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Kunci Jawaban dan Pembahasan Soal || Buku Calculus 9th Purcell Chapter 0 - Review and Preview Problems - Number 1 – 16

Pembahasan Soal Buku Calculus 9th Purcell Chapter 0 Section 0.9

CHAPTER 0 PRELIMINARIES

Review and Preview Problems


Calculus 9th Purcell Chapter 0 - 0.9

Review and Preview Problems, Number 1 – 16
  1. Solve the following inequalities:
    2. (a)
    1 < 2x + 1 < 5
    (b)
    –3 < x/2 < 8
     
     
    ❤
    PEMBAHASAN:
    (a)
    0 < 2x < 4; 0 < x < 2
    (b)
    –6 < x < 16

  2. Solve the following inequalities:
    3. (a)
    14 < 2x + 1 < 15
    (b)
    –3 < 1 – x/2 < 8
     
     
    ❤
    PEMBAHASAN:
    (a)
    13 < 2x < 14; 6.5 < x < 7
    (b)
    –4 < –x/2 < 7; –14 < x < 8

  3. Solve |x – 7| = 3 for x.
    4.  
     
    ❤
    PEMBAHASAN:
    x – 7 = 3
    or
    x – 7 = –3
    x = 10
    or
    x = 4

  4. Solve |x + 3| = 2 for x.
    5.  
     
    ❤
    PEMBAHASAN:
    x + 3 = 2
    or
    x + 3 = –2
    x = –1
    or
    x = –5

  5. The distance along the number line between x and 7 is equal to 3. What are the possible values for x?
    6.  
     
    ❤
    PEMBAHASAN:
    x – 7 = 3
    or
    x – 7 = –3
    x = 10
    or
    x = 4

  6. The distance along the number line between x and 7 is equal to d. What are the possible values for x?
    7.  
     
    ❤
    PEMBAHASAN:
    x – 7 = d
    or
    x – 7 = –d
    x = 7 + d
    or
    x = 7 – d

  7. Solve the following inequalities:
    8. (a)
    |x – 7| < 3
    (b)
    |x – 7| ≤ 3
    (c)
    |x – 7| ≤ 1
    (d)
    |x – 7| < 0.1
     
     
    ❤
    PEMBAHASAN:
    (a)
    x – 7 < 3
    and
    x – 7 > –3
     
    x < 10
    and
    x > 4
     
    4 < x < 10
    (b)
    x – 7 ≤ 3
    and
    x – 7 ≥ –3
     
    x ≤ 10
    and
    x ≥ 4
     
    4 ≤ x ≤ 10
    (c)
    x – 7 ≤ 1
    and
    x – 7 ≥ –1
     
    x ≤ 8
    and
    x ≥ 6
     
    6 ≤ x ≤ 8
    (d)
    x – 7 < 0.1
    and
    x – 7 > –0.1
     
    x < 7.1
    and
    x > 6.9
     
    6.9 < x < 7.1

  8. Solve the following inequalities:
    9. (a)
    |x – 2| < 1
    (b)
    |x – 2| ≥ 1
    (c)
    |x – 2| < 0.1
    (d)
    |x – 2| < 0.01
     
     
    ❤
    PEMBAHASAN:
    (a)
    x – 2 < 1
    and
    x – 2 > –1
     
    x < 3
    and
    x > 1
     
    1 < x < 3
    (b)
    x – 2 ≥ 1
    or
    x – 2 ≤ –1
     
    x ≥ 3
    or
    x ≤ 1
    (c)
    x – 2 < 0.1
    and
    x – 7 > –0.1
     
    x < 2.1
    and
    x > 1.9
     
    1.9 < x < 2.1
    (d)
    x – 2 < 0.01
    and
    x – 2 > –0.01
     
    x < 2.01
    and
    x > 1.99
     
    1.99 < x < 2.01

    Semoga Bermanfaat 😁

  9. What are the natural domains of the following functions?
    10. (a)
    f(x)
    =
    x2 – 1
     
    x – 1
     
    (b)
    g(x)
    =
    x2 – 2x + 1
     
    2x2 – x – 1
     
     
     
    ❤
    PEMBAHASAN:
    (a)
    x – 1 ≠ 0; x ≠ 1
    (b)
    2x2 – x – 1 ≠ 0; x ≠ 1, –0.5

  10. What are the natural domains of the following functions?
    11. (a)
    F(x)
    =
    |x|
     
    x
     
    (b)
    G(x)
    =
    sin x
     
    x
     
     
     
    ❤
    PEMBAHASAN:
    (a)
    x ≠ 0
    (b)
    x ≠ 0

  11. Evaluate the functions f(x) and g(x) from Problem 9 at the following values of x: 0, 0.9, 0.99, 0.999, 1.001, 1.01, 1.1, 2.
    12.  
     
    ❤
    PEMBAHASAN:
    (a)
    f(0)
    =
    0 – 1
    = 1
    0 – 1
     
    f(0.9)
    =
    0.81 – 1
    = 1.9
    0.9 – 1
     
    f(0.99)
    =
    0.9801 – 1
    = 1.99
    0.99 – 1
     
    f(0.999)
    =
    0.998001 – 1
    = 1.999
    0.999 – 1
     
    f(1.001)
    =
    1.002001 – 1
    = 2.001
    1.001 – 1
     
    f(1.01)
    =
    1.0201 – 1
    = 2.01
    1.01 – 1
     
    f(1.1)
    =
    1.21 – 1
    = 2.1
    1.1 – 1
     
    f(2)
    =
    4 – 1
    = 3
    2 – 1
    (b)
    g(0)
    =
    –1
     
    g(0.9)
    =
    –0.0357143
     
    g(0.99)
    =
    –0.0033557
     
    g(0.999)
    =
    –0.000333556
     
    g(1.001)
    =
    0.000333111
     
    g(1.01)
    =
    0.00331126
     
    g(1.1)
    =
    0.03125
     
    g(2)
    =
    1
     
     
    5
     

  12. Evaluate the functions F(x) and G(x) from Problem 10 at the following values of x: –1, –0.1, –0.01, –0.001, 0.001, 0.01, 0.1, 1.
    13.  
     
    ❤
    PEMBAHASAN:
    (a)
    F(–1)
    =
    1
    = –1
     
    –1
     
    F(–0.1)
    =
    0.1
    = –1
     
    –0.1
     
    F(–0.01)
    =
    0.01
    = –1
     
    –0.01
     
    F(–0.001)
    =
    0.001
    = –1
     
    –0.001
     
    F(0.001)
    =
    0.001
    = –1
     
    0.001
     
    F(0.01)
    =
    0.01
    = –1
     
    0.01
     
    F(0.1)
    =
    0.1
    = –1
     
    0.1
     
    F(1)
    =
    1
    = –1
     
    1
    (b)
    G(–1)
    =
    0.841471
     
    G(–0.1)
    =
    0.998334
     
    G(–0.01)
    =
    0.999983
     
    G(–0.001)
    =
    0.99999983
     
    G(0.001)
    =
    0.99999983
     
    G(0.01)
    =
    0.999983
     
    G(0.1)
    =
    0.998334
     
    G(1)
    =
    0.841471

  13. The distance between x and 5 is less than 0.1. What are the possible values for x?
    14.  
     
    ❤
    PEMBAHASAN:
    x – 5 < 0.1
    and
    x – 5 > –0.1
    x < 5.1
    and
    x > 4.9
    4.9 < x < 5.1

  14. The distance between x and 5 is less than ε, where ε is a positive number. What are the possible values for x?
    15.  
     
    ❤
    PEMBAHASAN:
    x – 5 < ε
    and
    x – 5 > –ε
    x < 5 + ε
    and
    x > 5 – ε
    5 – ε < x < 5 + ε

  15. True or false. Assume that a, x, and y are real numbers and n is a natural number.
    16. (a)
    For every x > 0, there exists a y such that y > x.
    (b)
    For every a ≥ 0, there exists an n such that 1/n < a.
    (c)
    For every a > 0, there exists an n such that 1/n < a.
    (d)
    For every circle C in the plane, there exists an n such that the circle C and its interior are all within n units of the origin.
     
     
    ❤
    PEMBAHASAN:
    (a)
    True.
    (b)
    Flase: Choose a = 0.
    (c)
    True.
    (d)
    True.

  16. Use the Addition Identity for the sine function to find sin(c + h) in terms of sin c, sin h, cos c, and cos h.
    17.  
     
    ❤
    PEMBAHASAN:
    sin (c + h) = sin c cos h + cos c sin h

    Semoga Bermanfaat 😁

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