Home page for Math 317/001:

Advanced Calculus of One Variable

Spring Semester 2006


Holger Kley	Time: MWF 10:00--10:50
Hours: Thursdays at 4:30, Fridays at 3, and by appointment	Place: Engineering E105
Problem Session: Thursdays 10:00--10:50, Engineering E105


Announcements	Description	Schedule	Text	Homework	Grading	Handouts	Links

Announcements:


Office Hours: Monday, 5/8: 2:30--4:00. Tueday 5/9: 12:30--5:00.
Solutions 6
Solutions 5	4/25
Office hours: 4/13 3--4pm, 4/14 9--10 am, 4/17 3--4 pm. Back to normal on 4/20.	4/11
Test 2 Solutions	4/11
Sample Test 2 is now available	3/29
Solution 4	3/27
Solution 3	3/7
Test 1 Solutions	2/21
Thursday, 2/16: Office hours 2--3 pm. Friday, 2/17, office hours 3--5 pm.	2/15
Sample Test 1 is now available	2/15
Solution 2 is now available	2/14
Starting 2/16, Thursday office hours will be 4:30--5:30.	2/9
Solution 1 is now available	1/31

Description:

The prerequisites for this course are one-variable calculus, meaning M 160 and M 161. The emphasis in those courses is on conceptual understanding, problem-solving and computation. While simple proofs are occasionally carried out, this are not emphasized, and students are not expected to develop skill at proving results.

By contrast, in M317, the emphasis is almost entirely on proof. Essentially, we will be exploring the foundations of mathematical analysis in one (real) variable, as establish by Cauchy, Weierstrass and others in the 19th century and proving almost every one of the major theorems on which the techniques of M 160-1 rely will be proved in this course.

Schedule of Lectures:

For the most part, we will be following the text. The titles of the lectures give a theme or focus and are not meant to describe everything which may be covered on any given day. I've added roughly corresponding section number from the text, but this certainly does not mean that I will cover every word in those sections. Since I've undoubtedly misestimated how long it will take to cover certain topics, this schedule is subject to change.

	Date	Lecture	Text
Week 1	Jan. 17	Introduction; familiar number systems.	1, 2
	Jan. 18	Algebra and order among the real numbers	3, 5
	Jan. 20	(continuation)	3, 5
Week 2	Jan. 23	The completeness axiom	4
	Jan. 24	(continuation)	4
	Jan. 25	Sequences and their limits	7
	Jan. 27	Epsilon-N proofs	7, 8
Week 3	Jan. 30	(continuation) Assignment 1 due. Solution 1	7, 8
	Jan. 31	Limit theorems	9
	Feb. 1	(continuation)	9
	Feb. 3	Some sequences that must converge	10
Week 4	Feb. 6	(continuation)	10
	Feb. 7	Subsequences	11
	Feb. 8	(continuation)	11
	Feb. 10	lim sup, lim inf	12
Week 5	Feb. 13	Series and the sequence of partial sums Assignment 2 due. Solution 2	14
	Feb. 14	Test for convergent series	14
	Feb. 15	(continuation)	14
	Feb. 17	Catch-up and review
Week 6	Feb. 20	Test 1 Test 1 Solutions
	Feb. 21	The alternating series test	15
	Feb. 22	Limits of functions: epsilon-delta proofs	20
	Feb. 24	(continuation)	20
Week 7	Feb. 27	The definition of continuous function	17
	Feb. 28	Maximum value theorem and the intermediate value theorem	18
	Mar. 1	(continuation)	18
	Mar. 3	Uniform continuity	19
Week 8	Mar. 6	(continuation) Assignment 3 due. Solution 3.	19
	Mar. 7	Power series	23
	Mar. 8	(continuation)	23
	Mar. 10	Uniform convergence	24
Week 9	Mar. 20	(continuation) Last day to W drop	24
	Mar. 21	Derivatives	28
	Mar. 22	(continuation)	28
	Mar. 24	The mean value theorem	29
Week 10	Mar. 27	(continuation) Assignment 4 due. Solution 4.	29
	Mar. 28	Taylor's theorem	31
	Mar. 29	(continuation)	31
	Mar. 31	Catch-up and review
Week 11	Apr. 3	Test 2
	Apr. 4	l'Hospital's rule	30
	Apr. 5	Definition of the Riemann integral	32
	Apr. 7	(continuation)
Week 12	Apr. 10	(continuation)
	Apr. 11	Properties of the Riemann integral	33
	Apr. 12	(continuation)	33
	Apr. 14	The Fundamental theorem of calculus	34
Week 13	Apr. 17	(continuation)	34
	Apr. 18	Improper integrals and the integral test Assignment 5 due. Solutions 5.	36, 15
	Apr. 19	Back to uniform convergence	25
	Apr. 21	(continuation)	25
Week 14	Apr. 24	Integration and differentiation of power series	26
	Apr. 25	(continuation)	26
	Apr. 26	The Weierstrass approximation theorem	27
	Apr. 28	Logarithmic, exponential, and trigonometric functions	37
Week 15	May 1	Optional material or catch-up	?
	May 2	Optional material or catch-up	?
	May 3	Review Assignment 6 due. Solutions 6
	May 5	Review
Finals Week	May 10	Final Exam, 7:00--9:00 a.m.

Text:

Ross, Kenneth A. Elementary Analysis: The Theory of Calculus, Springer, New York, 2000. ISBN 0-387-90459. The text (affectionately known as Ross) is available at the CSU bookstore.

Homework:

There will be two kinds of problems: discussion problems and assignments.

Discussion problems will be assigned along with reading from each section of the text we cover. These will form the basis of our Thursday problems session and will be discussed (in detail, if needed) in office hours and (if needed) in class. You are encouraged to work together on discussion problems, and to write down rought drafts of solutions. Discussion problems will not be turned in for a grade. A list of discussion problems is posted below.

Assignments will consist of problems similar to discussion problems. Work on assignments must be your own, and you should not expect more than a vague hint from the me. If you like, you may think of these as take-home tests. There will be six assignments, due as indicated in the schedule above. Each assignment will be given a week before the due date. Work that you submit for assignments should be neat and carefully done -- you will probably want to rewrite and polish your first attempts at a solution. Please leave sufficient space in the margins and between problems for the grader's comments. I will post solutions to assignments after they are due. Once a solution is posted, late submissions will not be accepted for credit.


Week	Reading	Discussion Problems	Week	Reading	Discussion Problems
1	Section 1	2, 3, 7, 11	2	Section 4	1, 2, 3, 4, 5
1	Section 3	1, 2, 4, 5, 7	2	Section 7
3	Section 7	3	4	Section 10	1, 6, 11
	Section 8	2, 4, 7		Section 11	3, 4
	Section 9	1, 3, 4, 8, 12
5	Section 14	5, 6, 14	6	Section 14	2, 3, 7
5			6	Section 15	2
7	Section 20	9, 11, 20	8	Section 18	2, 5, 9, 10
7	Section 17	3, 8, 10, 13	8	Section 19	1, 2, 7
9	Section 23	1, 4, 7	10	Section 28	2, 4, 7
9	Section 24	2, 5, 10.	10	Section 29	2, 5
11	Section 32	1, 6, 7, 8	12	Section 34	2, 3, 6, 10
11	Section 33	3, 4, 9, 10	12	Section 36	2, 6
13	Section 26	2, 4	14	Section 37	5, 6, 7, 8
13	Section 31	2, 4, 5	14
15	Section 27

Grading:

Course grades will be computed as follows:

Assignments 1--6	7% each
Tests 1 and 2	14% each
Final	30%
Total	100%

Handouts:


Class Policies	1/17	Assignment 1	1/23
Solution 1	1/31	Assignment 2	2/6
Solution 2	2/14	Sample Test 1	2/25
Test 1 Solutions	2/21	Assignment 3	2/28
Solution 3	3/6	Assignment 4	3/20
Solution 4		Assignment 5
Test 2 Solutions		Solutions 5	4/25
Assignment 6

Links:

If you know of any good resources that should appear here, please let me know.

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