Math 431.02: Quiz # 5 Solutions Instructor: Dr. Herzog
February 12th, 2014
1. Let f be the function on [0, 1] given by f (x) = 0 1 if x is rational if x is irrational.
Explain why UP (f ) = 1 and LP (f ) = 0 for every partition P . Is f Riemann integrable
SOLUTION SET FOR THE HOMEWORK PROBLEMS
Page 5. Problem 8. Prove that if x and y are real numbers, then
2xy x2 + y 2 .
Proof. First we prove that if x is a real number, then x2 0. The product
of two positive numbers is always positive, i.e., if x 0 and y 0
Math 431.02: Exam #1 Instructor: Dr. Herzog
February 19th, 2014
1. Let cfw_an and cfw_bn be bounded sequences and define sets A, B, and C by A = cfw_an , B = cfw_bn , and C = cfw_an + bn . Prove that sup C sup A + sup B. Give an example to show that str
Math 431.01 Spring 2014: Quiz # 2 Instructor: Dr. Herzog
January 24th, 2014
1. Suppose that an a and that an b for each n. Prove that a b. Proof. To obtain a contradiction, we suppose that a < b. Let = (b - a)/2 > 0. Since an a, we may pick N N such that
Math 431.01 Spring 2014: Quiz # 1 Solutions Instructor: Dr. Herzog
January 13th, 2013
1. Prove that if x and y are real numbers, then 2xy x2 + y 2 . Proof. Let x, y R. Observe that 0 (x - y)2 = x2 - 2xy + y 2 . Adding 2xy to both sides of the previous ine
Math 431.01 Spring 2014: Quiz # 3 Solutions Instructor: Dr. Herzog
January 27th, 2014
1. Suppose that cfw_an is a Cauchy sequence. Prove that cfw_a2 is also a n Cauchy sequence. Is the converse true? Proof. Since an is Cauchy, an a for some a R. By the
Math 431.01 Spring 2014: Quiz # 4 Instructor: Dr. Herzog
February 5th, 2014
1. Let cfw_dn be a sequence of limit points of the sequence cfw_an and suppose that dn d. Prove that d is also a limit point of cfw_an . Proof. Let > 0 and N N be given. We must
Math 431: Quiz # 7 Instructor: Dr. Herzog
March 5th, 2014
1. Suppose f is twice continuously differentiable on [0, 1] and that 1 f (0) = 1, f (0) = 2, and |f (x)| .3 for all x [0, 1]. Compute 0 f (x) dx as best you can. Solution. Since f is twice continuo
Math 431.01 Spring 2014: Quiz # 6 Solutions Instructor: Dr. Herzog
February 26th, 2014
1. Suppose that f (x) 0 for all x R. Assume that f (x)2 is differentiable. Is f (x) necessarily differentiable? Prove your conclusion is correct. Proof. No, this assert
Math 139.2 Spring 2012: Quiz # 12 Solutions
Instructor: Dr. Herzog
April 24th, 2012
1. Given the following conditions on
cfw_aj , what can you say
Pthe coefficients
j
about the radius of convergence of j aj x ?
(a) 2j aj 3j .
(b) j 2 aj j 3 .
Proof. Let R