Math 142B
Final Examination
September 2, 2011
Instructions
1. You may use any type of calculator, but no other electronic devices during this exam.
2. You may use one page of notes, but no books or other assistance during this exam.
3. Write your Name, PI
Midterm 1
Math 142A, Lecture C
Fall 2016
Time allowed: 50mins
(40 points total)
Name:
PID:
1
1. (3 points) State the Completeness Axiom for R.
Every nonempty subset S R which is bounded above has a
least upper bound.
2. Carefully state the following defin
Math 142B
Midterm Exam 1
August 11, 2011
Instructions
1. You may use any type of calculator, but no other electronic devices during this exam.
2. You may use one page of notes, but no books or other assistance during this exam.
3. Write your Name, PID, an
Math 142B
Midterm Exam 2
August 25, 2011
Instructions
1. You may use any type of calculator, but no other electronic devices during this exam.
2. You may use one page of notes, but no books or other assistance during this exam.
3. Write your Name, PID, an
Math 142B
Midterm Exam 2 Solution
1. Let f : R R be continuous. Dene
x
(x t) f (t) dt for all x.
G(x) =
0
Use the Second Fundamental Theorem to show that G (x) = f (x) for all x. (Hint:
Use the linearity property of the integral to rewrite it in a more co
Math 142B
Summer 2011 Midterm Exam 1 Solution
1. Let f : [0, 1] R be dened by
f (x) =
1
x if x = n for some n N,
0 otherwise.
Show the f is integrable on [0, 1] and determine the value of
1
0
f.
1
1
Given a natural number n, f (x) = 0 for each x in ( k ,
MATH 142B
Homework 2 Solutions
Jonathan Conder
Section 6.2
6. (a) Given n N, let Pn be the regular partition of [a, b] into n intervals. Dene f : R R by f (x) = x. Now
n
U (f, Pn ) =
a+i
i=1
ba
=
n
n
i=1
ba
n
ba
n
(b a)2
a+
n2
n
i
i=1
(b a)2 n(n + 1)
n2
2
MATH 142B
Homework 3 Solutions
Jonathan Conder
Section 6.4
1. (a) This is false. For example f could be nonzero at a single point.
(b) This is false, with the same counterexample as part (a).
(c) This is true, by monotonicity.
(d) This is false. For examp