calculus-2014-05-21

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DRAFT c○ 2014 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse 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 3 . 22. What is the length of the interval removed in constructing C 3 1? 23. What is the total length of the intervals removed in constructing C 3 2 from C 3 1? (Note that your answer should not include the length you identified in Investigation 22.) 24. What is the total length of the intervals removed in constructing C 3 3 from C 3 2? (Note that your answer should not include the lengths you identified in Investigations 22-23.) 25. What is the total length of the intervals removed in constructing C 3 4 from C 3 3? (Note that your answer should not include the lengths you identified in Investigations 22-24.) 26. What is the total length of the intervals removed in constructing C 3 5 from C 3 4? (Note that your answer should not include the lengths you identified in Investigations 22-25.) 27. What patterns do you notice in your answers to Investigations 22-26? Explain. 28. Use your answer to Investigation 27 to complete the table in Table 3.1. Stage C 3 1 Total Length Removed 1 3 C 3 2 C 3 3 C 3 4 C 3 5 C 3 6 C 3 7 C 3 8 . . C 3 n Table 3.1. Total Length of Intervals Removed in Constructing C 3 n for n = 1, 2, 3, . . . , 8 29. Use your answer for Investigation 28 to write down an infinite series that represents the total length of open intervals that were removed in constructing C 3 . 30. Use the methods for evaluating infinite geometric series to determine the sum in Investigation 29. 31. Using your answers to Investigations 1 and 30 to determine the measure of the Cantor Ternary Set, C 3 . 30
DRAFT c○ 2014 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse 32. In light of your answer to Investigation 18, are you surprised by your answer to Investigation 31? Explain. One surprising aspect of the Cantor Ternary Set that makes the set important, is that there are exactly as many numbers in C 3 as there are in the interval [0, 1]. That is, we can match each number in C 3 with exactly one number in [0, 1] and vice versa. The proof of this fact is beyond the scope of this book, so we will accept this fact without proof. 33. Explain why the above result makes your answer to Investigation 31 even more surprising. 2.2. The Cantor Quinary Set. We now consider another Cantor Set the Cantor Quinary Set, C 5 . This set is constructed a manner similar to that of C 3 , except this time in each stage we partition every interval into fifths and then remove the middle fifth (this is why our notation for the Cantor Ternary Set has a superscript 5). Stage 0: We start with the unit interval [0, 1], which we will denote by C 5 0: Figure 3.8. Stage 0: The interval [0, 1] Stage 1: We then remove the open interval ( 2 5 , 3 5 ); that is, we remove all numbers that strictly bigger than 2 5 and strictly less than 3 5 but leaving the numbers 2 5 and 3 5 . We denote this stage by C5 1: Figure 3.9. Stage 1: [ 0, 2 5 ] ∪ [ 3 5 , 1] Stage 2: The next stage, denoted by C2, 5 is obtained by removing the open intervals ( 2 25 , 4 19 25 ) and ( 25 , 21 25 ), again leaving the end points 2 25 , 4 25 , 19 21 25 and 25 : Figure 3.10. Stage 2: [ 0, 4 25] ∪ [ 6 25 , 2 5] ∪ [ 3 5 19 25 ] ∪ [ 21 25 , 1] Stage 3: For the next stage, C3, 5 we remove the open intervals ( 8 125 , 12 125 8 again leaving the end points 125 , 12 125 , 38 125 , 42 125 , 83 125 , 87 125 , 113 125 ), ( 38 125 , 42 125 117 and 125 : 34. What patterns do you observe in the construction of stages C 5 1 - C 5 3? Explain. 35. What are the open intervals that will be removed in stage C 5 4? 31 83 ), ( 125 , 87 113 125 ) and ( 125 , 117 125 ),
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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Index address, 19 algebraic curves,
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