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Math 142 - Advanced Calculus - Spring, 2008 - Updated on
May 10, 2008
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Later (hopefully more complete) versions of this syllabus will
be on the course website:
http://math.ucsd.edu/~aterras/ma142.htm.
Keep looking here for the latest info!
There are some new things below - a new calendar and more
homework assignments.
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Lecture on Weierstrass Approximation Theorem, Section 8.7: weierstrass
approx.pdf
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Midterm exam 1 solutions:
exam
1 solutions
Practice midterm 1 practice
exam 1
Corrected Solutions to practice exam 1 practice
exam 1 solutions
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Homework 1 & 2 solutions are on the website.
http://math.ucsd.edu/~aterras/ma142 hw1 solns.pdf
http://math.ucsd.edu/~aterras/ma142 hw2 solns.pdf
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Time of Lecture: MWF 12:00p -
12:50p Place: WLH
2111
Instructor: Audrey Terras email
address: aterras@ucsd.edu
Office Hours: (tentative) Wed. 3:30-4:30, Thurs.
3-4 in 7408 AP&M and by appointment
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Text:
Patrick M. Fitzpatrick, Advanced Calculus, 2nd Edition.
This text should be on reserve in S&E library.
a free text on the web:
http://www.freetechbooks.com/one-variable-advanced-calculus-t701.html
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Course Description. The Riemann
integral. Transcendental functions. Limits and continuity. Infinite series.
Sequences and series of functions. Uniform convergence.
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Sections are Thursday: Th 09:00a - 09:50a CENTR 201
and Th 08:00a - 08:50a CENTR 201
Teaching Assistant:
office
hours: 2-3pm on Tuesdays and 2-3pm pm on
Wednesdays
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Grades: Midterms 1+2: 30%
each, Final 20%, text homework 20%
Exams will be closed book, no notes, no calculators, no computers,
no headphones.
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New Calendar
|
week |
ending |
Monday |
Wednesday |
Friday |
|
1 |
April 4 |
introduction |
6.1 |
6.1 |
|
2 |
April 11 |
6.2 |
6.2 |
6.3 |
|
3 |
April 18 |
6.3
|
6.4 |
6.5 |
|
4 |
April 25 |
6.6 |
Review |
Exam 1 |
|
5 |
May 2 |
8.5 |
8.2 |
8.1,3 |
|
6 |
May 9 |
8.4 |
8.6 |
8.7 |
|
7 |
May 16 |
9.1 |
9.2 |
9.3 |
|
8 |
May 23 |
9.4 |
Review |
Exam 2 |
|
9 |
May 30 |
holiday |
Improper
Integrals |
Gamma
& Beta Functions |
|
10 |
June 6 |
Fourier
Series |
Fourier
Series |
Review |
|
finals week |
June 13 |
|
final: 11:30a - 2:29p |
|
Warning: It is very important to stay caught up in this class! It is
like a language course.
Lots
of new vocabulary to practice with! Also, please ask questions!!!
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Homework Assignments: Although
you are required to turn in only the assigned HW problems, you are
strongly advised to attempt solving as many problems from each section as
possible. Keep a notebook of homework solutions. I will look
at it the last day of class. I don’t care what sort of notebook you
use. It can be loose lined pages held together in a binder. It should just be neat. You can use the
graded homework papers or copies thereof and add any extra homework problems
you may do.
The Homework Drop Box is located on the 6th floor of
AP&M. The homework must be in the drop box by noon of the day after
the homework is due; Friday at noon. Make sure you print your name and student
ID number on your homework. Also make sure you staple the homework together.
Neatness counts !!!!!!!!!!!!!!!!! It is important to justify every step
of a proof either by referring to some earlier result in the text or by giving
a convincing explanation. A main goal of this course is for you to be able to
give a rigorous proof of the main results about integrals.
The optional problems assigned in class will be graded for extra
credit.
Homework Assignment #1. (due Thursday, April 3, 2008) Section
6.1 # 3,4
optional: (sorry for the
inability to type math; this computer is not displaying equations
correctly. Here <= means less than or equal).
1) Suppose f(x)
<= M, for all x in [a,b]. Show that U(f,P) <= M(b-a).
2) Prove that if
partition Q is a refinement of partition P, then U(f,Q) <= U(f,P).
Homework Assignment #2. (due Thursday, April 10, 2008) Section
6.1 # 1a, 5;
Section 6.2 # 1,3,4a,5a
optional:
1) Prove the
squeezing lemma. This means show that
if 0 <= a <= bn, for all n =1,2,3, ...
; and Limn->infinity bn =0,
then a=0.
2) Assume that f is
integrable on [a,b] as in the Archimedes-Riemann Thm. on p. 143.
Show that Limn->infinity U(f,Pn) = Integral of f from a to b .
Homework Assignment #3. (due Thursday, April 17, 2008) Section
6.2 # 7, 8;
Section 6.3 # 2, 4, 5; Section 6.4 # 1, 5
optional:
1) Prove if f and g
are bounded on [a,b], then L(f,P)+L(g,P) <= L(f+g,P).
Delete the 2nd optional problem as it is part of
#4 in Section 6.3. Sorry
2) Show that if a<0 and S is a bounded set of real
numbers, then
sup{ ax | x in S} = a inf{x | x in S}.
Use this to show U(af,P)=aL(f,P).
Homework Assignment #4. (due Thursday, April 24, 2008) Section
6.4 # 6, 7, 9;
Section 6.5 # 5, 6; Section 6.6 # 1, 3
Homework Assignment #5. (due Thursday, May 8, 2008) Section 8.5
# 2,3,5,6;
Section 8.2 # 3,4,7; Section 8.1 #2b,d; Section 8.3
# 1,2; Section 8.4 # 1,2
Optional:
1) We wrote
f(b)=f(a)+f’(a)(b-a)/1!
+ ... + f(n-1)(a)(b-a)n-1/(n-1)! +Rn.
Prove the Lagrange form of the remainder Rn follows
from the Cauchy form in the case that b<a. We did the case that b>a
in class.
Homework Assignment #6. (due Thursday, May 15, 2008) Section 8.6
#1; Section 8.7 #5;
Section 9.1 #1 a,c,e,g; 4, 7; Section 9.2 # 3,4
Optional: See the lecture
above on Weierstrass approximation (Section 8.7).
Homework Assignment #8. (due Thursday, May 22, 2008) Section 9.3
# 3,4,5,7;
Section 9.4 # 1,3
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Some
motivation:
Joseph
Fourier from Wikipedia
Around the early 1800’s Fourier was studying heat flow in wires or metal
plates. He wanted to model this mathematically and came up with the heat
equation. Suppose that we have a wire stretched out on the x-axis from x=0
to x=1. Let u(x,t) represent the temperature of the wire at position x
and time t. The heat equation is the PDE
holding
for t>0 and 0<x<1. Here c is a constant depending on the metal.
If you are given an initial heat distribution f(x) on the wire at time 0, then
we have the initial condition: u(x,0)=f(x) also.
Fourier plugged in the function u(x,t)=X(x)T(t) and found that to
fit the initial condition he needed to express f(x) as a Fourier series:
Note
that eix=cosx+isinx, where i=(-1)1/2 (which is not a real number). This
means you can rewrite the series of complex exponentials as 2 series - one
involving cosines and the other involving sines. Fourier made the
claim that any function f(x) has such an expression as a sum of cnsin(nx)
and dncos(nx). People took issue with this although they did
believe in power series expressions of functions (
Fourier tells us that the Fourier coefficients are
If
you believe that it is legal to interchange sum and integral, then a bit of
work will make you believe this, but unfortunately, that isn’t always
legal. This left mathematicians in an uproar in the early 1800’s.
And it took at least 50 years to bring some order to the subject.
Part of the problem was that in the early 1800’s people viewed integrals as
antiderivatives. And they had no precise meaning for the convergence of a
series of functions of x such as the Fourier series above. They argued a
lot. They would not let Fourier publish his work until many years had
passed. False formulas abounded. Confusion reigned supreme.
So this course was invented.
We will end up with a precise formulation of Fourier’s theorems. And we
will be able to do many more things of interest in applied mathematics.
The integral itself provides definitions of arc length, area, volume,
probability. Many mathematicians have considered ways of defining the
integral. Riemann was perhaps the first (mid 1800’s). That will be
our definition. About 50 years later Lebesgue found another definition
that has proved to be extremely useful. You have to take Math 140 to
learn about that version of integrals.
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