Math 140, Winter Instructor: Professor Ni 1. Page 114: Exercise 4, Exercise 6. 2. Page 115: Exercise 11, 15. 3. Page 117: Exercise 22. 4. Page 138: Exercise 5.
Practice problems
April, 2010
5. Let f (x) = 1 x 3 . Check that f (1) = f (1) = 0. Explain why
HW #1 SOLUTIONS
1. Let f : [a, b] R and let v : I R, where I is an interval containing f ([a, b]). Suppose that f is continuous at x [a, b]. Suppose that limsf (x) v(s) = v(f (x). Using the definition of limit, prove that
limtx v(f (t) = v(f (x).
Solution
HW #2 SOLUTIONS
1. Let f, g : [a, b] R be nonnegative bounded functions.
(a) Is it necessarily true that
Z
b
Z
f dx +
b
Z
g dx =
a
a
b
(f + g) dx?
a
Either prove this or show that there is a counterexample.
(b) Suppose that f (x) > g(x) + 1 for all x [a,
Math 140B
HW2, due Friday January 27 at 2pm
#1. Let f; g : [a; b] ! R be nonnegative bounded functions.
(a) Is it necessarily true that
Z
b
f dx +
a
Z
b
gdx =
a
Z
b
(f + g)dx ?
a
Either prove this or show that there is a counterexample.
(b) Suppose that f
Math 140B
HW1 (Part I), due Friday January 20 at 2pm
#1. Let f : [a; b] ! R and let v : I ! R, where I is an interval containing f ([a; b]). Suppose that
f is continuous at x 2 [a; b]. Suppose that lims!f (x) v (s) = v(f (x). Using the "- denition of limi
Math 140B - Spring 2013 - Midterm I
Name:
Student ID:
Instructions:
There are 4 questions which are worth 40 points. You have 50 minutes to complete the test.
Question
Score
Maximum
1
10
2
10
3
10
4
10
Total
40
Problem 1. [10 points.]
Consider a twice die
MATH 140B - SPRING 2013 - SOLUTIONS
Problem 1.
Consider dierentiable functions fn : [0, 1] R for n 1, such that the sequences cfw_fn and
cfw_fn are uniformly bounded. Show that cfw_fn has a uniformly convergent subsequence.
Solution: By Arzel`-Ascoli,
Math 140B - Homework 3. Due Wednesday, May 1.
1. The mean value theorem for integrals
(i) Let f : [a, b] R be a continuous function. Show that there exists x0 (a, b) such that
1
ba
f (x0 ) =
b
f (t) dt.
a
(ii) If f is the derivative of a continuously dier
Math 140B - Spring 2013 - Final Exam
Name:
Student ID:
Instructions:
Please print your name and student ID.
During the test, you may not use books, calculators or telephones. Read each question carefully,
and show all your work. Answers with no explanatio
Math 140B - Spring 2013 - Midterm I
Name:
Student ID:
Instructions:
There are 4 questions which are worth 40 points. You have 50 minutes to complete the test.
Question
Score
Maximum
1
10
2
10
3
10
4
10
Total
40
Problem 1. [10 points.]
Consider a twice die
Problem 1.
Give an example of a sequence of continuous functions fn : [0, 1] R converging pointwise to 0
1
as n , such that 0 fn (x) dx does not converge to 0, as n .
Solution:There are many possible answers here. You can take for instance
fn (x) = nx(1 x
Review Problems for Final Exam
Make sure you review all homework problems, denitions, theorems and proofs.
Problem 1. (Fourier Series.)
Prove Poincars inequality:
e
2
2
|f (x)|2 dx
0
|f (x)|2 dx
0
for any 2 -periodic complex valued continuously dierentia
Review Problems for Midterm II
Make sure you review all homework problems, denitions, theorems and proofs.
Problem 1.
Consider C 1 ([a, b]) the set of continuously dierentiable real valued functions endowed with the
norm
|f | = sup|f (x)| + sup |f (x)|.
(
Math 140B - Spring 2013 - Midterm II
Name:
Student ID:
Instructions:
Read each question carefully, and show all your work. Answers with no explanation will receive
no credit, even if they are correct.
There are 4 questions which are worth 40 points. You h
Review Problems for Midterm I
Make sure you review all homework problems, denitions, theorems and proofs.
Problem 1.
For what values of > 0, is the function f (x) = |x| sin x twice dierentiable?
Problem 2.
A function f : R R is said to be Lipschitz if the
Math 140B - Homework 6. Due Wednesday, May 29.
1. Rudin, Chapter 7, solve problems 20, 21, 23.
2. Show that power series can be integrated term by term within the radius of convergence.
That is, assume that
cn xn
f (x) =
n=0
has radius of convergence R >
Math 140B - Homework 2. Due Wednesday, April 17.
1. Consider the function
1
f (t) =
(i)
(ii)
(iii)
(iv)
e t
0
for t > 0
.
for t 0
Calculate the derivative f (t) for all values of t. Conrm that f (0) = 0.
Calculate the derivative f (t) for all values of t.
Math 140B
HW 6, May 13 Chapter 7, Page 165-167: #1, 9, 11, 12.
Denitions for #1 :
We say that ffn gn 1 is a sequence of bounded functions on a set E if for each n
exists Mn such that jfn (x)j Mn for all x 2 E:
n
We say that ffn gn
1 and all x 2 E:
1
is un