Solutions To Selected Problems

Chapter 2

Exercises 2.0

  1. 1.

    80x12y17

  2. 2.

    a16b7

  3. 3.

    x316y22z35

  4. 4.

    x2y4z5z4=x2y4z21/4

  5. 5.

    3x(x2+9x+3)

  6. 6.

    5(x-1)3x13

  7. 7.

    -5x+42x12(x+4)2

  8. 8.

    6x(3x2+2)3(x2-5)2(7x2-18)

  9. 9.
    (a) 8 (b) 44 (c) x2-6x+8 (d) x2+2x-4
  10. 10.
    (a) -13 (b) undefined (c) 1x-2-5 (d) 1x-5-2
  11. 11.
    (a) Possible solution: f(x)=5x and g(x)=x+4 (b) Possible solution: f(x)=|x| and g(x)=4-x2 (c) Possible solution: f(x)=x-5 and g(x)=(x+2)2
  12. 12.
    (a) Possible solution: f(x)=x3, g(x)=x2, and h(x)=2x+1 (b) Possible solution: f(x)=2x+1, g(x)=x3, and h(x)=x2

Exercises 2.1

  1. 1.

    T

  2. 3.

    Answers will vary.

  3. 5.

    (a) f(x)=0, (b) y=6

  4. 7.

    (a) f(x)=-3, (b) y=4-3x

  5. 9.

    (a) h(x)=2-2x (b) y=1

  6. 11.

    (a) g(x)=12x+3, (b) y=x4+74

  7. 13.

    (a) h(x)=-32xx, (b) y=-3x16+94

  8. 15.

    f(x)=x, c=16.

  9. 17.

    f(x)=1x, c=2

  10. 19.

    y=8.1(x-3)+16

  11. 21.

    y=-0.099(x-9)+1

  12. 23.

    y=.49(x-2)+ln2

  13. 25.
    (a) Approximations will vary; they should match (c) closely. (b) f(x)=2x (c) At (-1,0), slope is -2. At (0,-1), slope is 0. At (2,3), slope is 4.
  14. 27.
    -2-11234-1123xy
  15. 29.
    -2-112-55xy
  16. 31.
    -2-112-33-22-44-66xy
  17. 33.
    (a) Approximately on (-2,0) and (2,). (b) Approximately on (-,-2) and (0,2). (c) Approximately at x=0,±2. (d) Approximately on (-,-1) and (1,). (e) Approximately on (-1,1). (f) Approximately at x=±1.
  18. 35.
  19. 37.

    Approximately 0.54.

  20. 39.
    (a) 1 (b) 3 (c) Does not exist (d) (-,-3)(3,)

Exercises 2.2

  1. 1.

    Velocity

  2. 3.

    Linear functions.

  3. 5.

    -17

  4. 7.

    f(10.1) is likely most accurate, as accuracy is lost the farther from x=10 we go.

  5. 9.

    6

  6. 11.

    ft/s2

  7. 13.
    (a) thousands of dollars per car (b) It is likely that P(0)<0. That is, negative profit for not producing any cars.
  8. 15.

    f(x)=g(x)

  9. 17.

    g(x)=f(x)

  10. 19.

    f(6)=1, f(6)=-34

  11. 21.
    Answers vary. Possible solution123-1123xy
  12. 23.

    f(x)=10x

  13. 25.

    f(π)0.

Exercises 2.3

  1. 1.

    Power Rule.

  2. 3.

    One answer is f(x)=10ex.

  3. 5.

    f(x), g(x), h(x), and m(x)

  4. 7.

    One possible answer is f(x)=17x-205.

  5. 9.

    f(x) is a velocity function, and f′′(x) is acceleration.

  6. 11.

    f(x)=14x-5

  7. 13.

    m(t)=45t4-38t2+3

  8. 15.

    f(r)=6er

  9. 17.

    f(x)=2x-1

  10. 19.

    h(t)=et-cost+sint

  11. 21.

    f(t)=0

  12. 23.

    g(x)=24x2-120x+150

  13. 25.

    f(x)=18x-12

  14. 27.

    f(x)=32x-12xx

  15. 29.
  16. 31.

    a is f, b is f, c is f′′

  17. 33.

    f(x)=6x5 f′′(x)=30x4 f′′′(x)=120x3 f(4)(x)=360x2

  18. 35.

    h(t)=2t-et h′′(t)=2-et h′′′(t)=-et h(4)(t)=-et

  19. 37.

    f(θ)=cosθ+sinθ f′′(θ)=-sinθ+cosθ f′′′(θ)=-cosθ-sinθ f(4)(θ)=sinθ-cosθ

  20. 39.
    (a) v(t)=4t3-8t, a(t)=12t2-8 (b) a(1.5)=19ft/s2 (c) t=0 sec and t=2 sec
  21. 41.

    Tangent line: y=2(x-1)

  22. 43.

    Tangent line: y=x-1

  23. 45.

    Tangent line: y=2(x-π4)-2

  24. 47.

    n=-3,2

  25. 49.

    The tangent line to f(x)=x4 at x=3 is y=108(x-3)+81; thus (3.01)4y(3.01)=108(.01)+81=82.08.

Exercises 2.4

  1. 1.

    F

  2. 3.

    T

  3. 5.

    F

  4. 7.
    ddx(cotx) =ddx(cosxsinx) =sinx(-sinx)-(cosx)(cosx)(sinx)2 =-[(sinx)2+(cosx)2](sinx)2 =-1(sinx)2=-csc2x
  5. 9.
    (a) f(x)=(x2+3x)+x(2x+3) (b) f(x)=3x2+6x (c) They are equal.
  6. 11.
    (a) h(s)=2(s+4)+(2s-1)(1) (b) h(s)=4s+7 (c) They are equal.
  7. 13.
    (a) f(x)=x(2x)-(x2+3)1x2 (b) f(x)=1-3x2 (c) They are equal.
  8. 15.
    (a) h(s)=4s3(0)-3(12s2)16s6 (b) h(s)=-9/4s-4 (c) They are equal.
  9. 17.

    f(x)=sinx+xcosx

  10. 19.

    H(y)=(y5-2y3)(14y+1)+(5y4-6y2)(7y2+y-8)

  11. 21.

    g(x)=-12(x-5)2

  12. 23.

    g(x)=x+82(x+4)2

  13. 25.

    h(x)=-csc2x-ex

  14. 27.

    f(x)=(x+2)(4x3+6x2)-(x4+2x3)(1)(x+2)2

  15. 29.

    y=-2x-5-10x2=-2x3+5x2+10x2

  16. 31.

    p(x)=-1x2-2x3-3x4=-x2+2x+3x4

  17. 33.

    f(t)=5t4(sect+et)+t5(secttant+et)

  18. 35.

    g(x)=0

  19. 37.

    f(y)=y(2y3-5y-1)(12y)+y(6y2-5)(6y2+7)+1(2y3-5y-1)(6y2+7)=72y5-64y3-18y2-70y-7

  20. 39.

    h(x)=(t2cost+2)(2tsint+t2cost)-(t2sint+3)(2tcost-t2sint)(t2cost+2)2

  21. 41.

    g(x)=2sinxsecx+2xcosxsecx+2xsinxsecxtanx=2tanx+2x+2xtan2x=2tanx+2xsec2x

  22. 43.

    y=2x+2

  23. 45.

    y=4

  24. 47.

    x=3/2

  25. 49.

    f(x) is never 0.

  26. 51.

    f′′(x)=2cosx-xsinx

  27. 53.

    f′′(x)=cot2xcscx+csc3x

  28. 55.

    1

  29. 57.

    -4

  30. 59.

    -125

  31. 61.

    (a) -72 (b) 118 (c) -92 (d) 152

Exercises 2.5

  1. 1.

    T

  2. 3.

    F

  3. 5.

    T

  4. 7.

    f(x)=10(4x3-x)9(12x2-1)=(120x2-10)(4x3-x)9

  5. 9.

    g(θ)=3(sinθ+cosθ)2(cosθ-sinθ)

  6. 11.

    f(x)=4(x+1x)3(1-1x2)

  7. 13.

    f(x)=-3sin(3x)

  8. 15.

    h(x)=(2θ+4)sec2(θ2+4θ)

  9. 17.

    h(t)=8sin3(2t)cos(2t)

  10. 19.

    g(x)=2(tanxsec2x-xsec2(x2))

  11. 21.

    f(x)=-tanx

  12. 23.

    f(x)=2/x

  13. 25.

    r(x)=-6(x-1)x34x-3

  14. 27.

    h(x)=200(2x+1)9[(2x+1)10+1]9

  15. 29.

    F(x)=2(2x+1)(2x+3)2(24x2+26x+3)

  16. 31.

    f(x)=5(x2+x)4(2x+1)(3x4+2x)3+(x2+x)53(3x4+2x)2(12x3+2)

  17. 33.

    g(t)=10tcos(1t)e5t2+1t2sin(1t)e5t2

  18. 35.

    f(x)=tan(5x)8(4x+1)-(4x+1)25sec2(5x)tan2(5x)

  19. 37.

    y=-cosxsinxcos(cos2x)sin(cos2x)

  20. 39.

    f(x)=12x-1/2-12x-3/2=12x-12x3

  21. 41.

    f(t)=-t1-t2

  22. 43.

    h(x)=1.5x0.5=1.5x

  23. 45.

    g(x)=x(1)-(x+7)(1/2x-1/2)x=12x-72x3

  24. 47.

    15

  25. 49.

    (a) 6   (b) 1   (c) -3   (d) 1.5

  26. 51.

    y=0

  27. 53.

    y=-3(θ-π/2)+1

  28. 55.

    In both cases the derivative is the same: 1/x.

  29. 57.
    Let g(x)=-x. Then (a) fg=f, so f(-x)=fg(x)=-fg(x)g(x)=-(fg)(x)=-f(x) (b) fg=-f, so f(-x)=fg(x)=-fg(x)g(x)=-(fg)(x)=f(x)
  30. 59.

    [f(g(x))]′′=[f(g(x))g(x)]=[f(g(x))]g(x)+f(g(x))g′′(x)=f′′(g(x))g(x)g(x)+f(g(x))g′′(x)=f′′(g(x))[g(x)]2+f(g(x))g′′(x)

  31. 61.

    2xexcotx+x2excotx-x2excsc2x

Exercises 2.6

  1. 1.

    Answers will vary.

  2. 3.

    T

  3. 5.

    dydx=-4x32y+1

  4. 7.

    dydx=sinxsecy

  5. 9.

    dydx=yx

  6. 11.

    -2sin(y)cos(y)x

  7. 13.

    12y+2

  8. 15.

    -cos(x)(x+cos(y))+sin(x)+ysin(y)(sin(x)+y)+x+cos(y)

  9. 17.

    -2x+y2y+x

  10. 19.

    3x2ycos(x3)-sin(y3)3xy2cos(y3)-sin(x3)

  11. 21.

    dydx=y(y-2x)x(x-2y)

  12. 23.
    (a) y=0 (b) y=-1.859(x-0.1)+0.281
  13. 25.
    (a) y=4 (b) y=0.93(x-2)-1084
  14. 27.
    (a) y=-13(x-72)+6+332 (b) y=3(x-4+332)+32
  15. 29.

    d2ydx2=(2y+1)(-12x2)+4x3(2-4x32y+1)(2y+1)2

  16. 31.

    d2ydx2=cosxcosy+sin2xtanycos2y

  17. 33.

    In each, dydx=-yx.

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