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
HOMEWORK #5
Solutions
1. (a) Write the equation of the plane tangent to the hyperbolic paraboloid z =
x2 y 2 at the point (0, 0, 0).
(b) What is the intersection of this tangent plane with the hyperboloid (provide a
geometric description of this intersect
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
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
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
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
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
HOMEWORK #4
Solutions
1. (8.2.2 (b). Find an equation of a 3-dimensional plane in R4 which contains lines
(t, t, t, 1) and (1, t, 1 + t, t). (Explanation. A 3-dimensional plane in R4 has an equation
A1 x1 + A2 x2 + A3 x3 + A4 x4 = B determined up to a mul
HOMEWORK #7
Solutions
1. Prove that any finite subset of Rn is a Jordan region of volume 0.
Solution. It is obvious that A finite subset E of Rn is its own boundary. Let > 0.
Take any rectangle R E and any greed G on R such that no point from E lies on th
HOMEWORK #6
Solutions
1. Write out the expression in powers of (x + 1) and (y 1) for f (x, y) = x2 + xy + y 2 .
Hint: this is the Taylor formula for f (x, y) for a = (1, 1) and p = 3 (or p > 3).
Solution. According to the Taylor formula,
f
f
(1, 1)(x + 1)
PRACTICE FINAL
1. Let a1 , a2 , . . . be a bounded sequence of real numbers (that is, m an M for all
n and some m, M ), and let b be a real number. Prove that the function f : [0, 1] R,
f (x) = an for 2n < x 21n , n = 1, 2, . . .
f (0) = b
is Riemann inte
MIDTERM #2
Solutions
p
Problem 1. Is the function f (x, y) = x4 + y 4 differentiable at the point (0, 0)?
Justify your answer.
Solution. For (x, y) 6= (0, 0),
f
2x3
f
2y 3
(x, y) = p
,
(x, y) = p
.
x
x4 + y 4 y
x4 + y 4
p
f
f
Since f (x, 0) = x4 = x2 and
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
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
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
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
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