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 function indeed has the specied
property for each problem.
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 the Taylor series:
(a) f (x) = (1 x)2 at x = 1.
(b) f
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
0, x y
1. Prove that A is convex.
Solution. Take any (
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 , . For a vector x RN , the numbers x1, , xN are
x y
cal
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. The above functional equations is called the Cauchy equ
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 partition of the interval [a, b] is a set of points
P =
Taylor expansion
Taylor polynomial (expansion with Peano form of the remainder)
Denition 1. Let f (x) be kth dierentiable on (a, b) for k = 1, 2,
Then the polynomial
Pn(x) 7 f (x0) + f (x0) (x x0) +
f (x0)
(x x0)2 +
2
, n 1 and f (n)(x0) exists for x0 (a,
Denitions
Second order partial derivatives
f
Denition 1. Let f : RN R. If the j-th partial derivative of x : RN
i
f
a second order partial derivative for the function f at x0.
x
x
j
R exists at x0 RN, then we call
i
Remark 2. Clearly we can dene second or
Properties of Riemann integrals
Fundamental Theorem of Calculus
Theorem 1. (FTC 1st Version) Let f be integrable on [a, b]. If F is continuous on [a, b] and is an
antiderivative of f, that is F = f, on (a, b), then
b
f (x) dx = F (b) F (a).
(1)
a
Exercise
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 this weeks lecture notes before working on the problem
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 function indeed has the specied
property for each problem.
Math 217 Fall 2013 Homework 5
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
property for each problem.
Please read this weeks lecture notes bef
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 indeed has the specied
property for each problem.
Please r
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 indeed has the specied
property for each problem.
Questio
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 indeed has the specied
property for each problem.
Please rea
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 indeed has the specied
property for each problem.
Please re
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
property for each problem.
Please read this weeks lecture
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 indeed has the specied
property for each problem.
Pleas
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 function indeed has the specied
property for each probl
Motivation
Consider the equation for the unit circle S:
x2 + y 2 = 1.
(1)
We see that if we consider y > 0, we can write y as a function of x:
y=
1 x2 .
(2)
Similarly, for y < 0, we can write
y = 1 x2 .
(3)
Summarizing, given any (x0, y0) S with y0 0, the
Matrix representation of Df (x0), Partial derivatives
In this section we study the matrix representation of Df (x0).
Jacobian matrix
Denition 1. Let f : RN RM be dierentiable at x0 RN. Then the matrix representation of its derivative
Df (x0) is called the
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 when m = n = 1,
sup(S) = 2.
We claim that inf(S) = 0. Firs
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 series
an xn is 1/L if L = limn |an+1 |/|an | exists.
H
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 cover of S. Then
Sk
Ui
iI
and iI Ui is an open cover of
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 + y + z < 1.
(c) cfw_(x, y) R2 : x + y Q.
(d) cfw_(x, y)
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 an < 1 for n > N . And since b 1,
ab an when n > N . Th
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 function indeed has the specied
property for each prob
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 function indeed has the specied
property for each probl
Math 217 Fall 2013 Homework 8 Solutions
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 this weeks lecture notes before working on t
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 S1 +S2 = cfw_v1 +v2 : v1 S1 , v2 S2
and we take S1 + S2