Homework for Math 4318
Due March 29 (hand in class)
If you have questions on the homework, feel free to come to office hours.
You can discuss the homework with other students, but each one of you have to write
it separately.
In case that you use reference
6
Let bk = ak+m, for positive integers k and m. We need to prove that 22:1 bk if and only
if 22:1 ah converges.
Let 22:1 bk converge. This means that for some S E R, given any 6 > 0 there exists
N > 0 such that for any n > N we have
:bkS'
k1=l
<6.
No
(a)
M) ={ 8(5) ::8
i is a differentiable function on R\{0}, and sin(x) is a differentiable function on R.
Hence, sin(%) is a differentiable function on R\{O} since it is a composite function of two
differentiable functions. x is a differentiable functi
Math 4318 : Real Analysis II
Mid-Term Exam 2
28 March 2013
Name:
Denitions:
True/False:
Proofs:
1.
2.
3.
4.
5.
6.
Total:
Denitions and Statements of Theorems
1. (3 points) For a sequence of real numbers cfw_an , state the denition for the series
to conver
Math 4318 : Real Analysis II
Mid-Term Exam 1
14 February 2013
Name:
Denitions:
True/False:
Proofs:
1.
2.
3.
4.
5.
6.
Total:
Denitions and Statements of Theorems
1. (2 points) For a function f (x) dened on (a, b) and for x0 (a, b) state the denition
of f (
Math 4318 : Real Analysis II
Group Problem Set 1
January 22, 2013
1. Suppose that f is dened in a neighborhood of x and that f (x) exists. Show that
f (x + h) + f (x h) 2f (x)
= f (x).
h0
h2
lim
Give an example of a function such that the limit exists eve
Math 4318 : Real Analysis II
Mid-Term Exam 1
14 February 2013
Name:
Denitions:
True/False:
Proofs:
1.
2.
3.
4.
5.
6.
Total:
Denitions and Statements of Theorems
1. (2 points) For a function f (x) dened on (a, b) and for x0 (a, b) state the denition
of f (
Final Exam Review
by Laura Starkston (lstarkst@fas.harvard.edu)
1
One dimensional Integrals
Definition 1. A partition, P of an interval [a, b] is an ordered set of points: a = x0 < x1 < <
xn = b.
Definition 2. Given a partition, P , for each 1 i n define
Math 4318 : Real Analysis II
Mid-Term Exam 1
14 February 2013
Name:
Definitions:
True/False:
Proofs:
1.
2.
3.
4.
5.
6.
Total:
Definitions and Statements of Theorems
1. (2 points) For a function f (x) defined on (a, b) and for x0 2 (a, b) state the definit
Homework for Math 4318
Due Feb 9 (hand in class)
If you have questions on the homework, feel free to come to office hours.
You can discuss the homework with other students, but each one of you have to write
it separately.
It is important that you write le
Homework for Math 4318
Due Feb 16 (hand in class)
If you have questions on the homework, feel free to come to office hours.
You can discuss the homework with other students, but each one of you have to write
it separately.
It is important that you write l
Homework for Math 4318
Due March 14 (hand in class)
If you have questions on the homework, feel free to come to office hours.
You can discuss the homework with other students, but each one of you have to write
it separately.
It is important that you write
Homework for Math 4318
Due April 26 (hand in class)
If you have questions on the homework, feel free to come to office hours.
You can discuss the homework with other students, but each one of you have to write
it separately.
It is important that you write
Homework for Math 4318
Due March 3 (hand in class)
If you have questions on the homework, feel free to come to office hours.
You can discuss the homework with other students, but each one of you have to write
it separately.
It is important that you write
Math 4318 : Real Analysis II
Mid-Term Exam 1
14 February 2013
Name:
Denitions:
True/False:
Proofs:
1.
2.
3.
4.
5.
6.
Total:
Denitions and Statements of Theorems
1. (2 points) For a function f (x) dened on (a, b) and for x0 (a, b) state the denition
of f (
Math 4318 : Real Analysis II
Mid-Term Exam 2
28 March 2013
Name:
Denitions:
True/False:
Proofs:
1.
2.
3.
4.
5.
6.
Total:
Denitions and Statements of Theorems
1. (3 points) For a sequence of real numbers cfw_an , state the denition for the series
to conver
Math 4317
Practice Final Exam
Fall 2010
1) Show that ex 1 + x for all x R and that equality holds if and only if x = 0.
2) Let f : [0, 1] R be an integrable function. Suppose that for every 0 a < b 1 there
1
is a point c [a, b] such that f (c) = 0. Show t
Math 4318
Midterm Exam 1
Solutions
Spring 2011
1) For a xed c [a, b] dene the function
c : C 0 ([a, b]) R : f f (c).
If we give C 0 ([a, b]) the usual sup-norm
Solution: Notice that
,
then show that c is continuous.
|c (f ) c (g )| = |f (c) g (c)| f g
So
Math 4318
Midterm Exam 1 Solutions
Spring 2011
1) Let f : [a, b] R be Riemann integrable and set
x
F (x) =
f (t) dt.
a
Prove that F is Lipschitz.
Solution: Since f is Riemann integrable it is bounded, that is there is some M such
that |f (x)| M for all x
Math 4318 - Spring 2011
Homework 6
Work all these problems and talk to me if you have any questions on them, but carefully write
up and turn in only problems 1, 4, 7, 8, 9, 10, 11, 12. Due: In class on April 21
1. A function f : Rn Rm is called homogeneou
Math 4318 - Spring 2011
Homework 5
Work all these problems and talk to me if you have any questions on them, but carefully write
up and turn in only problems 2, 3, 4, 11, 12, 13, 16, 17. Due: In class on March 31
2. For b > 0 and any a dene
x
f (y )exy dy
Math 4318 - Spring 2011
Homework 4
Work all these problems and talk to me if you have any questions on them, but carefully write
up and turn in only problems 2, 4, 6, 11, 12, 13, 14, 15. Due: In class on March 10
2. Let
x
1 + nx2
be a sequence of function
Math 4318 - Spring 2011
Homework 3
Work all these problems and talk to me if you have any questions on them, but carefully write
up and turn in only problems 1, 2, 4, 5, 7, 8, 9, 10. Due: In class on February 24
1. Let
f (x) =
x + 2x2 sin(1/x) x = 0
0
x =
Math 4318 - Spring 2011
Homework 2
Work all these problems and talk to me if you have any questions on them, but carefully write
up and turn in only problems 1, 2, 3, 5, 6, 7, 8, 14. Due: In class on February 10
3
1. Compute 1 x2 dx using the denition of