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Kunci Jawaban dan Pembahasan Soal || Buku Calculus 9th Purcell Chapter 0 - Sample Test Problems - Number 30 – 55

Pembahasan Soal Buku Calculus 9th Purcell Chapter 0 Section 0.8

CHAPTER 0 PRELIMINARIES

Sample Test Problems


Calculus 9th Purcell Chapter 0 - 0.9

Sample Test Problems, Number 30 – 55
  1. Find the equation of the line through the indicated point that is parallel to the indicated line, and sketch both lines.
    2. (a)
    (3, 2): 3x + 2y = 6
    (b)
    (1, –1): y = 2/3 · x + 1
    (c)
    (5, 9): y = 10
    (d)
    (–3, 4): x = –2
    💬 SOLUTION:
    (a)
    3x + 2y
    =
    6
     
    2y
    =
    –3x + 6
     
    y
    =
    –3/2 · x + 3
     
    m
    =
    –3/2
     
    y – 2
    =
    –3/2 · (x – 3)
     
    y
    =
    –3/2 · x + (13/2)

    Calculus 9th Purcell Chapter 0 - 0.9

    (b)
    m
    =
    2/3
     
    y + 1
    =
    2/3 · (x – 1)
     
    y
    =
    2/3 · x – (5/3)

    Calculus 9th Purcell Chapter 0 - 0.9

    (c)
    y
    =
    9

    Calculus 9th Purcell Chapter 0 - 0.9

    (d)
    x
    =
    –3

    Calculus 9th Purcell Chapter 0 - 0.9


  2. Write the equation of the line through (–2, 1) that
    3. (a)
    goes through (7, 3);
    (b)
    is parallel to 3x – 2y = 5;
    (c)
    is perpendicular to 3x + 4y = 9;
    (d)
    is perpendicular to y = 4;
    (e)
    has y-intercept 3.
    💬 SOLUTION:
    (a)
    m
    =
    3 – 1
    =
    2
    ;
     
     
    7 + 2
    9
     
    y – 1
    =
    2
    (x + 2)
     
     
     
    9
     
     
     
    y
    =
    2
    x +
    13
     
     
    9
    9
     
    (b)
    3x – 2y
    =
    5
     
     
    –2y
    =
    –3x + 5
     
     
    y
    =
    3
    x –
    5
    ;
     
     
    2
    2
     
     
    m
    =
    3
     
     
    2
     
     
    y – 1
    =
    3
    (x + 2)
     
     
    2
     
     
    y
    =
    3
    x + 4
     
     
    2
     
    (c)
    3x + 4y
    =
    9
     
     
    4y
    =
    –3x + 9;
     
     
    y
    =
    –
    3
    x +
    9
    ;
     
     
    4
    4
     
     
    m
    =
    4
     
     
    3
     
     
    y – 1
    =
    4
    (x + 2)
     
     
    3
     
     
    y
    =
    4
    x +
    11
     
     
    3
    3
     
    (d)
    x
    =
    –2
     
    (e)
    contains (–2, 1) and (0, 3);
     
     
    m
    =
    3 – 1
    ;
     
     
    0 + 2
     
     
    y
    =
    x + 3
     

  3. Show that (2, –1), (5, 3), and (11, 11) are on the same line.
    4. 💬 SOLUTION:
     
    m1
    =
    3 + 1
    =
    4
    ;
     
     
    5 – 2
    3
     
     
    m2
    =
    11 – 3
    =
    8
    =
    4
    ;
     
     
    11 – 5
    6
    3
     
     
    m3
    =
    11 + 1
    =
    12
    =
    4
    ;
     
     
    11 – 2
    9
    3
     
    m1 = m2 = m3 , so the point lie on the same line.

    Calculus 9th Purcell Chapter 0 - 0.9

  4. Figure 1 can be represented by which equation?
    5. (a)
    y = x3
    (b)
    x = y3
    (c)
    y = x2
    (d)
    x = y3
    💬 SOLUTION:
    The figure is a cubic with respect to y.
    The equation is (b) x = y3.

  5. Figure 2 can be represented by which equation?
    6. (a)
    y = ax2 + bx + c, with a > 0, b > 0, and c > 0
    (b)
    y = ax2 + bx + c, with a < 0, b > 0, and c > 0
    (c)
    y = ax2 + bx + c, with a < 0, b > 0, and c < 0
    (d)
    y = ax2 + bx + c, with a > 0, b > 0, and c < 0
    💬 SOLUTION:
    The figure is a quadratic, opening downward, with a negative y-intercept.
    The equation is (c) y = ax2 + bx + c, with a < 0, b > 0 , and c < 0.

    Semoga Bermanfaat 😁

    In Problems 35-38, sketch the graph of each equation.
  6. 3y – 4x = 6
    7. 💬 SOLUTION:

    Calculus 9th Purcell Chapter 0 - 0.9


  7. x2 – 2x + y2 = 3
    8. 💬 SOLUTION:
     
    x2 – 2x + y2
    =
    3
     
    x2 – 2x + 1 + y2
    =
    4
     
    (x – 1)2 + y2
    =
    4

    Calculus 9th Purcell Chapter 0 - 0.9


  8. y
    =
    2x
    x2 + 2
    9. 💬 SOLUTION:

    Calculus 9th Purcell Chapter 0 - 0.9


  9. x = y2 – 3
    10. 💬 SOLUTION:

    Calculus 9th Purcell Chapter 0 - 0.9


  10. Find the points of intersection of the graphs of y = x2 – 2x + 4 and y – x = 4
    11. 💬 SOLUTION:
     
    y
    =
    x2 – 2x + 4
    and
     
    y - x
    =
    4;
     
     
    x + 4
    =
    x2 – 2x + 4
     
     
    x2 – 3x
    =
    0
     
     
    x(x – 3)
    =
    0
     
     
    point of intersection: (0, 4) and (3, 7)

  11. Among all lines perpendicular to 4x – y = 2 find the equation of the one that, together with the positive x- and y-axes, forms a triangle of area 8.
    12. 💬 SOLUTION:
     
    4x – y
    =
    2
     
     
    y
    =
    4x – 2;
     
     
    m
    =
    –
    1
     
     
    4
     
     
    contains (a, 0), (0, b);
     
     
    ab
    =
    8
     
     
    2
     
     
    ab
    =
    16
     
     
    b
    =
    16
     
     
    a
     
     
    b - 0
    =
    –
    b
    =
    –
    1
    ;
     
    0 – a
    a
    4
     
    a
    =
    4b
     
     
    a
    =
    4 (
    16
    )
     
     
    a
     
     
    a2
    =
    64
     
     
    a
    =
    8
     
     
    b
    =
    16
     
     
    a
     
     
    b
    =
    2;
     
     
    y
    =
    –
    1
    x + 2
     
     
    4
     

  12. For f(x) = 1/(x + 1) – 1/x, find each value (if possible).
    13. (a)
    f (1)
    (b)
    f (–1/2)
    (c)
    f (–1)
    (d)
    f (t – 1)
    (e)
    f (1/t)
    💬 SOLUTION:
    (a)
    f (1)
    =
    1
    –
    1
    =
    –
    1
     
    1 + 1
    1
    2
    (b)
    f (–1/2)
    =
    1
    –
    1
    =
    4
     
    –1/2 + 1
    –1/2
    (c)
    f (–1) does not exist.
    (d)
    f (t – 1)
    =
    1
    –
    1
    =
    1
    –
    1
     
    t – 1 + 1
    t – 1
    t
    t – 1
    (e)
    f (1/t)
    =
    1
    –
    1
    =
    t
    –
    t
     
    1/t + 1
    1/t
    1 + t

    Semoga Bermanfaat 😁

  13. For g(x) = (x + 1)/x find and simplify each value.
    14. (a)
    g (2)
    (b)
    g (1/2)
    (c)
    g(2 + h) – g(2)
     
    h
     
    💬 SOLUTION:
    (a)
    g (2)
    =
    2 + 1
    =
    3
     
     
    2
    2
     
    (b)
    g (1/2)
    =
    1/2 + 1
    =
    3
     
     
    1/2
     
     
     
     
    2 + h + 1
    –
    2 + 1
    (c)
    g(2 + h) – g(2)
    =
    2 + h
    2
     
    h
    h
     
     
     
    2h + 6 – 3h – 6
     
     
     
    =
    2 (2 + h)
     
     
     
    h
     
     
     
     
    –
    h
     
     
     
    =
    2 (2 + h)
     
     
     
    h
     
     
    g(2 + h) – g(2)
    =
    –1
     
     
    h
    2 (2 + h)
     

  14. Describe the natural domain of each function.
    15. (a)
    g (2)
    =
    x
     
     
    x2 – 1
     
    (b)
    g (x)
    =
    √(4 – x2)
    💬 SOLUTION:
    (a)
    {x ∈ ℝ : x ≠ –1, 1}
    (b)
    {x ∈ ℝ : |x| ≤ 2}

  15. Which of the following functions are odd? Even? Neither even nor odd?
    16. (a)
    f (x)
    =
    3x
     
     
    x2 + 1
     
    (b)
    g (x)
    =
    |sin x| + cos x
    (c)
    h (x)
    =
    x3 + sin x
    (d)
    k (x)
    =
    x2 + 1
     
     
    |x| + x4
     
    💬 SOLUTION:
    (a)
    f (–x)
    =
    3(–x)
     
     
    (–x)2 + 1
     
     
     
    =
    –
    3(x)
    ; odd
     
     
    (x)2 + 1
     
    (b)
    g (–x)
    =
    |sin (–x)| + cos (–x)
     
     
     
     
    =
    |–sin x| + cos x
     
     
     
     
    =
    |sin x| + cos x
    ; even
     
    (c)
    h (–x)
    =
    (–x)3 + sin (–x)
     
     
     
    =
    –x3 – sin x
    ; odd
    (d)
    k (x)
    =
    (–x)2 + 1
     
     
    |–x| + (–x)4
     
     
     
    =
    x2 + 1
    ; even
     
     
    |x| + x4
     

  16. Sketch the graph of each function.
    17. (a)
    f (x)
    =
    x2 – 1
     
    (b)
    g (x)
    =
    x
     
     
    x2 + 1
     
    (c)
    h (x)
    = {
    x2
    if 0 ≤ x ≤ 2
     
    6 – x
    if x > 2
    💬 SOLUTION:
    (a)
    f (x)
    =
    x2 – 1

    Calculus 9th Purcell Chapter 0 - 0.9

    (b)
    g (x)
    =
    x
     
    x2 + 1

    Calculus 9th Purcell Chapter 0 - 0.9

    (c)
    h (x)
    = {
    x2
    if 0 ≤ x ≤ 2
     
    6 – x
    if x > 2

    Calculus 9th Purcell Chapter 0 - 0.9


  17. Suppose that f is an even function satisfying f(x) = –1 + √x for x ≥ 0. Sketch the graph of f for –4 ≤ x ≤ 4.
    18. 💬 SOLUTION:

    Calculus 9th Purcell Chapter 0 - 0.9


  18. An open box is made by cutting squares of side x inches from the four corners of a sheet of cardboard 24 inches by 32 inches and then turning up the sides. Express the volume V(x) in terms of x. What is the domain for this function?
    19. 💬 SOLUTION:
     
    V(x) = x(32 – 2x)(24 – 2x)
     
    Domain [0, 12]

  19. Let f(x) = x – (1/x) and g(x) = x2 + 1. Find each value.
    20. (a)
    (f + g)(2)
    (b)
    (f ⋅ g)(2)
    (c)
    (f ◦ g)(2)
    (d)
    (g ◦ f)(2)
    (e)
    f 3 (–1)
    (f)
    f 2 (2) + g2 (2)
    💬 SOLUTION:
    (a)
    (f + g)(2)
    =
    (2 –
    1
    ) + (22 + 1)
    =
    13
     
    2
    2
    (b)
    (f ⋅ g)(2)
    =
    (
    3
    ) (5)
    =
    15
     
    2
    2
    (c)
    (f ◦ g)(2)
    =
    f(5)
    =
    5 –
    1
    =
    24
     
    5
    5
    (d)
    (g ◦ f)(2)
    =
    g(
    3
    )
    =
    (
    3
    )2 + 1 =
    13
     
    2
    2
    4
    (e)
    f 3 (–1)
    =
    (–1 +
    1
    )3
    =
    0
     
    1
    (f)
    f 2 (2) + g2 (2)
    =
    (
    3
    )2 + (5)2
     
    2
     
     
    =
    9
    + 25
     
     
    4
     
     
     
    =
    109
     
     
    4
     

    Semoga Bermanfaat 😁

  20. Sketch the graph of each of the following, making use of translations.
    21. (a)
    y = 1/4 x2
    (b)
    y = 1/4 (x + 2)2
    (c)
    y = –1 + 1/4 (x + 2)2
    💬 SOLUTION:
    (a)
    y = 1/4 x2

    Calculus 9th Purcell Chapter 0 - 0.9

    (b)
    y = 1/4 (x + 2)2

    Calculus 9th Purcell Chapter 0 - 0.9

    (c)
    y = –1 + 1/4 (x + 2)2

    Calculus 9th Purcell Chapter 0 - 0.9


  21. Let f (x) = √(16 – x) and g(x) = x4. What is the domain of each of the following?
    22. (a)
    f
    (b)
    f ◦ g
    (c)
    g ◦ f
    💬 SOLUTION:
    (a)
    f = (–∞, 16]
    (b)
    f ◦ g = √(16 – x4); domain [–2, 2]
    (c)
    g ◦ f = [√(16 – x)] 4 = (16 – x)2;
     
    domain (–∞, 16]
     
    (note: the simplification [√(16 – x)] 4 = (16 – x)2 is only true given the restricted domain)

  22. Write F(x) = √(1 + sin2 x) as the complosite of four functions, f ◦ g ◦ h ◦ k.
    23. 💬 SOLUTION:
    f(x) = √x, g(x) = 1 + x, h(x) = x2, k(x) = sin x, F(x) = √(1 + sin2 x) = f ◦ g ◦ h ◦ k.

  23. Calculate each of the following without using a calculator.
    24. (a)
    sin 570°
    (b)
    cos 9π/2
    (c)
    cos (–13π/6)
    💬 SOLUTION:
    (a)
    sin (570°) = sin (570°) = –1/2
    (b)
    cos (9π/2) = cos (9π/2) = 0
    (c)
    cos (–13π/6) = cos (–π/6) = √3/2

  24. If sin t = 0.8 and cos t < 0, find each value.
    25. (a)
    sin (–t)
    (b)
    cos (t)
    (c)
    sin (2t)
    (d)
    tan (t)
    (e)
    cos [(π/6) – t]
    (f)
    sin (π + t)
    💬 SOLUTION:
    (a)
    sin (–t) = –sin (t) = –0.8
    (b)
    sin2 t + cos2 t = 1
     
    cos2 t = 1 – (0.8)2 = 0.36
     
    cos t = –0.6
    (c)
    sin (2t) = 2 sin t cos t = 2(0.8)(–0.6) = –0.96
    (d)
    tan (t) = sin t/cos t = 0.8/–0.6 = –4/3 ≈ –1.333
    (e)
    cos [(π/6) – t] = sin t = 0.8
    (f)
    sin (π + t) = –sin t = –0.8

  25. Write sin 3t in terms of sin t. Hint: 3t = 2t + t.
    26. 💬 SOLUTION:
     
    sin 3t = sin (2t + t) = sin 2t cos t + cos 2t sin t
     
    = 2 sin t cos2 t + (1 – 2 sin2 t) sin t
     
    = 2 sin t (1 – 2 sin2 t) + sin t – 2 sin3 t
     
    = 2 sin t – 2 sin3 t + sin t – 2 sin3 t
     
    = 3 sin t – 4 sin3 t

  26. A fly sits on the rim of a wheel spinning at the rate of 20 revolutions per minute. If the radius of the whee; is 9 inches, how far does the fly travel in 1 second?
    27. 💬 SOLUTION:
     
    s = rt
     
    = 9 (20 rev/min)(2π rad/rev)(1 min/60 sec)(1 sec) = 6π
     
    ≈ 18.85 in.

Semoga Bermanfaat 😁

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