Math 042
Solutions to Homework 1
June 27, 2011
2.3 Number 3
By denition,
lim inf xk = lim bk
k
k
where
bk = inf cfw_x : k .
Similarly,
lim inf xnk = lim bk
k
k
where
bk = inf cfw_xn : k .
Since nk k , it follows that
cfw_xn : k cfw_x : k
so bk bk , and
Math 0420 Practice Test 2 Solutions
Page 1
Question 1. For each positive integer n, define hn : R R by
hn (x) = nex/n n. Show that this sequence of functions converges pointwise
on R and find its limit function.
Solution. Fix x0 R. Note that making a subs
Math 0420 Practice Test Sequences/Series of Functions and Power Series
1. Determine whether or not the following sequences converge.
(a) fn (x) =
1
1+nx
on [0, 1]
(b) gn (x) = n2 xn on [0, 1]
(c) hn (x) = cosn (x) on [/2, /2]
(d) fn (x) =
xn
1+xn
on
(i) [
3., Lu AtpruS
Math
Exam 2
O42A
e This exam has 5 questions, for a total of 55 points.
I
April 5, 2016
hlarne:
lt[o books, calculators, or other electronic equipment al
lowed.
Score:
155
points
g and
gt: f
c Answer the questions in the spaces provided. I
Math 0420
Quiz 2
Spring 2014
Solutions
1. By using the denition of the uniform continuity show that the function f : [0, 1] R, dened
by f (x) = x3 is uniformly continuous. [Do not use the theorem 3.4.4 saying that if a function
dened on a closed interval
Math 0420
Quiz 1
Spring 2014
Solutions
1. Find an example of functions f and g such that both limits lim f (x) and lim g(x) do not exist
x1
x1
but lim f (x)g(x) exists.
x1
Solution:
f (x) =
For example,
x, if x = 1
2, if x = 1
g(x) =
x, if x = 1
1
2 , if
Math 0420
Spring 2014
Quiz 3
Your name:
1. (a) (5 points) State the Mean Value Theorem.
(b) (5 points) Prove the Mean Value Theorem.
2. (a) (3 points) Say precisely what is meant that a function f is uniformly continuous on a set S.
(b) (3 points) Say pre
Math 0420
Spring 2014
Quiz 4
Your name:
1. To pass part 1 you have to get at least 4 points for each question. In this case your overall score
for part 1 will be 12. If you do not pass this part then your score will be 0.
(a) (5 points) Give precise denit
Quiz 5
Math 0420
Spring 2014
Your name:
1. To pass part 1 you have to get at least 4 points for each question. In this case your overall score
for part 1 will be 12. If you do not pass this part then your score will be 0.
(a)
Give precise denition of unif
Math 0420
Spring 2014
Quiz 2
Your name:
1. By using the denition of the uniform continuity show that the function f : [0, 1] R, dened
by f (x) = x3 is uniformly continuous. [Do not use the theorem 3.4.4 saying that if a function
dened on a closed interval
Math 0420
Final Exam
Spring 2014
Your name:
No calculators, no books. Show all your work (no work = no credit). Write neatly.
The exam contains two parts. There are six problems in Part 1 and nine problems in Part 2.
Problems in Part 1 are 5 points each.
Quiz 1
Math 0420
Spring 2014
Your name:
No calculators, no notes, no books. Show all your work (no work = no credit). Write neatly.
1. (5 points) Find an example of functions f and g such that both limits lim f (x) and lim g(x) do
x1
not exist but lim f (
Math
t
I
I
r
Exam
0420
February 16,
L
This exam has 5 questions, for a total of ?s points.
2A1C
Narne:
No books, calculators, or other electronic equipment allowed.
Answer the questions in the spaces provided. If you need more room,
continue on the back o
MATH 0420 Midterm 2 Sample Exam
Tom Everest
November 8, 2015
1. a) (5 pts) Write the definition of the derivative of f : I R exists at
c I.
b) (10 pts) Use the definition to find f 0 (1) for f (x) = x x.
2. (15 pts) Show that f : R R defined by f (x) = 1
Math 042
Solutions to Homework 2
July 3, 2011
Supplement Number 4
(a) The set of cluster points is [0, 1]
(b) There are no cluster points.
(c) The set of cluster points is R.
(d) The only cluster point is 0.
Supplement Number 5
Only (b) is closed.
Supplem
Math 042
Solutions to Homework 3
July 4, 2011
3.3 Number 1
There are lots of examples. Heres one:
f (x ) =
1
1
if x = 0
if x > 0.
Then f (0) < 0 < f (1), but f never assumes the value 0.
3.3 Number 2
Again, there are many examples. Heres one:
f (x ) =
x
1
Math 042
Solutions to Homework 4
July 19, 2011
4.1 Number 5
For dierentiability at 0, we have, for x = 0,
x
0
f (x) f (0)
=
x0
if x Q
otherwise.
Thus for all nonzero real numbers x we have
f (x) f (0)
x 
x0
so, by the Squeeze Theorem,
lim
x 0
f (x) f
Math 042
Solutions to Homework 5
July 27, 2011
5.1 Number 7
By denition of integrability, there are partitions P1 and P2 such that
b
L( P 1 , f ) >
a
and
f
b
f + .
U ( P2 , f ) <
a
Let P be the partition obtained by combining P1 and P2 . Then
b
b
a
f < L
Math 042
Solutions to Homework 6
July 27, 2011
6.1 Number 2
a) For any xed x R, the sequence (x/n) converges to 0. Since the exponential function is continuous, ex/n e0 = 1, so
ex/n
0.
n
b) Let xn = n log n. Then
exn /n /n = 1
so the sequence does not co
Math 042 Midterm Exam
July 11, 2011
Solutions
Instructions: No books or notes may be used during the exam. Attempt all problems, giving complete and clear explanations of your answers.
1. (3 points each) Answer true or false. If you answer false, give a c
L
S'uyyos"
+h,j,
f 'l^,U] IR is rw'onotonz ov1 fo,b1
th"L I is tvlo'^ol.amt irtctcasinfl
&od so
ir
,
c,r^llrwLov'S
Tf
l1nt^
ir
c,onsta/& ,

t
ov1 fa'Ll
I is Kit,^n 'vLtJtoLQ5r, t^f (0 54 't h^'(. t is nof on JLant ' Tn f*hu'Jo'r'
Wtoe ,
ASSL,trvtr
C
Solurt DNs
Math 0420
I
.EJXAIn.
February 15, 201C
o This ex&m has 4 questions, for a total of G0 points.
o No books,
.
Name:
calculators? or other electronic equipment allowed.
Answer the questions in the spaces provided. If you need more
continue on the
MATH 0420 Practice Test Integration and Sequences of Functions
1. Show that if f : [a, b] R is monotone on [a, b], then f is Riemann integrable
on [a, b].
2. Let f : [a, b] R and let g : [c, d] [a, b].
(a) Suppose that f and g are continuous. Prove that f
MATH 0420 Practice Test Convergence of Sequences of Functions
1. For each positive integer n, define hn : R R by hn (x) = nex/n n.
Show that this sequence of functions converges pointwise on R and find its limit
function.
2. Consider the function f define
Math
o
o
Exarn 2
O42A
Aprit 4) ZO1C
This exam has 4 questions, for a total of 50 points.
I\o books, calculators, or other electronic equipment
Name:
al
lowed.
Score:
150 points
Answer the questions in the spaces provided. If you need
more room, continue
Math 0420
Spring 2014
Midterm Exam
Your name:
No calculators, no books. Show all your work (no work = no credit). Write neatly. Simplify your
answers.
For each problem (except the bonus) you will get 40% of the maximum points if you do not write a
solutio