Math 63 Winter 2009
Homework
Regular homework will be assigned at the end of each lecture. Homework assigned at a lecture will be generally due at the first lecture of the following week. Late Homework will not be accepted (except for emergency cases). Unexcused late and missing homework counts zero.
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Date |
Sections |
Written
Homework Assignment |
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Monday January 5 |
Chapter 1, pages 1-4 |
Exercise 2, page 22 due Wednesday January 14 |
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Wednesday January 7 |
Chapter 1, pages 5-8 |
Exercises 1, 5 page 22 due Wednesday January 14 |
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Friday January 9 |
Chapter 1, pages 9-14 |
Exercises 8, 9, 11 pages 22-23 due Wednesday January 14 |
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Monday January 12 |
Chapter 1, pages 14-17 |
Read about extended real number system on your own. Exercises 14, 17, 18 pages 22-23 due Wednesday January 21 |
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Wednesday January 14 |
Chapter 2, pages 24-30 |
Exercise 2 page 43 due Wednesday January 21 |
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Friday January 16 |
Chapter 2, pages 30-32 |
Exercises 4, 5 pages 43 due Wednesday January 21 |
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Monday January 19 Martin Luther King Jr.
Day. No class |
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Tuesday January 20 x-hour instead of the class on Monday January 19 Final day for electing
use of the Non-Recording option |
Chapter 2 pages 32-35 |
Exercises 6, 8 on page 43 due Wednesday January 28 |
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Wednesday January 21 |
Chapter 2, pages 35-37 |
Exercises 7.a and 7.b. on pages 43. These will count as separate exercises when graded due Wednesday January 28 |
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Friday January 23 |
Chapter 2 pages 37-39 |
Exercises 10, 12, 15 on page 44 due Wednesday January 28 |
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Monday January 26 |
Chapter 2 pages 40-43 |
Exercise 19 parts A and B only; Exercise 20 on page 44 due Wednesday February 4 |
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Tuesday January 27 x-hour |
Chapter 3, pages 47-49 |
Exercise 1, page 78 due Wednesday February 4 |
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Wednesday January 28 |
Chapter 3, pages 49-51 |
Exercise: Find an example of divergent sequences {sn}∞n=1 and {tn}∞n=1 of complex numbers such that {sn+ tn }∞n=1 converges. Justify your answer. Exercise: Find an example of a divergent sequence {sn}∞n=1 of complex numbers such that {sn3}∞n=1 converges. Justify your answer. Hint: there is more than one complex number whose cube is 1. Exercise: Let {sn}∞n=1 and {tn}∞n=1 be sequences of complex numbers such that {sn+ tn }∞n=1 converges to x and {tn}∞n=1 converges to y. Prove that the sequence {sn}∞n=1 converges and find its limit. Justify your answer. due Wednesday February 4 |
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Friday January 30 The takehome Midterm exam is
given out. It will be due on Wednesday February 4 |
Chapter 3, pages 51-53 |
Exercise : Let {sn}∞n=1 be a bounded increasing sequence of real numbers. Prove that the sequence converges to the supremum of the set that is the union of the elements of the sequence. Also Exercise 3, page 78. Due Wednesday February 4 |
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Monday February 2 |
Chapter 3, pages 53-57 |
Read and understand the proof of Theorem 3.20. Plus do the following exercises in writing: Exercise 5, page 78 Exercise: Let {sn}∞n=1 be sequence of real numbers. For a positive integer N put tN=sup{ sN, sN+1, sN+2, …}. Show that limN→∞ tN =limn→∞sup sn. Here the last quantity is introduced in definition 3.16 of the textbook. Please consider separately the case where the values are infinite. This case will be counted as a separate exercise when graded. Due Wednesday February 11 |
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Wednesday February 4 The Midterm Exam is due |
Chapter 3, pages 55-57, plus the squeezed sequence Theorem |
Read and understand the proof of Theorem 3.17 Page 78 Exercises 2, 4 in written form due Wednesday February 11 |
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Friday February 6 |
Chapter 3, pages 58-62 and 65-67 |
Read and understand the proofs of Theorems 3.27 and 3.28 Pages 78-79 Exercise 6 part A only, Exercise 8, and Exercise 11 part A only. Hint: you might want to consider separately the case where ∑∞n=1an diverges and limn→∞an=0, and the case where ∑∞n=1an diverges and limn→∞an does not exist or is nonzero. Due Wednesday February 11 |
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Monday February 9 |
Chapter 3, pages 69-78 except of summation by parts |
Read and understand the proofs of the Theorems 3.31 and 3.32 Pages 78-80 Exercise 9 part A only; Exercises 10, 13 Due Wednesday February 18 |
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Tuesday February 10 x-hour instead of the class on Friday February 13 |
Chapter 3, Theorem 3.50, Chapter 4 pages 83-85 |
Exercise: Let f(z)=z3+3 be a function C→C. Prove that limz→0f(z)=3 directly from the epsilon, delta definition of the limit. Exercise. Let g(z)=z2-9z+5 be a function C→C. Prove that limz→3g(z)=-13 directly from the epsilon, delta definition of the limit. Hint: you may want to rewrite the function g(z) differently and use properties of the absolute values of complex numbersunder arithmetic operations. Due Wednesday February 18 |
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Wednesday February 11 |
Chapter 4, pages 85-90 |
Page 98-99 Exercises 1, 3, 4 Due Wednesday February 18 |
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Friday Feburary 13 Winter Carnival! No class J Final day for
dropping a fourth course without a grade notation of "W" |
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Monday February 16 |
Chapter 4, pages 91-96 |
Pages 98-100 Exercises 14, 15, 18 Due Wednesday February 25 |
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Tuesday February 17, x-hour |
Chapter 4, pages 96-98 |
Page 98-100 Exercises 8, 9 Due Wednesday February 25 |
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Wednesday February 18 |
Theorem 4.19 and Chapter 5,103-105 |
Page 115 Exercise 12. Hint you might want to use the fact that f’(x) exists if and only if the right and the left handside derivatives f’(x+) and f’(x-) exist and are equal. Due Wednesday February 25 |
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Friday February 20 |
Pages 105-108 |
Pages 114-115 Exercises 1,2 and 4 Due Wednesday February 25 |
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Monday February 23 |
Pages 108 and 110-111 |
Pages 114-116 Exercises 17, 18 Due Wednesday March 4 |
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Tuesday February 24 x-hour The last day to withdraw from a course |
Pages 109 and 112 |
Exercise: Let f(x):R→R be a function defined as follows f(x)=0 for all x≤0, and f(x)=e-1/(x*x) for all x>0. Prove that f ’(0)=f ’’(0)=0. Note that you will have to use the limit definitions of the derivatives and the L’Hospital’s rule. Page 114, Exercise 7 Due Wednesday March 4 |
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Wednesday February 25 |
Pages 120-125 |
Page 138 Exercises 1 and 4 Hint: You might want to use Theorem 6.6 on page 124 Due Wednesday March 4 |
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Friday February 27 |
Pages 125-126 and 128-129 |
Page 138 Exercise 2 Exercise: Prove statement (b) of Theorem 6.12 on page 128 Exercise: Prove statement (d) of Theorem 6.12 on page 128 Due Wednesday March 4 |
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Monday March 2 Note that Tuesday March
3 is the final day to alter grade limit filed under the Non-Recording Option |
Pages 126-127 and 129-130 |
Read and understand the proof of Theorem 6.10 Page 138 Exercise 3 part A, Exericse 3 part B to be graded as separate exerises. Due Monday March 9 |
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Tuesday March 3, x-hour Instead of the class on Wednesday March 4 |
Pages 130-132 |
Page 138 Exercise 5 Due Monday March 9 |
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Wednesday March 4 NO CLASS |
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Friday March 6 |
Pages 132-135 |
Page 138 Exercise 7. Hint in part b you might want to use the fact that the series ∑∞n((-1)n/n) converges but ∑∞n(1/n) diverges Due Monday March 9 |
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Monday
March 9 The take home Final Exam will be distributed on this day. It will be due on Saturday March 14 |
Pages 143-154 |
Read and understand the formulations and the general ideas behind the proofs of uniform convergence Theorems on pages 143-154 |