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Kunci Jawaban dan Pembahasan Soal || Buku Calculus 9th Purcell Chapter 0 - 0.4 Number 22 – 41

Pembahasan Soal Buku Calculus 9th Purcell Chapter 0 Section 0.4

CHAPTER 0 PRELIMINARIES

SECTION 0.4 Graphs of Equations


Problem Set 0.4, Number 22 – 41.
In Problems 1–30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all x- and y-intercepts.
  1. y = [x]/[x2 + 1]
    2. 💬 SOLUTION:
    x-intercept = 0.
    Symmetric with respect to the origin.
    Number 22
    ■

  2. 2x2 – 4x + 3y2 + 12y = –2
    3. 💬 SOLUTION:
    2(x2 – 2x + 1) + 3(y2 + 4y + 4) = –2 + 2 + 12
    2(x – 1)2 + 3(y + 2)2 = 12
    y-intercepts = –2 ± ([√(30)]/[3])
    x-intercept = 1
    Number 23
    ■

  3. 4(x – 5)2 + 9(y + 2)2 = 36
    4. 💬 SOLUTION:
    4(x – 5)2 + 9(y + 2)2 = 36; x-intercept = 5
    Number 24
    ■

  4. y = (x – 1)(x – 2)(x – 3)
    5. 💬 SOLUTION:
    y = (x – 1)(x – 2)(x – 3); y-intercept = –6
    x-intercepts = 1, 2, 3
    Number 25
    ■

  5. y = x2(x – 1)(x – 2)
    6. 💬 SOLUTION:
    y = x2(x – 1)(x – 2); y-intercept = 0
    x-intercepts = 0, 1, 2
    Number 26
    ■

  6. y = x2(x – 1)2
    7. 💬 SOLUTION:
    y = x2(x – 1)2; y-intercept = 0
    x-intercepts = 0, 1
    Number 27
    ■

  7. y = x4(x – 1)4(x + 1)4
    8. 💬 SOLUTION:
    y = x4(x – 1)4(x + 1)4; y-intercept = 0
    x-intercepts = –1, 0, 1
    Symmetric with respect to the y-axis
    Number 28
    ■

  8. |x| + |y| = 1
    9. 💬 SOLUTION:
    |x| + |y| = 1; y-intercepts = –1, 1;
    x-intercepts = –1, 1
    Symmetric with respect to the x-axis, y-axis and origin
    Number 29
    ■

  9. |x| + |y| = 4
    10. 💬 SOLUTION:
    |x| + |y| = 4; y-intercepts = –4, 1;
    x-intercepts = –4, 1
    Symmetric with respect to the x-axis, y-axis and origin
    Number 30
    ■

Semoga Bermanfaat 😁

In Problems 31–38, plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs (see Example 4).
  1. y = –x + 1
    2. y = (x + 1)2
    💬 SOLUTION:
    –x + 1 = (x + 1)2
    –x + 1 = x2 + 2x + 1
    x2 + 3x = 0
    x(x + 3) = 0
    x = 0, –3
    Intersection points: (0, 1) and (–3, 4)
    Number 31
    ■

  2. y = 2x + 3
    3. y = –(x – 1)2
    💬 SOLUTION:
    2x + 3 = –(x – 1)2
    2x + 3 = –x2 + 2x – 1
    x2 + 4 = 0
    No Points of intersection
    Number 32
    ■

  3. y = –2x + 3
    4. y = –2(x – 4)2
    💬 SOLUTION:
    –2x + 3 = –2(x – 4)2
    –2x + 3 = –2x2 + 16x – 32
    2x2 – 18x + 35 = 0
    x = [18 ± √(324 – 280)]/[4]
    x = [18 ± 2√(11)]/[4]
    x = [9 ± √(11)]/[2];
    Intersection points:
    ([9 – √(11)]/[2], –6 + √(11)), ([9 + √(11)]/[2], –6 – √(11))
    Number 33
  4. y = –2x + 3
    5. y = 3x2 – 3x + 12
    💬 SOLUTION:
    –2x + 3 = 3x2 – 3x + 12
    3x2 – x + 9 = 0
    No points of intersection
    Number 34
  5. y = x
    6. x2 + y2 = 4
    💬 SOLUTION:
    x2 + x2 = 4
    x2 = 2
    x = ±√(2)
    Intersection points: (–√(2), –√(2)), (√(2), √(2))
    Number 35
  6. y = x – 1
    7. 2x2 + 3y2 = 12
    💬 SOLUTION:
    2x2 + 3(x – 1)2 = 12
    2x2 + 3x2 – 6x + 3 = 12
    5x2 – 6x – 9 = 0
    x = [6 ± √(36 + 180)]/[10]
    x = [6 ± 6√(6)]/[10]
    x = [3 ± 3√(6)]/[5]
    Intersection points:
    ([3 – 3√(6)]/[5], [–2 – 3√(6)]/[5]), ([3 + 3√(6)]/[5], [–2 + 3√(6)]/[5])
    Number 36

    Semoga Bermanfaat 😁

  7. y – 3x = 1
    8. x2 + 2x + y2 = 15
    💬 SOLUTION:
    y = 3x + 1
    x2 + 2x + (3x + 1)2 = 15
    x2 + 2x + 9x2 + 6x + 1 = 15
    10x2 + 8x – 14 = 0
    2(5x2 + 4x – 7) = 0
    x = [–2 ± √(36)]/[5] ≈ –1.65, 0.85
    Intersection points:
    ([–2 – √(39)]/[5], [–1 – 3√(39)]/[5]) and ([–2 + √(39)]/[5], [–1 + 3√(39)]/[5])
    [or roughly (–1.65, –3.95) and (0.85, 3.55)]
    Number 37
  8. y = 4x + 3
    9. x2 + y2 = 81
    💬 SOLUTION:
    x2 + (4x + 3)2 = 81
    x2 + 16x2 + 24x + 9 = 81
    17x2 + 24x – 72 = 0
    x = [–12 ± √(38)]/[17] ≈ –2.88, 1.47
    Intersection points:
    ([–12 – √(38)]/[17], [3 – 24√(38)]/[17]) and ([–12 + √(38)]/[17], [3 + 24√(38)]/[17])
    [or roughly (–2.88, –8.52) and (1.47, 8.88)]
    Number 38
  1. Choose the equation that corresponds to each graph in Figure 8.
    2. (a) y = ax2, with a > 0
    (b) y = ax3 + bx3 + cx + d, with a > 0
    (c) y = ax3 + bx3 + cx + d, with a < 0
    (d) y = ax3, with a > 0
    Number 39
    💬 SOLUTION:
    (a) y = ax2, with a > 0 ; (2)
    (b) y = ax3 + bx3 + cx + d, with a > 0 ; (1)
    (c) y = ax3 + bx3 + cx + d, with a < 0 ; (3)
    (d) y = ax3, with a > 0 ; (4)

  2. Find the distance between the points on the circle x2 + y2 = 13 with the x-coordinates –2 and 2. How many such distances are there?
    3. 💬 SOLUTION:
    x2 + y2 = 13; (–2, –3), (–2, 3), (2, –3), (2, 3)
    d1 = √[(2 + 2)2 + (–3 + 2)2] = 4
    d2 = √[(2 + 2)2 + (–3 – 3)2] = √(54) = 2√(13)
    d3 = √[(2 – 2)2 + (3 + 3)2] = 6
    Three such distances.

  3. Find the distance between the points on the circle x2 + 2x + y2 – 2y = 20 with the x-coordinates –2 and 2. How many such distances are there?
    4. 💬 SOLUTION:
    x2 + 2x + y2 – 2y = 20; (–2, 1 + √(21)), (–2, 1 – √(21)), (2, 1 + √(13)), (2, 1 – √(13))
    d1 = √[(–2 – 2)2 + [1 + √(21) – (1 + √(13))]2]
    d1 = √[16 + (√(21) – √(13))2]
    d1 = √[50 – 2√(273)] ≈ 4.12
    d2 = √[(–2 – 2)2 + [1 + √(21) – (1 – √(13))]2]
    d2 = √[16 + (√(21) + √(13))2]
    d2 = √[50 + 2√(273)] ≈ 9.11
    d3 = √[(–2 + 2)2 + [1 + √(21) – (1 – √(13))]2]
    d3 = √[0 + (√(21) + √(21))2] = √[(2√(21))2]
    d3 = 2√(21) ≈ 9.17
    d4 = √[(–2 – 2)2 + [1 – √(21) – (1 + √(13))]2]
    d4 = √[16 + (–√(21) – √(13))2]
    d4 = √[50 + 2√(273)] ≈ 9.11
    d5 = √[(–2 – 2)2 + [1 – √(21) – (1 – √(13))]2]
    d5 = √[16 + (√(13) – √(21))2]
    d5 = √[50 – 2√(273)] ≈ 4.12
    d6 = √[(2 – 2)2 + [1 + √(13) – (1 – √(13))]2]
    d6 = √[0 + (√(13) + √(13))2] = √[(2√(13))2]
    d6 = 2√(13) ≈ 7.21
    Four such distances (d2 = d4 and d1 = d5).

Semoga Bermanfaat 😁

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