Solutions To Selected Problems

Chapter 10

Exercises 10.0

  1. 1.

    y=12(x-3)2+32

  2. 2.

    y=-112(x+1)2-1

  3. 3.

    x=-14(y-5)2+2

  4. 4.

    x=y2

  5. 5.

    y=-14(x-1)2+2

  6. 6.

    x=-112y2

  7. 7.

    y=4x2

  8. 8.

    x=-18(y-3)2+2

  9. 9.
    -22424xy
  10. 10.
    -55-6-4-2xy
  11. 11.

    (x+1)29+(y-2)24=1

  12. 12.

    (x-1)21/4+y29=1

  13. 13.

    (x-1)22+(y-2)2=1

  14. 14.

    x23+y25=1

  15. 15.

    x24+(y-3)26=1

  16. 16.

    (x-2)24+(y-2)24=1

  17. 17.

    x2-y23=1

  18. 18.

    y2-x224=1

  19. 19.

    (y-3)24-(x-1)29=1

  20. 20.

    (x-1)29-(y-3)24=1

  21. 21.
    -55-5xy
  22. 22.
    -10-55510xy
  23. 23.

    x24-y23=1

  24. 24.

    x23-(y-1)29=1

  25. 25.

    (y-2)2-x210=1

  26. 26.

    4y2-x24=1

Exercises 10.1

  1. 1.

    T

  2. 3.

    2

  3. 5.

    4/3

  4. 7.

    109/2

  5. 9.

    12/5

  6. 11.

    -ln(2-3)1.31696

  7. 13.

    011+4x2dx

  8. 15.

    011+14xdx

  9. 17.

    -111+x21-x2dx

  10. 19.

    121+1x4dx

  11. 21.

    1.4790

  12. 23.

    Simpson’s Rule fails, as it requires one to divide by 0. However, recognize the answer should be the same as for y=x2; why?

  13. 25.

    Simpson’s Rule fails.

  14. 27.

    1.4058

  15. 29.

    2π012x5dx=2π5

  16. 31.

    2π01x1+1/(4x)dx=π/6(55-1)

Exercises 10.2

  1. 1.

    T

  2. 3.

    rectangular

  3. 5.
    510-5xy
  4. 7.
    1212xy
  5. 9.
    -10-55102468xy
  6. 11.
    -55-55xy
  7. 13.
    -1-0.50.51-1-0.50.51xy
  8. 15.
    510-1010xy
  9. 17.
    -11-11xy
  10. 19.
    (a) Traces the parabola y=x2, moves from left to right. (b) Traces the parabola y=x2, but only from -1x1; traces this portion back and forth infinitely. (c) Traces the parabola y=x2, but only for 0<x. Moves left to right. (d) Traces the parabola y=x2, moves from right to left.
  11. 21.

    Possible Answer: x=t, y=9-4t

  12. 23.

    Possible Answer: x=-9+7cost, y=4+7sint

  13. 25.

    Possible Answer: x=54t+114, y=t, [-3,1]

  14. 27.

    Possible Answer: x=t, y=t2+2t, (-,-1]

  15. 29.

    x=(t+11)/6, y=(t2-97)/12. At t=1, x=2, y=-8.

    y=6x-11; when x=2, y=1.

  16. 31.

    x=cos-1t, y=1-t2. At t=1, x=0, y=0.

    y=cosx; when x=0, y=1.

  17. 33.
    Possible answers: (a) x=sint, y=cost, [π/2,5π/2] (b) x=cost, y=sint, [0,2π] (c) x=sint, y=cost, [π/2,9π/2] (d) x=cost, y=sint, [0,4π]
  18. 35.

    x=4t, y=-16t2+64t

  19. 37.

    x=10t, y=-16t2+320t

  20. 39.

    x=3cos(2πt)+1, y=3sin(2πt)+1; other answers possible

  21. 41.

    x=5cost, y=24sint; other answers possible

  22. 43.

    x=2tant, y=±6sect; other answers possible

  23. 45.

    y=-1.5x+8.5

  24. 47.

    (x-1)216+(y+2)29=1

  25. 49.

    y=2x+3

  26. 51.

    y=e2x-1

  27. 53.

    x2-y2=1

  28. 55.

    y=ba(x-x0)+y0; line through (x0,y0) with slope b/a.

  29. 57.

    (x-h)2a2+(y-k)2b2=1; ellipse centered at (h,k) with horizontal axis of length 2a and vertical axis of length 2b.

  30. 59.

    t=±1

  31. 61.

    t=π/2,3π/2

  32. 63.

    t=-1

  33. 65.

    t=kπ for integer values of k

Exercises 10.3

  1. 1.

    F

  2. 3.

    F

  3. 5.
    (a) dydx=2t (b) Tangent line: y=2(x-1)+1; normal line: y=-1/2(x-1)+1
  4. 7.
    (a) dydx=2t+12t-1 (b) Tangent line: y=3x+2; normal line: y=-1/3x+2
  5. 9.
    (a) dydx=csct (b) t=π/4: Tangent line: y=2(x-2)+1; normal line: y=-1/2(x-2)+1
  6. 11.
    (a) dydx=costsin(2t)+2sintcos(2t)-sintsin(2t)+2costcos(2t) (b) Tangent line: y=x-2; normal line: y=-x
  7. 13.

    horizontal: t=0; vertical: none

  8. 15.

    horizontal: t=-1/2; vertical: t=1/2

  9. 17.

    horizontal: none; vertical: t=0

  10. 19.

    The solution is non-trivial; use identities sin(2t)=2sintcost and cos(2t)=cos2t-sin2t=1-2sin2t to rewrite dy/dt=2sint(2cos2t-sin2t) and dx/dt=2cost(1-3sin2t). Horizontal: sint=0 when t=0,π,2π, and 2cos2t-sin2t=0 when t=tan-1(2),π±tan-1(2), 2π-tan-1(2). Vertical: cost=0 when t=π/2,3π/2, and 1-3sin2t=0 when t=sin-1(1/3),π-sin-1(1/3).

  11. 21.

    t0=0; limt0dydx=0.

  12. 23.

    t0=1; limt1dydx=.

  13. 25.

    d2ydx2=2; always concave up

  14. 27.

    d2ydx2=-4(2t-1)3; concave up on (-,1/2); concave down on (1/2,).

  15. 29.

    d2ydx2=-cot3t; concave up on (-π/2,0); concave down on (0,π/2).

  16. 31.

    d2ydx2=4(13+3cos(4t))(cost+3cos(3t))3, obtained with a computer algebra system; concave up on (-tan-1(12),tan-1(12)), concave down on (-π2,-tan-1(12)); (tan-1(12),π2)

  17. 33.

    L=6π

  18. 35.

    L=234

  19. 37.

    2π

  20. 39.

    -103+ln(3+10)+2-ln(1+2)

  21. 41.

    L2.4416 (actual value: L=2.42211)

  22. 43.

    L4.19216 (actual value: L=4.18308)

  23. 45.

    The answer is 16π for both (of course), but the integrals are different.

  24. 47.

    6πa25

  25. 49.

    SA8.50101 (actual value SA=8.02851

  26. 51.

    12sinhθcoshθ-12θ

Exercises 10.4

  1. 1.

    Answers will vary.

  2. 3.

    T

  3. 5.
    12OABCD
  4. 7.
    A(2.5,π/4) and A(-2.5,5π/4); B(-1,5π/6) and B(1,11π/6); C(3,4π/3) and C(-3,π/3); D(1.5,2π/3) and D(-1.5,5π/3)
  5. 9.
    A=(2,2); B=(2,-2); C=(5,-0.46); D=(5,2.68)
  6. 11.
    1212xy
  7. 13.
    -22-2-112xy
  8. 15.
    -22-22xy
  9. 17.
    -22-22xy
  10. 19.
    -11-11xy
  11. 21.
    -11-11xy
  12. 23.
    -22231xy
  13. 25.
    -2-1-11xy
  14. 27.
    -11-1-0.50.51xy
  15. 29.
    -55-4-224xy
  16. 31.
    -55-4-224xy
  17. 33.

    (x-1)2+y2=1

  18. 35.

    x2+(y-32)2=94

  19. 37.

    (x-1/2)2+(y-1/2)2=1/2

  20. 39.

    x=3

  21. 41.

    x4+x2y2-y2=0

  22. 43.

    x2+y2=4

  23. 45.

    θ=π/4

  24. 47.

    r=5secθ

  25. 49.

    r=cosθ/sin2θ

  26. 51.

    r=7

  27. 53.

    P(3/2,π/6), P(0,π/2), P(-3/2,5π/6)

  28. 55.

    P(0,0)=P(0,π/2), P(2,π/4)

  29. 57.

    P(2/2,π/12), P(-2/2,5π/12), P(2/2,3π/4), and the origin.

  30. 59.
    For all points, r=1; θ=π12,5π12,7π12,11π12,13π12,17π12,19π12,23π12.
  31. 61.

    Answers will vary. If m and n do not have any common factors, then an interval of 2nπ is needed to sketch the entire graph.

Exercises 10.5

  1. 1.

    Using x=rcosθ and y=rsinθ, we can write x=f(θ)cosθ, y=f(θ)sinθ.

  2. 3.
    (a) dydx=-cotθ (b) tangent line: y=-(x-2/2)+2/2; normal line: y=x
  3. 5.
    (a) dydx=cosθ(1+2sinθ)cos2θ-sinθ(1+sinθ) (b) tangent line: x=33/4; normal line: y=3/4
  4. 7.
    (a) dydx=θcosθ+sinθcosθ-θsinθ (b) tangent line: y=-(2/π)x+π/2; normal line: y=(π/2)x+π/2
  5. 9.
    (a) dydx=4sin(θ)cos(4θ)+sin(4θ)cos(θ)4cos(θ)cos(4θ)-sin(θ)sin(4θ) (b) tangent line: y=53(x+3/4)-3/4; normal line: y=-1/53(x+3/4)-3/4
  6. 11.

    horizontal: θ=π/2,3π/2;

    vertical: θ=0,π,2π

  7. 13.
    horizontal: θ=tan-1(1/5),π/2, π-tan-1(1/5),π+tan-1(1/5), 3π/2, 2π-tan-1(1/5);

    vertical: θ=0,tan-1(5),π-tan-1(5),π,π+tan-1(5), 2π-tan-1(5)

  8. 15.

    In polar: θ=0θ=π

    In rectangular: y=0

  9. 17.
    In polar: θ=π4 and θ=-π4 In rectangular: y=x and y=-x.
  10. 19.

    area = 4π3+23

  11. 21.

    area = π/12

  12. 23.

    area = 3π/2

  13. 25.

    area = 2π+33/2

  14. 27.

    area = 1

  15. 29.

    area = 132(4π-33)

  16. 31.

    x(θ)=f(θ)cosθ-f(θ)sinθ, y(θ)=f(θ)sinθ+f(θ)cosθ. Square each and add; applying the Pythagorean Theorem twice achieves the result.

  17. 33.

    4π

  18. 35.

    L2.2592; (actual value L=2.22748)

  19. 37.

    SA=16π

  20. 39.

    SA=32π/5

  21. 41.

    SA=36π

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