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Kunci Jawaban dan Pembahasan Soal || Buku Calculus 9th Purcell Chapter 0 - 0.1 Number 1 – 42

Pembahasan Soal Buku Calculus 9th Purcell Chapter 0 Section 0.1

◀ CHAPTER 0 PRELIMINARIES ▶

◀ SECTION 0.1 Real Numbers, Estimation, and Logic ▶


Concepts Review
  1. Numbers that can be written as the ratio of two integers are called ______. (rational numbers)
  2. Between any two real numbers, there is another real number. This is what it means to say that the real numbers are ______. (dense)
  3. The contrapositive of "If P then Q" is ______. (if not Q then not P)
  4. Axioms and definitions are taken for granted, but ______ require proof. (theorems)

Calculus 9th Purcell Chapter

Problem Set 0.1, Number 1 - 42.
In Problems 1-16, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions.
  1. 4 – 2 (8 – 11) + 6
    Answer:
    4 – 2 (8 – 11) + 6
    = 4 – 2 (–3) + 6
    = 4 + 6 + 6
    = 16
     
  2. 3 [2 – 4 (7 – 12)]
    Answer:
    3 [2 – 4 (7 – 12)]
    = 3 [2 – 4(–5)]
    = 3 [2 + 20]
    = 3 (22)
    = 66
     
  3. –4 [ 5 (–3 + 12 – 4) + 2 (13 – 7)
    Answer:
    –4 [ 5 (–3 + 12 – 4) + 2 (13 – 7)
    = –4 [5(5) + 2(6)]
    = –4 [25 + 12]
    = –4 (37)
    = –148
     
  4. 5 [–1 (7 + 12 – 16) + 4] + 2
    Answer:
    5 [–1 (7 + 12 – 16) + 4] + 2
    = 5 [–1 (3) + 4] + 2
    = 5 (–3 + 4) + 2
    = 5 (1) + 2
    = 5 + 2
    = 7
     
  5. (5/7) – (1/13)
    Answer:
    (5/7) – (1/13)
    = (65/91) – (7/91)
    = (58/91)
     
  6. (2/(4 – 7)) + (3/21) – (1/6)
    Answer:
    (2/(4 – 7)) + (3/21) – (1/6)
    = (3/–3) + (3/21) – (1/6)
    = –(42/42) + (6/42) – (7/42)
    = –(43/42)
     
  7. (1/3) [½ (¼ – (1/3)) + (1/6)]
    Answer:
    (1/3) [½ (¼ – (1/3)) + (1/6)]
    = (1/3) [½ ((3– 4)/12) + (1/6)]
    = (1/3) [½ (–(1/12)) + (1/6)]
    = (1/3) [–(1/24) + (4/24)]
    = (1/3) (3/24)
    = 1/24
     
  8. (–1/3) [(2/5) – ½ ((1/3) – (1/5))]
    Answer:
    (–1/3) [(2/5) – ½ ((1/3) – (1/5))]
    = (–1/3) [(2/5) – ½ (2/15)]
    = (–1/3) [(2/5) – (1/15)]
    = (–1/3) [(6/15) – (1/15)]
    = (–1/3) (5/15)
    = –1/9
     
  9. (14/21) [2/(5 – (1/3))]2
    Answer:
    (((14/21) [2/(5 – (1/3))]2
    = (14/21) [2/(14/3)]2
    = (14/21) [6/14]2
    = (14/21) [3/7]2
    = (2/3) [9/49]
    = 6/49
     
  10. [(2/7) – 5] / [1 – (1/7)]
    Answer:
    [(2/7) – 5] / [1 – (1/7)]
    = [(2/7) – (35/7)] / [(7/7) – (1/7)]
    = (–33/7) / (6/7)
    = – (33/6)
    = – (11/2)
     
  11. [(11/7) – (12/21)] / [(11/7) + (12/21)]
    Answer:
    [(11/7) – (12/21)] / [(11/7) + (12/21)]
    = [(11/7) – (4/7)] / [(11/7) + (4/7)]
    = (7/7) / (15/7)
    = 7/15
     
  12. [(½) – (¾) + (7/8)] / [(½) + (¾) – (7/8)]
    Answer:
    [(½) – (¾) + (7/8)] / [(½) + (¾) – (7/8)]
    = [(4/8) – (6/8) + (7/8)] / [(4/8) + (6/8) – (7/8)]
    = (5/8) / (3/8)
    = 5/3
     
  13. 1 – [1 / (1 + (½))]
    Answer:
    1 – [1 / (1 + (½))]
    = 1 – [1 / (3/2)]
    = 1 – (2/3)
    = (3/3) – (2/3)
    =  1/3
     
  14. 2 + [3 / (1 + (5/2))]
    Answer:
    2 + [3 / (1 + (5/2))]
    = 2 + [3 / ((2/2) + (5/2))]
    = 2 + [3 / (7/2)]
    = 2 + (6/7)
    = (14/7) + (6/7)
    = 20/7
     
  15. (√5 + √3) (√5 – √3)
    Answer:
    (√5 + √3) (√5 – √3)
    = (√5)2 – (√3)2
    = 5 – 3
    = 2
     
  16. (√5 – √3)2
    Answer:
    (√5 – √3)2
    = (√5)2 – 2 (√5)(√3) + (√3)2
    = 5 – 2 (√15) + 3
    = 8 – 2 (√15)
     
    17. In Problems 17-28, perform the indicated operations and simplify.
  17. (3x – 4) (x + 1)
    Answer:
    (3x – 4) (x + 1)
    = 3x2 + 3x – 4x – 4
    = 3x2 – x – 4
     
  18. (2x – 3)2
    Answer:
    (2x – 3)2
    = (2x – 3) (2x – 3)
    = 4x2 – 6x – 6x + 9
    = 4x2 – 12x + 9
     
  19. (3x – 9) (2x + 1)
    Answer:
    (3x – 9) (2x + 1)
    = 6x2 + 3x – 18x – 9
    = 6x2 – 15x – 9
     
  20. (4x – 11) (3x – 7)
    Answer:
    (4x – 11) (3x – 7)
    = 12x2 – 28x – 33x + 77
    = 12x2 – 61x + 77
     
  21. (3t2 – t + 1)2
    Answer:
    (3t2 – t + 1)2
    = (3t2 – t + 1) (3t2 – t + 1)
    = 9t4 – 3t3 + 3t2 – 3t3 + t2 – t + 3t2 – t + 1
    = 9t4 – 6t3 + 7t2 – 2t + 1
     
  22. (2t + 3)3
    Answer:
    (2t + 3)3
    = (2t + 3) (2t + 3) (2t + 3)
    = (4t2 + 12t + 9) (2t + 3)
    = 8t3 + 12t2 + 24t2 + 36t + 18t + 27
    = 8t3 + 36t2 + 54t + 27
     
  23. (x2 – 4) / (x – 2)
    Answer:
    (x2 – 4) / (x – 2)
    = [(x + 2) (x – 2)] / (x – 2)
    = x + 2, x ≠ 2
     
  24. (x2 – x – 6) / (x – 3)
    Answer:
    (x2 – x – 6) / (x – 3)
    = [(x – 3) (x + 2)] / (x – 3)
    = x + 2, x ≠ 3
     
  25. (t2 – 4t – 21) / (t + 3)
    Answer:
    (t2 – 4t – 21) / (t + 3)
    = [(t + 3) (t – 7)] / (t + 3)
    = (t – 7), t ≠ –3
     
  26. (2x – 2x2) / (x3 – 2x2 + x)
    Answer:
    (2x – 2x2) / (x3 – 2x2 + x)
    = 2x (1 – x) / [x (x2 – 2x + 1)]
    = –2x (x – 1) / [x (x – 1) (x – 1)]
    = – [2/(x – 1)]
     
  27. [12 / (x2 + 2x)] + [4/x] + [2 / (x + 2)]
    Answer:
    [12 / (x2 + 2x)] + [4 / x] + [2 / (x + 2)]
    = [12 / (x (x + 2))] + [(4 (x + 2)) / (x (x + 2))] + [2x / (x (x + 2))]
    = (12 + 4x + 8 + 2x) / (x (x + 2))
    = (6x + 20) / (x (x + 2))
    = [2 (3x + 10)] / (x (x + 2))
     
  28. [2 / (6y – 2)] + [y / (9y2 – 1)]
    Answer:
    [2 / (6y – 2)] + [y / (9y2 – 1)]
    = [2 / (2 (3y – 1))] + [y / ((3y + 1) (3y – 1))]
    = [(2 (3y + 1)) / (2 (3y + 1) (3y – 1))] + [2y / (2 (3y + 1) (3y – 1))]
    = (6y + 2 + 2y) / [2 (3y + 1) (3y – 1)]
    = (8y + 2) / [2 (3y + 1) (3y – 1)]
    = [2 (4y + 1)] / [2 (3y + 1) (3y – 1)]
    = (4y + 1) / [(3y + 1) (3y – 1)]
     
  29. Find the value of each of the following; if undefined, say so.
    (a) 0 · 0
    Answer:
    0
    (b) 0/0
    Answer:
    undefined
    (c) 0/17
    Answer:
    0
    (d) 3/0
    Answer:
    undefined
    (e) 05
    Answer:
    0
    (f) 170
    Answer:
    1
     
  30. Show that division by 0 is meaningless as follows: Suppose that a ≠ 0. If a/0 = b, then a = 0 · b = 0, which is a contradiction. Now find a reason why 0/0 is also meaningless.
    Answer:
    If 0/0 = a, then 0 = 0 · a, but this is meaningless because a could be any real number. No single value satisfies 0/0 = a.
     
    31. In Problems 31-36, change each rational number to a decimal by performing long division.
  31. 1/12
    Answer:
    12 /¯1.000 = 0.083
            9 6 –
               40
               36 –
                 4
     
  32. 2/7
    Answer:
    7 /¯2.000000 = 0.285714
         1 4 –
            60
            56 –
              40
              35 –
                50
                49 –
                  10
                    7 –
                    30
                    28 –
                      2
     
  33. 3/21
    Answer:
    21 /¯3.000000 = 0.142857
            21 –
              90
              84 –
                60
                42 –
                120
                105 –
                  150
                  147 –
                      3
     
  34. 5/17
    Answer:
    17 /¯5.000000… = 0.294117 → 0.2941176470588235
            34 –
            160
            153 –
                70
                68 –
                  20
                  17 –
                    30
                    17 –
                    130
                    119 –
                      11
                       ⁝
     
  35. 11/3
    Answer:
    3 /¯11.0 = 3.6
           9 –
           20
           18 –
             2
     
  36. 11/13
    Answer:
    13 /¯11.000000 = 0.846153
             104 –
                 60
                 52 –
                   80
                   78 –
                     20
                     13 –
                       70
                       65 –
                         50
                         39 –
                         11
     
    37. In Problem 37-42, change each repeating decimal to a ratio of two ontegers (see Example 1).
  37. 0.123123123…
    Answer:
    x = 0.123123123…
    1000x = 123.123123…
            x = 0.123123…      –
      999x = 123
            x = 123/999
               = 41/333
     
  38. 0.217171717…
    Answer:
    x = 0.217171717…
    1000x = 217.171717…
        10x = 2.171717…      –
      990x = 215
            x = 215/990
               = 43/198
     
  39. 2.56565656…
    Answer:
    x = 2.56565656…
    100x = 256.565656…
          x = 2.565656…      –
      99x = 254
          x = 254/99
     
  40. 3.929292…
    Answer:
    x = 3.929292…
    100x = 392.929292…
          x = 3.929292…      –
      99x = 389
          x = 389/99
     
  41. 0.199999…
    Answer:
    x = 0.199999…
    100x = 19.99999…
      10x = 1.99999…      –
      90x = 18
          x = 18/90
             = 1/5
     
  42. 0.399999…
    Answer:
    x = 0.399999…
    100x = 39.99999…
      10x = 3.99999…      –
      90x = 36
          x = 36/90
             = 2/5
     

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