Math 633, Advanced Calculus 1
Fall 2002, Ref: 16900
Todd Cochrane, CW 209, 532-0565,
cochrane@math.ksu.edu
http://www.math.ksu.edu/~cochrane/advcalc1/advancedcalculusf02.html
Text: Advanced Calculus, An Introduction to Analysis- Third
Edition, by Watson Fulks, Wiley & Sons, New York, 1978.
Goals of the course:
1) To obtain a deeper understanding of Calculus 1 and 2. (Calculus
3 will be reviewed in Advanced Calculus II next semester.)
2) To review and reinforce concepts and theorems that
you have already seen in Calc 1 and Calc 2, as well as to introduce a
few new concepts that underlie these ideas.
3) To develop skill at reading, constructing and writing
proofs. You will be expected to write clean and rigorous arguments
with correct logic and grammar. This will set the foundation
for graduate level courses in mathematics and elsewhere, in particular,
for Analysis and Topology.
Grading:
1) Weekly Homework Assignments worth 16 points each. Probably
12 assignments altogether, worth a total of 192 points. Assignments
will be due on Friday in class: Sep 6, Sep 13, Sep 20, Sep 27, Oct 11, Oct
16, Oct 25, Nov 1, Nov 15, Nov 22, Dec 6, Dec 13
2) Two Midterms worth 100 points each, October 4 and November
8 (tentative dates.)
3) Final Exam worth 160 points.
Advanced Help Session:
Beginning September 4, Cardwell 144, with John Rapalino
and Mark Norfleet. Monday 6:30-7:30 p.m.
Wednesday 5:30 - 7:30 p.m.
Old Exams: These are from Fall 1987, the last
time I taught Advanced Calc. Test
1 Test 2 Final
Solutions to this semesters exams: Test 1 Test 2
Final
Solutions to homework: HW2
HW5
HW6 HW7 HW10
Hints for HW 7: #2b) Proof by contradiction. If the limit is
not L then there is an epsilon> 0 and subsequence {a_{n_k}} all of whose
terms are not in the epsilon NBD of L (why?) Now apply Bolzano-Weierstrass
to this subsequence.
#4a) Use Cauchy-Criterion, 4b) Let L be the limit from part
a. Let delta be such that abs(x-y)<delta implies abs(f(x)-f(y))<epsilon/2.
Let N_1 be such that n>N_1 implies abs(a_n-b)<delta, and N_2 be such
that n>N_2 implies abs(f(a_n)-L)<epsilon/2. Suppose b-delta<x<b.
Use triangle inequality to estimate abs(f(x)-L)=abs(f(x)-f(a_n)+f(a_n)-L),
with n sufficiently large.