Math 0420
Exam 1
October L2r 2A15
SoLUTt p N S
Name:
/65 points
Instructions:
This exam has 4 questions, for a total of 65 points.
Answer the questions in the spaces provided. If you need more room, continue on the back of the page. For full
credit, arlsw
MATH 0420 - HOMEWORK 2 SELECTED SOLUTIONS
3.2.3. Let f : R R be dened by
x if x Q;
x2 if x R\Q.
Using the denition of continuity directly prove that f is continuous at 1 and
discontinuous at 2.
f (x) :=
Proof. First we show that f is continuous at 1. Let
Math 0420 - HW5 Solutions
Page 1
Exercise (3.3.3). Let f : (0, 1) R be a continuous function such that
lim f (x) = lim f (x) = 0. Show that f achieves either an absolute maximum or
x0
x1
an absolute minimum on (0, 1) (but perhaps not both).
Proof. Define
Math 0420 - HW2 Solutions
Page 1
Exercise (2.3.4). Prove that a bounded sequence cfw_xn is convergent and converges to x if and only if every convergent subsequence cfw_xnk converges to x.
Proof. First suppose that cfw_xn converges to x. Let cfw_xnk b
Math 0420 - HW6 Solutions
Page 1
Exercise (3.5.3). Prove that a continuous function on an interval is injective
if and only if it is strictly monotone.
Proof. Let f be a continuous function defined on an interval I.
First, assume that f is strictly monoto
Math 0420 - HW9 Solutions
Page 1
Exercise (5.1.1). Let f : [0, 1] R be defined by f (x) := x3 and let P :=
cfw_0, 0.1, 0.4, 1. Compute L(P, f ) and U (P, f ).
Solution.
L(P, f ) = m1 x1 + m2 x2 + m3 x3
= (0)(0.1) + (0.001)(0.3) + (0.064)(0.6)
= 0.0387.
U
Math 0420 Homework 3
Exercise 3.1.1 Find the limit or prove the limit does not exit.
(a) lim
xc
x for c 0.
Solution: First, we might guess that the limit is
c.
Scratch Work:For
c > 0, according to the definition of limit, we want to find to bound
the ter
5’0 1‘. VTIOA/S
Math 0420 Exam 2 November 16, 2015
o This exam has 6 questions, for a total of 100 points. Name:
o No books, calculators, or other electronic equipment allowed.
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Sample final exam answers
Math 0420, Fall 2014
These solutions are minimal. Please fill in details for yourself.
1. Explain how the exponential and logarithm functions can be used to define bx for all
x in R.
We didnt cover exponential and logarithmic fun
Math 0420 - HW10 Solutions
Page 1
Exercise (5.2.1). Let f be in R[a, b]. Prove that f is in R[a, b] and
Z b
Z b
f (x) dx =
f (x) dx.
a
a
Proof. Note that for any set A R, sup(A) = inf A and inf(A) = sup A,
where A = cfw_a : a A.
Let P = cfw_x0 , x1 , x2
MATH 0420 - HOMEWORK 5 SELECTED SOLUTIONS
4.3.3. Suppose that f : R R is a dierentiable function such that f is a
bounded function. Then show that f is a Lipschitz continuous function.
Proof. Suppose that |f (x)| K , for all x R. Let x, y R be arbitrary.
MATH 0420 - HOMEWORK 4 SELECTED SOLUTIONS
3.6.10. Let f be a bounded, monotone function on the interval (a, b). Show that
limxb f (x) exists.
Proof. Without loss of generality, assume that f is monotone increasing. Then,
since f is bounded, s = supcfw_f (
MATH 0420 - HOMEWORK 3 SELECTED SOLUTIONS
3.3.7. Suppose that f : [a, b] R is a continuous function. Prove that the direct
image f ([a, b]) is a closed and bounded interval.
Proof. The Min-Max Theorem tells us that f attains both an absolute maximum
value
MATH 0420 - HOMEWORK 1 SELECTED SOLUTIONS
Problem 1. (1) Find the following limit or show that it does not exist.
lim
n
1
1
1
+
+ +
n2 + 1
n2 + 2
n2 + n
Claim 1.
lim
n
Proof. Let xn :=
1
n2 +1
1
n2
+
+1
+
1
n2 +2
1
n2
+2
+ +
+ +
1
.
n2 +n
1
n2
+n
=1
N
Math 0420: Review Questions for Final
You should not assume that the nal exam will resemble this review sheet. The
problems below were chosen because they seemed interesting or because they illustrated
an important concept. They are not meant to be typica
Math 0420 - HW 11 Solutions
Page 1
Exercise (6.1.2). a) Find the pointwise limit
ex/n
for x R.
n
b) Is the limit uniform on R?
c) Is the limit uniform on [0, 1]?
ex/n
Solution. a) Let fn (x) :=
. For a fixed x0 R, fn (x0 ) 0 as n .
n
Hence the pointwise l
Math 0420 - HW7 Solutions
Page 1
Exercise (4.2.1). The function f (x) = 3x5 10x3 +15x+1 is strictly increasing,
and therefore invertible, on R. Find (f 1 )0 (1).
Proof. Note that since f (0) = 1, f 1 (1) = 0, and so
(f 1 )0 (1) =
1
f 0 (0)
=
1
.
15
Exerci
Math 0420 - HW4 Solutions
Page 1
Exercise (3.2.2). Using the definition of continuity directly prove that f :
(0, ) R defined by f (x) := 1/x is continuous
Proof. Let c (0, ) be arbitrary. We will show that f is continuous at c.
o
n
2
Let > 0 be given. Ch
Math 0420 - HW8 Solutions
Page 1
Exercise (4.3.7). Suppose f : (a, b) R is a differentiable function such that
f 0 (x) 6= 0 for all x (a, b). Suppose that there exists a point c (a, b) such that
f 0 (c) > 0. Prove that f 0 (x) > 0 for all x (a, b).
Proof.
Math 0420 Homework 4
Exercise 3.2.1 Using the definition of continuity directly prove that f :
R R defined by f (x) := x2 is continuous.
Scratch Work: we know, |x c| < ,
|f (x) f (c)| = |x2 c2 |
= |x c|x + c|
(|x| + |c|)|x c|
(|c| + 1 + |c|)|x c|
In the
Math 0420 - HW1 Solutions
Page 1
Exercise (2.2.3). Prove that if cfw_xn is a convergent sequence, k N, then
lim xkn =
n
lim xn
k
n
.
Proof. Let P (k) be the statement
lim xkn =
n
lim xn
n
k
.
Note that P (1) is trivially true, and P (2) is true by the al
Math 0420: Review Questions for Midterm
You should not assume that the midterm exam will resemble this review sheet. The
problems below were chosen because they seemed interesting or because they illustrated
an important concept. They are not meant to be