Math 217 Fall 2013 Homework 6 Solutions
Due Thursday Oct. 31, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your fun
Solutions for Math 217 Assignment #10
(1) Show that the Taylor series of the following functions f (x) at
the given point x = x0 converges to f (x) for |xx0 | < R, where
R is the radius of converge of
Applications of single variable calculus: Additive functions
Positive results
Denition 1. Let f : R
R. f is said to be additive if the following holds:
x, y R
f (x + y) = f (x) + f (y).
(1)
Remark 2.
Math 217 Fall 2013 Homework 1 Solutions
This homework consists of 10 problems of 3 points each. The total is 30.
You need to fully justify your answer prove that your function indeed has the specied
p
Math 217 Fall 2013 Homework 3 Solutions
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your function indeed has the specied
pr
Math 217 Fall 2013 Homework 4 Solutions
Due Thursday Oct. 10, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your fun
Denitions and Properties of RN
RN as a set
As a set Rn is simply the set of all ordered n-tuples (x1, , xN ), called vectors. We usually denote the
vector (x1, , xN ), (y1, , yN ), by x, y,
or R , R ,
Math 217 Fall 2013 Midterm Solutions
NAME:
ID#:
There are four questions. Last two pages are scrap paper.
Please write clearly and show enough work.
Question 1. (5 pts) Let A
7 cfw_(x, y) R O x
2
0, y
Math 217 Fall 2013 Homework 8
Due Thursday Nov. 14, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer for each problem.
Please read
Math 217 Fall 2013 Homework 4 Solutions
Due Thursday Oct. 10, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your fun
Math 217 Fall 2013 Homework 9 Solutions
by Due Thursday Nov. 21, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your
Math 217 Fall 2013 Homework 10
by Due Thursday Nov. 28, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your function
Math 217 Fall 2013 Homework 7 Solutions
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your function indeed has the specied
pr
Math 217 Fall 2013 Homework 6
Due Thursday Oct. 31, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your function inde
Math 217 Fall 2013 Homework 3
Due Thursday Oct. 3, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your function indee
Math 217 Fall 2013 Homework 1
Due Thursday Sept. 19, 2013 5pm
This homework consists of 10 problems of 3 points each. The total is 30.
You need to fully justify your answer prove that your function in
Math 217 Fall 2013 Homework 2
Due Thursday Sept. 26, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your function ind
Math 217 Fall 2013 Homework 2 Solutions
Due Thursday Sept. 26, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your fu
Math 217 Fall 2013 Homework 5 Solutions
Due Thursday Oct. 17, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your fun
Riemann integration
Partition and integration of step functions
Integration theory is developed through the eort of making the ideas of area volume exact.
Denition 1. (Partition) Let a, b R, a < b. A
Math 217 Final Review1
Materials covered: Sec. 1.1-2.4, 3.1-3.2, 3.4-3.6, 7.1, 8.1-8.2
Some sample problems:
(1) Show that the sums of two open sets in Rn is open. Here the
sum S1 +S2 is dened to be S
Math 217 Midterm Solution1
3 + 2 is not a rational number.
Proof. Suppose that r = 3 3 + 2 is rational. Then
3 = (r 2)3 = r3 3r2 2 + 6r 2 2 = (r3 + 6r) (3r2 + 2) 2.
(1) (25 points) Show that
3
Since 3
Solutions for Math 217 Assignment #1
(1) Find sup(S) and inf(S) for
S=
1
1
+ : m, n Z+ .
m n
Solution. For all m, n Z+ , m, n 1 and hence
1
1
1 1
+ + 2
m n
1 1
So sup(S) 2. And since 2 is achieved whe
Solutions for Math 217 Assignment #9
(1) Show that if the limit
L = lim
n
exists, limn
n
an+1
an
|an | exists and
lim
n
n
|an | = L.
Use this to conclude that the radius R of convergence of the
power
Solutions for Math 217 Assignment #5
(1) Show that the union of nitely many compact sets in Rn is
compact.
Proof. Let
m
S=
Sk
k=1
where Sk are compact sets for k = 1, 2, ., m. Let iI Ui S
be an open c
Solutions for Math 217 Assignment #3
(1) Which of the following sets in Rn are open? Which are closed?
Which are neither open nor closed?
(a) cfw_(x, y) R2 : x2 y 2 = 1.
(b) cfw_(x, y, z) R3 : 0 < x +
Solutions for Math 217 Assignment #7
n=1
(1) Let an R+ for all n Z+ . If
converges for all b 1.
an converges, then
n=1
ab
n
Proof. Since
n=1 an converges, limn an = 0. Therefore,
there is N such that
Math 217 Fall 2013 Homework 10 Solutions
by Due Thursday Nov. 28, 2013 5pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answer prove that your