

156. 0.946

157. 0.499

158. 1!7!5!3

1limsin

lim642

00=

+−+−=

→→L

xxx

x

xxx

159. 1315

17

15

2

31lim

tanlim

642

00=

+−+−=

→→L

xxx

x

xxx

160.

1

!7!5!31

!4!3!21

lim

!7!5!3

!4!3!2limsin

1lim

642

32

0

753

432

00

=+−+−

++++

=+−+−

++++=−

→

→→

L

L

L

L

xxx

xxx

xxxx

xxxx

x

e

x

x

x

x

161.82

163 xx −+

162.82

163 xx −−

163. 432

128

35

16

5

8

3

21 xxx

x ++++

164. 432

243

35

81

14

9

2

3

11 xxxx ++++

165. 432

62521

1256

252

51

1 xxxx −+−+

166. 432

62544

12511

253

51

1 xxxx +−+−

167. 10.0995

168. 1.9475

149

CHAPTER 12

Vectors

12.1 Cartesian Space Coordinates

1. Plot points A(2, 7, 8) and B(3, 9, 7) on a right-handed coordinate system. Then calculate the length of the linesegment AB and find the midpoint.

2. Plot points A(–3, –2, 4) and B(9, 7, 2) on a right-handed coordinate system. Then calculate the length of theline segment AB and find the midpoint.

3. Plot points A(–1, 1, 1) and B(–1, 4, 4) on a right-handed coordinate system. Then calculate the length of theline segment AB and find the midpoint.

4. Find an equation for the plane through (2, –1, –2) that is parallel to the xy-plane.

5. Find an equation for the plane through (–3, 2, –1) that is perpendicular to the z-axis.

6. Find an equation for the plane through (–2, –4, 3) that is parallel to the yz-plane.

7. Find an equation for the sphere centered at (2, 1, 3) with radius 4.

8. Find an equation for the sphere that is centered at (–4, 0, 6) and passes through (2, 2, 3).

9. Find an equation for the sphere that is centered at (5, 1, –4) and passes through (3, –5, –1).

10. Find an equation for the sphere that has the line segment joining (4, 3, 0) and (2, 4, –4) as a diameter.

11. Find an equation for the sphere that is centered at (–2, 1, 4) and is tangent to the plane x = 2.

12. The points P(a, b, c) and Q(3, 2, –1) are symmetric about the xy-plane. Find a, b, c.

13. The points P(a, b, c) and Q(–3, 2, –1) are symmetric about the yz-plane. Find a, b, c.

14. The points P(a, b, c) and Q(–3, –2, 1) are symmetric about the xz-plane. Find a, b, c.

15. The points P(a, b, c) and Q(1, 2, –4) are symmetric about the z-axis. Find a, b, c.

16. The points P(a, b, c) and Q(2, –1, 3) are symmetric about the plane x = 2. Find a, b, c.

17. The points P(a, b, c) and Q(–2, 1, –3) are symmetric about the plane y = –3. Find a, b, c.

18. The points P(a, b, c) and Q(4, 2, 2) are symmetric about the point (0, 2, 1). Find a, b, c.

12.3 Vectors

19. Simplify (3 i – j + 2 k) – 2(i – 2 j + k).

20. Simplify 2(i – 3 k) – 3(2 i + j – k).

21. Calculate the norm of the vector 4 i – 3 j.

22. Calculate the norm of the vector 3 i – j + k.


23. Calculate the norm of 2(2 i – j + k) – (–2 i – j).

24. Let a = (–2, 3, 5), b = (3, 5, –2), c = (2, 1, 2), d = (–3, 0, –1). Express a – 2 b + 2 c + 3 d as a linearcombination of i, j, k.

25. Given that a = (1, 2, 5) and b = (–1, 0, 3), calculate(a) ||a||(b) ||b||(c) ||2a – 3b||(d) ||3a + b||

26. Find α given that 3 i + 2 j and –2 i + α j have the same length.

27. Find the unit vector in the direction of 2 i – j + 2 k.

28. Given that a = 3 i – 5 k and b = – i + 2 j + 3 k, find the unit vector in the direction of a – 2 b.

29. Given that a = 2 i + 9 j + k and b = i + 7 j + 8 k, find the unit vector in the direction of 2 a + b.

30. Find the vector of norm 2 in the direction of 3 i + 4 j + 2 k.

31. Find the vector of norm 2 parallel to 5 i – 12 j + k.

12.4 The Dot Product

32. Simplify (2 a • 2 b) + a • (a + 2 b).

33. Simplify (a – 2 b) • c + b • (a – c) – 2 a • (b – 3c).

34. Taking a = i + 2 j, b = 2 i – j + 3 k, c = –2 j + k, calculate:(a) the three dot products a • b, a • c, b • c(b) the cosines of the angles between these vectors.(c) the component of a (i) in the b direction, (ii) in the c direction(d) the projection of a (i) in the b direction, (ii) in the c direction

35. Taking a = i – 2 j + 4 k and b = 3 i – 2 j + k, c = –2 j – k, calculate:(e) the three dot products a • b, a • c, b • c(f) the cosines of the angles between these vectors.(g) the component of a (i) in the b direction, (ii) in the c direction(h) the projection of a (i) in the b direction, (ii) in the c direction

36. Find the angle between the vectors 2 i + 2 j – k and i + 2 j + k.

37. Find the angle between the vectors 2 i + 3 j + k and 3 i + j + 8 k.

38. Find the direction angles of the vector 2 i – j + k.

39. Find the direction angles of the vector 3 i – 2 k.

40. Find the unit vectors u that are perpendicular to both i + 2 j + k and i – 2 j + 2 k.

41. Find the cosine of the angle between u = 2i – 2 j + k and v = –i + 4 j + 2 k.

42. A 100 Newton force is applied along a rope making a 30° angle with the horizontal to pull a box a distance of5 meters along the ground. What is the work done?

Vectors 151

43. Find the work done by the force F = 3 i + 5 j + 2 k in moving an object from the point P(2, 0, 2) to thepoint Q(1, 4, 5).

12.5 The Cross Product

44. Calculate (i – k) × (j + k).

45. Calculate (j × k) • j.

46. Calculate (k × j) × i.

47. Calculate (i – 4 j – 2 k) × (2 i + j).

48. Calculate [(i + 2 j – k) × (i + j + k)] × (i + 2 j + 2 k).

49. Calculate (3 i – 4 j – 4 k) × [(2 i – 6 j) × (i – 2 j + 2 k)].

50. Calculate (i – j) • [(3 i – 4 j) × (i – 2 j + 2 k)].

51. Calculate (2 i + 3 j – 4 k) • [(–i + j + 2 k) × (i – j + k)].

52. Calculate (3 i + 2 k) × [(2 i + 2 j – k) × (i – 2 j + 3 k)].

53. Use a cross product to find the area of triangle PQR, P(1, 2, 3), Q(–1, 0, 1), R(2, –2, –1).

54. Use a cross product to find the area of triangle PQR, P(1, 1, 1), Q(2, –1, 3), R(2, 3, –4).

55. Find the volume of the parallelepiped with edges determined by 3 i – 4 j – k, i – 2 j + 2 k, i + j.

56. Find the volume of the parallelepiped with vertices A(0, 0, 0), B(1, –1, 1), C(2, 1, –2) and D(–1, 2, –1).

57. Find the volume of the parallelepiped with edges determined by i + 2 k, 4 i + 6 j + 2 k, 3 i + 3 j – 6 k.

58. Find the volume of the parallelepiped with edges determined by 2 i + k, 3 i + 2 j + 5 k, –i + 2 k.

59. Find the area of the triangle with vertices P(1, –2, 3), Q(2, 4, 1), R(2, 0, 1).

60. Find the area of the triangle with vertices P(1, 2, 1), Q(2, 4, 3), R(5, –1, 4).

12.6 Lines

61. Which of the points P(–1, 3, –1), Q(3, 2, –1), R(3, 0, –2) lie on the line l: r(t) = (2 i + j) + t(3 i – 2 j + k)?

62. Determine whether the lines are parallel.l1: r1(t) = (i – 2 k) + t(i – 2 j – 3k)l2: r2(u) = (3 i + 2 j – 3 k) + u(i + 2 j – k)

63. Find a vector parametrization for the line that passes through P(2, 3, 3) and is parallel to the liner(t) = (2 i – j) + t k.

64. Find a vector parametrization for the line that passes through the origin and P(3, 1, 8).

65. Find a vector parametrization for the line that passes through P(4, 0, 5) and Q(2, 3, 1).

66. Find a vector parametrization for the line that passes through P(3, 3, 1) and Q(4, 0, 2).


67. Find a set of scalar parametric equations for the line that passes through P(1, 4, 6) and Q(2, –1, 3).

68. Find a set of scalar parametric equations for the line that passes through P(–3, –1, 0) and Q(–1, 2, 1).

69. Find a set of scalar parametric equations for the line that passes through P(4, –2, –1) and is perpendicular tothe xy-plane.

70. Find a set of scalar parametric equations for the line that passes through P(–1, 2, –3) and is perpendicular tothe xz-plane.

71. Give a vector parametrization for the line that passes through P(1, –2, 3) and is parallel to the line 3(x – 2) =2(y + 2) = 5z.

72. Find the point where l1 and l2 intersect and give the angle of intersection:l1: x1(t) = 3 – t, y1(t) = 5 + 3t, z1(t) = –1 – 4tl2: x2(u) = 8 + 2u, y2(u) = –6 – 4u, z2(u) = 5 + u.

73. Where does the line that passes through (1, 4, 2) and is parallel to 3 i + 2 j – 2 k intersect the xy-plane?

74. Where does the line that passes through (3, 5, –1) and is parallel to i – j + k intersect the xz-plane?

75. Find scalar parametric equations for all lines that are perpendicular to the line x(t) = 5 + 2t, y(t) = –5t,z(t) = –t and intersect the line at the point P(–3, 2, 2).

76. Find the distance from P(4, –3, 1) to the line through the origin parallel to 4 i – 3 j + k.

77. Find the distance from P(3, –4, 1) to the line r(t) = 2 i – j + t(i – 2 j + 2 k).

78. Find the standard vector parametrization for the line through P(–1, 2, 4) parallel to i – 2 j + 3 k.

79. Find the cosine of the angle between the lines x1(t) = 2 + t, y1(t) = 3 + t, z(t) = –1 + 2t and x2(u) = 2 + 2u,y2(u) = 3 – u, z2(u) = –1 + 3u.

80. Find the cosine of the angle between the line x(t) = 2t, y(t) = 3t, z(t) = t and the y-axis.

12.7 Planes

81. Which of the points P(–2, 3, –1), Q(2, 3, 4), R(3, 4, 1) lie on the plane 2(x – 2) + 3(y – 2) – 2(z + 3) = 0?

82. Which of the points P(4, 1, 0), Q(2, 1, –3), R(4, 1, –2), S(0, 2, –1) lie on the plane N • (r – r0) = 0if N = 2 i – 4 j + k and r0 = i + 2 j – 3 k?

83. Write an equation for the plane that passes through the point P(2, 1, 3) and is perpendicular to 3 i + j – 5 k.

84. Write an equation for the plane that passes through the point P(5, –2, –1) and is perpendicular to the plane3x – y + 6z + 8 = 0.

85. Find the unit normals for the plane 3x + 3y – 5z – 6 = 0.

86. Write the equation of the plane 5x – 3y – 2z – 1 = 0 in intercept form.

87. Where does the plane 4x + 3y – 2z + 4 = 0 intersect the coordinate axes?

88. Find the angle between the planes 3(x – 1) – 2(y – 5) + 2(z + 1) = 0 and 2x + 5(y – 1) + (z + 4) = 0.

89. Find the angle between the planes x – 2y + 3z = 5 and 2x + y – z = 7.

Vectors 153

90. Determine whether or not the vectors are coplanar: i + 2 j – 3 k, i – 2 j, 4 i + j – 2 k.

91. Find an equation in x, y, z for the plane that passes through the points P1(1, 1, 1), P2(2, 4, 3), P3(–1, –2, –1).

92. Find a set of scalar parametric equations for the line formed by the two intersecting planes:P1: 3x – 2y + z = 0, P2: 8x + 2y + z – 11 = 0.

93. Let l be the line determined by P1, P2, and let p be the plane determined by Q1, Q2, Q3. Where, if anywhere,does l intersect p?P1(2, 5, –2), P2(1, –2, 2); Q1(2, 1, –4), Q2(1, 2, 3), Q3(–1, 2, 1).

94. Find an equation in x, y, z for the plane that passes through (1, 2, –3) and is perpendicular to the linex(t) = 1 + 2t, y(t) = 2 + t, z(t) = –3 – 5t.

95. Find an equation in x, y, z for the plane that passes through (2, 1, 5) and the line x(t) = –1 + 3t, y(t) = –2,z(t) = 2 + 4t.

96. Find a vector equation for the line through (1, 1, 1) that is parallel to the line of intersection of the planes3x – 4y + 2z – 2 = 0 and 4x – 3y – z – 5 = 0.

97. Find parametric equations for the line through (2, 0, –3) that is parallel to the line of intersection of the planesx + 2y + 3z + 4 = 0 and 2x – y – z – 5 = 0.

98. Find an equation for the plane that passes through (3, 0, 1) and is perpendicular to the line x(t) = 2t,y(t) = 1 – t, z(t) = 4 – 3t.

99. Find an equation for the plane that contains the point (–2, 1, 1) and the liner(t) = 2 i + j + k + t(–i + 4 j + 4 k).

100. Find an equation for the plane that contains P1(1, 1, 1) and P2(–1, 2, 1) and is parallel to the line ofintersection of the planes 2x + y – z – 4 = 0 and 3x – y + z – 2 = 0.

101. Find an equation for the plane that contains P1(3, 1, 2) and P2(–1, 2, –1) and is parallel to the line ofintersection of the planes 2x – y – z – 2 = 0 and 3x + 2y – 2z – 4 = 0.

102. Sketch the graph of 20x + 12y + 15z – 60 = 0.

103. Find the equation of the plane pictured below.



1.

215

,8,25

;6

2.

3,

25

,3;229

3.

−

25

,25

,1;23

4. z = –2

5. z = –1

6. x = –2

7. (x – 2)2 + (y – 1)2 + (z – 3)2 = 16

8. (x + 4)2 + y2 + (z – 6)2 = 49

9. (x – 5)2 + (y – 1)2 + (z + 4)2 = 49

10. (x – 3)2 + (y – 7/2)2 + (z + 2)2 = 21/4

11. (x + 2)2 + (y – 1)2 + (z – 4)2 = 16

12. (3, 2, 1)

13. (3, 2, –1)

14. (–3, 2, 1)

15. (–1, –2, –4)

16. (2, –1, 3)

17. (–2, –7, –3)

18. (–4, 2, 1)

19. i + 3j

20. –4i – 3j – 3k

21. 5

22. 11

23. 73

24. –13i – 5j + 10k

25. (a) 30 (b) 10 (c) 42 (d) 912

26. ±3

27.32

i + 31

j + 32

k

28.162

5i –

162

4j –

162

11k

29.30

1i +

30

5j +

30

2k

30.29

6i +

29

8j +

29

4k

31.170

10i –

170

24j +

170

2k

32. a • a + 6a • b

33. –a • b + 7a • c – 3b • c

34. (a) 0, –4, 5(b) cos (a, b) = 0; cos (a, c) = –4/5

cos (b, c) = 1470

(c) (i) 0; (ii) 5

4−

(d) (i) 0; (ii) 58

j + 54

k

35. (a) 11, –8, –5

(b) cos (a, b) = 1421

11; cos (a, c) =

521

8−

cos (b, c) = 514

5−

(c) (i) 14

11; (ii)

5

8−

(d) (i)

−

141

,1422

,1433

; (ii)

−

58

,516

,0

36. ≈ 47.12° or 0.8225 radians

37. ≈ 58.12° or 1.014 radians

38. π/4, 2π/3, π/3

39. π/6, 0, π/3

Vectors 155

40.53

6i –

53

1j –

53

4k or

53

6−i +

53

1j +

53

4k

41.213

8−

42. 3250 Joules

43. W = 23

44. i – j + k

45. 0

46. 0

47. 2i – 4j + 9k

48. –2i – 7j + 8k

49. –12i + 6j – 54k

50. –2

51. 15

52. 14i + 26j – 21k

53. 25

54. 10121

55. 17

56. 4

57. 54

58. 10

59. 52

60.2290

61. P(–1, 3, –1)

62. No

63. r(t) = 2i + 3j + 3k + tk

64. r(t) = t(3i + j + 8k)

65. r(t) = 4i + 5k + t(–2i + 3j – 4k)

66. r(t) = 3i + 3j + k + t(i – 3j + k)

67. x(t) = 1 + t; y(t) = 4 – 5t; z(t) = 6 – 3t

68. x(t) = –3 + 2t; y(t) = –1 + 3t; z(t) = t

69. x(t) = 4; y(t) = –2; z(t) = –1 + t

70. x(t) = –1; y(t) = 2 + t; z(t) = –3

71. r(t) = i – 2j + 3k + t(10i + 15j + 6k)

72. (4, 2, 3);

−= −

273

3cos 1θ

73. (4, 6, 0)

74. (8, 0, 4)

75. x(t) = −3 + tay(t) = 2 + tbz(t) = 2 + t(2a – 5b); a, b ∈ ℜℜ

76. 0

77. 2

78. r(t) = –i + 2j + 4k + t(i – 2j + 3k)

79.212

7

80.14

3

81. R

82. S

83. 3x + y – 5z + 8 = 0

84. 3x + y + 6z – 7 = 0

85.

−−

−

43

5,

43

3,

43

3,

43

5,

43

3,

43

3

86. 12/13/15/1

=−

+−

+zyx

87. x = –1, y = –4/3, z = 2


88. °≈− 92.84510

2cos 1

89. °≈− 89.70212

3cos 1

90. No

91. 2y – 3z + 1 = 0

92. x = 1 – 4t, y = 3/2 + 5t, z = 22t

93.

−

612

,6195

,6192

94. 2x + y – 5z – 19 = 0

95. 10x – 3y – 9z + 28 = 0

96. i + j + k + t(10i + 11j + 7k)

97. x = 2 + t, y = 7t, z = –3 – 5t

98. 2x – y – 3z – 3 = 0

99. y – z = 0

100. x + 2y – 2z – 1 = 0

101. 5x + 8y – 4z – 15 = 0

102.

103. 3x + z = 3

157

CHAPTER 13

Vector Calculus

13.1 Vector Functions

1. Differentiate f(t) = (3 – 4t) i + 5t j + (2 – 5t) k.

2. Differentiate f(t) = (1 – t) –1/2 i + 221 t+ j – t3/2 k.

3. Differentiate f(t) = e–t i + ln (2t2 – t) j + sin–1 t k.

4. Differentiate f(t) = sin2 t2 i + cos 2t j + t2 e–2t k.

5. Calculate ∫3

1f(t) dt for f(t) = (1 + 2t2 ) i – 5t k.

6. Calculate ∫2/

4/

π

πr(t) dt for r(t) = t sin 2t2 i + sin 3t j + e2t k.

7. Calculate ∫1

0g(t) dt for g(t) =

22tte− i + 342 tet j –

32

cost

k.

8. Find 1

lim→t

f(t) if it exists. f(t) = ln t i – 3 t j + e4t k.

9. Find 0

lim→t

r(t) if it exists. r(t) = t cos t i – e–t j + te

tt 123 2 +− k.

10. Find a vector-valued function f that traces out the curve 16x2 + 4y2 = 64 in (a) a counterclockwise directionand (b) a clockwise direction.

11. Find a vector-valued function f that traces out the curve y = 2(x – 1)2 in a direction from (a) left to right and(b) right to left.

12. Find a vector-valued function f that traces out the directed line segment from (2, –1, 3) to (1 , 4, –2).

13. Find f(t) given that f ′ (t) = (t2 + 1) i + 2t2(1 + t3)–1 j + 2/2 3tet k and f(0) = 2 i – 3 j + ½ k.

14. Find f(t) given that f ′ (t) = 2t2 i – 3(t + 1) j + et(t + 1) k and f(0) = i – 2 j + 2 k.

15. Sketch the curve represented by r(t) = 3t i + t3 j and indicate the orientation.

16. Sketch the curve represented by r(t) = 5 sin t i + 3 cos t j and indicate the orientation.

13.2 Differentiation Formulas

17. Find f ′ (t) and f ′ ′ (t) for f(t) = et i + e2t j + k.

18. Find f ′ (t) and f ′ ′ (t) for f(t) = sin t i + sinh 2t j + sech 2t k.

19. Find f ′ (t) and f ′ ′ (t) for f(t) = tt 22 + i + ln tt 22 + j + t k.


20. Find f ′ (t) and f ′ ′ (t) for f(t) = t i + ln cos 2t j + ln sin 2t k.

21. Find f ′ (t) and f ′ ′ (t) for f(t) = (cos t + t sin t) i + (sin t – t cos t) j + t2 k.

22. Find f ′ (t) and f ′ ′ (t) for f(t) = (t2 i + cos t j ) × (et j + sin t k).

23. Find f ′ (t) and f ′ ′ (t) for f(t) = [( t i – t–3/2 j) • (sin t j – t k)] j.

24. Find the derivative 2

2

dtd

[et sin2 t i + 2t2 j].

25. Given g(t) = 2t i + 32

t3 j – (1 + t2) k and u(t) = 41

t2, f(t) = t i + t1

j + e3t k, find

(a) (f + g) ′ (t)(b) (2f) ′ (2t)(c) (uf) ′ (t)(d) (f • g) ′ (t)(e) (f × g) ′ (t)(f) (g × f) ′ (t)(g) (f 0 u) ′ (t)

26. Find r(t) given that r ′ (t) = t i – j + e2t k and r(0) = i – 41

j + 2k.

27. Find r(t) given that r ′ (t) = sin 2t i + cos 2t j – t k and r(0) = i + j + k.

28. Find r(t) given that r ′ (t) = 21

1

t+ i + 2 tan 2t j + e–t k and r(0) = i + j + k.

29. Calculate r(t) • r ′ (t) and r(t) × r ′ (t) given that r(t) = (sin t + t cos t) i + t j.

30. Find f ′ (π / 3) for f(t) = t i + ln sin 2t j + cos2 2t k.

31. Find f ′ (t) for f(t) = sin–1 2t i + tan–1 2t j.

13.3 Curves

32. Find the tangent to the vector r ′ (t) and the tangent line for r(t) = 3t cos t i + 3t sin t j + 4t k at t = π.

33. Find the tangent to the vector r ′ (t) and the tangent line for r(t) = 2

sin 1 t− i + tan–1 3t j – 3t k at t = 1.

34. Find the tangent to the vector r ′ (t) and the tangent line for r(t) = 6 sin 2t i + 6 cos 2t j + π

22tk at t = π/4.

35. Find the tangent to the vector r ′ (t) and the tangent line for r(t) = t2 i + 4t–2 j + (1 – t3) k at t = 1.

36. Find the tangent to the vector r ′ (t) and the tangent line for r(t) = t2 i + 2t j + e2t k at t = 1.

37. Find the points on the curve r(t) = t j at which r(t) and r ′ (t) are perpendicular.

38. Find the point at which the curvesr1(t) = (1 – t) i + (1 + t) j + (1 – t) k andr2(u) = 2u3 i + (1 – u2) j + (1 + u2) k intersect and find the angle of intersection.

Vector Calculus 159

39. Find a vector parametrization for the curve x2 = y – 1, x ≥ 1.

40. Find a vector parametrization for the curve r = 2 cos 2θ, θ ∈ [0, π] (polar coordinates).

41. Find an equation in x and y for the curve r(t) = t2i + 2t j. Draw the curve. Does the curve have a tangentvector at the origin? If so, what is the unit tangent vector?

42. At t = π/4 find the unit tangent vector for the curve r(t) = sin 2t i + cos 3t j + tan t k.

43. At t = π/6 find the unit tangent vector for the curve r(t) = cos 2t i + sin 2t j – 3t k.

44. Find the tangent vector r ′ (t) and the tangent line for r(t) = sec 2t i + cos 2t j + 2t k at t = 0.

45. Find the unit tangent vector, the principal normal vector, and an equation in x, y, z for the osculating plane ofthe curve r(t) = t2 i + 2t j + t k at t = 1.

46. Find the unit tangent to the curve r(t) = ln t i + t j + sin πt k at t = 2.

47. Find the unit tangent vector, the principal normal vector, and an equation in x, y, z for the osculating plane ofthe curve r(t) = et sin 2t i + 2et cos 2t j + 2et k at t = 0.

13.4 Arc Length

48. Find the length of the curve r(t) = 3 cos t i + 3 sin t j + t k from t = 0 to t = 2π.

49. Find the length of the curve r(t) = 5t i + 4 sin 3t j + 4 cos 3t k from t = 0 to t = 2π.

50. Find the length of the curve r(t) = 3

3ti +

2

2tj + t k from t = 0 to t = 3.

51. Find the length of the curve r(t) = 6 sin 2t i + 6 cos 2t j + 5t k from t = 0 to t = π.

52. Find the length of the curve r(t) = cos t i + sin t j + t3/2 k from t = 0 to t = 20/3.

53. Find the length of the curve r(t) = (2 + cos 3t) i + (3 – sin t) j + 4t k from t = 0 to t = 2π/3.

54. Find the length of the curve r(t) = sin3 2t i + cos3 2t j from t = 0 to t = π/4.

55. Find the length of the curve r(t) = 2e–t i + (4 – 2t) j + et k from t = 0 to t = 2.

56. Find the curvature of y = e2x.

57. Find the curvature of y = ln sin 2x.

58. Find the curvature of y = 2x2 – x + 1.

59. Find the curvature of y = 4 sin 3x.

60. Find the radius of curvature of y = x2/4 at x = 1.

61. Find the radius of curvature of y2 = 4x at (1, 2).

62. Find the radius of curvature of xy = 6 at x = 2.

63. Find the radius of curvature of y = ex at x = 0.


64. Find the curvature of r(t) = 2 cos t i + cos 2t j at t = π/4.

65. Find the curvature of r(t) = t3 i + 2t2 j at t = 1.

66. Find the curvature of r(t) = (t2 + 1) i + (t – 2) j at t = 2.

67. Find the curvature of r(t) = 6 cos 2t i + sin 2t j + 3t k at t = π.

68. Find the curvature of r(t) = sin t i + cos t j + ln cos t k at t = 0.

69. Find the curvature of r(t) = et i + et cos t j + et sin t k at t = 0.

70. Interpret r(t) as the position of a moving object at time t. Find the curvature of r(t) = t2i + t1

j at t = ½ and

determine the tangential and normal components of acceleration.

71. Interpret r(t) as the position of a moving object at time t. Find the curvature of r(t) = 2et i + 2e–t j at t = 0and determine the tangential and normal components of acceleration.

72. Find the curvature of r(t) = ln t i + t j at t = 2.

73. Find the radius of curvature of r(t) = et i + 2 t j + e-t k at t = 0.

74. Find the radius of curvature of r(t) = 4 sin t i + (2t – sin 2t) j + cos 2t k at t = π/2.

75. Find the radius of curvature of r(t) = 2 cos t i + 3 sin t j at t = π/2.

76. Find the curvature of x2 + y2 = 10x at (2, –4).

77. Find the curvature of y = 3 cosh x/3 at x = 0.

13.5 Curvilinear Motion; Vector Calculus in Mechanics

78. A particle moves so that r(t) = t2 i – 2t j. Find the velocity, speed, acceleration, and the magnitude of theacceleration at the time t = 2.

79. An object moves so that r(t) = 4 cos t i + sin t j. Sketch the curve and then compute and sketch the velocityand acceleration vectors at t = π/2.

80. An object moves so that r(t) = t2 i + t3 j. Sketch the curve and then compute and sketch the velocity andacceleration vectors at t = 1.

81. An object moves so that r(t) = 6 cos 2t i + 6 sin 2t j + 5t k. Find the velocity, speed, acceleration, and themagnitude of the acceleration at the time t = π.

82. An object moves so that r(t) = 2t i + 4 sin 3t j + 4 cos 3t k. Find the velocity, speed, acceleration, and themagnitude of the acceleration at the time t = π/2.

83. An object moves so that r(t) = cos t i + sin t j + t3/2 k. Find the velocity, speed, acceleration, and themagnitude of the acceleration at the time t = π/2.

84. An object moves so that r(t) = et i + et cos t j + et sin t k, t ≥ 0. Find(a) the initial velocity(b) the initial position(c) the initial speed(d) the acceleration throughout the motion(e) the acceleration at t = 0.

Vector Calculus 161

85. Find the force required to propel a particle of mass m so that r(t) = t2 i + t3 j.

86. At each point P(x(t), y(t), z(t)) of its motion, an object of mass m is subject to a forceF(t) = mπ2[2 cos π t i + 3 sin π t j]. Given that v(0) = 2 i – 3π/2 j + ½ k and r(0) = 3 i + 2 j, find thefollowing at time t = 1:(a) velocity(b) speed(c) acceleration(d) momentum(e) angular momentum(f) torque.

87. At each point P(x(t), y(t), z(t)) of its motion, an object of mass m is subject to a force

F(t) = 2mπ2[ tπcos23

i – 2 sin π t j]. Given that v(0) = 3 i + 2π j – k and r(0) = 3 j – 2 k, find the

following at time t = 1:(a) velocity(b) speed(c) acceleration(d) momentum(e) angular momentum(f) torque.

88. Show that the position vector and the velocity vector of the particle whose position is given byr(t) = sin t cos t i + cos2 t j + sin t k are at right angles.

89. Solve the initial values problem: F(t) = m r ′ ′ (t) = m(et i + e2t j)r0 = r(0) = 2 iv0 = v(0) = 3 j + k

90. Solve the initial values problem: F(t) = mr ′ ′ (t) = m(3 cos 2t i + 3 sin 2t j)r0 = r(0) = i – 2 j

v0 = v(0) = 2 i – 32

k



1. –4 i + 5 j – 5 k

2. 2/3)1(21 −− t i +

221

2

t

t

+j – 2/1

23

t k

3. te−− i + tt

t−−

2214

j – 21

1

t−k

4. 4t sin t2 cos t2 i – 2 sin 2t j + 2te–2t(1 – t) k

5.358

i – 20 k

6.

−−

8cos

2cos

41 22 ππ

i + 62

j + ( )tt ee )2/(

21 ππ − k

7. )1(41 2−− e i + )1(

121 4 −e j –

32

sin23

k

8. –j + e4 k

9. –j + k

10. (a) f(t) = 2 cos t i + 4 sin t j(b) g(t) = 2 cos t i – 4 sin t j

11. (a) f(t) = (t + 1) i + 2t2 j(b) g(t) = (1 – t) i + 2t2 j

12. f(t) = (2 – t) i + (–1 + 5t) j + (3 – 5t) k, t ∈ [0, 1]

13. f(t) =

++ 2

3

3

tt

i +

−+ 31ln

32 3t j

+

−

61

32 2/3te k

14. f(t) =

+1

32 2t i –

+

+ 2

23

2

tt

j + (tet + 2) k

15.

16.

17. f ′ (t) = et i + 2e2t jf ′ ′ (t) = et i + 4e2t j

18. f ′ (t) = cos t i + 2 cosh 2t j – 2 sech 2t tanh 2t kf ′ ′ (t) = –sin t i + 4 sinh 2t j +

(4 sech 2t tanh2 2t – 4 sech3 2t) k

19. f ′ (t) = tt

t

2

12 +

+i –

tt

t

2

12 +

+j + k

f ′ ′ (t) = 2/32 )2(

1

tt +−

i – 22

2

)2(1

tttt

+++

j

20. f ′ (t) = i – 2 tan 2t j – 2 cot 2t kf ′ ′ (t) = –4 sec2 2t j – 4 csc2 2t k

21. f ′ (t) = t cos t i + t sin t j + 2t kf ′ ′ (t) = (cos t – t sin t) i + (sin t + t cos t) j + 2 k

22. f ′ (t) = cos 2t i – (2t sin t + t2 cos t) j + et(t2 + 2t) kf ′ ′ (t) = –2 sin 2t i – (2 sin t + 4t cos t + t2 sin t) j

+ et(t2 + 4t + 2) k

23. f ′ (t) =

−− t

tt

t cos2sin32/3 j

f ′ ′ (t) =

++

−− tt

tt

tt sin

cos34

sin152

2/3 j

24. et(sin2 t + 2 sin 2t + 2 cos 2t) i + 4 j

25. (a) 3 i +

+

− 22

21

tt

j + (3e3t – 2t) k

(b) 2 i – 2

2

tj + 6e3t k

(c) 2

43

t i + 41

j +

+ tt e

tet 332

243

k

(d) 4t – t34

– e3t(3 + 2t + 3t2)

Vector Calculus 163

(e)

−−+ tt etet

t3332

222

11 i +

(2e3t + 6te3t + 1 + 3t2) j + 3

38

t k

(f)

++−− tt etet

t3332

222

11 i +

(–2e3t – 6te3t – 1 – 3t2) j – 3

38

t k

(g) 2t

i + 3

8

tj + 4/3 2

23 tet

k

26. r(t) =

+1

2

2ti +

+

41

t j +

+

23

21 2te k

27. r(t) =

− t2cos

21

23

i +

+ t2sin

21

1 j +

−

21

2tk

28. r(t) = (1 + tan–1 t) i + (1 – ln cos 2t) j + (2 – e–t) k

29. r(t) • r ′ (t) = 3t cos2 t + tt

2sin2

12

−

r(t) × r ′ (t) = [(t2 + 1)sin t – t cos t] k

30. i + 3

2 j + 3 k

31.241

2

t−i +

241

2

t+ j

32. r ′ (t) = 3(cos 2 – t sin t) i + 3(sin t + t cos t) j + 4 ktangent line: –(3π + 3t) i – 3πt j + (4π + 4t) k

33. r ′ (t) =24

1

t−i +

291

3

t+j – 3 k

tangent line:

− t

15

46π

i +

+− t

103

3tan 1 j

– 3(1 + t) k

34. r ′ (t) = 12cos 2t i – 12sin 2t j + πt4

k

tangent line: 6 i – 12t j +

+ t

8π

k

35. r ′ (t) = 2t i – 3

8

t j – 3t2 k

tangent line: (1 + 2t) i + (4 – 8t) j – 3t k

36. r(t) = 2t i + 2 j + 2e2t ktangent line: (1 + t) i + (2 + t) j + e2(1 + t) k

37. (x, y) = (0, 0), (1/4, –1/2)

38. (2, 0, 2); °≈−

= − 5.2933

5cos 1θ

39. r(t) = t i + (t2 + 1) j

40. r(t) = t uθ + (2 + cos 2t) ur

41. x = y2/4, tangent vector at origin is j

42. T(π/4) = 17

3−j +

17

22k

43. T(π/6) = 4

3−i +

21

j + 43

k

44. r ′ (t) = (2 sec 2t tan 2t) i – 2sin 2t j + 2 ktangent line: i + j + 2t k

45. T(1) = 32

i + 32

j + 31

k

N(1) = 53

5i +

53

4j +

53

2k

osculating plane: 5x – 4y – 2z + 5 = 0

46. T(2) = 245

1

π+ i +

245

2

π+j +

245

2

π

π

+k

47. T(0) = 3

1i +

3

1j +

3

1k

N(0) = 14

2i +

14

3j +

14

1k

osculating plane: 2x – 3y + z + 4 = 0

48. 102π

49. 26π

50. 12

51. 13π

52. 56/3

53. 10π/3

54. 3/2


55. e2 + 1 – e–2

56. 2/34

2

)41(4

x

x

ee

+

57.2/32

2

)2cot41(2csc4

xx

+

58.2/32 )1682(

4

xx ++

59. 0

60.455

61. 24

62.12

1313

63. 22

64.33

1

65.12512

66.1717

12

67.16924

68. 2

69.32

70.1717

24, aT =

17

62−, aN =

17

24

71.8

1, aT = 0 aN = 8

72.55

2

73. 22

74. 22

75. 4/3

76. 1/5

77. 1/3

78. v(2) = 4 i + 2 j ; ||v(2)|| = 52a(2) = 2 i ; ||a(2)|| = 2

79. v(π/2) = –4 i ; a(π/2) = – j

80. v(1) = 2 i + 3 j ; a(1) = 2 i + 6 j

81. v(π) = 12 j + 5 k ; ||v(π)|| = 13a(π) = –24 i ; ||a(π)|| = 24

82. v(π/2) = 2 i + 12 k ; ||v(π/2)|| = 372a(π/2) = 36 j ; ||a(π/2)|| = 36

83. v(π/2) = –i + 22

3 πk ; ||v(π/2)|| = π

89

1+

a(π/2) = –j + π2

43

k ; ||a(π/2)|| = π89

1+

84. (a) i + j + k(b) (1, 1, 0)

(c) 3(d) a(t) = et i – 2et sin t j + 2et cos t k(e) a(0) = i + 2 k

Vector Calculus 165

85. F(t) = 2m i + 6mt j

86. (a) 2 i + 9π/2 j + ½ k

(b) 2811721

π+

(c) –2π2 i

(d) 2m i + mπ29

j + 2m

k

(e) m

−

23

1π

i + m21

j + m

− 4

275π

k

(f) –mπ2 j

87. (a) 3 i – 6π j – k

(b) 23610 π+(c) –3π2 i(d) 3m i – 6πm j + m k(e) –(3 + 16π)m i – (9 + 48π)m k(f) 9π2m j + (9π2 – 6π3)m k

88. r(t) = ½ sin 2t i + ½ (1 + cos 2t) j + cos t kv(t) = cos 2t i – sin 2t j + sin t kr(t) • v(t) = 0

89. v(t) = (et – 1) i + ½ (e2t + 5) j + kr(t) = (et – t + 1) i + ¼ (e2t + 10t – 1) j + t k

90. v(t) =

+ 22sin

23

t i +

−12cos

23

t j – 32

k

r(t) =

++−

47

22cos43

tt i –

+−

38

22sin43

tt j

– t32

k

167

CHAPTER 14

Functions of Several Variables

14.1 Elementary Examples

1. Find the domain and range of )ln(tan2),,( 221 zxxy

zyxf ++= − ; find f(1, 1, 1) and f(1, –1, 1).

2. Find the domain and range of f(x, y, z) = zex cos y; find f(ln 2, 0, –1).

3. Find the domain and range of )ln( 22),,( yxyzezyxf += ; find f(1, –1, 2) and f(0, 1, 4).

4. Find the domain and range of

+

= −−

yx

yx

yxf 11 tansin),( ; find f(1, 1).

5. Find the domain and range of yxyxf21

32

4),( −−= ; find f(2, 3).

6. Find the domain and range of f(x, y) = 1 – y2; find f(2, 0).

7. Find the domain and range of 2224),,( zyxzyxf −−−= ; find f(1, 1, ½).

8. Find the domain and range of yxyx

yxf−+

=),( ; find f(3, 1).

9. Find the domain and range of xy

xyyxf −= 2),( ; if x(t) = 2t and y(t) = t2.

10. Determine a function f of x and y giving the volume of a cone of base diameter x and height y.

11. Determine a function f of x, y, and z giving the surface area of a box of length x, width y, and volume z.

14.2 A Brief Catalogue of Quadratic Surfaces; Projections

12. Identify the surface x2 + 4y2 + 9z2 + 2x + 16y – 18z – 10 = 0.

13. Identify the surface 5x2 + 4y2 + 20z2 – 20x + 32y + 40z + 56 = 0.

14. Identify the surface 9x2 + 4y2 – 54x – 16y – 36z + 277 = 0.

15. Identify the surface 3x2 – 2y2 + 3z2 + 30x – 8y – 24z + 131 = 0.

16. Identify the surface 6x2 + 4y2 – 3z2 + 36x – 16y – 6z + 55 = 0.

17. Identify the surface 6x2 + 4y2 – 2z2 – 6x – 4y + z = 0.

18. Identify the surface 3x2 – 2y2 – z2 – 6x + 8y – 2z + 6 = 0.

19. Identify the surface x2 + y2 – z2 – 2x + 4y – 2z = 0.


20. Sketch the cylinder 9x2 + 4z2 – 36 = 0.

21. Identify and sketch the surface z = 4x2 + y2.

22. Identify and sketch the surface z2 = x2 + y2.

23. Identify and sketch the surface 11694

222

=++zyx

24. Identify the surface and find the traces: 2x2 + y2 – 4z = 0.

25. Identify the surface and find the traces: x2 – y2 + z2 + 2y = 1.

26. Identify the surface and find the traces: x2 + y2 + z – 5 = 0.

27. Identify the surface and find the traces: x2 + 4y2 + z2 – 8y = 0.

28. Write an equation for the surface obtained by revolving the parabola y – 4z2 = 0 about the y-axis.

29. The planes 2x + y + z = 4 and x + y – z = 1 intersect in a space curve C. Determine the projection of C ontothe xy-plane.

30. The sphere x2 + (y – 1)2 + z2 = 6 and the hyperboloid x2 – y2 + z2 = 1 intersect in a space curve C. Determinethe projection of C onto the xy-plane.

31. The sphere x2 + (y – 2)2 + z2 = 2 and the cone y2 + z2 = 5x2 intersect in a space curve C. Determine theprojection of C onto the xy-plane.

32. The cone x2 + y2 = 2z2 and the plane y + 4z = 5 intersect in a space curve C. Determine the projection of Conto the xy-plane.

14.3 Graphs; Level Curves and Level Surfaces

33. Sketch the graph of 2216),( yxyxf −−= .

34. Sketch the graph of 22 216),( yxyxf −−= .

35. Identify the level curves and sketch some of them. f(x, y) = x2 + y2

36. Identify the level curves and sketch some of them. f(x, y) = 4x2 + y2

37. Identify the level curves and sketch some of them. yxyx

yxf−+

=),(

38. Identify the level curves and sketch some of them. f(x, y) = xy

39. Identify the c-level surface for f(x, y, z) = 2x – 3y + z and c = 1.

40. Identify the c-level surface for f(x, y, z) = x2 – y2 + z2 and c = 1.

41. Identify the c-level surface for f(x, y, z) = 249

222 zyx++ and c = 1.

Functions of Several Variables 169

14.4 Partial Derivatives

42. Calculate the partial derivatives of )sin( 2xyyx

z = .

43. Calculate the partial derivatives of f(x, y) = xy.

44. Calculate the partial derivatives of z = x3 + xy – y cos xy.

45. Calculate the partial derivatives of z = y2e–x + y.

46. Calculate the partial derivatives of 22 9416),( yxyxf −−=

47. Calculate the partial derivatives of f(x, y) = sin (x2y).

48. Calculate the partial derivatives of f(x, y) = (1 + x2 + y)5/3.

49. Find fx(1, 1) and fy(1, 1) given that 1),( 22++−= xeyyxf xy .

50. Find fx(4, 2) and fy(4, 2) given that xexyyxf y+−= )1ln(),( .

51. Find fx(4, 2) and fy(4, 2) given that xyyxyyxf 22 )ln(),( ++= .

52. Find fx(x, y) and fy(x, y) by forming the appropriate difference quotient and taking the limit as h tends to zero(Definition 14.4.1). f(x, y) = xy2.

53. Find fx(x, y, z), fy(x, y, z), and fz(x, y, z) by forming the appropriate difference quotient and taking the limit ash tends to zero (Definition 14.4.2). f(x, y, z) = x2yz.

54. Let C be the curve of intersection of the surface z = 3xy2 with the plane y = 2. Find an equation for thetangent line to C at the point (1, 2, 12).

55. Let C be the curve of intersection of the surface 22 yxz −= with the plane x = 3. Find an equation for the

tangent line to C at the point (3, 1, )22 .

56. Use implicit differentiation to find xz

∂∂

and yz

∂∂

given that x2z2 – 2xyz + y2z3 = 3.


∂∂

at (1, –2, 1) given that x3z – 3xy2 – (yz)3 = –3.


∂∂

and yz

∂∂

given that x2 + y2 + z2 – 2xyz = 5.


∂∂

and yz

∂∂

given that x1/3 – y1/3 + z1/3 = 16.


∂∂

and yz

∂∂

given that (x+ y)2 = (y – z)3.


61. Find xz

∂∂

and yz

∂∂

given that x3z2 – 2xyz2 + z3y2 = 2z.

62. Evaluate xz

∂∂

and yz

∂∂

at (3, 3, 2) given that x3 + y3 + z3 – 3xyz = 8.

63. Evaluate xz

∂∂

and yz

∂∂

at (1, 0, π/6) given that x2 cos2 z – y2 sin z = sin2 2z.

64. Verify that zyz

yxz

x =∂∂

+∂∂

given that xyyeyx

xz /sin +

= .

65. Verify that zyz

yxz

x 3=∂∂

+∂∂

given that z = x3 + 2x2y – 3xy2 + y3.

66. Verify that 034 =∂∂

−∂∂

yz

xz

given that z = (3x + 4y)4.

67. The area of a triangle is given by the formula A = ½ bc sin θ. At time t0 we have b0 = 5 cm, c0 = 10 cm,and θ0 = π/6 radians.(a) Find the area of the triangle at time t0.(b) Find the rate of change of the area with respect to b at time t0 if c and θ remain constant.(c) Find the rate of change of the area with respect to θ at time t0 if b and c remain constant.(d) Using the rate found in (c), calculate (by differentials) the approximate change in area if the angle is

increased by one degree.(e) Find the rate of change of c with respect to b at time t0 if the area and the angle are to remain

constant.

14.5 Open Sets and Closed Sets

68. Specify the interior and the boundary of the set (x, y) : 1 ≤ x ≤ 3, 2 ≤ y ≤ 4. State whether the set is open,closed, or neither. Then sketch the set.

69. Specify the interior and the boundary of the set (x, y) : 4 < x2 + y2 < 9. State whether the set is open,closed, or neither. Then sketch the set.

70. Specify the interior and the boundary of the set (x, y, z) : x2 + y2 ≤ 2, 0 ≤ z ≤ 2. State whether the set isopen, closed, or neither. Then sketch the set.

71. Specify the interior and the boundary of the set (x, y, z) : x2 + (y – 1)2 + z2 < 1. State whether the set isopen, closed, or neither. Then sketch the set.

72. Specify the interior and the boundary of the set y : 2 < y < 4. State whether the set is open, closed, orneither.

73. Specify the interior and the boundary of the set x : x ≤ 2. State whether the set is open, closed, or neither.

74. Specify the interior and the boundary of the set x : x > –1. State whether the set is open, closed, or neither.


14.6 Limits and Continuity; Equality of Mixed Partials

75. Find the second partials of 22 4916),( yxyxf −−= .

76. Find the second partials of f(x, y) = cos (xy2).

77. Find the second partials of f(x, y) = (1 + x + y2)4/3.

78. Find the second partials of )ln(tan2),,( 221 zxxy

zyxf ++= − .

79. Find the second partials of f(x, y, z) = xz2 cosh (ln y).

80. Find the second partials of 4

1),(

22 −+=

yxyxf .

81. Find the second partials of f(x, y, z) = ln (2x + 3y + 2z).

82. Find the second partials of f(x, y) = (x2 + xy)5/2.

83. Find the second partials of zyxzyxf 2),,( 22 ++= .

84. Evaluate )3(lim 2

)2,1(),(yx

yx+

→.

85. Evaluate yxyx

yx 334

lim)1,1(),( −+

−+→

.

86. Evaluate xy

xyx

sinlim 3

)2/,1(),( π→.

87. Evaluate 33)0,0(),(

2lim

yx

yxyx +

+→

.

88. Evaluate 22

22

)0,0(),(

)tan(lim

yxyx

yx ++

→.

89. Evaluate )43(tan

)2(sinlim

1

1

)1,1(),( −−

−

−

→ xyxy

yx.

90. Show that 22

2

)0,0(),(lim

yxx

yx +→ does not exist.

91. Show that the function f(x, y) = e–2y cos 2x is harmonic.

92. For what value of c > 0 is f(x, t) = sin (2x + 3t) a solution of the wave equation ?02

22

2

2

=∂∂

−∂∂

x

fc

t

f



1. dom (f) = (x, y, z) : x ≠ 0ran (f) = (–∞, ∞)f(1, 1, 1) = π/2 + ln 2f(1, –1, 1) = –π/2 + ln 2

2. dom (f) = (x, y, z) : x, y, z ∈ ℜran (f) = (–∞, ∞)f(ln 2, 0, –1) = –2

3. dom (f) = (x, y, z) : x2 + y2 ≠ 0ran (f) = (–∞, ∞)f(1, –1, 2) = –4f(0, 1, 4) = 4

4. dom (f) = (x, y) : |x/y| ≤ 1ran (f) = [–3π/4, 3π/4]f(1, 1) = 3π/4

5. dom (f) = (x, y) : x, y ∈ ℜran (f) = (–∞, ∞)f(2, 3) = 7/6

6. dom (f) = (x, y) : x, y ∈ ℜran (f) = (–∞, 1]f(2, 0) = 1

7. dom (f) = (x, y, z) : x2 + y2 + z2 ≤ 4ran (f) = [0, 2]

f(1, 1, ½ ) = 27

8. dom (f) = (x, y) : –x ≤ y ≤ x, x > 0;or x ≤ y ≤ –x, x < 0

ran (f) = [0, ∞)

f(3, 1) = 2

9. dom (f) = (x, y) : x ≠ 0 = t : t ≠ 0ran (f) = (–∞, ∞)

10. yxyxf 2

121

),( π=

11.xz

yz

xyzyxf 222),,( ++=

12. ellipsoid

13. ellipsoid

14. elliptic paraboloid

15. hyperboloid of two sheets

16. hyperboloid of one sheet




20. 194

22

=+zx

21. 22

4/1y

xz += ; elliptic paraboloid

22. circular cone



24. elliptic paraboloid

traces: origin, 2

,4

22 xz

yz ==

25. circular conetraces: z = ±(y – 1), x2 + z2 = 1, x = ± (y – 1)

26. circular paraboloidtraces: z = 5 – y2, z = 5 – x2, x2 + y2 = 5

27. ellipsoidtraces: (y – 1)2 + z2/4= 1, origin, x2/4 + (y – 1)2 = 1

28. 4x2 + 4z2 = y

29. 3x + 2y = 5

30. x = 0

31.

−=

21

322 yx

32. 149/200

)7/5(7/25

22

=+

=yx

; an ellipse

33.

34.

35. The level curves for z ≥ 0 are concentric circlesin the xy plane.

36. The level curves for z ≥ 0 are concentricellipses in the xy plane.

37. The level curves are staight lines through theorigin.


38. hyperbolas

39. plane


41. ellipsoid

42. )sin(1

)cos( 22 xyy

xyxyzxz

x +==∂∂

)sin()cos(2 22

22 xyy

xxyxz

yz

y +==∂∂

43. 1−= yx yxf ; xxf y

y ln=

44. zx = 3x2 + y + y2 sin xyzy = x + xy sin xy – cos xy

45. zx = –y2e–x

zy = 2ye–x + 1

46.22 9416

4

yx

xf x

−−

−=

22 9416

9

yx

xf y

−−

−=

47. fx = 2xy cos (x2y)fy = x2 cos (x2y)

48. 3/222 )1(3

10yx

xf x ++=

3/222 )1(35

yxf x ++=

49.22

)1,1( +−= ef x

fy (1,1) = 1 – 2e

50.47

2)2,4(

2ef x += ; 22

74

)2,4( ef y +=

51. fx(4, 2) = 5/4 ; fy(4, 2) = ln 8 + 9

52. fx(x, y) = y2 ; fy(x, y) = 2xy

53. fx(x, y, z) = 2xyz ; fy(x, y, z) = x2zfz(x, y, z) = x2y

54. r(t) = t i + 2 j + 12t k

55. r(t) = 3 i + t j + )9(42

t− k

56. 222

2

32222

zyxyzxxzyz

xz

+−−

=∂∂

222

3

32222

zyxyzxyzxz

yz

+−−

=∂∂

57.259

=∂∂xz

58.zxy

yzxxz

−−

=∂∂

, xyzyxz

yz

−−

=∂∂

59.3/2

−=

∂∂

xz

xz

,3/2

−=

∂∂

yz

yz

60.2)(3

)(2

zy

yxxz

−+−

=∂∂

, 2)(3

)(21

zy

yxyz

−+−

=∂∂

61.223

222

342223

yzxyzzxyzzx

xz

−+−−

=∂∂

223

23

342223

yzxyzzxyzyz

yz

−+−−

=∂∂

62.53

=∂∂xz

, 53

=∂∂yz

63.3

1=

∂∂xz

, 0=∂∂yz

64. +

−+ xye

y

xyx

yx

yx

x /2

2

cossin

zyeyx

xexy

eyx

y

xy xyxyxy =+=

++

− ///2

2

sincos

65. x(3x2 + 4xy + 3y2) + y(2x2 + 6xy + 3y2) = z


66. 4[12(3x + 4y)3] – 3[16(3x + 4y)3] = 0

67. (a) 25/2 cm2 (d) 72

35πcm2

(b) 5/2 cm2/cm (e) –c/b

(c) 2

325cm2/rad

68. Interior: (x, y): 1 < x < 3, 2 < y < 4Boundary: (x, y): 1 ≤ x ≤ 3, y = 2 or y = 4

U (x, y): 2 < y < 4, x = 1 or x = 3closed set

69. Interior: (x, y): 4 < x2 + y2 < 9Boundary: (x, y): x2 + y2 = 4

U (x, y): x2 + y2 = 9open set

70. Interior: (x, y, z): x2 + y2 < 2, 0 < z < 2Boundary: (x, y, z): x2 + y2 = 2, 0 < z < 2

U (x, y, z): x2 + y2 ≤ 2, z = 0 or 2closed set

71. Interior: (x, y): x2 + (y – 1)2 + z2 < 1Boundary: (x, y): x2 + (y – 1)2 + z2 = 1open set

72. Interior: (y: 2 < y < 4Boundary: y = 2, y = 4open set

73. Interior: (x: x < 2Boundary: x = 2closed set

74. Interior: (x: x > –1Boundary: x = –1open set

75.2/322

2

)4916(14436

yxy

f xx −−−

=

2/322

2

)4916(6436yx

xf yy −−

−=

2/322 )4916(

36

yx

xyf xy −−

−=

76. fxx = –y4 cos (xy2)fyy = –4x2y cos (xy2) – 2x sin (xy2)fxy = fyx = –2xy3 cos (xy2) – 2y sin (xy2)

77. 3/22 )1(94 −++= yxf xx

3/22

2

)1(9402424

yxyx

f yy ++++

=

3/22 )1(9

8

yx

yff yxxy ++

==

78.222

22

222 )(22

)(4

yxxz

yxxy

f xx +−

++

=

222 )(4

yxxy

f yy +−

= ,222

22

)(22zx

zxf zz +

−=

222

22

)(22

yxyx

f xy ++−

= ,222 )(

4

zx

xzf xz +

−=

fyz = 0

79. fxx = 0 , )sinh(ln2

yy

zf xy =

)]sinh(ln)[cosh(ln2

2

yyyxz

f yy −=

)cosh(ln2 yzf xz = , )cosh(ln2 yxf zz =

)][sinh(ln2

yyxz

f yz =


80.322

22

)4(826

−++−

=yx

yxf xx ,

322

22

)4(826

−++−

=yx

xyf yy

322 )4(8

−+=

yxxy

f xy

81.2)232(

4

zyxf xx ++

−= ,

2)232(

9

zyxf yy ++

−=

2)232(

4

zyxf zz ++

−= ,

2)232(

6

zyxf xy ++

−=

2)232(

4

zyxf xz ++

−= ,

2)232(

6

zyxf yz ++

−=

82. 2/3222/12 )(5)2()(4

15xyxyxxyxf xx ++++=

2/322/12 )(25

)2()(4

15xyxyxxyx

xf xy ++++=

2/122

)(4

15xyx

xf yy +=

83.2/322

2

)2(2

zyxzy

f xx +++

= ,2/322

2

)2(2

zyxzx

f yy +++

=

2/322 )2( zyx

xyf xy ++

−= ,

2/322 )2( zyx

xf xz ++

−=

2/322 )2( zyx

yf yz ++

−= ,

84. 7

85. 4

86. 1

87. does not exist

88. 1

89. 2

90. along y = 0 1lim2

2

)0,0(),(=

→ x

xyx

along y = x21

2lim

2

2

)0,0(),(=

→ x

xyx

limit does not exist in either case

91.2

22

2

2

2cos4y

fxe

xf y

∂∂

−=−=∂∂ −

so 02

2

2

2

=∂∂

+∂∂

yf

yf

92. c = 3/2

177

CHAPTER 15

Gradients; Extreme Values; Differentials

15.1 Differentiability and Gradient

1. Find the gradient of f(x, y) = x2exy.

2. Find the gradient of f(x, y) = 2x3 + 5xy – 2y2.

3. Find the gradient of f(x, y) = xy sin xy2.

4. Find the gradient of f(x, y, z) = 2x2yz.

5. Find the gradient of f(x, y, z) =zyx ++

1.

6. Find the gradient of f(x, y, z) = x3y + yz2 – xy2z.

7. Find the gradient of f(x, y, z) = ey ln xz.

8. Find the gradient of f(x, y) = (x2 – y) cos (x + y).

9. Find the gradient of f(x, y, z) = xex + y + z.

10. Find the gradient of f(x, y) = 22 yx + .

11. Find the gradient vector at (1, 2) of f(x, y) = 3x2 – 2xy + 4y2.

12. Find the gradient vector at (1, 2) of f(x, y) =22

2

yx

yx

++

.

13. Find the gradient vector at (½, 1) of f(x, y) = (x + y) sin πx.

14. Find the gradient vector at (1, –1, 2) of f(x, y, z) = ex sin (y + z2).

15. Find the gradient vector at (1, 1, 2) of f(x, y, z) = ln (x + y + z2).

16. Calculate (a) ∇ (ln r3) ; (b) ∇ (sin 2r) ; (c) )(22 rre +∇

where 222 zyxr ++=

15.2 Gradients and Directional Derivatives

17. Find the directional derivative of f(x, y) = ex sin y at (0, π/3) in the direction of 5 i – 2 j .

18. Find the directional derivative of f(x, y) = 3 22ln yx + at (3, 4) in the direction of 4 i + 3 j .

19. Find the directional derivative of f(x, y) = 916

22 yx+ at (4, 3) in the direction of i + j .


20. Find the directional derivative of f(x, y) = ex cos y at (2, π) in the direction of 2 i + 3 j .

21. Find the directional derivative of f(x, y) = 3xy2 – 4x3y at (1, 2) in the direction of 3 i + 4 j .

22. Find the directional derivative of f(x, y) = ex sin πy at (0, 1/3) toward the point (3, 7/3).

23. Find the directional derivative of f(x, y) = x tan–1 y/x at (1, 1) in the direction of a = 2 i – j .

24. Find the rate of change of yx

xyxf

−=

2),( at (1, 0) in the direction of a vector at an angle of 60° from the

positive x-axis.

25. Find the rate of change of yxyx

yxf−

+=

2),( at (1, 1) in the direction of a vector at an angle of 150° from the

positive x-axis.

26. Find the rate of change of xy

xyyxf −= 2),( at (1, 2) in the direction of a vector at an angle of 120° from the

positive x-axis.

27. Find the directional derivative of f(x, y, z) = x sin (πyz) + yz tan (πx) at (1, 2, 3) in the direction of 2 i + 6 j– 9 k.

28. Find the directional derivative of f(x, y, z) = x2 – 2y2 + z2 at (3, 3, 1) in the direction of 2 i + j – k.

29. Find the directional derivative of xzyxzyxf += 32),,( at (1, –2, 3) in the direction of 5 j + k.

30. Find the directional derivative of f(x, y, z) = x2y + xy2 + z2 at (1, 1, 1) toward the point (3, 1, 2).

31. The temperature, T, at a point (x, y) on a semi-circular plate is given by T(x, y) = 3x2y – y3 + 273 degreesCelsius.(a) Find the temperature at (1, 2).(b) Find the rate of change of temperature at (1, 2) in the direction of i – 2 j.(c) Find a unit vector in the direction in which the temperature increases most rapidly at (1, 2) and find this

maximum rate of increase in temperature at (1, 2).

32. The temperature, T, at a point (x, y) in the xy-plane is given by T(x, y) = xy – x. Find a unit vector in thedirection in which the temperature increases most rapidly at (1, 1) and find this maximum rate of increase intemperature at (1, 1).

33. Find the unit vector in the direction in which f(x, y, z) = 4exy cos z decreases most rapidly at (0, 1, π /4) andfind the rate of decrease of f in that direction.

34. Find the unit vector in the direction in which f(x, y, z) = ln (1 + x2 + y2 – z2) increases most rapidly at (1, –1,1) and find the rate of increase of f in that direction.

15.3 The Mean Value Theorem; Chain Rules

35. Find the rate of change of f(x, y) = xy2 with respect to t along the curve r(t) = et i + t2 j.

36. Find the rate of change of f(x, y, z) = x2 + y2 + z2 with respect to t along the curve r(t) = t i + t2 j – t3 k.

37. Find the rate of change of f(x, y, z) = xy + yz – xz with respect to t along the curve r(t) = 2 cos ω t i +3 sin ω t j – 2ω t k.

Gradients; Extreme Values; Differentials 179

38. Find the rate of change of f(x, y, z) = x2 cos (y + z) with respect to t along the curve r(t) = t i – sin t j – t2 k.

39. Use the chain rule to find tu

∂∂

if 22 yxu += ; x = et, y = sin t.


∂∂

if u = y2ex ; x = cos t, y = t3.


∂∂

if u = xy and x = ex cos x.


∂∂

if u = xy and x = y sin y.


∂∂

at t = 1 if u = x3y2 ; x = t2 + 1, y = t3 + 2.

44. Use the chain rule to find t∂

∂ω at ω = tan–1 (xyz) and x = t2, y = t3, z = t–4.


∂ω at ω = sin xy + ln xz + z and x = et, y = t2, z = 1.

46. At t = 0, the position of a particle on a rectangular membrane is given by P(x, y) = sin πx/3 sin πy/5. Find therate at which P changes if the particle moves from (3/4, 15/4) in a direction of a vector at an angle of 30°from the positive x-axis.

47. Use the chain rule to find sz

∂∂

and tz

∂∂

if z = x sin y, x = set, y = se–t.

48. Use the chain rule to find uz

∂∂

and vz

∂∂

if z = x2 tan y, x = u2 + v3, y = ln (u2 + v2).

49. Use the chain rule to find rz

∂∂

and θ∂

∂z if

22 yx

xyz

+= , x = r cos θ, y = r sin θ.


∂∂

and vz

∂∂

if z = x cos y + y sin x, x = uv2, y = u + v.

51. Use the chain rule to find sz

∂∂

and tz

∂∂

if z = x2 + y3, x = s + t, y = s – t.


∂∂

and vz

∂∂

if z = x3 + xy + y2, x = 2u + v, y = u – 2v.

53. Verify that z = f(x3 – y2) satisfies the equation 032 2 =∂∂

+∂∂

yz

xxz

y .

54. Verify that z = f(y/x) satisfies the equation 0=∂∂

+∂∂

yz

yxz

x .

55. Use the chain rule to find r∂

∂ω and

s∂∂ω

if ω = ln (x2 + y2 + 2z), x = r + s, y = r – s, z = 2rs.



∂ω,

θω

∂∂

, and z∂

∂ω if ω = xy + yz, x = r cos θ, y = r sin θ, z = z.


∂ω and

s∂∂ω

if ω = ln (x2 + y2 + z2), x = er cos s, y = er sin s, z = es.


∂ω if ω = x2 + y2 + z2, x = et cos t, y = et sin t, z = et.


∂ω,

u∂∂ω

, and v∂

∂ω if ω = 2x + y – z, x = t2 + u2, y = u2 + v2, z = v2 + t2.

60. Use the chain rule to find u∂

∂ω and

v∂∂ω

if ω = 2x – 3y + z, x = u sin v, y = v sin u, z = sin u sin v.


∂ω and

s∂∂ω

if ω = 222 zyx ++ , x = r cos s, y = r sin s, z = r tan s.


∂ω and

s∂∂ω

if ω = x2 + y2 + z2, x = r cos s, y = r sin s, z = rs.

63. Evaluatexz

∂∂

andyz

∂∂

at (½ , –1, 2) if exz + ln (yz + 3) = y + 1 + e and z is a differentiable function of x and y.


∂∂

and vz

∂∂

if z = x2y3 + x sin y, x = u2, y = uv.

65. Show that 222

1

zyxu

++= satisfies 0

2

2

2

2

2

2

=∂∂

+∂∂

+∂∂

zu

yu

xz

.

66. Let z = f(x, y) with x = r cos θ and y = r sin θ. Show that 222

2

21

∂∂

+

∂∂

=

∂∂

+

∂∂

yz

xzz

rrz

θ.

15.4 The Gradient as a Normal; Tangent Lines and Tangent Planes

67. Write an equation for the plane tangent to the surface 4x2 + 9y2 + z = 17 at the point (–1, 1, 4) .

68. Write an equation for the plane tangent to the surface z = ex sin πy at the point (2, 1, 0) .

69. Write an equation for the plane tangent to the surface z = e–x y2 + y at the point (0, 2, 6) .

70. Write an equation for the plane tangent to the surface z = x2 + y2 at the point (2, –1, 5) .

71. Write an equation for the plane tangent to the surface z = xesin y at the point (2, π, 2) .

72. Write an equation for the plane tangent to the surface z = 3x2 + 2y2 at the point (2, –1, 4) .

73. Find a normal vector at the point indicated. Write an equation for the normal line and an equation for the

plane tangent to the surface xy

z −=3

2

at the point (0, 0, 0).


74. Find a normal vector at the point indicated. Write an equation for the normal line and an equation for theplane tangent to the surface zx2 – xy2 – yz – 18 = 0 at the point (0, –2, 3).

75. Find a normal vector at the point indicated. Write an equation for the normal line and an equation for theplane tangent to the surface xyz + x + y + z – 3 = 0 at the point (1, –2, 2).

76. Find a normal vector at the point indicated. Write an equation for the normal line and an equation for the

plane tangent to the surface 13694

222

=++zyx

at the point (2, 3, 6).

77. The surfaces 2x2 – 2y2 + z3 = 4 and 3x2 – 2y2 + 3z = 1 intersect in a curve that passes through the point(1, 2, 2). What are the equations of the respective tangent planes to the two surfaces at this point?

78. Find a point on the surface z = 16 – 4x2 – y2 at which the tangent plane is perpendicular to the line x = 3 + 4t,y = 2t, z = 2 – t.

79. Find a point on the surface z = 9 – x2 – y2 at which the tangent plane is parallel to the plane 2x + 3y + 2z = 6.

15.5 Maximum and Minimum Values

80. Find the stationary points and determine the local extreme values for the function f(x, y) = 2x2 – 3y + y2.

81. Find the stationary points and determine the local extreme values for the function f(x, y) = x2 – 2x – y3.

82. Find the stationary points and determine the local extreme values for the functionf(x, y) = x2 + xy + y2 – 2x – 2y + 6.

83. Find the stationary points and determine the local extreme values for the functionf(x, y) = x2 – xy + y2 – 2x – 2.

84. Find the stationary points and determine the local extreme values for the functionf(x, y) = x3 + 3x – 2y.

85. Find the stationary points and determine the local extreme values for the functionf(x, y) = 3xy – 5x2 – y2 + 5x – 2y.

86. Find the stationary points and determine the local extreme values for the functionf(x, y) = x2 – xy + y2 – 5x + 2.

87. Find the stationary points and determine the local extreme values for the functionf(x, y) = 2x2 + y2 + 4y.

88. Find the stationary points and determine the local extreme values for the functionf(x, y) = x2 – xy + y2 + 5x + 1.

89. Find the stationary points and determine the local extreme values for the functionf(x, y) = y sin 2x, –π/2 < x < π/2.

90. Find the absolute extreme values taken on by 22),( yxyxf += on the set D = (x, y): 2 ≤ x ≤ 4, 1 ≤ y ≤ 4.

91. Find the absolute extreme values taken on by f(x, y) = 2x2 – 3y2 on the set D = (x, y): –2 ≤ x ≤ 2.

92. Find the absolute extreme values taken on by f(x, y) = (x – 2)2 + (y – 1)2 on the set D = (x, y): x2 + y2 ≤ 1.


93. Find the absolute extreme values taken on by f(x, y) = (x – 2)2 + y2 on the set D = (x, y): 0 ≤ x ≤ 1,x2 ≤ y ≤ 4x.

94. Find the absolute extreme values taken on by f(x, y) = (x – y)2 on the set D = (x, y): 0 ≤ x ≤ 2,0 ≤ y ≤ 4 –x.

95. Find the distance from the point (1, 1, 1) to the sphere x2 + y2 + z2 = 4.

96. Find the distance from the point (½, 1, ½) to the sphere (x – 1)2 + y2 + z2 = 2.

97. Find the stationary points and the local extreme values for f(x, y) = 5xy – 7x2 – y2 + 3x – 6y + 2.

98. Find the stationary points and the local extreme values for f(x, y) = x3 + y2 – 12x – 6y + 7.

99. Find the stationary points and the local extreme values for f(x, y) = x2 + 3xy + 3y2 – 6x + 3y – 6.

100. Find the stationary points and the local extreme values for f(x, y) = x2 – xy + y2 + 2x + 2y – 4.

101. Find the stationary points and the local extreme values for f(x, y) = x2 – 2y2 – 6x + 8y + 3.

102. Find the stationary points and the local extreme values for f(x, y) = x2 + 3xy + y2 – 10x – 10y.

103. Find the stationary points and the local extreme values for f(x, y) = 2x2 + y2 – 4x – 6y.

104. Find the stationary points and the local extreme values for f(x, y) = x3 – 9xy + y3.

105. Find the stationary points and the local extreme values for f(x, y) = x2 + 31

y3 – 2xy – 3y.

106. A rectangular box, open at the top, is to contain 256 cubic inches. Find the dimensions of the box for whichthe surface area is a minimum.

107. Find the point on the plane 2x – 3y + z = 19 that is closest to (1, 1, 0).

108. Find the shortest distance to 2x + y – z = 5 from (1, 1, 1).

109. An open rectangular box containing 18 cubic inches is to be constructed so that the base material costs 3cents per square inch, the front face costs 2 cents per square inch, and the sides and back each cost 1 cent persquare inch. Find the dimensions of the box for which the cost of construction will be a minimum.

110. Find the points on z2 = x2 + y2 that are closest to (2, 2, 0).

111. Find the point on x + 2y + z = 1 that is closest to the origin.

112. Find the maximum product of x, y, and z where x, y, and z are positive numbers such that 4x + 3y + z = 108.

113. Find the maximum sum of 9x + 5y + 3z if x, y, and z are positive numbers such that xyz = 25.

114. Find the maximum product of x2yz if x, y, and z are positive numbers such that 3x + 2y + z = 24.

15.6 Maxima and Minima with Side Conditions

115. Maximize xy on the ellipse 194

22

=+yx

.


116. Minimize xy on the ellipse 194

22

=+yx

.

117. Maximize x2y on the circle x2 + y2 = 4.

118. Minimize x2y on the circle x2 + y2 = 4.

119. Maximize xyz2 on the sphere x2 + y2 + z2 = 4.

120. Use the Lagrange multiplier method to find the point on the surface z = xy + 1 that is closest to the origin.

121. Use the Lagrange multiplier method to find the point on the plane x + 2y + z = 1 that is closest to the point(1, 1, 0).

122. Use the Lagrange multiplier method to find three positive numbers whose sum is 12 for which x2yz is amaximum.

123. Use the Lagrange multiplier method to find the maximum for x2 + y2 + z2 if x + 2y + 2z = 12.

124. An open rectangular box is to contain 256 cubic inches. Use the Lagrange multiplier method to find thedimensions of the box that uses the least amount of material.

125. An open rectangular box containing 18 cubic inches is constructed of material costing 3 cents per square inchfor the base, 2 cents per square inch for the front face, and 1 cent per square inch for the sides and back. Usethe Lagrange multiplier method to find the dimensions of the box for which the cost of construction is aminimum.

126. Use the Lagrange multiplier method to find the maximum possible volume for a rectangular box inscribed inthe ellipsoid 2x2 + 3y2 + 4z2 = 12.

127. Use the Lagrange multiplier method to find the maximum possible volume for a rectangular box inscribed inthe ellipsoid 2x2 + 3y2 + 6z2 = 18.

128. Use the Lagrange multiplier method to find the point on the plane 2x – 3y + z = 19 that is closest to (1, 1, 0).

129. Use the Lagrange multiplier method to find the shortest distance from (1, 1, 1) to 2x + y – z = 5.

130. Use the Lagrange multiplier method to find the points on z2 = x2 + y2 that are closest to (2, 2, 0).

131. Use the Lagrange multiplier method to find three positive numbers whose sum is 36 and whose product is aslarge as possible.

132. Use the Lagrange multiplier method to find three positive numbers whose product is 64 and whose sum is assmall as possible.

133. Use the Lagrange multiplier method to find three positive numbers x, y, and z whose product is as large aspossible given that 2x + 2y + z = 84.

134. The base of a rectangular box costs three times as much per square foot as do the sides and top. Use theLagrange multiplier method to find the dimensions of the box with least cost if the box is to contain 54 cubicfeet.

15.7 Differentials

135. Find the differential df given that 3 22 2),,( yxezyxf z += .

136. Find the differential df given that f(x, y, z) = x2 + 3xy – 2y2 + 3xz + z2.


137. Find the differential df given that f(x, y, z) = z4 – 3yz2 + y sin z.

138. Find the differential df given that f(x, y, z) = 3x2 + y2 + z2 – 3xy + 4xz – 15.

139. Find the differential df given that f(x, y, z) = x sin–1 y + x2y.

140. Compute ∆u and du for u = 2x2 + 3xy – y2 at x = 2, y = –2, ∆x = –0.2, ∆y = 0.1.

141. The radius and height of a right-circular cylinder are measured with errors of at most 0.1 inches. If the heightand radius are measured to be 10 inches and 2 inches, respectively, use differentials to approximate themaximum possible error in the calculated value of the volume.

142. The power consumed in an electrical resistor is given by P = E2 / R watts. Suppose E = 200 volts and R = 8ohms, approximate the change in power if E is decreased by 5 volts and R is decreased by 0.20 ohm.

143. Let yxyxf 2),( += . Use a total differential to approximate the change in f(x, y) as (x, y) varies from (3, 5)

to (2.98, 5.1).

144. The legs of a right triangle are measured to be 6 and 8 inches with a maximum error of 0.10 inches in eachmeasurement. Use differentials to estimate the maximum possible error in the calculated value of thehypotenuse and the area of the triangle.

145. Find the differential df given that 3 1ln),( xyyxf += .

15.8 Restructuring a Function from its Gradient

146. Determine whether the vector function x2y3 i + x3y2 j is the gradient ∇f(x, y) of a function everywheredefined. If so, find such a function.

147. Determine whether the vector function e2x i – e–2y j is the gradient ∇f(x, y) of a function everywheredefined. If so, find such a function.

148. Determine whether the vector function 2yx

i –

+ 2

3

2

yyx

j is the gradient ∇f(x, y) of a function

everywhere defined. If so, find such a function.

149. Determine whether the vector function sin (x + y – 2z) i – sin (x + y – 2z) j + 2 cos (x + y – 2z) k is thegradient ∇f(x, y, z) of a function everywhere defined. If so, find such a function.

150. Determine whether the vector function )3(22 zyex z − i – )1(

23 +zex j + )2(23 xyzex z − k is the gradient

∇f(x, y, z) of a function everywhere defined. If so, find such a function.

151. Determine whether the vector function

++

++ 22222 11

2

zyx

yz

yx

xi +

+

+++ 22222 11

2zyx

xzyxy

j +

2221 zyx

xy

+k is the gradient ∇f(x, y, z) of a function everywhere defined. If so, find such a function.

15.9 Exact Differential Equations

152. Verify that the equation 2xy + (1 + x2)y′ = 0 is exact, then solve it.

153. Verify that the equation yexy + xexyy′ = 0 is exact, then solve it.


154. Verify that the equation (cos y + y cos x) + (sin x – x sin y) y′ = 0 is exact, then solve it.

155. Verify that the equation 0')3(33 23 =+− yexyxe xx is exact, then solve it.

156. Verify that the equation 0')ln23(2

=+++ yy

xxxyx is exact, then solve it.

157. Solve (y + x4) – xy′ = 0.

158. Solve (x2 + y2 + x) + xyy′ = 0.

159. Solve (2xy4ey + 2xy3 + y) + (x2y4ey – x2y2 – 3x)y′ = 0.

160. Solve (y + ln x) – xy′ = 0.

161. Find the integral curve of (x + sin y) + x(cos y – 2y)y′ = 0 that passes through (2, π).

162. Find the integral curve of (y2 – y) + xy′ = 0 that passes through (–1, 2).

163. Find the integral curve of (2y – 3x) + xy′ = 0 that passes through (–5, 3).



1. (2x + x2y)exy i + x3exy j

2. (6x2 + 5y) i + (5x – 4y) j

3. [y sin(xy2) + xy3 cos(xy2)] i+ [x sin(xy2) + 2x2y2 cos(xy2)] j

4. 4xyz i + 2x2z j + 2x2y k

5.2/3)(2

1

zyx ++−

(i + j + k)

6. (3x2y – y2z) i + (x3 + z2 – 2xyz) j + (2yz – xy2) k

7.x

e y

i + ey ln xz j + z

e y

k

8. [2x cos(x + y) – (x2 – y) sin(x + y)] i– [cos(x + y) – (x2 – y) sin(x + y)] j

9. ex + y + z (i + j + k)

10.22 yx

x

+i +

22 yx

y

+j

11. 2 i + 14 j

12.252

i – 2511

j

13. i + j

14. e sin 3i + e cos 3j + 4e cos 3k

15.61

i + 61

j + 32

k

16. (a) 2

3

r(xi + yj + zk)

(b) rr

2cos2

(xi + yj + zk)

(c)

+

+

re

rr 14

22 (xi + yj + zk)

17.292

235 −

18.12516

19.26

7

20.13

2 2e−

21. –4/5

22.132

233 π+

23.52

3

52−

π

24. 3

25.2

333 +

26.2

63 −

27.11

12π

28.3

6−

29.782

1

26

60+

30.5

58

31. (a) 271 °C

(b) 56

(c) u = 54

i – 53

j ; 5

32. u = j ; 1

33. u = 2

2−i –

22

k ; 4

34. u = 3

1(i – j – k); 3

35. et(t4 + 4t3)

36. 2t + 4t3 + 6t5


37. 6ω cos 2ω t – 4ω2 t – 6ω2 t cos ω t– 6ω sin ω t – 4ω cos ω t

38. 2t cos(t2 – sin t) + (t2 cos t – 2t3) sin (t2 – sin t)

39.te

ttet

t

22

2

sin

sincos

+

++

40. 6t5ecos t – t6ecos t sin t

41. ex cos x + xex cos x – xex sin x

42. y2 cos y + 2y sin y

43. 360

44.21

1

t+

45. t2et cos t2et + 2tet cos t2et + 3t2

46.( )

60335 π−

47. )cos()sin( ttt sesseesz −− +=

∂∂

)cos()sin( 2 ttt sessesetz

+=∂∂ −

48. 22

2232 )ln(tan)(4vu

vuvuuuz

+++

=∂∂

22

22232 )ln(sec)(2vu

vuvuu+

+++

22

22322 )ln(tan)(6vu

vuvuvvz

+++

=∂∂

22

22232 )ln(sec)(2vu

vuvuv+

+++

49. 0=∂∂rz

; θθ

2cos=∂∂z

50.22

2

sin)sin(

cos)(2)cos(2

uvvuuv

uvvuuvvuuvvz

++−

+++=∂∂

22

222

sin)sin(

cos)()cos(

uvvuuv

uvvuvvuvuz

++−

+++=∂∂

51. 2)(3)(2 tstssz

−++=∂∂

; 2)(3)(2 tststz

−−+=∂∂

52. vuvuuz

76)2(6 2 −++=∂∂

vuvuvz

47)2(3 2 +−+=∂∂

53. 2y • 3x2 f ′(x3 – y2) + 3x2(–2y) f ′(x3 – y2) = 0

54. 0=

′+

′−

=∂∂

+∂∂

xy

fxy

xy

fxy

yz

yxz

x

55.srssrr +

=∂∂

+=

∂∂ 2

;2 ωω

56. θθω

sin2sin zrr

+=∂∂

θθθω

cos2cos2 rzr +=∂∂

θω

sinrz

=∂∂

57.sr

s

rr

r

ee

esee

er 22

2

22

2 2,

2

+=

∂∂

+=

∂∂ ωω

58. 4e2t

59. 0,6,2 =∂∂

=∂∂

=∂∂

vu

ut

tωωω

60. vuuvur

cossinsin3cos2 −=∂∂ω

vuuvvu

sincoscos3sin2 +−=∂∂ω

61. sr

sec=∂∂ω

; sss

sectan=∂∂ω

62. 222 rsrr

+=∂∂ω

; srs

22=∂∂ω

63.e

exz

−=

∂∂

24

;e

eyz

−=

∂∂

22

64. uvvuuvuvuuz

cossin27 236 ++=∂∂

uvuvuvz

cos3 327 +=∂∂

65.

0)(

2222222/3222

222222

2

2

2

2

2

2

=++

−+−+−=

∂∂

+∂∂

+∂∂

zyx

zzyyxx

zu

yu

xu


66. fx2(cos2 θ + sin2 θ) + fy

2(cos2 θ + sin2 θ) =22

∂∂

+

∂∂

yz

xz

67. 8x – 18y – z + 30 = 0

68. πe2y + z – πe2 = 0

69. 4x – 5y + z + 4 = 0

70. 4x – 2y – z – 5 = 0

71. x – 2y – z + 2π = 0

72. 12x – 4y – z – 14 = 0

73. normal vector: i + knormal line: x = t, z = ttangent plane: x + z = 0

74. normal vector: 4 i + 3 j – 2 knormal line: x = 4t, y = –2 + 3t, z = 3 – 2ttangent plane: 4x + 3 y – 2z + 12 = 0

75. normal vector: 3 i – 3 j + knormal line: x = 1 + 3t, y = –2 – 3t, z = 2 + ttangent plane: 3x + 3y – z – 11 = 0

76. normal vector: 3 i + 2 j – knormal line: x = 2 + 3t, y = 3 + 2t, z = 6 – ttangent plane: 3x + 2y – z – 6 = 0

77. 4x – 3y + 6z – 10 = 06x – 8y + 3z + 4 = 0

78. (–½ , –1, 14)

79. (½ , ¾ , 131/16)

80. stationary point: (0, 3/2, –9/4) local minimum

81. stationary point: (1, 0 –1) local minimum

82. stationary point: (2/3, 2/3, 14/3) local minimum

83. stationary point: (4/3, 2/3, –10/3) local minimum

84. No stationary points

85. stationary point: (4/11, –5/11, 15/11)local maximum

86. stationary point: (10/3, –5/3, -44/3)local minimum

87. stationary point: (1, –2, –4) local minimum

88. stationary point: (–10/3, –5/3, –22/3) localminimum

89. stationary points: (–π/2, 0), (–π/4, 0), (0, 0),(π/4, 0). (π/2, 0)no local extrema

90. absolute minimum at (0, 0, 0)

absolute maximum at (4, 4, 24 )

91. absolute maximum at (–2, 0, 8) and (2, 0, 8)no absolute minimum

92. absolute maximum at (5

2−,

5

1−, 526 + )

absolute minimum at (5

2,

5

1, 526 − )

93. absolute maximum at (1, 4, 17)absolute minimum at (0.83512, 0.6974,1.8433)

94. absolute maximum at (0, 4, 16)absolute minimum at (x, x, 0), 0 ≤ x ≤ 2

95.3

32=== zyx , distance: 32 −

96.2

622 −

97. f(–8, –23) = 59, relative maximum

98. f(2, –3) = –32, relative minimum;(–2, –3) is a saddle point

99. f(15, –8) = –63, relative minimum

100. f(–2, –2) = –8, relative minimum

101. (3, 2) is a saddle point

102. (2, 2) is a saddle point

103. f(1, 3) = –11, relative minimum

104. f(3, 3) = –27, relative minimum;(0, 0) is a saddle point

105. f(3, 3) = –9, relative minimum;(–1, –1) is a saddle point

106. 8 in. by 8 in. by 4 in.

107. (27/7, –23/7, 10/7)


108.26

109. 2 in. by 3 in. by 3 in.

110. (1, 1, 2 ) and (1, 1, 2− )

111. (1/6, 1/3, 1/6)

112. x = 9, y = 12, z = 36; product = 3888

113. x = 5/3, y = 3, z = 5; sum = 45

114. x = 4, y = 3, z = 6; product = 288

115. (x, y) =

−−

2

3,2 or

−−

2

3,2 ; xy = 3

116. (x, y) =

−2

3,2 or

−−

2

3,2 ; xy = –3

117. (x, y) =

−−

3

2,

3

22or

−3

2,

3

22;

x2y = 9

316

118. (x, y) =

−−

3

2,

3

22or

−3

2,

322

;

x2y = 9

316−

119. (x, y, z) = (–1, –1, 2± ), (1, 1, 2± ); xyz2 = 2

120. (0, 0, 1)

121. (2/3, 1/3, –1/3)

122. x = 6, y = 3, z = 3; product = 324

123. x = 4/3, y = 8/3, z = 8/3; sum = 16

124. 8 in. by 8 in. by 4 in.

125. 2 in. by 3 in. by 3 in.

126. x = 2 , y = 3

2, z = 1; volume =

3616

127. x = 2 , y = 3 , z = 1; volume = 68

128. [27/7, –23/7, 10/7]

129.26

130. (1, 1, 2 ) and (1, 1, 2− )

131. x = y = z = 12; product = 1728

132. x = y = z = 4; sum = 12

133. x = 14, y = 14, z = 28; product = 5488

134. 3 ft. by 3 ft. by 6 ft.

135. dyyx

yedx

yxxe zz

3/222

2

3/222

2

)(32

)(32

++

+

+ 2e2z(x2 + y2)1/3 dz

136. (2x + 3y + 3z)dx + (3x – 4y)dy + (3x + 2z)dz

137. 0dx + (sin z – 3z2)dy + (4z3 – 6yz + y cos z)dz

138. (6x – 3y + 4z)dx + (2y – 3x)dy + (2z + 4x)dz

139. (sin–1 y + 2xy)dx + dyxy

x

+

−

2

21

140. ∆u = 0.61; du = 0.60

141. 4.4π

142. decreased by 125 watts

143. ≈ 0.025

144. error in hypotenuse = 0.14; error in area = 0.7

145. dyxy

ydx

xyy

)1(3)1(3 ++

+

146. Cyx

yxf += 33

3),(

147. Ceeyxf yx ++= 22

21

21

),(

148. It is the gradient of Cy

yx

yxf +−=32

),(3

2

2

,

but f is only defined for y ≠ 0.

149. not a gradient

150. Cyxzyexzyxf z ++−=23),,(


151. f(x, y, z) = ln (x2 + y2 + 1) + tan –1 (xyz) + C

152. 12 +

=x

Cy

153. exy = C

154. x cos y + y sin x = C

155.313

+= −xCey

156. 3x + x2 ln xy = C

157. Cxxy −= 4

31

158. 3x4 + 4x3 + 6x2y2 = C

159. Cyx

yx

ex y =++3

22

160. y = Cx – ln x – 1

161. ½ x2 + x sin y – y2 = 2 – π2

162.2/1+

=x

xy

163.2

3 200

x

xy

+=

191

CHAPTER 16

Double and Triple Integrals

16.1 Multiple-Sigma Notation

1. Let P1 = x0, x1, … , xm be a partition of [a1, a2]Let P2 = y0, y1, … , yn be a partition of [b1, b2]Let P3 = z0, z1, … , zq be a partition of [c1, c2]Take ∆xi = xi – xi – 1, ∆yj = yj – yj – 1, ∆zk = zk – zk – 1

Evaluate the sum,∑∑= =

m

i

q

kki zx

1 1

∆∆∆∆


Evaluate the sum,∑∑∑= = =

m

i

n

j

q

kkji zyx

1 1 1

∆∆∆∆∆∆



−+n

j

q

kkjkk zyzz

1 11 )( ∆∆∆∆



−m

i

n

jji yx

1 1

)34( ∆∆∆∆


Evaluate the sum,∑∑∑= = =

−+m

i

n

j

q

kkjijj zyxyy

1 1 11 )( ∆∆∆∆∆∆

16.2 The Double Integral

6. Take f(x, y) = 3x – 2y on R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, and P as the partition P = P1 × P2. Find Lf (P) and Uf (P) ifP1 = 0, ¼, ½, ¾, 1 P2 = 0, ½ ,1, 3/2, 2.

7. Take f(x, y) = 3x – 2y on R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, and P as the partition P = P1 × P2. P1 = x0, x1, … , xm is anarbitrary partition of [0, 1], and P2 = y0, y1, … , yn is an arbitrary partition of [0, 2].(a) Find Lf (P) and Uf (P)

(b) Evaluate the double integral ∫∫ −R

dydxyx )23( using part (a).


8. Take f(x, y) = 2x(y – 1) on R: 0 ≤ x ≤ 2, 0 ≤ y ≤ 1, and P as the partition P = P1 × P2. Find Lf (P) and Uf (P) ifP1 = 0, 1, 3/2, 2 P2 = 0, ½ ,1.

9. Take f(x, y) = 2x(y – 1) on R: 0 ≤ x ≤ 2, 0 ≤ y ≤ 1, and P as the partition P = P1 × P2. P1 = x0, x1, … , xm isan arbitrary partition of [0, 2], and P2 = y0, y1, … , yn is an arbitrary partition of [0, 1].(a) Find Lf (P) and Uf (P)


dydxyx )1(2 using part (a).

10. Take f(x, y) = 2x2 – 3y2 on R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and P as the partition P = P1 × P2. Find Lf (P) and Uf (P) ifP1 = 0, ½ , ¾, 1 P2 = 0, ½ ,1.

11. Take f(x, y) = 2x2 – 3y2 on R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and P as the partition P = P1 × P2. P1 = x0, x1, …, xm is anarbitrary partition of [0, 2], and P2 = y0, y1, … , yn is an arbitrary partition of [0, 1].(a) Find Lf (P) and Uf (P)


dydxyx ))32( 22 using part (a).

16.3 The Evaluation of a Double Integral by Repeated Integrals

12. Evaluate ∫∫ −ΩΩ

dydxyx )( 22 taking Ω: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.

13. Evaluate ∫∫ΩΩ

dydxxyy cos taking Ω: 0 ≤ x ≤ 1, 0 ≤ y ≤ π.

14. Evaluate ∫∫ΩΩ

dydxyx 22 taking Ω: –1 ≤ x ≤ 1, 0 ≤ y ≤ π/2.


dydxxy )3( 2 taking Ω: 2 ≤ x ≤ 4, 0 ≤ y ≤ 3.


dxdyx )1( taking Ω: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2.

17. Evaluate ∫∫ +

ΩΩ

dxdye yx taking Ω: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.

18. Evaluate ∫∫ +ΩΩ

dydxyxx 2 taking Ω: 0 ≤ x ≤ 1, 0 ≤ y ≤ 3.


dydxyx )( 2 taking Ω: 0 ≤ x ≤ 3, 1 ≤ y ≤ 4.

20. Evaluate ∫∫−ΩΩ

dxdyx

1

12

taking Ω: 0 ≤ x ≤ ½, 0 ≤ y ≤ 2.


dxdyy

1

12

taking Ω: 0 ≤ x ≤ 4, 0 ≤ y ≤ 1.

22. Evaluate ∫∫−ΩΩ

dydxx

4

12

taking Ω: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2.

Double and Triple Integrals 193

23. Calculate the average value of f(x, y) = x cos y over the region Ω: 0 ≤ x ≤ 1, 0 ≤ y ≤ π/4.

24. Calculate the average value of f(x, y) = ex over the region Ω: 0 ≤ x ≤ ln y, 0 ≤ y ≤ e.


dydxx )2( 2 taking Ω: the bounded region between y2 = 2x and y2 = 8 – 2x.


dydxxxy )2( 2 taking Ω: the bounded region between y = x3 and y = x2.

27. Evaluate ∫ ∫1

0

1

2

y

x dydxe by first sketching the region of integration Ω then changing the order of integration.

28. Evaluate ∫ ∫ +1

0

222

)(x

xdydxyx by first sketching the region of integration Ω then changing the order of

integration.

29. Evaluate ∫ ∫−1

0

12

0

2

y

dydxx by first sketching the region of integration Ω then changing the order of

integration.


1 0

2 lnx

dxdyxy .


0

2

2

2 )cos(y

dydxx by first expressing it as an equivalent double integral with order of integration

reversed.

32. Evaluate ∫ ∫ +1

0 0

22 x

dxdyyxy .

33. Sketch the region of integration Ω and express ∫ ∫4/

0

cos

sin ),(

π x

xdxdyyxf as an equivalent double integral with

order of integration reversed.

34. Sketch the region of integration Ω and express ∫ ∫−

−

1

0

2

1 ),(

y

ydydxyxf as an equivalent double integral with

order of integration reversed.

35. Use a double integral to find the area bounded by y = x2 and y = x .

36. Use a double integral to find the area bounded by x = y – y2 and x + y = 0.

37. Find the volume of the solid bounded by y = x2 – x, y = x, z = 0, and z = x + 1.

38. Find the volume of the solid in the first octant bounded by y = x2/4, z = 0, y = 4, x = 0, and x – y + 2z = 2.

39. Find the volume of the solid bounded by x = 0, z = 0, z = 4 – x2, y = 2x, and y = 4.

40. Find the volume of the solid bounded by y = x2 – x + 1, y = x + 1, z = 0, and z = x + 1.

41. Find the volume of the solid in the first octant bounded by z = x2 + y2, z = 0, and x + y = 1.

42. Find the volume of the solid in the first octant bounded by z = 4 – y2, z = 0, x = 0, and y = x.

43. Find the volume of the solid in the first octant bounded by x2 + y2 = 4, y = z, and z = 0.


44. Find the volume bounded by x2 + y2 = 1 and y2 + z2 = 1.

16.4 Double Integrals in Polar Coordinates

45. Calculate ∫ ∫−

− +−2

2

4

0

)(2

22

x yx dxdye by changing to polar coordinates.


−− +

3

0

9

9 22

2

2

1y

ydydx

yx by changing to polar coordinates.


−3

3

9

0

2

x

dxdyy by changing to polar coordinates.


+2

0

4

0

222

)(x

dxdyyx by changing to polar coordinates.

49. Integrate f(x, y) = 2(x + y) over Ω, the region bounded by x2 + y2 = 9 and x ≥ 0.

50. Find the volume in the first octant bounded by x = 0, y = 0, and z = 0, the plane z + y = 3, and the cylinderx2 + y2 = 4.

51. Use a double integral in polar coordinates to find the volume in the first octant of the solid bounded byx2 + y2 = 4, y = z, and z = 0.

52. Use a double integral in polar coordinates to find the volume of the solid bounded by x2 + y2 = 5 – z, andz = 1.

53. Use a double integral in polar coordinates to find the volume of the solid between the sphere x2 + y2 + z2 = 9and the cylinder x2 + y2 = 1.

54. Use a double integral in polar coordinates to find the volume of the solid bounded by the paraboloidz = 4 – x2 – y2 and z = 0.

55. Use a double integral in polar coordinates to find the volume of the solid in the first octant bounded by the

ellipsoid 9x2 + 9y2 + 4z2 = 36 and the planes x = y3 , x = 0, and z = 0.

56. Use a double integral in polar coordinates to find the volume bounded by the sphere x2 + y2 + z2 = 16 andthe cylinder (x – 2)2 + y2 = 4.

57. Use a double integral in polar coordinates to find the volume bounded by z = 0, x + 2y – z = –4, and thecylinder x2 + y2 = 1.

58. Use a double integral in polar coordinates to find the volume that is inside the sphere x2 + y2 + z2 = 9, outsidethe cylinder x2 + y2 = 4, and above z = 0.

59. Use a double integral in polar coordinates to find the area bounded by the limaςon r = 4 + sin θ.

60. Use a double integral in polar coordinates to find the area that is inside r = 1 + cos θ and outside r = 1.

61. Use a double integral in polar coordinates to find the area that is inside r = 3 sin 3θ.


16.5 Some Applications of Double Integration

62. Find the center of mass of a plate of mass M bounded by x = 0, x = 4, y = 0, and y = 3 if its density is givenby λ(x, y) = k(x + y2).

63. Find the center of mass of a plate of mass M bounded by y2 = 4x, x = 4, and y = 0 if its density is given byλ(x, y) = ky.

64. Find the center of mass of a homogeneous plate of mass M bounded by x = 0, x = 4, y = 0, and y = 3 if itsdensity is given by λ(x, y) = kx2y.

65. Find the center of mass of a plate of mass M bounded by y = sin x, y = 0, and 0 ≤ x ≤ π if its density isproportional to the distance from the x-axis.

66. Find the center of mass of a plate of mass M bounded by r = a cos θ, 0 ≤ θ ≤ π/2 if its density isproportional to the distance from the origin.

67. Find the mass of a homogeneous plate of mass M in the first quadrant that is inside r = 8 cos θ and outsider = 4 if the density of the region is given by λ(r, θ) = sin θ.

68. Find the mass of a homogeneous plate of mass M cut from the circle x2 + y2 = 36 by the line x = 3 if its

density is given by22

2

),(yx

xyx

+=λ .

69. Find the centroid of the region bounded by x = 4y – y2 and the y-axis.

70. Find the centroid of the region bounded by y = 4 – x , x = 0, and y = 0.

71. Find the centroid of the region bounded by y = x2 and the line y = 4.

72. Find the centroid of the region bounded by y = x3, x = 2, and the y = 0.

73. Find the centroid of the region bounded by y = x2 – 2x and y = 0.

74. Find the centroid of the region bounded by x2 = 8y, y = 0, and x = 4.

75. Find the centroid of the region bounded by 24 xy −= and y = 0.

76. Find the centroid of the region enclosed by the cardiod r = 2 – 2 cos θ .

77. Find the moments of inertia Ix, Iy, Iz of the plate of mass density λ(x, y) = 6x + 6y + 6 occupying the regionΩ: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2x.

78. Find the moments of inertia Ix, Iy, Iz of the plate of mass density λ(x, y) = y + 1 occupying the region Ωbounded by y = x, y = –x, y = 1.

16.6 Triple Integrals

79. Evaluate ∫ ∫ ∫1

0 0 0

z yzdzdydxx .

80. Evaluate ∫∫∫T

dzdydxx where T is the solid in the first octant bounded by x + y + z = 3 and the coordinate

planes.


16.7 Reduction to Repeated Integrals


dzdydxy z where T is the solid in the first octant bounded by y = 0, 21 xy −= , and z = x.


dzdydxy where T is the solid in the first octant bounded by y = 1, y = x, z = x + 1, and the

coordinate planes.

83. Use a triple integral to find the volume of the solid bounded by z = 0, y = 4 – x2, y = 3x, and z = x + 4.

84. Use a triple integral to find the volume of the solid whose base is the region in the xy-plane bounded by y = x2 – x + 1 and y = x + 1, and whose height is given by z = x + 1.

85. Use a triple integral to find the volume of the solid bounded by z = x2 + y2, y = x2, z = 0, and y = x.

86. Use a triple integral to find the volume of the solid bounded by y = x2, x = y2, z = 0, and z = 3.

87. Use a triple integral to find the volume of the solid bounded by 1

42 +

=y

z z = 0, y = x, y = 3, and x = 0.

88. Use a triple integral to find the volume of the solid bounded by z = 0, y = x2 – x, y = x, and z = x + 1.

89. Use a triple integral to find the volume of the solid bounded by x2 = 4y, y + z = 1, and z = 0.

90. Use a triple integral to find the volume of the solid bounded by y2 = 4x, z = 0, z = x, and x = 4.

91. Find the centroid of the tetrahedron bounded by 2x + 2y + z = 6 and the coordinate planes.

92. Use a triple integral to find the volume of the solid in the first octant bounded by z = y, y2 = x, and x = 1.

93. Use a triple integral to find the volume of the solid in the first octant bounded by the cylinder x = 4 – y2, andthe planes z = y, x = 0, and z = 0.

94. Use a triple integral to find the volume of the solid in the first octant bounded by z = x2 + y2, y = x, andx = 1.

95. Use a triple integral to find the volume of the solid in the first octant bounded by the cylinder x = 4 – y2, andthe planes y = x, z = 0, and x = 0.

96. Use a triple integral to find the volume of the tetrahedron bounded by the plane 3x + 6y + 4z = 12 and thecoordinate planes.

97. Find the centroid of the solid bounded below by the paraboloid z = x2 + y2 and above by the plane z = 4.

98. Find the centroid of the solid bounded by z = 4y2, z = 4, x = –1, and x = 1.

99. Find the center of mass of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1) if the massdensity is proportional to the distance from the yz-plane.

100. Find the moments of inertia about its three edges of a homogeneous box of mass M with edges of lengths a,b, and c.

101. Find the moments of inertia Ix, Iy, Iz of the homogeneous tetrahedron bounded by the coordinate planes andthe plane x + y + z = 1.


102. Find the moment of inertia about the y-axis of the homogeneous solid bounded by z = 1 – x2, z = 0, y = –1,and y = 1.

103. Find the moment of inertia about the z-axis of the cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 if the mass density isλ(x, y, z) = kz.

16.8 Triple Integrals in Cylindrical Coordinates

104. Find the cylindrical coordinates (r, θ, z) of the point with rectangular coordinates (2, 1, 2).

105. Find the rectangular coordinates of the point with cylindrical coordinates (2, π/4, 2).

106. Evaluate ∫ ∫ ∫4/

0

cos

1 1 22 1π θ

θr

ddrdzzr

.

107. Evaluate ∫ ∫ ∫π

θ2

0

2

1

5

0 ddrdzrez .

108. Use cylindrical coordinates to find the volume of the solid in the first octant bounded by the coordinateplanes, the cylinder x2 + y2 = 4, and the plane z + y = 3.

109. Use cylindrical coordinates to find the volume and centroid of the cylinder bounded by x2 + y2 = 4, z = 0, andz = 4.

110. Use cylindrical coordinates to find the volume inside x2 + y2 = 4x, above z = 0, and below x2 + y2 = 4z.

111. Use cylindrical coordinates to find the volume of the solid cut from the sphere x2 + y2 + z2 = 4, boundedbelow z = 0, and on the sides by the cylinder x2 + y2 = 1.

112. Use cylindrical coordinates to evaluate ∫∫∫ +T

dzdydxyx 22 where T is the solid bounded by z = x2 + y2

and z = 8 – x2 – y2.

113. Use cylindrical coordinates to find the volume and centroid of the solid bounded by the paraboloidz = x2 + y2 and the plane z = 4.

114. Use cylindrical coordinates to find the volume and centroid of the solid bounded by 22 yxz += , and the

plane z = 1.

16.9 Triple Integrals in Spherical Coordinates

115. Find the spherical coordinates (ρ, θ, φ) of the point with rectangular coordinates (2, –1, 1).

116. Find the rectangular coordinates of the point with spherical coordinates (2, 2π/3, π/4).

117. Find the spherical coordinates of the point with cylindrical coordinates (1, π/6, 2).

118. Find the cylindrical coordinates of the point with spherical coordinates (3, π/3, π/6).

119. Evaluate ∫ ∫ ∫2/

0 0

2

sin2

2 coscossin π θ

φθφρθφφρ ddd .

120. Evaluate ∫ ∫ ∫π π

θφρφφρ2

0 0

2

1

24 sincos ddd .


121. Use spherical coordinates to find the mass of the ball bounded by x2 + y2 + z2 ≤ 9 if its density is given by

λ(x, y, z) = 222

2

zyxz

++.

122. Use spherical coordinates to find the mass of the ball bounded by x2 + y2 + z2 = 2z if its density is given by

λ(x, y, z) = 222 zyx ++ .

123. Use spherical coordinates to find the mass and center of mass of the ball bounded by x2 + y2 + z2 ≤ 4 if itsdensity is given by λ(x, y, z) = x2 + y2.

124. Use spherical coordinates to find the mass of a ball of radius 4 if its density is proportional to the distancefrom its center. Take k as the constant of proportionality.

16.10 Jacobians; Changing Variables in Multiple Integration

125. Find the Jacobian of the transformation x = 2u + 3v, y = –u + 4v.

126. Find the Jacobian of the transformation x = u2v, y = u2 + v2.

127. Find the Jacobian of the transformation x = u ln v, y = ln u + v.

128. Find the Jacobian of the transformation x = u2 – v2, y = uv.

129. Take Ω as the parallelogram bounded by x + y = 0, x + y = 1, x – y = 0, x – y = 2.

Evaluate ∫∫ +ΩΩ

dydxyx )( .

130. Take Ω as the parallelogram bounded by x – y = 0, x – y = π, x + 2y = 0, x + 2y = ½π.

Evaluate ∫∫ −ΩΩ

dydxyx )( 22 .

131. Take Ω as the parallelogram bounded by x – y = 0, x – y = π, x + 2y = 0, x + 2y = ½π.

Evaluate ∫∫ΩΩ

dydxyx 2 2 .

132. Take Ω as the parallelogram bounded by x + y = 0, x + y = 1, x – y = 0, x – y = 2.

Evaluate ∫∫ΩΩ

dydxx 2sin .



1. (a2 – a1)(c2 – c1)

2. (a2 – a1)(b2 – b1)(c2 – c1)

3. (b2 – b1)(c22 – c1

2)

4. 4n(a2 – a1) – 3m(b2 – b1)

5. (a2 – a1)(b22 – b1

2)(c2 – c1)

6. Lf (P) = –11/4 ; Uf (P) = 3/4

7. (a) Lf (P) = −−− )()(23

122

12 bbaa

−−− 21212 ))(( bbaa

−−− xbbaa ∆∆))((23

1212

ybbaa ∆∆))(( 1212 −−(b) –1

8. Lf (P) = –33/8 ; Uf (P) = –5/8

9. (a) ∑ ∑= =

− −=m

ij

n

jjiif yyxxPL

1 11 )1(2)( ∆∆∆∆

∑ ∑= =

− −=m

ij

n

jjiif yyxxPU

1 11 )1(2)( ∆∆∆∆

(b) –2

10. Lf (P) = –47/32 ; Uf (P) = 21/32

11. (a) ∑ ∑ ∑∑= = ==

− −=m

ij

m

i

n

jji

n

jjiif yyxyxxPL

1 1 1

2

1

21 32)( ∆∆∆∆∆∆∆∆

∑ ∑ ∑∑= = =

−=

−=m

ij

m

i

n

jji

n

jjiif yyxyxxPU

1 1 1

21

1

2 32)( ∆∆∆∆∆∆∆∆

(b) –1/3

12. 2/3

13. 2

14. π3/36

15. 84

16. –1

17. e2 – 2e + 1

18.5

361562

−

19. 9/2

20. π/3

21. π

22. π/3

23. π2

24. e2/2 – e + ½

25. 7744/105

26. 1/120

27. )1(21 −e

28. 3/35

29. 4/3

30. 2 ln 2 – 3/4

31. 4sin41

32.12

122 −

33. ∫ ∫∫ ∫−−

+1

22

cos

0

2

2

0

sin

0

11

),( ),(y

dydxyxfdydxyxf

34. ∫ ∫∫ ∫−

−+

2

1

2

0

1

0

1

1 ),( ),(

x

xdxdyyxfdxdyyxf


35. 1/3

36. 4/3

37. 8/3

38. 232/15

39. 40/3

40. 8/3

41. 1/6

42. 4

43. 8/3

44. 16/3

45. )1(2

4−− eπ

46. 3π

47. 18

48. 2π

49. 36

50.38

3 −π

51. 8/3

52. 8π

53. 23

64π

54. 8π

55. 4π/3

56. )43(9

128−π

57. 4π

58. π3

510

59. 33π/2

60. 2 + π/4

61. 9π/4

62. (34/15, 39/20)

63. (8/3, 32/75)

64. (3, 2)

65. (π/2, 16/9π)

66. (3a5, 9a/40)

67. 16/3

68.2

393 +π

69. (8/5, 2)

70. (4/3, 4/3)

71. (0, 12/5)

72. (8/5, 16/7)

73. (1, −2/5)

74. (3, 3/5)

75. (0, 8/3π)

76.

−= 0 ,

35

),( yx

77. Ix = 12, Iy = 39/5, Iz = 99/5

78. Ix = 9/10, Iy = 3/10, Iz = 6/5

79. 1/16

80. 27/8

81. 1/30

82. 11/24

83. 625/12

84. 8/3

85. 3/35

86. 1

87. 2 ln 10

88. 8/3


89. 16/15

90. 256/5

91. (3/4, 3/4, 3/2)

92. 1/4

93. 4

94. 1/3

95. 4

96. 4

97. (0, 0, 8/3)

98. (0, 0, 12/5)

99. (2/5, 1/5, 1/5)

100. )(3

),(3

),(3

222222 baM

IcaM

IcbM

I cba +=+=+=

101. Ix = Iy = Iz = 1/30

102. Iy = 8/7

103. k/3

104.

− 2 ,

21

tan ,5 1

105. ( )2 ,2 ,2

106. ( )12ln82

1 +−+ π

107. 3π ( e5 – 1)

108. 3π − 8/3

109. 16π ; (0, 0, 2)

110. 6π

111. ( )3383

2−

π

112. 16π

113. 8π ; (0, 0, 8/3)

114. π/3 ; (0, 0, 3/4)

115.

− −−

6

1cos ,

21

tan ,6 11

116.

− 2 ,

26

,22

117.

−

5

2cos ,

6 ,5 1π

118.

233

,3

,23 π

119. 1/10

120. 124π/15

121. 12π

122. 8π/5, (0, 0, 8/7)

123. 256π/15

124. 256πk

125. 11

126. 4uv2 – 2u3

127. ln v – 1/v

128. 2u2 + 2v2

129. ½

130. 7π4/216

131. 5

961

π−

132. )1sin2sin3(sin21 −−−

203

CHAPTER 17

Line Integrals and Surface Integrals

17.1 Line Integrals

1. Integrate h (x, y) = x2 i + y j over(a) r(u) = u2 i – 2u j , u ∈ [0, 1](b) the line segment from (2, 3) to (1, 2).

2. Integrate h (x, y) = 2xy i + 3y2 j over(a) r(u) = eu i + e–u j , u ∈ [0, 2](b) the line segment from (1, 2) to (2, 3).

3. Integrate h (x, y) = (2xy – y) i + 3xy j over(a) r(u) = (1 – u) i + 2u j , u ∈ [0, 1](b) the line segment from (1, 1) to (2, 2).

4. Integrate h (x, y) = x–2 y–2 i – x–1 y–1 j over

(a) r(u) = 12 −u i + u+2 j, u ∈ [1, 3](b) the line segment from (2, 3) to (4, 5).

5. Integrate h (x, y) = 2y i – 3x j over the triangle with vertices (–2, 0), (2, 0), (0, 2) traversedcounterclockwise.

6. Integrate h (x, y) = ex – 2y i – e2x + y j over the line segment from (–1, 1) to (1, 2).

7. Integrate h (x, y) = (x2 + y) i + (2y2 – xy) j over the closed curve that begins at (–2, 0), goes along the x-axisto (2, 0), and returns to (–2, 0) by the upper part of the circle.

8. Integrate h (x, y) = 2xy2 i + (xy2 – 2x3) j over the square with vertices (0, 0), (1, 0), (1, 1), (0, 1) traversedcounterclockwise.

9. Integrate h (x, y, z) = xz2 i + y2z j + xy k over(a) r(u) = 2u i – u2 j + u3 k, u ∈ [0, 1](b) the line segment from (0, 0, 0) to (1, 1, 1).

10. Integrate h (x, y, z) = xex i + ez j + e–y k over(a) r(u) = 2u i – 3u j + 2u k, u ∈ [0, 1](b) the line segment from (0, 0, 0) to (1, 1, 1).

11. Calculate the work done by the force F(x, y, z) = xy i – y2 j – xyz k applied to an object that moves in astraight line from (0, 2, –1) to (2, 1, 1).

12. Calculate the work done by the force F(x, y, z) = x2 i + xy j + z k applied to an object that moves in astraight line from (–1, 1, 2) to (2, –1, –1).

13. An object of mass m moves from time t = 0 to t = 1 so that its position at time t is given by the vectorfunction r(t) = 2t i – t2 j. Find the total force acting on the object at time t and calculate the work done bythat force during the time interval [0, 1].


17.2 The Fundamental Theorem for Line Integrals

14. Calculate the line integral of h (x, y) = 2xy i + (x2 + y) j over the curve r(u) = 2 cos u i + sin u j,u ∈ [0, 2π].

15. Calculate the line integral of h (x, y) = (y2 + 2xy) i + (x2 + 2xy) j over the curve r(u) = sin u i + (2 – 2 cosu) j, u ∈ [0, π/2].

16. Calculate the line integral of h (x, y) = (sin y + y cos x) i + (sin x + x cos y) j over the straight-line segmentfrom (π/2, π/2) to (π, π).

17. Calculate the line integral of h (x, y, z) = 8xz i – 2yz j + (4x2 – y2) k over the curve r(u) = (1 + 3u) i +2u5/2 j + (2 + u) k, u ∈ [0, 1].

18. Calculate the line integral of h (x, y, z) = (y + z) i + (x + z) j + (x + y) k over the curve r(u) = u4 i +

u2

sin2π

j + 3u2 k, u ∈ [0, 1].

19. Calculate the work done by the force F(x, y, z) = z i + y j + x k applied to a particle that moves along thecurve r(u) = i + sin u j + cos u k for 0 ≤ u ≤ π/3.

20. Calculate the work done by the force F(x, y, z) = 3x2 i + yz2

j + 2z ln y k applied to a particle that moves

from the point (0, 1, 1) to the point (2, 2, 1).

17.4 Line Integrals with Respect to Arc Length

21. Evaluate ∫ ++C

x dyxedxxy )( 2 2 , where C is the line segment from (0, 0) to (1, 1).

22. Evaluate ∫ −C

dyxdxy 22 , where C is the line segment from (0, 1) to (1, 0).

23. Evaluate ∫ −C

dyydxxy 2 , where C is the line segment from (0, 0) to (2, 1).

24. Evaluate ∫ −−C

dyxydxyx 2 )( 22 , where C is the parabola y = 2x2 from (0, 0) to (1, 2).


dyxydxyx 4 )3( 2 , where C is the broken line path from (0, 0) to (2, 0) to (0, 4) to (0, 0).

26. Evaluate ∫ ++−C

yx dyxedxye )6( )3( , where C is the broken line path from (0, 0) to (1, 0) to (0, 2) to (0, 0).

27. Evaluate ∫ ++−−C

dzxydyxzdxyz )1( , where C is the circular helix r(t) = 2 cos t i + 2 sin t j + 3 t k

from (2, 0, 0) to (2, 0, 6π).

28. Evaluate ∫ +C

dydxyx 4 2 , where C is the curve r(t) = et i + e–t j for 0 ≤ t ≤ 1.


dzydyxdxz , where C is the helix r(t) = sin t i + 3 sin t j + sin2 t k for 0 ≤ t ≤ π/2.

30. Evaluate ∫ +−−C

dzzdyxdxy , where C is the circle x = cos t, y = sin t for 0 ≤ t ≤ 2π.

Line Integrals and Surface Integrals 205

31. Evaluate ∫ +−C

dzdyydxxy 3 8 4 , where C is the curve given by y = 2x, z = 3 from (0, 0, 3) to (3, 6, 3).

32. Evaluate ∫ +++C

dzyxdyydxyx )( cos sin , where C is the straight line x = y = z from (0, 0, 0) to (1, 1, 1).

33. Find the length and centroid of a wire shaped like the helix x = 3 cos u, y = 3 sin u, z = 4u, u ∈ [0, 2π].

34. Find the mass, center of mass, and moments of inertia Ix, Iy of a wire shaped like the first-quadrant portionof the circle x2 + y2 = a2 with mass density λ(x, y) = kxy.

35. Find the moment of inertia about the z-axis of a thin homogeneous rod of mass M that lies along the interval0 ≤ x ≤ L of the x-axis.

36. Find the center of mass and moments of inertia Ix, Iy, Iz of a wire shaped like the curve r(u) = (u2 – 1) j +

2u k, u ∈ [0, 1] if the mass density is λ(x, y, z) = 2+y .

17.5 Green’s Theorem

37. Use Green’s Theorem to evaluate ∫ ++C

dyxydxyx 4 )3( 2 , where C is the triangular region with vertices

(0, 0), (2, 0), and (0, 4). Assume that the curve is traversed in a counterclockwise manner.

38. Use Green’s Theorem to evaluate ∫ −+−C

dyyxdxyxy )( )2( 222 , where C is the boundary of the region

enclosed by y = x and y = x2. Assume that the curve C is traversed in a counterclockwise manner.

39. Use Green’s Theorem to evaluate ∫ ++C

dyydxyx 4 )3( 22 , where C is the boundary of the region enclosed

by x = y2 and y = x/2 traversed in a counterclockwise manner.

40. Use Green’s Theorem to evaluate ∫ +−C

dyxdxxy cos )sin( , where C is the boundary of the region with

vertices (0, 0), (π/2, 0), and (π/2, 1) traversed in a counterclockwise manner.

41. Use Green’s Theorem to evaluate ∫ ++−C

yx dyxedxye )6( )3( , where C is the boundary of the triangular

region with vertices (0, 0), (1, 0), and (0, 2) traversed in a counterclockwise manner.


dyxdxyxy )2( 22 , where C is the boundary of the region enclosed

by y = x + 1 and y = x2 + 1 traversed in a counterclockwise manner.


dyyxdxyx )sin( )3( 3 , where C is the boundary of the triangular



dyxydxyx )sin( )cosh( 2 , where C is the boundary of the region

enclosed by 0 ≤ x ≤ π and 0 ≤ y ≤ 1 traversed in a counterclockwise manner.


dyyxdxxy ) ( 22 , where C is the boundary of the region in the first

quadrant enclosed by y = 1 – x2 traversed in a counterclockwise manner.


46. Use Green’s Theorem to evaluate ] )3( [ 233∫ ++C

dyxyxdxy , where C is the boundary of the region

enclosed by y = x2 and y = x traversed in a counterclockwise manner.

47. Use Green’s Theorem to evaluate ) ( 22∫ +−C

dyxydxyx , where C is the circle x2 + y2 = 16 traversed in a

counterclockwise manner.

48. Use Green’s Theorem to evaluate ] )( 2[ 2∫ ++C

x dyxedxxy , where C is the boundary of the triangular


49. Use a line integral to find the area of the region in the first quadrant enclosed by y = x and y = x3.

50. Use a line integral to find the area of the region enclosed by y = 1 – x4 and y = 0.

51. Use Green’s Theorem to evaluate ∫ ++−

Cdyyxdx

xy

] )ln( tan2[ 221 , where C is the boundary of the circle

(x – 2)2 + y2 = 1 traversed in a counterclockwise manner.

52. Use a line integral to find the area of the region enclosed by x2 + 4y2 = 4.

53. Use a line integral to find the area of the region enclosed by y = x and y = x2.

54. Use a line integral to find the area of the region enclosed by y = sin x, y = cos x, and x = 0.

17.6 Parameterized Surfaces; Surface Area

55. Find the surface area cut from the plane z = 4x + 3 by the cylinder x2 + y2 = 25.

56. Find the surface area of that portion of the paraboloid z = x2 + y2 that lies below the plane z = 1.

57. Find the surface area cut from the plane 2x – y – z = 0 by the cylinder x2 + y2 = 4.

58. Find the surface area of that portion of the plane 3x + 4y + 6z = 12 that lies in the first octant.

59. Find the surface area of that portion of the paraboloid z = 25 – x2 – y2 for which z ≥ 0.

60. Find the surface area of that portion of the sphere x2 + y2 + z2 = 4 that lies inside the cylinder x2 + y2 = 2xand above the xy-plane.

61. Find the surface area of that portion of the paraboloid z = 25 – x2 – y2 that lies inside the cylinder x2 + y2 = 9and above the xy-plane.

62. Find the surface area of the surface )(1 22 xya

z −= cut by the cylinder x2 + y2 = a2 that lies above the xy-

plane.

63. Find the surface area of that portion of the cylinder y2 + z2 = 4 in the first octant cut out by the planes x = 0and y = x.

64. Find the surface area of that portion of the cylinder x2 + z2 = 25 that lies inside the cylinder x2 + y2 = 25.

65. Find the surface area of the surface z = 2x + y2 that lies above the triangular region with vertices at (0, 0, 0),(0, 1, 0), and (1, 1, 1).


66. Find the surface area of that portion of the plane z = x + y in the first octant that lies inside the cylinder4x2 + 9y2 = 36.

67. Find the surface area of that portion of the cylinder y2 + z2 = 4 that lies above the region in the xy-planeenclosed by the lines x + y = 1, x = 0, and y = 0.

68. Find the surface area of that portion of the cylinder z = y2 that lies above the triangular region with verticesat (0, 0, 0), (0, 1, 0), and (1, 1, 0).

69. Find the surface area of that portion of the cylinder x2 = 1 – z that lies above the triangular region withvertices at (0, 0, 0), (1, 0, 0), and (1, 1, 0).

70. Find the surface area of that portion of the sphere x2 + y2 + z2 = 4 that lies inside the cylinder x2 + y2 = 2y.

71. Find the surface area of that portion of the sphere x2 + y2 + z2 = 4 that lies in the first octant between theplanes y = 0, and y = x.

72. Find the surface area of that portion of the sphere x2 + y2 + z2 = 4 that lies in the first octant between the

planes y = 0, and y = x3 .

17.7 Surface Integrals

73. Evaluate the surface integral ∫∫ +S

dyx σ )( 22 where S is the portion of the cone )(3 22 yxz += for

0 ≤ z ≤ 3.

74. Evaluate the surface integral ∫∫S

dx σ 8 where S is the surface enclosed by z = x2, 0 ≤ x ≤ 2, and

–1 ≤ y ≤ 2.


dyx σ sin3 3 where S is the surface enclosed by z = x3, 0 ≤ x ≤ 2, and

0 ≤ y ≤ π.


dyx σ )sin(cos where S is that portion of the plane x + y + z = 1 that lies

in the first octant.

77. Evaluate the surface integral ∫∫ −

S

dxy

σ tan 1 where S is that portion of the paraboloid z = x2 + y2 enclosed by

1 ≤ z ≤ 9.


dx σ where S is that portion of the plane x + 2y + 3z = 6 that lies in the first

octant.


dyx σ )( 22 where S is that portion of the plane z = 4x + 20 intercepted by

the cylinder x2 + y2 = 9.


dy σ where S is that portion of the plane z = x + y inside the elliptic

cylinder 4x2 + 9y2 = 36 that lies in the first octant.



dy σ where S is that portion of the cylinder y2 + z2 = 4 that lies above the

region in the xy-plane enclosed by the lines x + y = 1, x = 0, and y = 0.


dy σ 4 where S is that portion of the surface z = y4 that lies above the

triangle in the xy-plane with vertices (0, 0), (0, 1), and (1, 1).


dx σ 2 where S is that portion of the surface z = x3 that lies above the

triangle in the xy-plane with vertices (0, 0), (1, 0), and (1, 1).


dx σ 2 where S is that portion of the plane x + y + z = 1 that lies inside the

cylinder x2 + y2 = 1.


dy σ 2 where S is that portion of the plane x + y + z = 1 that lies in the first

octant.


dy σ 2 where S is that portion of the cylinder y2 + z2 = 1 that lies above the

xy-plane between x = 0 and x = 5.


dyx σ )( 22 where S is that portion of the cylinder x2 + z2 = 1 that lies

above the xy-plane enclosed by 0 ≤ y ≤ 5.


dzy σ )( 22 where S is the portion of the cone )(3 22 zxx += for

0 ≤ x ≤ 3.


dx σ 8 where S is the surface enclosed by y = x2 , 0 ≤ x ≤ 2, and

–1 ≤ z ≤ 2.


dzy σ )cos(sin where S is that portion of the plane x + y + z = 1 that lies

in the first octant.

91. Evaluate ∫∫ •S

dσ ) ( nv where v = y i – x j + 8 k and S is that portion of the paraboloid z = x2 + y2 that

lies below the plane z = 4. Take n as the downward unit normal.


dσ ) ( nv where v = y i – x j + 9 k and S is that portion of the paraboloid z = 4 – x2 – y2

that lies above z = 0. Take n as the upward unit normal.


dσ ) ( nv where v = x i – y j + z k and S is that portion of the plane 2x + 3y + 4z = 12 that

lies in the first octant. Take n as the upward unit normal.



dσ ) ( nv where v = yz i – xz j + xy k and S is that portion of the hemisphere

224 yxz −−= that lies above the xy-plane. Take n as the upward unit normal.


dσ ) ( nv where v = y i – x j – 4z2 k and S is that portion of the cone 22 yxz += that

lies above the square in the xy-plane with vertices (0, 0), (1, 0), (1, 1) and (0, 1). Take n as the downwardunit normal.


dσ ) ( nv where v = y i – x j – k and S is that portion of the hemisphere 224 yxz −−−=

that lies above the plane z = 0. Take n as the downward unit normal.


dσ ) ( nv where v = z i + x j + y k and S is that portion of the cylinder x2 + y2 = 4 in the

first octant between z = 0 and z = 4. Take n as the outward unit normal.


dσ ) ( nv where v = x i + y j + z k and S is that portion of the cone 22 yxz += that lies

in the first octant between z = 1 and z = 2. Take n as the downward unit normal.


dσ ) ( nv where v = –xy2 i + z j + xz k and S is that portion of the surface z = xy bounded

by 0 ≤ x ≤ 3 and 0 ≤ y ≤ 2. Take n as the upward unit normal.


dσ ) ( nv where v = y i + 2x j + xy k and S is that portion of the cylinder x2 + y2 = 9 in

the first octant between z = 1 and z = 4.


dσ ) ( nv where v = y i |+ z j + y k and S is that portion of the cone 22 zyx += that lies

in the first octant between x = 1 and x = 3. Take n as the unit normal that points away from the yz-plane.

102. Calculate the flux of v = x i + y j – 2z k across the portion of the sphere x2 + y2 + z2 = 9 that lies abovethe xy-plane, with upward unit normal.


dσ ) ( nv where v = x i + 4 j + 2x2 k and S is that portion of the paraboloid z = x2 + y2

that lies above the xy-plane enclosed by the parabolas y = 1 – x2 and y = x2 – 1. Take n as the downwardunit normal.


dσ ) ( nv where v = 2 i – z j + y k and S is that portion of the paraboloid x = y2 + z2

between x = 0 and x = 4. Take n as the unit normal that points away from the yz-plane.

105. Calculate the flux of v = 9 i – z j + y k across the portion of the paraboloid x = 4 – y2 – z2 for whichx ≥ 0, in the direction pointing away from the xy-plane.


dσ ) ( nv where v = –x i – 2x j + (z – 1) k and S is the surface enclosed by that portion of

the paraboloid z = 4 – y2 that lies in the first octant and is bounded by the coordinate planes and the planey = x. Take n as the upward unit normal.


17.8 The Vector Differential Operator ∇∇

107. Given that v(x, y) = x2 i + 2y j , find ∇∇ • v and ∇∇ × v.

108. Given that v(x, y) = 3y i – 2x2 j , find ∇∇ • v and ∇∇ × v.

109. Given that v(x, y, z) = yz i – xz j + 3xy k, find ∇∇ • v and ∇∇ × v.

110. Given that v(x, y, z) = –2xy2 i + z j + xz k, find ∇∇ • v and ∇∇ × v.

111. Given that v(x, y, z) = –x i + j – 2x2 k, find ∇∇ • v and ∇∇ × v.

112. Given that v(x, y, z) = –x i – 2x j + (z – 1) k, find ∇∇ • v and ∇∇ × v.

113. Given that v(x, y, z) = (2x + cos z) i + (y – ex) j – (2z – ln y) k, find ∇∇ • v and ∇∇ × v.

114. Given that v(x, y, z) = ex i – yex j + 3yz k, find ∇∇ • v and ∇∇ × v.

115. Given that v(x, y, z) = (x3 + 3xy2) i + z3 k, find ∇∇ • v and ∇∇ × v.

116. Given that v(x, y, z) = –y3 i + x3 j – (x + z) k, find ∇∇ • v and ∇∇ × v.

117. Given that f(x, y, z) = x3 + y3 + z3, calculate the Laplacian ∇∇2f.

118. Given that f (x, y, z) = 2x2y3z, calculate the Laplacian ∇∇2f.

119. Given that f(x, y, z) = 2(x2 + y2), calculate the Laplacian ∇∇2f.

120. Given that f(r) = sin r, calculate the Laplacian ∇∇2f.

17.9 The Divergence Theorem

121. Find the divergence of v(x, y, z) = x2y i + xy2 j + xyz k.

122. Find the divergence of v(x, y, z) = cosh x i + sinh y j + ln (xy) k.

123. Find the divergence of v(x, y, z) = ex cos y i + ex sin y j + z k.

124. Use the divergence theorem to evaluate ∫∫ •S

dσ ) ( nv where v(x, y, z) = x i + y j + z k, n is the outer

unit normal to S, and S is the surface of the paraboloid z = x2 + y2 that is inside the cylinder x2 + y2 = 1.



unit normal to S, and S is the surface of the cube –1 ≤ x ≤ 1, –1 ≤ y ≤ 1, –1 ≤ z ≤ 1.



unit normal to S, and S is the surface formed by the intersection of two paraboloids z = x2 + y2 andz = 4 – (x2 + y2).


dσ ) ( nv where v(x, y, z) = (2x + z) i + y j – (2z + sin x) k,

n is the outer unit normal to S, and S is the surface of the cylinder x2 + y2 = 4 enclosed between the planesz = 0 and z = 4.



dσ ) ( nv where v(x, y, z) = 3

3xi +

3

3yj –

3

3zk, n is the

outer unit normal to S, and S is the surface of the cylinder x2 + y2 = 1 enclosed between the planes z = 0and z = 1.


dσ ) ( nv where v(x, y, z) = x i + y j – z k, n is the outer

unit normal to S, and S is the surface of the solid bounded by x + y + z = 1, x = 0, y = 0, and z = 0.


dσ ) ( nv where v(x, y, z) = (x3 + 3xy2) i + z3 k, n is the outer

unit normal to S, and S is the surface of the sphere of radius a centered at the origin.


dσ ) ( nv where v(x, y, z) = ex i + yex j + 4x2z k, n is the

outer unit normal to S, and S is the surface of the solid enclosed by x2 + y2 = 4 and the planes z = 0 andz = 9.

132. Use the divergence theorem to calculate the total flux of v(x, y, z) = ex i – yex j + 3z k, out of the spherex2 + y2 + z2 = 9.

133. Use the divergence theorem to calculate the total flux of v(x, y, z) = x2 i – y2 j + z2 k, out of the cube0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.


dσ ) ( nv where v(x, y, z) = x3 i + x2y j + x2z k, n is the outer

unit normal to S, and S is the surface of the solid enclosed by the cylinder x2 + y2 = 2 and the planes z = 0and z = 2.


dσ ) ( nv where

v(x, y, z) = x(x2 + y2 + z2) i + y(x2 + y2 + z2) j + z(x2 + y2 + z2) k, n is the outer unit normal to S, and S isthe sphere x2 + y2 + z2 = 16.


dσ ) ( nv where v(x, y, z) = x3 i + x2y j + x2z k, n is the

outer unit normal to S, and S is the surface of the solid enclosed by the hemisphere 224 yxz −−= and

the xy-plane.


dσ ) ( nv where v(x, y, z) = x2 i + y2 j + z2 k, n is the outer

unit normal to S, and S is the surface of the solid enclosed by the cylinder x2 + y2 = 4 and the planes z = 0and z = 5.


dσ ) ( nv where v(x, y, z) = yz i + xy j + xz k, n is the outer

unit normal to S, and S is the solid enclosed by the cylinder x2 + z2 = 1 and the planes y = –1 and y = 1.


dσ ) ( nv where v(x, y, z) = y2x i + yz2 j + x2y2 k, n is the

outer unit normal to S, and S is the sphere x2 + y2 + z2 = 4.


17.10 Stokes’s Theorem

140. Find the curl of v(x, y, z) = x2y i + y2x j + xyz k.

141. Find the curl of v(x, y, z) = cosh x i + sinh y j + ln xy k.

142. Find the curl of v(x, y, z) = ex cos y i + ex sin y j + z k.

143. Verify Stokes’s Theorem if S is the portion of the sphere x2 + y2 + z2 = 1 for which z ≥ 0 andv(x, y, z) = (2x – y) i – yz2 j – y2z k.

144. Use Stokes’s Theorem to evaluate ∫ −+−+−C

dzxydyzxdxyz )()()( where C is the boundary, in the xy-

plane, of the surface given by z = 4 – (x2 + y2), z ≥ 0.

145. Use Stokes’s Theorem to evaluate ∫ +−+C

dzzxdyxdxy )(22 where C is the triangle in the xy-plane with

vertices (0, 0, 0), (1, 0, 0), and (1, 1, 0) with a counterclockwise orientation looking down the positive z-axis.

146. Use Stokes’s Theorem to evaluate ∫ ++−C

dzzdyxdxy 3 3 over the circle x2 + y2 = 1, z = 1 traversed

counterclockwise.

147. Use Stokes’s Theorem to evaluate ∫ ++C

dzydyxdxz over the triangle with vertices (1, 0, 0), (0, 1, 0), and

(0, 0, 1) traversed in a counterclockwise manner.

148. Use Stokes’s Theorem to evaluate ∫∫ •×∇S

dσ])[( nv where v(x, y, z) = x2 i + z2 j – y2 k and S is that

portion of the paraboloid z = 4 – x2 – y2 for which z ≥ 0 and n is the upper unit normal.


dσ])[( nv where v(x, y, z) = (z – y) i – (z2 + x) j + (x2 – y2) k

and S is that portion of the sphere x2 + y2 + z2 = 4 for which z ≥ 0 and n is the upper unit normal.


dσ])[( nv where v(x, y, z) = y k and S is that portion of the

ellipsoid 4x2 + 4y2 + z2 = 4 for which z ≥ 0 and n is the upper unit normal.

151. Use Stokes’s Theorem to evaluate ∫ +−C

dzydyxdxz sin cos sin over the boundary of rectangle 0 ≤ x ≤ π,

0 ≤ y ≤ 1, z = 2, traversed in a counterclockwise manner.

152. Use Stokes’s Theorem to evaluate ∫ ++−++C

dzzydyxdxyx )( )32( )( over the boundary of the triangle

with vertices (2, 0, 0), (0, 3, 0), and (0, 0, 6) traversed in a counterclockwise manner.

153. Use Stokes’s Theorem to evaluate ∫ +−C

dzxdyxdx 2 2 z4 where C is the intersection of the cylinder

x2 + y2 = 1 and the plane z = y + 1, traversed in a counterclockwise manner.


dzxydyxzdxyz over the circle x2 + y2 = 2, z = 1, traversed in a

counterclockwise manner.


dzzydyyzdxyx )24( 22 over the circle x2 + y2 = 4, z = 2,

traversed in a counterclockwise manner.


156. Use Stokes’s Theorem to evaluate ∫ −−− ++++−C

zyx dzedyxxzedxyze )2( )(22

over the circle x2 + y2 = 1,

z = 1, traversed in a counterclockwise manner.

157. Use Stokes’s Theorem to evaluate ∫ ++C

dzxdyydxxz 22 where C is the intersection of the plane

x + y + z = 5 and the cylinder 14

22 =+

yx , traversed in a counterclockwise manner.



1. (a) 7/3 (b) –29/6

2. (a) 2e2 – e–6 – 1 (b) –34/3

3. (a) 2 (b) 17/2

4. (a) ≈+−+− −

5

1tan

55

7455

752

55

2 1π–0.007326

(b) 23/60 + 3 ln 5/6

5. –20

6. )(51

2 143 −− −− eee

7. –16/3 – 2π

8. –8/3

9. (a) –13/18 (b) 5/6

10. (a) 6

1134

23 322 −++ −− eee

(b) 1 + e – e–1

11. 13/3

12. ½

13. F = –2m j ; work = 2m

14. 0

15. 6

16. –π

17. 172

18. 11

19. –1/8

20. 8 + ln 2

21. e

22. 2/3

23. 1

24. –11/3

25. 52/3

26. 9

27. 6π

28. e2/2 + 4/e – 9/2

29. 23/6

30. 2π

31. –72

32. 2 sin 1 – cos 1 + 1

33. length = 10π, centroid (0, 0, 4π)

34. mass = 3

21

ka , center of mass (2a/3, 2a/3),

Ix = Iy = 5

41

ka

35. L2M/3

36. center of mass (0, –3/5, 9/8),Ix = 192/35, Iy = 65/15, Iz = 128/105

37. 52/3

38. 2/15

39. 4/3

40. –2/π – π/4

41. 9

42. 7/15

43. 4

44. π(cosh 1 – 1)

45. 1/3

46. 3/20

47. 128π

48. 1

49. 1/4

50. 8/5

51. 0

52. 2π

53. 1/6


54. 12 −

55. π1725

56. ( )1556

−π

57. π64

58. 61

59. ( )11011016

−π

60. 4π

61. ( )137376

−π

62. ( )1556

2

−aπ

63. 4

64. 200

65.12

5527−

66.233 π

67. 4323

−+π

68. ( )155121

−

69. ( )155121

−

70. 8π − 16

71. π

72. 4π/3

73. 9π

74. ( )117172 −

75. ( )114514591

−

76. )1sin1cos2(3 −−

77. ( )5537376

2

−π

78. 146

79. π21781

80. 34

81.3

234

π−−

82. ( )11717144

1−

83. ( )11010541

−

84. π43

85.12

3

86. 5π/2

87. 265π/6

88. 9π

89. ( )117172 −

90. )1sin1cos2(3 −−

91. −32π

92. 36π

93. 36

94. 0

95. 8/3


96. 4π

97. 24

98. ( )1223

−π

99. 18

100. 81/2

101. 0

102. 0

103. 0

104. 8π

105. 36π

106. –6

107. 2x + 2 ; 0

108. 0 ; –(4x + 3) k

109. 0 ; 4x i – 2y j – 2z k

110. –2y2 + x ; – i – z j + 4xy k

111. –1 ; 4x j

112. 0 ; –2 k

113. 1 ; y1

i – sin z j – ex k

114. 3y ; 3z i – yex k

115. 2x + 3y2 + 3z2 ; –6xy k

116. –1 ; j + (3x2 + 3y2) k

117. 6(x + y + z)

118. 4y3z + 12x2yz

119. 8

120. –sin x – sin y – sin z

121. 5xy

122. sinh x + cosh y

123. 2ex cos y + 1

124. π/2

125. 24

126. 4π

127. 16π

128. 5π/6

129. ½

130. 12πa5/5

131. 144π

132. 108π

133. 3

134. 10π

135. 4096π

136. 64π/5

137. 100π

138. 0

139. 256π/15

140. xz i – yz j + (y2 – x2) k

141. 1/y i – 1/x j

142. 2ex sin y k

143. π

144. 8π

145. 1/3

146. 6π

147. 3/2

148. 0

149. 8π

150. 0

151. 2

152. 12


153. –4π

154. 4π

155. 8π

156. 4π

157. 0

219

CHAPTER 18

Elementary Differential Equations

18.1 Introduction

1. Classify the differential equation 0232

2

=++ ydxdy

dxyd

as ordinary or partial and give its order.

2. Classify the differential equationdydz

xzxz

+=∂∂


3. Classify the differential equation y′′′ + 2(y′′)2 + y′ = cos x as ordinary or partial and give its order.

4. Classify the differential equation (y′)4 + 3xy – 2y′′ = x2 as ordinary or partial and give its order.

5. Classify the differential equation 042

2

2

2

=−∂∂

dxyd

ty


6. Determine whether y1(x) = 3ex, y2(x) = 5e–x are solutions of y′ + y = 0.

7. Determine whether y1(x) = 1/x, y2(x) = 2/x are solutions of y′ + y2 = 0.

8. Determine whether u1(x, t) = xet, u2(x, t) = tex are solutions of dtdu

xu

x =∂∂

.

9. For what values of C is y = C e–x a solution of y′ + y = 0 with side condition y(3) = 2?

10. For what values of C1, C2 is y = C1 sin 2x + C2 cos 2x a solution of y′′ + 4y = 0 with side conditionsy(0) = 0 and y′(0) = 1?

11. For what values of r is y = erx a solution of y′′ + y′ – 6y = 0?

12. For what values of r is y = xr a solution of x3y′′ – 2xy = 0?

18.2 First-Order Linear Differential Equations; Numerical Methods

13. Find the general solution of y′ – 3y = 6.

14. Find the general solution of y′ – 2xy = x.

15. Find the general solution of 44xy

xy =+′ .


2=

++′ y

xy .

17. Find the general solution of y′ – y = –ex.

18. Find the general solution of x ln xy′ + y = ln x.


19. Find the particular solution of y′ + 10y = 20 determined by the side condition y(0) = 2.

20. Find the particular solution of y′ – y = –ex determined by the side condition y(0) = 3.

21. Find the particular solution of xy′ – 2y = x3 cos 4x determined by the side condition y(π) = 1.

22. A 100-gallon mixing tank is full of brine containing 0.8 pounds of salt per gallon. Find the amount of saltpresent t minutes later if pure water is poured into the tank at the rate of 4 gallons per minute and themixture is drawn off at the same rate.

23. Determine the velocity at time t and the terminal velocity of a 2 kg object dropped with a velocity 3 m/s, ifthe force due to air resistance is –50v Newtons.

24. Use a suitable transformation to solve the Bernoulli equation y′ + xy = xy2.

25. Use a suitable transformation to solve the Bernoulli equation xy′ + y = x3y6.

26. Find the general solution of y′ = y2x3.


2

37yy

xy

−+

=′ .

28. Find the general solution of y′ = y2 + 1.

29. Find the general solution of x(y2 + 1)y′ + y3 – 2y = 0.

30. Find the particular solution of exdx – ydy = 0 determined by the side condition y(0) = 1.

31. Find the particular solution of y′ = y(x – 2) determined by the side condition y(2) = 5.

32. Verify that the equation x

xyy

+=′ is homogeneous, then solve it.

33. Verify that the equation 3

442xy

xyy


34. Verify that the equation xyx

yy


35. Verify that the equation of [2x sinh (y/x) + 3y cosh (y/x)]dx – 3x cosh (y/x)dy = 0 is homogeneous, thensolve it.

36. Find the orthogonal trajectories for the family of curves x2 + y2 = C.

37. Find the orthogonal trajectories for the family of curves x2 + y2 = Cx.

18.3 The Equation y′′′′ + ay′′ + by = 0

38. Find the general solution of y′′ – 5y = 0.

39. Find the general solution of y′′ – 60y′ + 900y = 0.

40. Find the general solution of y′′ – 6y′ + 25y = 0.

Elementary Differential Equations 221

41. Find the general solution of 16y′′ + 8y′ + y = 0.

42. Find the general solution of 2y′′ – 5y′ + 2y = 0.

43. Find the general solution of y′′ + 4y′ + 5y = 0.

44. Solve the initial value problem y′′ + 25y = 0, y(0) = 3, y′(0) = 10.

45. Solve the initial value problem y′′ + y′ – 6y = 0, y(0) = 4, y′(0) = 13.

46. Solve the Euler equation x2y′′ + 5xy′ + 4y = 0.

47. Solve the Euler equation x2y′′ – 5xy′ + 25y = 0.

18.4 The Equation y′′′′ + ay′′ + by = φφ(x)

48. Find a particular solution of y′′ – y′ – 2y = 4x2.

49. Find a particular solution of y′′ + 5y′ + 6y = 3e–2x.

50. Find a particular solution of y′′ + 4y′ + 8y = 16 cos 4x.

51. Find a particular solution of y′′ + 6y′ + 9y = 16 e–x cos 2x.

52. Find the general solution of y′′ – y′ – 2y = e2x.

53. Find the general solution of y′′ – 7y′ = (3 – 36x)e4x.

54. Find the general solution of y′′ + 4y = 8x sin 2x.

55. Use variation of parameters to find a particular solution of y′′ – 2y′ + y = ex/x.

56. Use variation of parameters to find a particular solution of y′′ + y = sec x.

57. Use variation of parameters to find a particular solution of y′′ + 4y = sin2 2x.

18.5 Mechanical Vibrations

58. An object is in simple harmonic motion. Find an equation for the motion given that the period is π/2 and, attime t = 0, x = 2 and v = 1. What is the amplitude? What is the frequency?

59. An object is in simple harmonic motion. Find an equation for the motion given that the frequency is 4/π and,at time t = 0, x = 3 and v = –3. What is the amplitude? What is the period?

60. An object in simple harmonic motion passes through the central point x = 0 at time t = 2 and every 4seconds thereafter. Find the equation of motion given that v(0) = 3.

61. Find an equation for the oscillatory motion given that the period is 3π/4 and at time t = 0, x = 2 and v = 5.

62. Find an equation for the oscillatory motion given that the period is 5π/6 and at time t = 0, x = 1 and v = 4.



1. ordinary, order 2

2. partial, order 1



5. partial, order 2

6. y1 is not, y2 is

7. y1 is, y2 is not

8. u1 is, u2 is not

9. C = 2e3

10. C1 = ½, C2 = 0

11. r = 2, –3

12. r = 2, –1

13. y = Ce–3x

14.212

−= xCey

15. 54 9

1x

xC

y +=

16.x

Cxxy

210440 2

+++

=

17. y = (C – x)ex

18.xCx

yln2

ln 2 +=

19. y = 2 (identically)

20. y = (3 – x)ex

21.2

2 4sin41

+=

πx

xxy

22. 80e–0.04t pounds

23. v = 0.392 + 2.608e–25t ;terminal velocity 0.392 m/s

24.2/2

1

1xCe

y+

=

25.5/1

53

25

−

+= Cxxy

26.Cx

y+

−=

4

4

27. Cxyyy =+−+ 731

53

101 3510

28. y = tan (x + C)

29. (y2 – 2)3x4 = Cy2

30.21

ln,12 >−= xey x

31. 2/)2( 25 −= xey

32. y = x ln |Cx|

33. x8 = C(y4 + x4)

34. Cyyx =+− ||ln/2

35. x2 = C sinh3 (y/x)

36. y = Kx

37. x2 + y2 = Ky

38. xx eCeCy 52

51

−+=

39. y = (C1 + C2x)e30x

40. y = e3x(C1 cos 4x + C2 sin 4x)

41. y = (C1 + C2x)e–x/4

42. y = C1ex/2 + C2e

2x

43. y = e–2x(C1 cos x + C2 sin x)

44. y = 3 cos 5x + 2 sin 5x

45. y = 5e2x – e–3x

46. 221 ln

xxCC

y+

=

47. y = x3[C1 cos (ln x4) + C2 sin (ln x4)]

48. yp = –2x2 + 2x – 3

Elementary Differential Equations 223

49. yp = 3xe–2x

50. xxy p 4cos52

4sin54 −=

51. yp = 2e–x sin 2x

52. xxx xeeCeCy 2221 3

1++= −

53. y = C1 + C2e7x + 3xe4x

54. xxxxxCxCy 2cos2sin21

2sin2cos 221 −++=

55. yp = –xex + xex ln | x |

56. yp = (ln | cos x | ) cos x + x sin x

57. xxy p 2sin121

2cos61 22 −=

58. tttx 4sin41

4cos2)( +=

amplitude = 465

; frequency = 4

59. tttxπ

ππ

4sin

434

cos3)( −=

amplitude = 4

9144 2π+; period = π2/2

60. ttx2

sin6

)(π

π=

61. tttx38

sin8

1538

cos2)( −=

62. tttx5

12sin

35

512

cos)( +=

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