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.
Date 
Sections 
Written
Homework Assignment 
Monday January 5 
Chapter 1, pages 14 
Exercise 2, page 22 due Wednesday January 14 
Wednesday January 7 
Chapter 1, pages 58 
Exercises 1, 5 page 22 due Wednesday January 14 
Friday January 9 
Chapter 1, pages 914 
Exercises 8, 9, 11 pages 2223 due Wednesday January 14 
Monday January 12 
Chapter 1, pages 1417 
Read about extended real number system on your own. Exercises 14, 17, 18 pages 2223 due Wednesday January 21 
Wednesday January 14 
Chapter 2, pages 2430 
Exercise 2 page 43 due Wednesday January 21 
Friday January 16 
Chapter 2, pages 3032 
Exercises 4, 5 pages 43 due Wednesday January 21 
Monday January 19 Martin Luther King Jr.
Day. No class 


Tuesday January 20 xhour instead of the class on Monday January 19 Final day for electing
use of the NonRecording option 
Chapter 2 pages 3235 
Exercises 6, 8 on page 43 due Wednesday January 28 
Wednesday January 21 
Chapter 2, pages 3537 
Exercises 7.a and 7.b. on pages 43. These will count as separate exercises when graded due Wednesday January 28 
Friday January 23 
Chapter 2 pages 3739 
Exercises 10, 12, 15 on page 44 due Wednesday January 28 
Monday January 26 
Chapter 2 pages 4043 
Exercise 19 parts A and B only; Exercise 20 on page 44 due Wednesday February 4 
Tuesday January 27 xhour 
Chapter 3, pages 4749 
Exercise 1, page 78 due Wednesday February 4 
Wednesday January 28 
Chapter 3, pages 4951 
Exercise: Find an example of divergent sequences {s_{n}}^{∞}_{n=1 }and {t_{n}}^{∞}_{n=1 }of complex numbers such that {s_{n}+ t_{n} }^{∞}_{n=1 }converges. Justify your answer. Exercise: Find an example of a divergent sequence {s_{n}}^{∞}_{n=1 }of complex numbers such that {s_{n}^{3}}^{∞}_{n=1 }converges. Justify your answer. Hint: there is more than one complex number whose cube is 1. Exercise: Let {s_{n}}^{∞}_{n=1 }and {t_{n}}^{∞}_{n=1 }be sequences of complex numbers such that {s_{n}+ t_{n} }^{∞}_{n=1 }converges to x and {t_{n}}^{∞}_{n=1 }converges to y. Prove that the sequence {s_{n}}^{∞}_{n=1 }converges and find its limit. Justify your answer. due Wednesday February 4 
Friday January 30 The takehome Midterm exam is
given out. It will be due on Wednesday February 4 
Chapter 3, pages 5153 
Exercise : Let {s_{n}}^{∞}_{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 
Monday February 2 
Chapter 3, pages 5357 
Read and understand the proof of Theorem 3.20. Plus do the following exercises in writing: Exercise 5, page 78 Exercise: Let {s_{n}}^{∞}_{n=1 }be sequence of real numbers. For a positive integer N put t_{N}=sup{ s_{N}_{, }s_{N+1, }s_{N+2, …}}. Show that lim_{N}_{→∞} t_{N} =lim_{n→∞}sup s_{n}_{. }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 
Wednesday February 4 The Midterm Exam is due 
Chapter 3, pages 5557, 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 
Friday February 6 
Chapter 3, pages 5862 and 6567 
Read and understand the proofs of Theorems 3.27 and 3.28 Pages 7879 Exercise 6 part A only, Exercise 8, and Exercise 11 part A only. Hint: you might want to consider separately the case where ∑^{∞}_{n=1}a_{n }diverges and lim_{n→∞}a_{n}=0, and the case where ∑^{∞}_{n=1}a_{n }diverges and lim_{n→∞}a_{n} does not exist or is nonzero. Due Wednesday February 11 
Monday February 9 
Chapter 3, pages 6978 except of summation by parts 
Read and understand the proofs of the Theorems 3.31 and 3.32 Pages 7880 Exercise 9 part A only; Exercises 10, 13 Due Wednesday February 18 
Tuesday February 10 xhour instead of the class on Friday February 13 
Chapter 3, Theorem 3.50, Chapter 4 pages 8385 
Exercise: Let f(z)=z^{3}+3 be a function C→C. Prove that lim_{z→0}f(z)=3 directly from the epsilon, delta definition of the limit. Exercise. Let g(z)=z^{2}9z+5 be a function C→C. Prove that lim_{z→3}g(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 
Wednesday February 11 
Chapter 4, pages 8590 
Page 9899 Exercises 1, 3, 4 Due Wednesday February 18 
Friday Feburary 13 Winter Carnival! No class J Final day for
dropping a fourth course without a grade notation of "W" 


Monday February 16 
Chapter 4, pages 9196 
Pages 98100 Exercises 14, 15, 18 Due Wednesday February 25 
Tuesday February 17, xhour 
Chapter 4, pages 9698 
Page 98100 Exercises 8, 9 Due Wednesday February 25 
Wednesday February 18 
Theorem 4.19 and Chapter 5,103105 
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 
Friday February 20 
Pages 105108 
Pages 114115 Exercises 1,2 and 4 Due Wednesday February 25 
Monday February 23 
Pages 108 and 110111 
Pages 114116 Exercises 17, 18 Due Wednesday March 4 
Tuesday February 24 xhour 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 
Wednesday February 25 
Pages 120125 
Page 138 Exercises 1 and 4 Hint: You might want to use Theorem 6.6 on page 124 Due Wednesday March 4 
Friday February 27 
Pages 125126 and 128129 
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 
Monday March 2 Note that Tuesday March
3 is the final day to alter grade limit filed under the NonRecording Option 
Pages 126127 and 129130 
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 
Tuesday March 3, xhour Instead of the class on Wednesday March 4 
Pages 130132 
Page 138 Exercise 5 Due Monday March 9 
Wednesday March 4 NO CLASS 


Friday March 6 
Pages 132135 
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 
Monday
March 9 The take home Final Exam will be distributed on this day. It will be due on Saturday March 14 
Pages 143154 
Read and understand the formulations and the general ideas behind the proofs of uniform convergence Theorems on pages 143154 