calculus-2014-05-21

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DRAFT c○ 2014 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse Figure 3.11. Stage 3: [ 8 0, 125 ] ∪ [ 12 [ 21 25 , ] [ 113 125 ∪ 117 125 , 1] 125 , 4 25 ] [ ∪ 6 25 , ] [ 38 125 ∪ 42 125 , ] [ 2 5 ∪ 3 5 , ] [ 83 125 ∪ 87 125 , 25] 19 ∪ 36. What are the closed intervals that will remain? 37. In your notebook draw a picture of C4. 5 38. What are the open intervals that will be removed in stage C5? 5 39. What are the closed intervals that will remain? 40. In your notebook draw a picture of stage C5. 5 41. Explain why we can continue this process for each value of n. The Cantor Quinary Set is what remains after we let n → ∞. That is, the Cantor Quinary ∞⋂ Set, C 5 , is defined by C 5 = Cn. 5 n=1 42. Are there any numbers in C 5 ? That is, are there any numbers that are in C 5 n for every n? Explain. 43. Based on your answer to Investigation 42 does C 5 have finitely many or infinitely many points? Explain. 44. What is happening to the lengths of the intervals in C 5 n for each n? Explain. 45. Based on your answers to Investigations 41 and 44 Does C 5 contain any interval of positive length? 46. Based on your answer to Investigation 45 what do you believe to be the measure of C 5 . Explain. We can use the same strategy as you used in Investigation 8 along with tools for evaluating infinite geometric series (see Chapter 3 in Discovering the Art of the Infinite for more details) to actually determine the measure of C 5 . 47. What is the length of the interval removed in constructing C 5 1? 48. What is the total length of the intervals removed in constructing C 5 2 from C 5 1? (Note that your answer should not include the length you identified in Investigation 47.) 49. What is the total length of the intervals removed in constructing C 5 3 from C 5 2? (Note that your answer should not include the lengths you identified in Investigations 47-48.) 50. What is the total length of the intervals removed in constructing C 5 4 from C 5 3? (Note that your answer should not include the lengths you identified in Investigations 47-49.) 51. What is the total length of the intervals removed in constructing C 5 5 from C 5 4? (Note that your answer should not include the lengths you identified in Investigations 47-50.) 52. What patterns do you notice in your answers to Investigations 47-51? Explain. 53. Use your answer to Investigation 52 to complete the table in Table 3.2. 54. Use your answer for Investigation 53 to write down an infinite series that represents the total length of open intervals that were removed in constructing C 5 . 55. Use the methods for evaluating infinite geometric series to evaluate the sum in Investigation 54. 56. Using your answers to Investigations 1 and 55 to determine the measure of the Cantor Quinary Set, C 5 . 57. In light of your answers to Investigation 43 and Investigation 31, are you surprised by your answer to Investigation 56? Explain. 58. Using the ideas from this section, describe some other Cantor sets that can be created and determine their measure. 32
DRAFT c○ 2014 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse Stage C 5 1 Total Length Removed 1 5 C 5 2 C 5 3 C 5 4 C 5 5 C 5 6 C 5 7 C 5 8 . . C 5 n Table 3.2. Total Length of Intervals Removed in Constructing C 5 n for n = 1, 2, 3, . . . , 8 33
Page 3: Discovering the Art of Mathematics
Page 7 and 8: Contents Acknowledgements v Preface
Page 9 and 10: Preface: Notes to the Explorer Yes,
Page 11 and 12: Navigating This Book Before you beg
Page 13 and 14: CHAPTER 1 What is Area? The simples
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Page 43 and 44: CHAPTER 4 String Art Not being able
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DRAFT c○ 2014 Julian Fleron, Phil
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Index address, 19 algebraic curves,
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