Math 549 Spring 2010
Homework 12
Due: Thursday April 23 in class or in mailbox by 4pm.
(1) Rudin pg. 244 # 31, pg. 332 # pg. 1, 2, 3, 5, 6
(2) Let f : U R be dierentiable with
it follow that f is independent of x?
f
x
= 0 for all p U . If U is connected,
Math 549 Spring 2010
Homework 11
Due: Thursday April 16 in class or in mailbox by 4pm.
(1) Rudin # 19, 21, 27, 28.
(2) Let f : R R be a dierentiable function (but f may not be continuous).
(a) If f (x) = 0 for all x R, show that f has an inverse.
(b) If f
Math 549 Spring 2010
Homework 10
Due: Thursday April 9 in class or in mailbox by 4pm.
(1) Lots of good problems in Rudin again. Do No. 5, 6, 7, 9, 14, 15 in the multivariable
chapter.
(2) Give an example that a mean value theorem like for functions of one
Math 549 Spring 2010
Homework 9
Due: Thursday April 2 in class or in mailbox by 4pm.
1
(1) Show that f (x) = e x2 if x = 0 and f (0) = 0 has derivatives of all order at
x = 0 and that they vanish. What does this say about the Taylor series? Show
that a fu
Math 549 Spring 2010
Homework 8
Due: Thursday March 18 in class or in mailbox by 4pm.
This week I actually liked some of the problems in Rudin.
(1) Do problems 4, 7, 20, 24 in Chapter 7 of Rudin.
(2) Let C 0 = C 0 ([a, b]) be the space of continuous funct
Math 549 Spring 2010
Homework 7
Due: Friday March 5 by 4pm in TAs mailbox.
(1) Formulate and prove a version of the substitution rule for integration.
(2) Let f be Riemann integrable on [a, b].
(a) Show that f is Riemann integrable on any subinterval.
b
c
Math 549 Spring 2010
Homework 6
Due: Friday February 26 by 4pm in TAs mailbox.
(1) Compute
b
a
xn dx directly from the denition of the integral.
(2) Dene a function f : [0, 1] [0, 1] by f (x) = 0 if x irrational, and f (p/q ) = 1/q
for rational x = p/q in
Math 549 Spring 2010
Homework 5
Due: Friday February 19 by 4pm in TAs mailbox or at the end of class on Thursday.
(1) Prove the following facts you know from calculus. Let f : (a, b) R and x0 (a, b)
such that f is dierentiable in (a, b) and f (x0 ) exists
Math 549 Spring 2010
Homework 4
Due: Thursday February 11 at the end of class.
(1) Show that sin(x) and cos(x) are continuous at all points. Are they uniformly continuous? Use the denitions in terms of angles and you can use any trigonometric
formula.
(2)
Math 549 Spring 2010
Homework 3
Due: Thursday February 4 at the end of class.
(1) Show that the harmonic series
1
n
diverges.
(2) Give an example of an alternating series (1)n an which diverges and such that
an 0, lim an = 0 but the sequence an is not nec
Math 549 Spring 2010
Homework 2
Due: Thursday January 28 at the end of class.
Present your solutions clearly and legibly. If you are using results from Rudins book,
you need to state them explicitly. An emphasis is placed on rigorous reasoning. A portion
Math 549 Spring 2010
Homework 1
Due: Thursday January 21 at the end of class.
Present your solutions clearly and legibly. If you are using results from Rudins book,
you need to state them explicitly. An emphasis is placed on rigorous reasoning. A portion