Worksheet for 8/6/14
1. Give an example of a function that exists and is bounded for all x in the interval [0, 1] but
which never achieves either its least upper bound or its greatest lower bound over
Worksheet for 8/6/14
1. Give an example of a function that exists and is bounded for all x in
the interval [0, 1] but which never achieves either its least upper bound
or its greatest lower bound over
Homework 2
Math 125B, Spring 2016
Due Tuesday, April 12
5.2.0.& 5.3.0. Decide whether the following are true or false then prove or provide a
counterexample.
a) If f and g are integrable on [a, b], th
Homework 4
Math 125B, Spring 2016
Due Tuesday, April 26
11.1.1. Compute all mixed second-order partial derivatives of each of the following
functions and verify the mixed partial derivatives are equal
Homework 1 Solutions
Math 125B, Spring 2016
Due Tuesday, April 5
R2
5.1.1. a) Let f (x) = 3 x2 . Compute U (f, P ), L(f, P ), and 0 f (x)dx, where P =
cfw_0, 1/2, 1, 2. Find out whether the lower sum
Homework 2
Math 125B, Spring 2016
Due Tuesday, April 12
5.2.0.& 5.3.0. Decide whether the following are true or false then prove or provide a
counterexample.
a) If f and g are integrable on [a, b], th
Sample Questions for Midterm 1
Math 125A, Fall 2012
Closed Book. Give complete proofs of all your answers. Unless stated otherwise, you can use any standard theorem provided you state it clearly.
1. (
Worksheet 3
n
x
1. For x 2 [0, 1), let fn (x) = 1+xn .
(a) Find f (x) = lim fn (x).
(b) Determine whether fn ! f uniformly on [0, 1].
(c) Determine whether fn ! f uniformly on [0, 1).
2. Show that the
Worksheet 3 Solutions
1. For x 2 [0, 1), let fn (x) =
xn
.
1+xn
(a) Find f (x) = lim fn (x).
If x < 1, then f (x) = 0. Clearly, f (1) = 1 . For x > 1, we have
2
1
fn (x) = x n +1 and so f (x) = 1. ~
(
1. Verify whether the following functions are uniformly continuous on
(0, 1):
(a) e1/x
No. Let xn =
1
.
ln n
Then cfw_xn is Cauchy but eln n = n is not. |
(b) e 1/x
Yes. We have that limx!0+ e 1/x =
Worksheet 5
1. Show the following inequalities using the Mean Value Theorem.
(a) ln(1 + x) x, for any x
(b) x +
x3
(c) x
x3
(d) 1
x2
2
3
6
0.
tan x, for 0 < x < /2.
< sin x < x for 0 < x /2.
< cos x
Worksheet 2
1. Verify whether the following functions are uniformly continuous on
(0, 1):
(a) e1/x
(b) e
1/x
(c) ln x
1
(d) sin x
2. Which of the following functions are uniformly continuous on [0, 1)
Worksheet 4
1. Let E Rk . Show that E is compact if and only if every sequence in E has a
subsequence converging in E.
2. Let (X, d) be a metric space.
(a) Show that if E is a closed subset of a compa
Homework 1
Math 125B, Spring 2016
Due Tuesday, April 5
R2
5.1.1. a) Let f (x) = 3 x2 . Compute U (f, P ), L(f, P ), and 0 f (x)dx, where P =
cfw_0, 1/2, 1, 2. Find out whether the lower sum or the upp