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Kunci Jawaban dan Pembahasan Soal || Buku Calculus 9th Purcell Chapter 0 - 0.1 Number 43 – 83

Pembahasan Soal Buku Calculus 9th Purcell Chapter 0 Section 0.1

◀ CHAPTER 0 PRELIMINARIES ▶

◀ SECTION 0.1 Real Numbers, Estimation, and Logic ▶


Problem Set 0.1, Number 43 - 83.
  1. Since 0.199999… = 0.200000… and 0.399999... = 0.400000... (see Problems 41 and 42), we see that certain rational numbers have two different decimal expansions. Which rational numbers have this property?
    2. Answer:
    Those rational numbers that can be expressed by a terminating decimal followed by zeros.

  2. Show that any rational number p/q, for which the prime factorization of q consists entirely of 2s and 5s, has a terminating decimal expansion.
    3. Answer:
    p/q = p · (1/q), so we only need to look at 1/q. If q = 2n · 5m, then 1/q = (1/2)n · (1/5)m = (0.5)n · (0.2)m. The product of any number of terminating decimals is also a terminating decimal, so (0.5)n and (0.2)m, and hence their product, 1/q, is a terminating decimal. Thus p/q has a terminating decimal expansion.

  3. Find a positive rational number and a positive irrational number both smaller than 0.00001.
    4. Answer:
    Answers will vary. Possible answer: 0.000001, 1/π12 ≈ 0.0000010819... .

  4. What is the smallest positive integer? The smallest positive rational number? The smallest positive irrational number?
    5. Answer:
    Smallest positive integer: 1; There is no smallest positive rational or irrational number.

    Pembahasan Soal Buku Calculus 9th Purcell

  5. Find a rational number between 3.14159 and π. Note that π = 3.141592… .
    6. Answer:
    Answers will cary. Possible answer: 3.14159101001... .

  6. Is there a number between 0.9999… (repeating 9s) and 1? How do you resolve this with the statement that between any two different real numbers there is another real number?
    7. Answer:
    There is no real number between 0.9999... (repeating 9's) and 1. 0.9999... and 1 represent the same real number.

  7. Is 0.1234567891011121314... rational or irrational? (You should see a pattern in the given sequence of digits.)
    8. Answer:
    Irrational

  8. Find two irrational numbers whose sum is rational.
    9. Answer:
    Answers will vary. Possible answers: –π and π, –√2 and √2.

    In Problems 51-56, find the best decimal approximation that your calculator allows. Begin by making a mental estimate.
  9. (√3 + 1)3
    10. Answer:
    (√3 + 1)3 ≈ 20.39230485

  10. (√2 – √3)4
    11. Answer:
    (√2 – √3)4 ≈ 0.0102051443

  11. ∜1.123 – ∛1.09
    12. Answer:
    ∜1.123 – ∛1.09 ≈ 0.00028307388

  12. (3.1415)–½
    13. Answer:
    (3.1415)–½ ≈ 0.5641979034

  13. √(8.9π2 + 1) – 3π
    14. Answer:
    √(8.9π2 + 1) – 3π ≈ 0.000691744752

  14. ∜(6π2 – 2) π
    15. Answer:
    ∜(6π2 – 2) π ≈ 3.661591807

  15. Show that between any two different real numbers there is a rational number. (Hint: If a < b, then b – a > 0, so there is a natural number n such that 1/n < b – a. Consider the set {k:k/n > b} and use the fact that a set of integers that is bounded from below contains a least element.) Show that between any two different real numbers there are infinitely many rational numbers.
    16. Answer:
    Let a and b be real numbers with a < b. Let n be a natural number that satisfies 1/n < b – a. Let S = {k:k/n > b}. Since a nonempty set of integers that is bounded below contains a least element, there is a k0 ∈ S such that k0/n > b but (k0 – 1)/n ≤ b. Then (k0 – 1)/n = [k0/n] – [1/n] > b – [1/n] > a. Thus, a < [(k0 – 1)/n] ≤ b. If [(k0 – 1)/n] < b, then choose r = [(k0 – 1)/n]. Otherwise, choose r = [(k0 – 2)/n]. Note that a < b – 1/n < r. Given a < b, choose r so that a < r1 < b. Then choose r2, r3 so that a < r2 < r1 < r3 < b, and so on.

  16. Estimate the number of cubic inches in your head.
    17. Answer:
    Answers will vary. Possible answer: ≈ 120 in3.

  17. Estimate the length of the equator in feet. Assume the radius of the earth to be 4000 miles.
    18. Answer:
    r = 4000 mi × 5280 ft/mi = 21,120,000 ft
    equator = 2πr = 2π(21,120,000)
    ≈ 132,700,874 ft

  18. About how many times has your heart beat by your twentieth birthday?
    19. Answer:
    Answers will vary Possible answer:
    70 beats/min × 60 min/hr × 24 hr/day × 365 day/year × 20 yr
    = 735,840,000 beats

  19. The General Sherman tree in California is about 270 feet tall and averages about 16 feet in diameter. Estimate the number of board feet (1 board foot equals 1 inch by 12 inches by 12 inches) of lumber that could be made from this tree, assuming no waste and ignoring the branches.
    20. Answer:
    V = πr2h = π[(16/2) • 12]2 × (270 • 12)
    ≈ 93,807,453.98 in.3
    volume of one board foot (in inches):
    1 × 12 × 12 = 144 in.3
    number of board feet:
    [93,807, 453.98/144] ≈ 651,441 board ft

  20. Assume that the General Sherman tree (Problem 61) produces an annual growth ring of thickness 0.004 foot. Estimate the resulting increase in the volume of its trunk each year.
    21. Answer:
    V = π(8004)2 (270) – π(8)2 (270) ≈ 54.3 ft.3

  21. Write the converse and the contrapositive to the follow ing statements.
    22. (a) If it rains today, then I will stay home from work.
    (b) If the candidate meets all the qualifications, then she will be hired.
    Answer:
    (a) If I stay home from work today then it rains. If I do not stay home from work, then it does not rain.
    (b) If the candidate will be hired then she meets all the qualifications. If the candidate will not be hired then she does not meet all the qualifications.

  22. Write the converse and the contrapositive to the follow ing statements.
    23. (a) If I get an A on the final exam, I will pass the course.
    (b) If I finish my research paper by Friday, then I will take off next week.
    Answer:
    (a) If I pass the course, then I got an A on the final exam. If I did not pass the course, thn I did not get an A on the final exam.
    (b) If I take off next week, then I finished my research paper. If I do not take off next week, then I did not finish my research paper.

  23. Write the converse and the contrapositive to the following statements.
    24. (a) (Let a, b, and c be the lengths of sides of a triangle.) If a2 + b2 = c2, then the triangle is a right triangle.
    (b) If angle ABC is acute, then its measure is greater than 0° and less than 90°.
    Answer:
    (a) If a triangle is a right triangle, then a2 + b2 = c2. If a triangle is not a right triangle, then a2 + b2 ≠ c2.
    (b) If the measure of angle ABC is greater than 0° and less than 90°, it is acute. If the measure of angle ABC is less than 0° or greater than 90°, then it is not acute.

  24. Write the converse and the contrapositive to the follow ing statements.
    25. (a) If the measure of angle ABC is 45°, then angle ABC is an acute angle.
    (b) If a < b then a2 < b2.
    Answer:
    (a) If angle ABC is an acute angle, then its measure is 45°. If angle ABC is not an acute angle, then its measure is not 45°.
    (b) If a2 < b2 then a < b. If a2 ≥ b2 then a ≥ b.

  25. Consider the statements in Problem 65 along with their converses and contrapositives. Which are true?
    26. Answer:
    (a) The statement, converse, and contrapositive are all true.
    (b) The statement, converse, and contrapositive are all true.

  26. Consider the statements in Problem 66 along with their converses and contrapositives. Which are true?
    27. Answer:
    (a) The statement and contrapositive are true. The converse is false.
    (b) The statement, converse, and contrapositive are all false.

  27. Use the rules regarding the negation of statements involving quantifiers to write the negation of the following statements. Which is true, the original statement or its negation?
    28. (a) Every isosceles triangle is equilateral.
    (b) There is a real number that is not an integer.
    (c) Every natural number is less than or equal to its square.
    Answer:
    (a) Some isosceles triangles are not equilateral. The negation is true.
    (b) All real numbers are integers. The original statement is true.
    (c) Some natural number is larger than its square. The original statement is true.

  28. Use the rules regarding the negation of statements involving quantifiers to write the negation of the following statements. Which is true, the original statement or its negation?
    29. (a) Every natural number is rational.
    (b) There is a circle whose area is larger than 9π.
    (c) Every real number is larger than its square.
    Answer:
    (a) Some natural number is not rational. The original statement is true.
    (b) Every circle has area less than or equal to 9π. The original statement is true.
    (c) Some real number is less than or equal to its square. The negation is true.

  29. Which of the following are true? Assume that x and y are real numbers.
    30. (a) For every x, x > 0 ⇒ x2 > 0.
    (b) For every x, x > 0 ⇔ x2 > 0.
    (c) For every x, x2 > x.
    (d) For every x, there exists a y such that y > x2.
    (e) For every positive number y, there exists another positive number x such that 0 < x < y.
    Answer:
    (a) True; If x is positive, then x2 is positive.
    (b) False; Take x = −2 . Then x2 > 0 but x < 0.
    (c) False; Take x = 1/2. Then x2 = 1/4 < x.
    (d) True; Let x be any number. Take y = x2 + 1. Then y > x2.
    (e) True; Let y be any positive number. Take x = y/2. Then 0 < x < y.

    Pembahasan Soal Buku Calculus 9th Purcell

  30. Which of the following are true? Unless it is stated otherwise, assume that x, y, and ɛ are real numbers.
    31. (a) For every x, x < x + 1.
    (b) There exists a natural number N such that all prime numbers are less than N. (A prime number is a natural number whose only factors are 1 and itself.)
    (c) For every x > 0, there exists a y such that y > 1/x.
    (d) For every positive x, there exists a natural number n such that 1/n < x.
    (e) For every positive ɛ, there exists a natural number n such that 1/2n < ɛ.
    Answer:
    (a) Thue; x + (–x) < x + 1 + (–x) : 0 < 1.
    (b) False; There are infinitely many prime numbers.
    (c) True; Let x be any number. Take y = 1/x + 1. Then y > 1/x.
    (d) True; 1/ n can be made arbitrarily close to 0.
    (e) True; 1/ n2 can be made arbitrarily close to 0.

  31. Prove the following statements.
    32. (a) If n is odd, then n2 is odd. (Hint: If n is odd, then there exists an integer k such that n = 2k + 1.)
    (b) If n2 is odd, then n is odd. (Hint: Prove the contrapositive.)
    Answer:
    (a) If n is odd, then there is an integer k such that n = 2k + 1. Then n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1.
    (b) Prove the contrapositive. Suppose n is even. Then there is an integer k such that n = 2k. Then n2 = (2k)2 = 4k = 2(2k2). Then n2 is even.

  32. Prove that n is odd if and only if n2 is odd. (See Problem 73.)
    33. Answer:
    Parts (a) and (b) prove that n is odd if and only if n2 is odd.

  33. According to the Fundamental Theorem of Arithmetic, every natural number greater than 1 can be written as the product of primes in a unique way, except for the order of the factors. For example, 45 = 3.3.5. Write each of the following as a product of primes.
    34. (a) 243
    (b) 124
    (c) 5100
    Answer:
    (a) 243 = 3 · 3 · 3 · 3 · 3.
    (b) 124 = 4 · 31 = 2 · 2 · 31 or 22 · 31.
    (c) 5100 = 2 · 2550 = 2 · 2 · 1275
    = 2 · 2 · 3 · 425 = 2 · 2 · 3 · 5 · 85
    = 2 · 2 · 3 · 5 · 5 · 17 or 22 · 3 · 52 · 17.

  34. Use the Fundamental Theorem of Arithmetic (Problem 75) to show that the square of any natural number greater than 1 can be written as the product of primes in a unique way, except for the order of the factors, with each prime occurring an even number of times. For example, (45)2 = 3 · 3 · 3 · 3 · 5 · 5.
    35. Answer:
    For example, let A = b · c2 · d3; then A2 = b2 · c4 · d6, so the square of the number is the product of primes which occur an even number of times.

  35. Show that √2 is irrational. Hint: Try a proof by contradiction. Suppose that √2 = p/q, where p and q are natural numbers (necessarily different from 1). Then 2 = p2/q2, and so 2q2 = p2. Now use Problem 76 to get a contradiction.
    36. Answer:
    √2 = p/q; 2 = p2/q2; 2q2 = p2; Since the prime factors of p2 must occur an even number of times, 2q2 would not be valid and p/q = √2 must be irrational.

  36. Show that √3 is irrational (see Problem 77).
    37. Answer:
    √3 = p/q; 3 = p2/q2; 3q2 = p2; Since the prime factors of p2 must occur an even number of times, 3q2 would not be valid and p/q = √3 must be irrational.

  37. Show that the sum of two rational numbers is rational.
    38. Answer:
    Let a, b, p, and q be natural numbers, so a/b and p/q are rational. a/b + a/b and p/q = a/b and [(aq) + (bp)]/bq. This sum is the quotient of natural numbers, so it is also rational.

  38. Show that the product of a rational number (other than 0) and an irrational number is irrational. Hint: Try proof by contradiction.
    39. Answer:
    Assume a is irrational, p/q ≠ is rational, and a · p/q = r/s is rational. Then a = [q · r]/[p · s] is rational, which is a contradiction.

  39. Which of the following are rational and which are irrational?
    40. (a) –√9
    (b) 0.375
    (c) (3 √2)(5 √2)
    (d) (1 + √3)2
    Answer:
    (a) –√9 = –3; rational
    (b) 0.375 = 3/8; rational
    (c) (3 √2)(5 √2) = 15 √4 = 30; rational
    (d) (1 + √3)2 = 1 + 2√3 + 3 = 4 + 2√3; Irrational

  40. A number b is called an upper bound for a set S of numbers if x ≤ b for all x in S. For example 5, 6.5, and 13 are upper bounds for the set S = {1, 2, 3, 4, 5). The number 5 is the least upper bound for S (the smallest of all upper bounds). Similarly, 1.6, 2, and 2.5 are upper bounds for the infinite set T = {1.4, 1.49, 1.499, 1.4999, …}, whereas 1.5 is its least upper bound. Find the least upper bound of each of the following sets.
    41. (a) S = {–10, –8, –6, –4, –2}
    (b) S = {–2, –2.1, –2.11, –2.111, –2.111, …}
    (c) S = {2.4, 2.44, 2.444, 2.4444, …}
    (d) S = {1 – 1/2, 1 – 1/3, 1 – 1/4, 1 – 1/5, …}
    (e) S = {x:x = (–1)n + 1/n, n a positive integer}; that is, S is the set of all numbers x that have the form x = (–1)n + 1/n, where n is a positive integer.
    (f) S = {x:x2 < 2, x arational number}
    Answer:
    (a) –2
    (b) –2
    (c)      x = 2.4444...?
          10x = 24.4444...
               x = 2.4444... –
             9x = 22
               x = 22/9.
    (d) 1
    (e) n = 1: x = 0, n = 2: x = 3/2, n = 3: x = –2/3, n = 4: x = 5/4. The upper bound is 3/2.
    (f) √2

  41. The Axiom of Completeness for the real numbers says: Every set of real numbers that has an upper bound has a least upper bound that is a real number.
    42. (a) Show that the italicized statement is false if the word real is replaced by rational.
    (b) Would the italicized statement be true or false if the word real were replaced by natural?
    Answer:
    (a) Answers will vary. Possible answer: An example is S = {x:x2 < 5; x a rational numer}. Here the least upper bound is √5, which is real but irrational.
    (b) True.

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