Math 25: Advanced Calculus
Fall 2010 Examples using the Field Axioms We proved the following Lemmas in class. You are free to use them on any homework assignment you wish, but you must write out the statement of the lemma that you are using. Lemma 1: If F
Math 25: Advanced Calculus
Fall 2010 Homework # 9 Due Date: Monday, November 29, 2010 Reading: Sections 2.102.13. Complete the following problems: 9.1: Let cfw_an be a sequence of numbers, and let cfw_ank be a subsequence. n=1 k=1 (a). If limn an = L, p
Math 25: Advanced Calculus
Fall 2010 Midterm 1 Your Name: Student ID: Signature: Instructions 1. You have 50 minutes to complete this exam. Complete the problems in a way that will optimize your total score. 2. There is plenty of room for you to do scratc
Math 25: Advanced Calculus
Fall 2010 Midterm 2 Your Name: Student ID: Signature: Instructions 1. You have 50 minutes to complete this exam. Complete the problems in a way that will optimize your total score. 2. There is plenty of room for you to do scratc
MAT 25 - A. Moon
Discussion 1
Exercise 1: Let f : R R be defined by f (x) = x2 . Let A = [0, 4] and
B = [1, 1].
(a) Find f 1 (A) and f 1 (B).
(b) Verify f 1 (A) f 1 (B) = f 1 (A B) and f 1 (A) f 1 (B) = f 1 (A B).
(c) Now let g : R R be an arbitrary funct
MAT 25 - A. Moon
Discussion 5
Exercise 1: Let 0 < 2 be a starting angle. Show that the sequence:
(xn )
xn = sin (en n )
contains a convergent subsequence. Give a guess for two possible subsequential
limits.
Exercise 2: Prove the direction of the Cauchy co
MAT 25 - A. Moon
Discussion 3
Exercise 1: Prove that if (xn ) converges to x, then (|xn |) converges to |x|.
(Hint: first prove that for all a, b R, |a| |b| |a b|.)
Exercise 2:
(a) Construct a sequence (xn ) such that for all > 0, there exists N such that
MAT 25 - A. Moon
Discussion 7
Exercise 1: (a) Internalize the statement: a sequence (xn ) of real numbers is
convergent if and only if it is Cauchy.
(b) Provide an example of a sequence which is not Cauchy.
Exercise 2:
(Towards proof of Alternating series
In the seventeenth century, the region of Chesapeake Bay underwent several
major transformations in politics, economy and social relations that shaped the southern
culture and led to a certain way of American development before the Civil War. One of
the m
The 17th century was not the best of times for the indentured servants, neither for
those who made the choice nor for those who did not, according to one source Only
about 40 percent of indentured servants lived to complete the terms of their contracts. 1
MAT 25 - A. Moon
Discussion 8
HW Review: Exercise 1: Prove or provide a counterexample:
(i) If
P
(ii) If
P
(iii) If
P
an converges absolutely, then
P
a2n converges absolutely.
an converges and (bn ) converges, then
an converges conditionally, then
P
P
an
MAT 25 - A. Moon
Discussion 9
Exercise 0: Prove that if (xn ) is a sequence and xn x, and x 6 cfw_xn , then
every point xn is an isolated point of cfw_xn .
Exercise 1: Give a sketch of a proof of the following facts:
(i) If A R is bounded then A is bounde
MAT 25 - A. Moon
Discussion 2
Exercise 1: (a) State the axiom of completeness.
(b) Let S R be a nonempty bounded set, i.e. there exist m, M such that for
all s S, m s M . Define:
S = cfw_s : s S
Prove that sup(S) = inf(S).
Exercise 2: Different form of Ar
Math 25: Advanced Calculus
Fall 2010 Homework # 8 Due Date: This assignment will not be collected. These problems are intended to help you in studying for the exam. Reading: Finish all reading through section 2.9 for the exam. Complete the following probl
Math 25: Advanced Calculus
Fall 2010 Homework # 7 Solutions
7.1: Prove that limn ( n + 1 - n) = 0. [Hint: What would you have done in calculus?] Solution: Multiplying by the conjugate of n + 1 - n gives ( n + 1 - n)( n + 1 + n) lim n + 1 - n = lim n n n+
Math 25: Advanced Calculus
Fall 2010 Homework # 1 Due Date: Friday, October 1 Reading: Read Appendix A of the textbook. Complete the following problems from the textbook 1 : A.5.2: Show that there are infinitely many prime numbers. A.6.1: Prove the follow
Math 25: Advanced Calculus
Fall 2010 Homework # 1 Solutions Due Date: Friday, October 1 A.5.2: Show that there are infinitely many prime numbers. Proof: We will use proof by contradiction. Suppose to the contrary that there are only finitely many primes,
Math 25: Advanced Calculus
Fall 2010 Homework # 2 Due Date: Friday, October 8 Reading: Finish reading Appendix A of the textbook. Read sections 1.1-1.3. Complete the following problems from the textbook: A.8.1, A.8.4, A.9.1, A.9.2, In addition, please com
Math 25: Advanced Calculus
Fall 2010 Homework # 2 Solutions Due Date: Friday, October 8 A.8.1: Prove by induction that for every n = 1, 2, 3, . . ., 12 + 22 + + n2 = n(n + 1)(2n + 1) . 6
Proof: First we establish the base case n = 1, where it is clear tha
Math 25: Advanced Calculus
Fall 2010 Homework # 3 Due Date: Friday, October 15 Reading: Finish reading sections 1.1-1.3. Read sections 1.4-1.6. Complete the following problems from the textbook: 1.3.7, 1.4.1, 1.4.3, 1.6.1, 1.6.2, 1.6.10, 1.6.17 In additio
Math 25: Advanced Calculus
Fall 2010 Homework # 3 Solutions 1.3.7: Which of the field axioms does Z6 fail to satisfy? Solution: By observation, Z6 satisfies A1,A2,M1,M2, and AM1. Furthermore, Z6 satisfies A3 because 0 + a = a for any a Z6 , and satisfies
Math 25: Advanced Calculus
Fall 2010 Homework # 4 Due Date: This assignment will not be collected. You are STRONGLY urged to complete these problems, as the material covered is fair game for the first midterm. Reading: Finish reading through section 1.9.
Math 25: Advanced Calculus
Fall 2010 Homework # 5 Due Date: Friday, October 29 Reading: Finish reading section 1.10. Read sections 2.2 and 2.4. Complete the following problems: 5.1: Problem 1.10.2 from the textbook. 5.2: In class, we saw that for all real
Math 25: Advanced Calculus
Fall 2010 Homework # 5 - Solutions 5.1: Show that maxcfw_x, y =
|x-y| 2
+
x+y . 2
What expression would give mincfw_x, y?
Solution 1: We claim that mincfw_x, y = x+y - |x-y| . 2 2 We know that either x y or x y. We will prove th
Math 25: Advanced Calculus
Fall 2010 Homework # 6 Due Date: Friday, November 5 Reading: Finish reading sections 2.2 and 2.4. Read sections 2.5-2.7. Complete the following problems: 6.1: Decide if each of the following sequences cfw_an converges or diverg
Math 25: Advanced Calculus
Fall 2010 Homework # 6 Solutions 6.1: Decide if each of the following sequences cfw_an converges or diverges. If the n=1 sequence converges, state its limit. In either case, you must use the appropriate definition prove that th
Math 25: Advanced Calculus
Fall 2010 Homework # 7 Due Date: Friday, November 12, 2010 Reading: Finish reading sections 2.5-2.7. Read sections 2.8-2.9. Complete the following problems: 7.1: Prove that limn ( n + 1 - n) = 0. [Hint: What would you have done
MAT 25 - A. Moon
Discussion 4
Exercise 1: (a) Separation property: Suppose lim xn = x and y 6= x. Prove
that there exists N N such that n > N implies
|xn y| >
|x y|
2
(b) Use this separation property to give (another) proof that limits of convergent
seque