MAT 125B Homework 2
M.Fukuda
Please submit your answers at the discussion session on January 29 Tuesday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1
MAT 125B Homework 4
M.Fukuda
Please submit your answers at the discussion session on February 12 Tuesday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers.
Practice Problems Easy problems:
1. The unit sphere in R3 is the set S 2 of all points (x, y, z) such that x2 + y 2 + z 2 = 1. For what points (x0 , y0 , z0 ) is it possible to nd a C 1 function z(x, y) dened near (x0 , y0 ) such that z(x0 , y0 ) = z0 and
1. Let 0 < c1 < c2 . . . < ck < 1. Dene f : [0, 1] R by f (cj ) = 1 for i = 1, . . . , k and f (x) = 0 for x = cj . Show that
1
f (x)dx
0
exists and evaluate the integral. 2. Suppose a < b. If f 3 is integrable on the interval [a, b], is f necessarily int
Real Analysis
Math 125A, Fall 2012
Sample Final Questions
1. Dene f : R R by
x3
1 + x2
Show that f is continuous on R. Is f uniformly continuous on R?
f ( x) =
2. Does there exist a dierentiable function f : R R such that f (0) = 0
but f (x) 1 for all x =
Real Analysis
Math 125A, Fall 2012
Solutions: Midterm 1
1. (a) Suppose that f : A R where A R and c R is an accumulation
point of A. State the - denition of limxc f (x).
(b) Prove from the denition that if f, g : A R and limxc f (x), limxc g (x)
exist, th
Solutions to Sample Questions
Midterm 1: Math 125A, Fall 2012
1. (a) Suppose that f : (0, 1) R is uniformly continuous on (0, 1). If (xn )
is a Cauchy sequence in (0, 1) and yn = f (xn ), prove that (yn ) is a Cauchy
sequence in R.
(b) Give a counter-exam
Real Analysis
Math 125A, Fall 2012
Solutions: Midterm 2
1. Suppose that f : (a, b) R is dierentiable at c (a, b) and f (c) > 0.
(a) Prove that there exists > 0 such that f (x) > f (c) for all c < x < c +
and f (x) < f (c) for all c < x < c.
(b) Does f ha
Real Analysis
Math 125A, Fall 2012
Final Solutions
1. (a) Suppose that f : [0, 1] R is continuous on the closed, bounded
interval [0, 1] and f (x) > 0 for every 0 x 1. Prove that the reciprocal
function 1/f : [0, 1] R is bounded on [0, 1].
(b) Does this r
1. Suppose F is a 1 1 continuously dierentiable mapping from unit disc in R2 into the unit circle in R2 . Use the Change of Variables Theorem to show that det DF (x) = 0 for all x R2 . 2. Use the Change of Variables Theorem to compute the volume of the un
DEFINITIONS
PARTITIONS
Let a, b R with a < b.
(i)
A partition of the interval [a, b] is a set of points P = cfw_ x 0 , , x n such that
a = x 0< < x n < = b.
(ii)
The norm of a partition P = P = cfw_ x 0 , , x n is the number
x jx j 1
|P|=max
1 j n
(i
Final Review
Here is a review for the final exam. I suggest writing out all the definitions, theorems, and
computations freshly.
Definitions: Be able to define the following, verify a function or set satisfies the
definition, and compute the associated ob
Name:
Student ID Number:
Midterm Exam
Math 125B, Spring 2016
Please answer all questions in the space provided. Write neatly and clearly. If you collaborate
with any other students or use an electronic device during the exam, you will receive a zero.
Ques
Sample Questions for Midterm 2: Solutions
Math 125A, Fall 2012
1. For R, dene f : R R by
f ( x) =
|x| sin(1/x) if x = 0,
0
if x = 0.
Determine, with proof, for what values of : (a) f is continuous at 0; (b) f
is dierentiable at 0; (c) f is continuously di
Math 125B: Winter 2013
Solution to Problem 9.6.11
As illustrated in Figure 1, we choose two constant vectors a, b R3 that
are linearly independent from (t0 ), which is possible since (t0 ) = 0. (For
example, we can use the normal and binormal vectors to t
MAT 125B Homework 5
M.Fukuda
Please submit your answers at the discussion session on March 3 Tuesday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1. F
MAT 125B Homework 1
M.Fukuda
Please submit your answers at the discussion session on January 22 Tuesday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1
MAT 125B Homework 3
M.Fukuda
Please submit your answers at the discussion session on February 5 Tuesday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1
DEPARTMENT OF MATHEMATICS
SYLLABUS
Course # & Name:
MAT 125B: Real Analysis
Recommended Text(s) & Price:
Prepared by:
R. Vershynin, D.
Fuchs, A. Krener, A.
Thompson, B. Temple
(Updated by Eric
Rains)
Lecture(s)
2
2
2
2
2
1
2
1
2
2
2
2
2
2
2
Sections
Chapt
Midterm 2: Sample questions
Math 125B: Winter 2013
1. Suppose that f : R3 R2 is dened by
f (x, y, z ) = x2 + yz, sin(xyz ) + z .
(a) Why is f dierentiable on R3 ? Compute the Jacobian matrix of f at
(x, y, z ) = (1, 0, 1).
(b) Are there any directions in