Math 5201 Real Variables
Fall 2011
Prof. Peckham
Homeworks and Tests
Text: Real Mathematical Analysis,
by Charles C. Pugh, Springer, 2002.
- Set 1 Due Wednesday Sept. 14:
- Review your calculus text for definitions of the limit of function, continuity, and
the limit of a sequence.
- A. Prove: lim as x->3 of x2 = 9.
- B. Prove: The derivative of f(x)=x2 at x=3 is 6.
- C. Prove: lim as n->infinity 1/(n^2+1) = 0.
- Read: Sections 1.1-1.2, Skim the rest of Chapter 1.
- Written HW: Ch 1: 2,3,5a,7d,9. Extra credit: 8.
- Set 2 Due Monday Sept. 26:
- Read Sections 1.3, 1.4. Look through 1.5 - 1.6 again.
- Show the sequence defined by an = log(n) is not Cauchy, but does satisfy
|an - an+1|->0.
- Written HW: Ch 1: 11, 17, 30, 35a, 36a, 39, 42
Change 42d to the following easier questions:
- Find sequences such that limsup (a_n+b_n) < limsup (a_n) + limsup (b_n)
- Find a sequence and a constant c such that limsup(c a_n) is NOT equal to c limsup(a_n).
Extra Credit: 40.
- Set 2.5 Due Friday Oct. 7:
- Redo the following problems from HW 2. For each problem, have a statement of the problem, and
clear explanations of each step you are performing:
- Show the sequence defined by an = log(n) is not Cauchy. Use espsilon-N definition directly. Do not directly use the fact that the sequence is known to be unbounded or that the sequence is known to diverge.
- Show directly using epsilon-delta arguments that a nondecreasing function defined on the reals
has a left limit at all points. (lim as x->a- of f(x) exists at any point a in the domain).
- Show directly from the definition of uniform continuity, using epsilon-delta arguments, that
f(x)=x^2 is not uniformly continuous.
- Set 3 Due Monday Oct. 10:
- Read Sections 2.1 - 2.2.
- Ch 2: 2,3, 6a,7,12 (w/o "more generally"), 30a
- Extra credit: 30b
Midterm 1: Friday, Oct. 14, 3-4:30.
See Topic list below. "Basic theorems" and proofs.
Definitions and short problems. Examples.
Test 1 topic list .
Set 4 Due Friday Oct. 28:
- Read Sections 2.1 - 2.3.
- Ch 2: 8,39, 89a, 91abdf, 92beg (for g, find an example with strict inclusion)
Set 5 Due Friday Nov. 11:
- Read Sections 2.2 - 2.5.
- Ch 2: 25a, 26, 27a, 29 (29b is asking: Is a Cauchy sequence necessarily bounded?) 43 (if true, give a formal proof; if false, give counterexample), 47, 48, 49, 54, 55.
- Extra Credit: 46.
Set 6 Due Monday Nov. 21:
- Read Ch 3: pp 139-142 (derivative), 146-149 (pathological functions), 179-185 (series)
- Ch 3: 5,14ab, 56, 59, 64a
For 14a, show only the first derivative at zero is zero and the first derivative
is continuous on the reals.
Extra Credit: 14a. Show all derivatives at zero are zero.
Reminder: Long classes Fri. Nov. 18 and Mon Nov 21. No class Wed Nov. 23 or Fri Nov 25 (Thanksgiving)
Midterm 2: Friday, Dec. 2, 3-4:30.
See Topic list below. "Basic theorems" and proofs. Definitions an short problems. Examples. Topic list will be updated soon.
Test 2 topic list.
Set 7 Due Monday Dec. 12:
- Read Chapter 4, Section 1. Skim section 5.
- A. Ch 4, #5a-d
B. Denote successive approximations via the Picard iteration method by yn(t). For the
initial value problem y'=y2, y(0) = 1, let y1(t)=1.
Compute y2 and y3 by hand, and y4 and y5
using Mathematica (or any other computational aid). Also compute the exact solution
using separation of variables. Rewrite the exact solution as a power series. Compare
your approximations y2, ..., y5 with the series solution.
Final Exam Tues. Dec. 20, 12-1:55PM. Cumulative. Ch's 1-5.
See midterm 1 and 2 topics lists above. Add to them:
- Prove that if a sequence of continuous functions converges uniformly, then the limit
function is also continuous.
- Prove that a contraction mapping on a metric space has a unique fixed point.
- Compute one or more Picard iterates for approximate solutions to an ODE.
- Determine the C^0 and/or L^1 distance between given functions.
- Determine whether a sequence of functions converges pointwise, in C^0, or in L^1.
This page is maintained by
Bruce Peckham (bpeckham@d.umn.edu)
and was last modified on
Friday, 09-Dec-2011 13:06:06 CST.