Final Exam - Math 139
Dec 16th
Instructor
Web page
Mauro Maggioni
www.math.duke.edu/ mauro/teaching.html
You have 3 hours. You may not use books, internet, notes, calculators. The exam should
be stapled, written legibly, with your name written at the top
Math 139.2 Spring 2012: Quiz # 1 Solutions
Instructor: Dr. Herzog
January 19th, 2012
1. Prove that all real numbers x and y satisfy
|x| |y | |x y |.
Hint: apply the triangle inequality to x = (x y ) + y and then reverse the
roles of x and y .
Proof. Note
Math 139.2 Spring 2012: Quiz # 2 Solutions
Instructor: Dr. Herzog
January 24th, 2012
1. Suppose that an a and that an b for all n. Prove that a b.
Proof. Suppose that, by way of contradiction, a < b. Since an a and
b a > 0, there exists N > 0 such that
|a
Math 139.2 Spring 2012: Quiz # 3 Solutions
Instructor: Dr. Herzog
January 31st, 2012
1. Suppose that cfw_an is a Cauchy sequence. Prove that cfw_a2 is Cauchy.
n
Show that the converse is not true by providing a counterexample.
Proof. By the Axiom of Com
Math 139.2 Spring 2012: Quiz # 4 Solutions
Instructor: Dr. Herzog
February 7th, 2012
1. Let cfw_dn be a sequence of limit points of the sequence cfw_an . Suppose
that dn d. Prove that d is a limit point of cfw_an .
Proof. Let > 0 and N N. We must show th
Math 139.2 Spring 2012: Quiz # 5 Solutions
Instructor: Dr. Herzog
February 14th, 2012
1. Let f be the function on [0, 1] given by
f ( x) =
0
1
if x is rational
if x is irrational.
Show that f is NOT Riemann integrable.
Proof. Let P = cfw_x0 , x1 , . . . ,
MATH 139.2 SPRING 2012: QUIZ # 6
INSTRUCTOR: DR. HERZOG
1. Let g be a twice continuously dierentiable function on [a, b].
Suppose that there are three distinct points x1 , x2 , x3 , in [a, b] such
that g (x1 ) = g (x2 ) = g (x3 ) = 0. Prove that there is
MATH 139.2 SPRING 2012: QUIZ # 7 SOLUTIONS
INSTRUCTOR: DR. HERZOG
1. For the following functions, compute how small h must be to
guarantee that the error in the left-hand endpoint Riemann sum
method is 104 :
(a)
2 2x
e
0
(b)
10
2
dx.
ln(x) dx.
Proof. (a)
MATH 139.2 SPRING 2012: QUIZ # 8 SOLUTIONS
INSTRUCTOR: DR. HERZOG
1. Let cfw_fn be a sequence of continuous functions that converges
uniformly on [0, 1]. Show that there exists an M such that
|fn (x)| M for all n and all x [0, 1].
Proof. Since fn f unifo
Math 139.2 Spring 2012: Quiz # 9
Instructor: Dr. Herzog
March 27th, 2012
1. Consider the integral equation:
1
xy (y ) dy.
( x) = 1 +
(0.1)
0
Suppose that | < 3 and 0 (x) = 1. Find . Show that there is no solution
if = 3. Hint: notice that any solution m
Math 139.2 Spring 2012: Quiz # 10 Solutions
Instructor: Dr. Herzog
April 10th, 2012
1. Suppose that xn x and yn y in a metric space (M, ). Prove that
limn (xn , yn ) = (x, y ).
Proof. First note that for all n N by the triangle inequality and symmetry
(xn
Math 139.2 Spring 2012: Quiz # 11 Solutions
Instructor: Dr. Herzog
April 17th, 2012
1. Show that the series
1
j2
sin
j =1
converges. Hint: Use the mean value theorem.
Proof. Since f (x) = sin(x) is continuously dierentiable on [0, 1], for each
j 1, we can
Math 139.2 Spring 2012: Test # 1 Solutions
Instructor: Dr. Herzog
February 21st, 2012
85 Total Points Possible
1. Suppose that an = 5
that an 5.
16
.
n
Prove (using only the denition of a limit)
Proof. Let > 0 be arbitrary and choose N 162 / 2 . Then for
Math 139.2 Spring 2012: Test # 2 Solutions
Instructor: Dr. Herzog
March 29th, 2012
80 Points Possible
1. (10 points) Suppose that f is continuously dierentiable on [a, b] and
f (x) = 0 for all x [a, b]. Prove that f is a constant function.
Proof. By the F
Note on Homework 5 and rst long assignment - Math 139
Instructor
Oce
Oce hours
Web page
Mauro Maggioni
293 Physics Bldg.
Monday 1:30pm-3:30pm.
www.math.duke.edu/ mauro/teaching.html
This is to clarify the connection between Homework 5, the rst long assign
Oce Problem 2 - Math 139
Due midnight Dec. 10th
Instructor
Oce
Web page
Mauro Maggioni
293 Physics Bldg.
www.math.duke.edu/ mauro/teaching.html
In this Oce Problem you will be constructing the Riemann integral for functions
f : Rn R with n 1.
(i) You will
Oce Problem 1 - Math 139
Due Oct 13th
Instructor
Oce
Oce hours
Web page
Mauro Maggioni
293 Physics Bldg.
Monday 1:30pm-3:30pm.
www.math.duke.edu/ mauro/teaching.html
Oce consultation: Oct. 7th, 8th, 1pm-3pm, usual oce hours on Oct. 11th. Possibly
other ti
Homework 6 - Math 139
Due Oct 13th
Instructor
Oce
Oce hours
Web page
Mauro Maggioni
293 Physics Bldg.
Monday 1:30pm-3:30pm.
www.math.duke.edu/ mauro/teaching.html
Reading: from Reeds textbook: Section 3.3
Problems: 3.1: #3,7,8,10
I suggest you pick and do
Homework 12 - Math 139
Due Thu Dec. 2nd
Instructor
Oce
Oce hours
Web page
Mauro Maggioni
293 Physics Bldg.
Monday 1:30pm-3:30pm.
www.math.duke.edu/ mauro/teaching.html
Reading: from Reeds textbook: Section 5.7, 6.1 (review), 6.2. I would suggest also
read