Solutions for Math 317 Assignment #7
(1) Consider the following equation
x2 + y + sin(xy) = 0.
(a) Prove that this equation has a unique continuous solution
y = y(x) such that y(0) = 0 in a small neighborhood of
(0, 0).
(b) Discuss the monotonicity of the
Solutions for Math 317 Assignment #5
(1) Find the volume of the set
B = cfw_x2 + y 2 + z 2 4, 3 x + 2y + 2z 3 R3 .
Solution. Let f : R3 R3 be a linear map given by
f1 (u, v, w)
u
f2 (u, v, w) = A v
f (u, v, w) =
f3 (u, v, w)
w
for some orthogonal matri
Math 317 Assignment #9
Due Apr 4, 2011
(1) Let fn (r) be the content of the n-dimension ball of radius r,
i.e.,
fn (r) = (Br (0) = (cfw_x2 + x2 + . + x2 r2 ).
1
2
n
(a) Show that
r
fn+1 (r) =
fn ( r2 x2 )dx
r
+
for all n Z and r 0.
(b) Show that fn (r) =
Solutions for Math 317 Assignment #3
(1) Compute the following integrals:
/3
tan2 xdx;
(a)
0
ex sin xdx;
(b)
0
(c)
0
1
dx
.
1 + x3
Solution. (a)
/3
/3
tan2 xdx =
0
(sec2 x 1)dx
0
/3
= (tan x x)
=
3
0
3
(b) Integrating by parts twice, we obtain
sin xdex =
Solutions for Math 317 Assignment #1
(1) Let S1 Rm and S2 Rn be two bounded sets. If either
(S1 ) = 0 or (S2 ) = 0, then (S1 S2 ) = 0 in Rm+n .
Proof. WLOG, we assume that (S1 ) = 0. So for all r > 0,
there exists compact intervals I1 , I2 , ., Il such th
PROOF OF CHANGE OF VARIABLES
We will give a proof of the following theorem.
Theorem 0.1. Let U Rn be an open set, C 1 (U, Rn ) and K U
be a compact set with content. Suppose that is injective and det J =
0 on K\Z for some Z K of content zero. Then (K) has
Math 317 Winter 2014 Homework 3 Solutions
Due Feb. 5 2p
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
Question 1. Calculate the radius of convergence of the power series
6
3
+ 2
n n
n=1
xn.
Math 317 Winter 2014 Homework 1 Solutions
Due Wednesday Jan. 15, 2014 2pm
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
Question 1. Are the following series convergent or divergent? Justify
Math 317 Winter 2014 Homework 2 Solutions
Due Jan. 29 2p
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
Question 1.
a) Prove the root test for
n=1
limsup |an |1/n < 1
an:
limsup|an |1/n > 1
Math 317 Winter 2014 Homework 4 Solutions
Due Feb. 26 2p
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
1 x>0
Question 1. Calculate the Fourier expansion of the function f (x) = 0 x = 0 on
ID:
Name:
Math 317 Quiz 2 Solutions
Jan. 20, 2014
The quiz has three problems. Total 10 + 1 points. It should be completed
in 20 minutes.
Question 1. (5 pts) Prove the convergence of the series
n=1
(n!)3
.
(3 n)!
(1)
Proof. We have
Thus
(n + 1)3
an+1
(n +
ID:
Name:
Math 317 Quiz 3 Solutions
Feb. 3, 2014
The quiz has three problems. Total 10 + 1 points. It should be completed
in 20 minutes.
Question 1. (5 pts) Prove by denition that
n x3 + 5
3 n2 + 7
fn(x) =
(1)
converges uniformly to 0 on [0, 1].
Solution.
ID:
Name:
Math 317 Quiz 1 Solutions
Jan. 8, 2014
The quiz has three problems. Total 10 + 1 points. It should be completed
in 20 minutes.
Question 1. (5 pts) Prove by denition or Cauchy criterion that
converges and nd its value.
n=1
2
3
n
Proof.
By denitio
Math 317 Winter 2014 Homework 5 Solutions
Due Mar. 12 2p
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
Question 1. Let x 2 (0; 1). Recall that it has decimal representation x = 0:a1a2a3:; a
Name:
ID:
Math 317 Quiz 4 Solutions
Mar. 3, 2014
The quiz has three problems. Total 10 + 1 points. It should be completed
in 20 minutes.
n
Question 1. (5 pts) Find a power series
n=1 an (x x0) such that
n an = 0 and its radius of convergence = (limsupn |a
Name:
ID:
Math 317 Quiz 5 Solutions
Mar. 17, 2014
The quiz has three problems. Total 10 + 1 points. It should be completed
in 20 minutes.
Question 1. (5 pts) Let A be the set of all nite rectangles in R2. Find its
cardinality. Justify your answer.
Solutio
Math 317 Winter 2014 Homework 6 Solutions
Due Mar. 26 2p
This homework consists of 6 problems of 5 points each. The total is 30.
You need to fully justify your answers.
Question 1. In the following a > 0 is a constant.
a) Calculate
a cos3t
a sin3t
with L: