Review Problems


This list of problems is for benefit in case you want some additional practice. The problems are grouped according to chapter and section. Answers start on page 283.

Chapter 1

Linear and Quadratic Functions

Find the slope of the graphs of the following equations:

  1. y equals 5 x minus 5
  2. 4 y minus 7 equals 5 x plus 2
  3. 3 y plus 7 x equals 2 y minus 5

Find the roots of the following:

  1. 4 x squared minus 2 x minus 3 equals 0
  2. x squared minus 6 x plus 9 equals 0

Trigonometry

  1. Show that sine theta cotangent theta slash StartRoot 1 minus sine squared theta EndRoot equals 1, left-parenthesis minus StartFraction pi Over 2 EndFraction less-than theta less-than StartFraction pi Over 2 EndFraction right-parenthesis.
  2. Show that cosine theta sine left-parenthesis StartFraction pi Over 2 EndFraction plus theta right-parenthesis minus sine theta cosine left-parenthesis StartFraction pi Over 2 EndFraction plus theta right-parenthesis equals 1.
  3. What is: (a) sine 135 Superscript ring (b) cosine StartFraction 7 pi Over 4 EndFraction, (c) sine StartFraction 7 pi Over 6 EndFraction?
  4. Show that cosine squared StartFraction theta Over 2 EndFraction equals one half left-parenthesis 1 plus cosine theta right-parenthesis.
  5. What is the cosine of the angle between any two sides of an equilateral triangle?

Exponentials and Logarithms

  1. What is left-parenthesis negative 1 right-parenthesis Superscript 13?
  2. Find left-bracket left-parenthesis 0.01 right-parenthesis cubed right-bracket Superscript negative 1 slash 2.
  3. Express log left-parenthesis x Superscript x Baseline right-parenthesis Superscript x in terms of log x.
  4. If log left-parenthesis log x right-parenthesis equals 0, find x.
  5. Is there any number for which x equals log x?

In the following five questions, make use of the log table below and the rules for manipulating logarithms.

xlog xxlog x
10.00 50.70
20.30 70.85
30.48101.00

Find

  1. log StartRoot 10 EndRoot
  2. log 21
  3. log StartRoot 14 EndRoot
  4. log 300
  5. log left-parenthesis 7 Superscript 3 slash 2 Baseline right-parenthesis

Chapter 2

Find the following limits, if they exist:

  1. limit Underscript x right-arrow 2 Endscripts StartFraction x squared minus 4 x plus 4 Over x minus 2 EndFraction
  2. limit Underscript theta right-arrow pi slash 2 Endscripts sine theta
  3. limit Underscript x right-arrow 0 Endscripts StartFraction x squared plus x plus 1 Over x EndFraction
  4. limit Underscript x right-arrow 1 Endscripts left-bracket 1 plus StartFraction left-parenthesis x plus 1 right-parenthesis squared Over x minus 1 EndFraction right-bracket
  5. limit Underscript x right-arrow 3 Endscripts left-bracket left-parenthesis 2 plus x right-parenthesis StartFraction left-parenthesis x minus 3 right-parenthesis squared Over x minus 3 EndFraction plus 7 right-bracket
  6. limit Underscript x right-arrow 1 Endscripts StartFraction x squared minus 1 Over x minus 1 EndFraction
  7. limit Underscript x right-arrow infinity Endscripts StartFraction 1 Over x EndFraction
  8. limit Underscript x right-arrow 0 Endscripts log x

Velocity

  1. What is the average velocity of a particle that goes forward 35 miles and backward for 72 miles, during the course of 1 hour?
  2. A particle always moves in one direction. Can its average velocity exceed its maximum velocity?
  3. A particle moves so that its position is given by upper S left-parenthesis t right-parenthesis equals upper S 0 sine left-parenthesis 2 pi t right-parenthesis, where upper S 0 is in meters, t is in hours. Find its average velocity from t equals 0 to
    1. t equals one fourth hour
    2. t equals one half hour
    3. t equals three fourths hour
    4. t = 1 hour
  4. Write an expression for the average velocity of a particle, which leaves the origin at t equals 0, whose position is given by upper S left-parenthesis t right-parenthesis equals italic a t cubed plus italic b t, where a and b are constants. The average is from t equals 0 to the present time t.
  5. Find the instantaneous velocity of a particle when t equals 2 whose position is given by upper S left-parenthesis t right-parenthesis equals italic b t cubed, where b is a constant.

Differentiation

Find the derivative of each of the following functions with respect to its appropriate variable, where a and b are constants.

  1. y equals x plus x squared plus x cubed
  2. y equals left-parenthesis a plus italic b x right-parenthesis plus left-parenthesis a plus italic b x right-parenthesis squared plus left-parenthesis a plus italic b x right-parenthesis cubed
  3. y equals left-parenthesis 3 x squared plus 7 x right-parenthesis Superscript negative 3
  4. p equals StartRoot a squared plus q squared EndRoot
  5. p equals StartFraction 1 Over StartRoot a squared plus q squared EndRoot EndFraction
  6. y equals x Superscript pi
  7. f equals theta squared sine theta
  8. f equals StartFraction sine theta Over theta EndFraction
  9. f equals left-parenthesis sine theta right-parenthesis Superscript negative 1
  10. f equals left-parenthesis StartRoot 1 plus cosine squared theta EndRoot right-parenthesis Superscript negative 1
  11. f equals sine squared theta plus cosine squared theta
  12. y equals sine left-parenthesis ln x right-parenthesis
  13. y equals x ln x
  14. y equals left-parenthesis ln x right-parenthesis Superscript negative 2
  15. y equals x Superscript x
    • (Hint: What is ln y? Use implicit differentiation, Appendix B1.)
  16. y equals a Superscript x squared
  17. f equals sine StartRoot 1 plus theta squared EndRoot
  18. y equals e Superscript minus x squared
  19. y equals pi Superscript x
  20. y equals pi Superscript x squared
  21. f equals ln left-parenthesis sine theta right-parenthesis
  22. f equals sine left-parenthesis sine theta right-parenthesis
  23. f equals ln e Superscript x
  24. f equals e Superscript ln x
  25. y equals StartRoot 1 minus sine squared theta EndRoot

Higher‐Order Derivatives

Evaluate each of the following:

  1. Find StartFraction d squared Over d theta squared EndFraction left-parenthesis cosine a theta right-parenthesis.
  2. Find StartFraction d Superscript n Baseline Over italic d x Superscript n Baseline EndFraction e Superscript italic a x Baseline equals (n is a positive integer).
  3. StartFraction d squared Over italic d x squared EndFraction left-parenthesis StartRoot 1 plus x squared EndRoot right-parenthesis
  4. StartFraction d squared Over d theta squared EndFraction left-parenthesis tangent theta right-parenthesis
  5. StartFraction d cubed Over italic d x cubed EndFraction left-parenthesis x squared e Superscript x Baseline right-parenthesis

Maxima and Minima

Find where the following functions have their maximum and/or minimum values. Either give the values of x explicitly, or find an equation for these values.

  1. y equals e Superscript minus x squared
  2. y equals StartFraction sine x Over x EndFraction
  3. y equals e Superscript negative x Baseline sine x
  4. y equals StartFraction ln x Over x EndFraction
  5. y equals e Superscript negative x Baseline ln x
  6. Find whether y has a maximum or a minimum for the function given in question 64.

Differentials

Find the differential italic d f of each of the following functions.

  1. f equals x
  2. f equals StartRoot x EndRoot
  3. f equals sine left-parenthesis x squared right-parenthesis
  4. f equals e Superscript sine x (Hint: Use chain rule.)

Chapter 3

You may find Table 2 on page 288 helpful in doing the problems in this section.

Integration

Find the following indefinite integrals. (Omit the constants of integration.)

  1. integral sine 2 x italic d x
  2. integral StartFraction italic d x Over x plus 1 EndFraction
  3. integral x squared e Superscript x italic d x (Try integration by parts.)
  4. integral x e Superscript minus x squared italic d x
  5. integral sine squared theta cosine theta d theta

Some Techniques of Integration and Definite Integrals

Evaluate the following definite integrals.

  1. integral Subscript negative 1 Superscript plus 1 Baseline left-parenthesis e Superscript x Baseline plus e Superscript negative x Baseline right-parenthesis italic d x
  2. integral Subscript minus infinity Superscript infinity Baseline StartFraction italic d x Over a squared plus x squared EndFraction
  3. integral Subscript minus infinity Superscript infinity Baseline StartFraction x italic d x Over StartRoot a squared plus x squared EndRoot EndFraction
  4. integral Subscript minus infinity Superscript 0 Baseline x squared e Superscript x italic d x   (Problem 76 may be helpful.)
  5. integral Subscript 0 Superscript plus pi slash 2 Baseline sine theta cosine theta d theta
  6. integral Subscript 0 Superscript 1 Baseline left-parenthesis x plus a right-parenthesis Superscript n Baseline italic d x
  7. integral Subscript negative 1 Superscript plus 1 Baseline StartFraction italic d x Over StartRoot 1 minus x squared EndRoot EndFraction
  8. integral Subscript negative 1 Superscript 1 Baseline left-parenthesis x plus x squared plus x cubed right-parenthesis italic d x

Answers to Review Problems

  1. 5
  2. five fourths
  3. −7
  4. left-parenthesis 1 plus-or-minus StartRoot 13 EndRoot right-parenthesis slash 4
  5. 3, 3 (roots are identical)
  6. No answer
  7. No answer
  8. (a) StartFraction StartRoot 2 EndRoot Over 2 EndFraction, (b)StartFraction StartRoot 2 EndRoot Over 2 EndFraction, (c) negative one half
  9. No answer
  10. ½
  11. −1
  12. 1000
  13. x squared log x
  14. x equals 10
  15. No
  16. 0.50
  17. 1.33
  18. 0.58
  19. 2.48
  20. 1.27
  21. 0
  22. 1
  23. No limit
  24. No limit
  25. 7
  26. 2
  27. 0
  28. No limit
  29. −37 mph
  30. No
  31. (a) 4 upper S 0 normal m slash h r, (b) 0 normal m slash h r, (c) minus four thirds upper S 0 normal m slash h r, (d) 0 normal m slash h r
  32. italic a t squared plus b
  33. 12 b
  34. 1 plus 2 x plus 3 x squared
  35. b plus 2 b left-parenthesis a plus italic b x right-parenthesis plus 3 b left-parenthesis a plus italic b x right-parenthesis squared
  36. minus 3 left-parenthesis 3 x squared plus 7 x right-parenthesis Superscript negative 4 Baseline left-parenthesis 6 x plus 7 right-parenthesis
  37. StartFraction italic d p Over italic d q EndFraction equals StartFraction q Over StartRoot a squared plus q squared EndRoot EndFraction
  38. StartFraction italic d p Over italic d q EndFraction equals StartFraction negative q Over left-parenthesis a squared plus q squared right-parenthesis Superscript 3 slash 2 Baseline EndFraction
  39. StartFraction italic d y Over italic d x EndFraction equals pi x Superscript pi minus 1
  40. StartFraction italic d f Over d theta EndFraction equals 2 theta sine theta plus theta squared cosine theta
  41. StartFraction italic d f Over d theta EndFraction equals StartFraction cosine theta Over theta EndFraction minus StartFraction sine theta Over theta squared EndFraction
  42. StartFraction italic d f Over d theta EndFraction equals minus StartFraction cosine theta Over sine squared theta EndFraction
  43. StartFraction italic d f Over d theta EndFraction equals StartFraction cosine theta sine theta Over left-parenthesis 1 plus cosine squared theta right-parenthesis Superscript 3 slash 2 Baseline EndFraction
  44. StartFraction italic d f Over d theta EndFraction equals 0
  45. StartFraction italic d y Over italic d x EndFraction equals StartFraction cosine left-parenthesis ln x right-parenthesis Over x EndFraction
  46. StartFraction italic d y Over italic d x EndFraction equals 1 plus ln x
  47. StartFraction italic d y Over italic d x EndFraction equals StartFraction negative 2 Over x EndFraction left-parenthesis ln x right-parenthesis Superscript negative 3
  48. StartFraction italic d y Over italic d x EndFraction equals x Superscript x Baseline left-parenthesis 1 plus ln x right-parenthesis
  49. StartFraction italic d y Over italic d x EndFraction equals 2 italic x a Superscript x squared Baseline ln a
  50. StartFraction theta Over StartRoot 1 plus theta squared EndRoot EndFraction cosine StartRoot 1 plus theta squared EndRoot
  51. minus 2 italic x e Superscript minus x squared
  52. pi Superscript x Baseline ln pi
  53. 2 x pi Superscript x squared Baseline ln pi
  54. cotangent theta
  55. left-bracket cosine left-parenthesis sine theta right-parenthesis right-bracket cosine theta
  56. 1
  57. 1
  58. minus sine theta
  59. minus a squared cosine left-parenthesis a theta right-parenthesis
  60. a Superscript n Baseline e Superscript italic a x
  61. StartFraction 1 Over StartRoot 1 plus x squared EndRoot EndFraction minus StartFraction x squared Over left-parenthesis 1 plus x squared right-parenthesis Superscript 3 slash 2 Baseline EndFraction
  62. 2 secant squared theta tangent theta
  63. left-parenthesis 6 plus 6 x plus x squared right-parenthesis e Superscript x
  64. x equals 0
  65. x equals tangent x (x equals 0 comma ellipsis)
  66. x equals tangent Superscript negative 1 Baseline 1 equals StartFraction pi Over 4 EndFraction plus-or-minus n pi, n equals 0 comma 1 comma 2 comma ellipsis
  67. x equals e x = e (ln x equals 1)
  68. StartFraction 1 Over x EndFraction equals ln x
  69. Maximum
  70. italic d f equals italic d x
  71. italic d f equals StartFraction italic d x Over 2 StartRoot x EndRoot EndFraction
  72. italic d f equals 2 x cosine left-parenthesis x squared right-parenthesis italic d x
  73. italic d f equals left-parenthesis cosine x right-parenthesis e Superscript sine x Baseline italic d x
  74. StartFraction negative 1 Over 2 EndFraction cosine left-parenthesis 2 x right-parenthesis
  75. ln left-parenthesis x plus 1 right-parenthesis
  76. x squared e Superscript x minus 2 italic x e Superscript x plus 2 e Superscript x
  77. one half e Superscript minus x squared
  78. one third sine cubed theta
  79. 2 left-parenthesis e minus StartFraction 1 Over e EndFraction right-parenthesis
  80. StartFraction pi Over a EndFraction
  81. 0
  82. 2
  83. ½
  84. StartFraction left-parenthesis 1 plus a right-parenthesis Superscript n plus 1 Baseline minus a Superscript n plus 1 Baseline Over n plus 1 EndFraction
  85. pi
  86. ⅔
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