Solutions To Selected Problems

Chapter 5

Exercises 5.1

  1. 1.

    Answers will vary.

  2. 3.

    Answers will vary.

  3. 5.

    Answers will vary.

  4. 7.

    velocity

  5. 9.

    3x4/4+C

  6. 11.

    10x3/3-2x+C

  7. 13.

    -1/(3t)+C

  8. 15.

    2x+C

  9. 17.

    -cosθ+C

  10. 19.

    5eθ+C

  11. 21.

    4/3t3+6t2+9t+C

  12. 23.

    x6/6+C

  13. 25.

    -x-3+C

  14. 27.

    29x9/2+C

  15. 29.

    5x-29x3+316x4+C

  16. 31.

    23u3+92u2+4u+C

  17. 33.

    2x+x+23xx+C

  18. 35.

    θ+tanθ+C

  19. 37.

    -cott-t+C

  20. 39.

    8u+4uu+C

  21. 41.

    6t1/3+34t4/3+C

  22. 43.

    -cosx+3

  23. 45.

    x4-x3+7

  24. 47.

    7x36-9x2+403

  25. 49.

    θ-sin(θ)-π+4

  26. 51.

    x-2+1

  27. 53.
    (a) x>0 (b) 1/x (c) x<0 (d) 1/x (e) ln|x|+C. Explanations will vary.
  28. 55.

    s(t)=241.67-16t2 ft, so s(t)=0 at t=3.89sec.

  29. 57.
    24-224xy Other antiderivatives are vertical shifts of this one.
  30. 59.

    dy=(2xexcosx+x2excosx-x2exsinx)dx

Exercises 5.2

  1. 1.

    Answers will vary.

  2. 3.

    0

  3. 5.
    (a) 3 (b) 4 (c) 3 (d) 0 (e) -4 (f) 9
  4. 7.
    (a) 4 (b) 2 (c) 4 (d) 2 (e) 1 (f) 2
  5. 9.
    (a) π (b) π (c) 2π (d) 10π
  6. 11.
    (a) -59 (b) -48 (c) -27 (d) -33 (e) 70 (f) 91
  7. 13.
    (a) 4 (b) 4 (c) -4 (d) -2 (e) 6 (f) 2
  8. 15.
    (a) 2ft/s (b) 2ft (c) 1.5ft
  9. 17.
    (a) 64ft/s (b) 64ft (c) t=2 (d) t=2+74.65 seconds
  10. 19.

    2

  11. 21.

    16

  12. 23.

    22

  13. 25.

    0

  14. 27.

    This is a triangle with base b and height mb.

  15. 29.

    1/4x4-2/3x3+7/2x2-9x+C

  16. 31.

    3/4t4/3-1/t+2t/ln2+C

Exercises 5.3

  1. 1.

    limits

  2. 3.

    Rectangles.

  3. 5.

    22+32+42=29

  4. 7.

    0-1+0+1+0=0

  5. 9.

    1+1/2+1/3+1/4+1/5=137/60

  6. 11.

    1/2+1/6+1/12+1/20=4/5

  7. 13.

    Answers may vary; i=153i

  8. 15.

    Answers may vary; i=14ii+1

  9. 17.

    50

  10. 19.

    1045

  11. 21.

    -8525

  12. 23.

    5050

  13. 25.

    155

  14. 27.

    24

  15. 29.

    0πsinx1+xdx

  16. 31.

    275x3-4x+7dx

  17. 33.

    limn[3ni=1n4-2(2+3in)]

  18. 35.

    limnπni=1nsin3(-π/2+πi/n)2+cos(-π/2+πi/n)

  19. 37.

    19

  20. 39.

    π/3+π/(23)1.954

  21. 41.

    0.388584

  22. 43.
    (a) Exact expressions will vary; (1+n)24n2. (b) 121/400, 10201/40000, 1002001/4000000 (c) 1/4
  23. 45.
    (a) 8. (b) 8, 8, 8 (c) 8
  24. 47.
    (a) Exact expressions will vary; 100-200/n. (b) 80, 98, 499/5 (c) 100
  25. 49.
    (a) Exact expressions will vary; 80.5. (b) 72.25 (c) 62.5
  26. 51.
    abkf(x)dx =limni=1nkf(ci)ΔxT5.3.2.2 =limnki=1nkf(ci)ΔxT5.3.1.3 =klimni=1nkf(ci)ΔxT1.3.1.4 =kabf(x)dxT5.3.2.2
  27. 53.

    F(x)=5tanx+4

  28. 55.

    G(t)=4/6t6-5/4t4+8t+9

  29. 57.

    G(t)=sint-cost-78

Exercises 5.4

  1. 1.

    Answers will vary.

  2. 3.

    T

  3. 5.

    20

  4. 7.

    0

  5. 9.

    1

  6. 11.

    23/2

  7. 13.

    e3-e

  8. 15.

    4

  9. 17.

    ln2

  10. 19.

    1/4

  11. 21.

    15

  12. 23.

    2

  13. 25.

    632

  14. 27.

    6π7

  15. 29.

    2

  16. 31.

    36

  17. 33.

    694

  18. 35.

    Explanations will vary. A sketch will help.

  19. 37.

    c=2/3

  20. 39.

    c=ln(e-1)0.54

  21. 41.

    2/π

  22. 43.

    2

  23. 45.

    16

  24. 47.

    (a) -300ft; (b) 312.5ft

  25. 49.

    (a) -1ft; (b) 3ft

  26. 51.

    -64ft/s

  27. 53.

    2ft/s

  28. 55.

    F(x)=(3x2+1)1x3+x

  29. 57.

    F(x)=2x(x2+2)-(x+2)

  30. 59.

    F(x)=lnx+4x2+7

  31. 61.

    F(x)=-15x2cos(5x3)+525x6+e5x3

  32. 63.
    (a) x 0 1 2 3 4 5 6 g(x) 0 .5 0 -.5 0 1.5 4 (b) g(7)5.7 (c) min at x=3; max at x=7 (d) Approximately246246xy
  33. 65.
    (a) bn=4/nπ for odd n and bn=0 for even n (b) answers will vary

Exercises 5.5

  1. 1.

    Chain Rule.

  2. 3.

    18(x3-5)8+C

  3. 5.

    118(x2+1)9+C

  4. 7.

    12ln|2x+7|+C

  5. 9.

    23(x+3)3/2-6(x+3)1/2+C=23(x-6)x+3+C

  6. 11.

    2ex+C

  7. 13.

    -12x2-1x+C

  8. 15.

    sin3(x)3+C

  9. 17.

    -16sin(3-6x)+C

  10. 19.

    12ln|sec(2x)+tan(2x)|+C

  11. 21.

    tan(x)-x+C

  12. 23.

    The key is to multiply cscx by 1 in the form (cscx+cotx)/(cscx+cotx).

  13. 25.

    ex33+C

  14. 27.

    x-e-x+C

  15. 29.

    e-3x3-e-x+C

  16. 31.

    (lnx)33+C

  17. 33.

    12ln|ln(x2)|+C

  18. 35.

    x33+x22+x+ln|x|+C

  19. 37.

    145(5x3+5x2+2)9+C

  20. 39.

    -13cot(x3+1)+C

  21. 41.

    -15cos(5x+1)+C

  22. 43.

    73ln|3x+2|+C

  23. 45.

    3ln|3x2+9x+7|+C

  24. 47.

    x2-6x+8+C

  25. 49.

    12sec2θ+C or 12tan2θ+C

  26. 51.

    -12(x2+1)+14(x2+1)2+C

  27. 53.

    111(x3+2)11-25(x3+2)10+49(x3+2)9+C

  28. 55.

    23sin6(x4)+C

  29. 57.

    -ln2

  30. 59.

    2/3

  31. 61.

    (1-e)/2

  32. 63.

    0

  33. 65.

    23

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