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 this interval.
2. Give an example of a bounded, contin
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 this interval.
Example: Let f (x) = 2x
1 for x 2 (0, 1
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], then f g is integrable on [a, b].
Proof. This is true. By
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.
a)f (x, y) = xey
b)f (x, y) = cos(xy)
c)f (x, y) =
x+
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 or the upper sum is a better approximation to the integ
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], then f g is integrable on [a, b].
b) If f is Riemann inte
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. (a) Suppose that f : (0, 1) R is uniformly continuous on
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 limit of a sequence of bounded functions that is unifo
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. ~
(b) Determine whether fn ! f uniformly on [0, 1].
No. By
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 = limy!1
may continuously extend e 1/x to [0, 1]. |
(c) l
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 < 1
x2
2
+
x4
,
24
for 0 < x /2.
2. Find the derivative
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)?
Feel free to remember everything you know about sin x
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 compact set F in X, then E is also
compact. Note how this co
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 upper sum is a better approximation to the integral. Graph