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Kunci Jawaban dan Pembahasan Soal || Buku Calculus 9th Purcell Chapter 0 - 0.8 Number 33 – 63

Pembahasan Soal Buku Calculus 9th Purcell Chapter 0 Section 0.8

CHAPTER 0 PRELIMINARIES

SECTION 0.8 Chapter Review


Calculus 9th Purcell Chapter 0 - 0.8

Problem Set 0.8, Number 33 – 63
  1. The distance between (a + b, a) and (a - b, a) is |2b|
    2. 💬 SOLUTION:
    True:
    d = √[((a + b) – (a – b))2 + (a – a)2]
     
    d = √(2b)2
     
    d = |2b|

  2. The equation of every line can be written in point-slope form.
    3. 💬 SOLUTION:
    False:
    The equation of a vertical line cannot be written in point-slope form.

  3. The equation of every line can be written in the general linear form Ax + By + C = 0
    4. 💬 SOLUTION:
    True:
    This is the general linear equation.

  4. If two nonvertical lines are parallel, they have the same slope.
    5. 💬 SOLUTION:
    True:
    Two non-vertical lines are parallel if and only if they have the same slope.

  5. It is possible for two lines to have positive slopes and be perpendicular.
    6. 💬 SOLUTION:
    False:
    The slopes of perpendicular lines are negative reciprocals.

  6. If the x- and y-intercepts of a line are rational and non-zero, then the slope of the line is rational.
    7. 💬 SOLUTION:
    True:
    If a and b are rational and (a, 0) (0, b) are the intercepts, the slope is – (b/a) which is rational.

  7. The lines ax + y = c and ax – y = c perpendicular.
    8. 💬 SOLUTION:
    False:
    ax + y = c ⇒ y = – ax + c
     
    ax – y = c ⇒ y = ax – c
     
    (a)(–a) ≠ –1
     
    (unless a = ± 1)

  8. (3x – 2y + 4) + m(2x + 6y – 2) = 0 is the equation of a line for each real number m.
    9. 💬 SOLUTION:
    True:
    The equation is (3 + 2m)x + (6m – 2)y + 4 – 2m = 0 which is the equation of a straight line unless 3 + 2m and 6m – 2 are both 0, and there is no real number m such that 3 + 2m = 0 and 6m – 2 = 0.

  9. The natural domain of
    10. f(x) = √[–(x2 + 4x + 3)]
    is the interval –3 ≤ x ≤ –1.
    💬 SOLUTION:
    True:
    f(x) = √[–(x2 + 4x + 3)]
     
    f(x) = √[–(x + 3)(x + 1)]
     
    –(x2 + 4x + 3) ≥ 0 on –3 ≤ x ≤ –1.

  10. The natural domain of T(θ) = sec(θ) + cos(θ) is (–∞, ∞).
    11. 💬 SOLUTION:
    False:
    The domain does not include nπ + (π/2) where nis an integer.

  11. The range of f(x) = x2 – 6 is the interval [–6, ∞).
    12. 💬 SOLUTION:
    True:
    The domain is (–∞, ∞) and the range is [–6, ∞).

    Semoga Bermanfaat 😁

  12. The range of the function f(x) = tan x – sec x is the set (–∞, -1] ∪ [1, ∞).
    13. 💬 SOLUTION:
    False:
    The range is (–∞, ∞).

  13. The range of the function f(x) = csc x – sec x is the set (–∞, -1] ∪ [1, ∞)
    14. 💬 SOLUTION:
    False:
    The range (–∞, ∞).

  14. The sum of two even functions is an even function.
    15. 💬 SOLUTION:
    True:
    If f(x) and g(x) are even functions, f(x) + g(x) is even. f(–x) + g(–x) = f(x) + g(x).

  15. The sum of two odd functions is an odd function.
    16. 💬 SOLUTION:
    True:
    If f(x) and g(x) are odd functions, f(–x) + g(–x) = –f(x) – g(x) = –[f(x) + g(x)] so f(x) + g(x) is odd.

  16. The product of two odd functions is an odd function.
    17. 💬 SOLUTION:
    False:
    If f(x) and g(x) are odd functions, f(–x)g(–x) = –f(x)[–g(x)] = f(x)g(x), so f(x)g(x) is even.

  17. The product of an even function with an odd function is an odd function.
    18. 💬 SOLUTION:
    True:
    If f(x) is even and g(x) is odd, f(–x)g(–x) = f(x)[–g(x)] = –f(x)g(x), so f(x)g(x) is odd.

  18. The composition of an even function with an odd function is an odd function.
    19. 💬 SOLUTION:
    False:
    If f(x) is even and g(x) is odd, f(g(–x)) = f(–g(x)) = f(g(x)); while if f(x) is odd and g(x) is even, f(g(–x)) = f(g(x)); so f(g(x)) is even.

  19. The composition of two odd functions is an even function.
    20. 💬 SOLUTION:
    False:
    If f(x) and g(x) are odd functions, f(g(–x)) = f(–g(x)) = –f(g(x)), so f(g(x)) is odd.

  20. The function f(x) = (2x3 + x) / (x2 + 1) is odd.
    21. 💬 SOLUTION:
    True:
    f(x) = (2(–x)3 + (–x))/(–x)2 + 1
     
    f(x) = (–2x3 – x)/(x2 + 1)
     
    f(x) = –[(2x3 + x)/(x2 + 1)]

  21. The function
    22. f(x)
    =
    (sin t)2 + cos t
    tan t csc t
    is even
    💬 SOLUTION:
    True:
    f(–t) = [(sin(–t))2 + cos(–t)]/[tan(–t) csc(–t)]
     
    f(–t) = [(–sin t)2 + cos t]/[–tan t (–csc t)]
     
    f(–t) = [(–sin t)2 + cos t]/[tan t csc t]

    Semoga Bermanfaat 😁

  22. If the range of a function consists of just one number, then its domain also consists of just one number.
    23. 💬 SOLUTION:
    False:
    f(x) = c has domain (–∞, ∞) and the only value of the range is c.

  23. If the domain of a function contains at least two numbers then the range also contains at least two numbers.
    24. 💬 SOLUTION:
    False:
    f(x) = c has domain (–∞, ∞), yet the range has only one value, c.

  24. If g(x) = [x/2] then g(–1.8) = –1
    25. 💬 SOLUTION:
    True:
    g(–1.8) = 〚(–1.8)/2〛 = 〚–0.9〛 = –1.

  25. If f(x) = x2 and g(x) = x3, then f ◦ g = g ◦ f.
    26. 💬 SOLUTION:
    True:
    (f ◦ g)(x) = (x3)2 = x6
     
    (g ◦ f)(x) = (x2)3 = x6

  26. If f(x) = x2 and g(x) = x3, then (f ◦ g)(x) = (g ◦ f)(x).
    27. 💬 SOLUTION:
    False:
    (f ◦ g)(x) = (x3)2 = x6
     
    f(x) ‧ g(x) = x2 ‧ x3 = x5

  27. If f and g have the same domain, then f/g also has that domain.
    28. 💬 SOLUTION:
    False:
    The domain of f/g excludes any values where g = 0.

  28. If the graph of y = f(x) has an x-intercept at x = a then the graph of y = f(x + h) has an x-intercept at x = a – h
    29. 💬 SOLUTION:
    True:
    f(a) = 0
     
    Let F(x) = f(x + h), then
     
    F(a – h) = f(a – h + h) = f(a) = 0

  29. The cotangent is an odd function.
    30. 💬 SOLUTION:
    True:
    cot x = (cos x)/(sin x)
     
    cot(–x) = [cos(–x)]/[sin(–x)]
     
    cot(–x) = [cos(–x)]/[sin(–x)]
     
    cot(–x) = [cos x]/[–sin x]
     
    cot(–x) = –cot x

  30. The natural domain of the tangent function is the set of all real numbers.
    31. 💬 SOLUTION:
    False:
    The domain of the tangent function excludes all nπ + (π/2) where n is an integer.

  31. If cos s = cos t, then s = t.
    32. 💬 SOLUTION:
    False:
    The cosine function is periodic, so cos s = cos t does not necessarily imply s = t; e.g., cos 0 = cos 2 π = 1, but 0 ≠ 2π.

Semoga Bermanfaat 😁

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