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 statem
Some Homework Solutions 9
Matthew Reed
Problem 13.2.3 Let (x1 , , xm ) be any finite sequence of points in a metric space
(X, d). Show that
d(x1 , xm )
m1
X
d(xi , xi+1 )
i=1
Solution: I will give an
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
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 electroni
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
MATH 125B Section A Advanced Calculus/Analysis, Spring Quarter 2016
Lecture: MWF 900-950pm Wellman 202
Discussion Sections: T 310-4 Wellman 115, T410-5 Wellman 115
Professor: Michael Bishop
Office: Ma
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_
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
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).
Solut
Some Homework Solutions 8
Matthew Reed
Problem 12.6.8 Note there were typos in this problem, the correct standard formulas
relating spherical to rectangular coordinates are
x = sin()cos(), y = sin()si
Some Hw Solutions 6
Matthew Reed
Problem 12.4.14 Consider the function below:
8
0,
if |y| x2
>
>
<
x,
if y = 0
f (x, y) =
1
2 ),
(y
x
if 0 < y < x2
>
>
: 1x
2
if
x2 < y < 0
x (y + x ),
(a) Verify anal
Some Hw Solutions 3
Matthew Reed
Extra Problem
Let f be a Riemann integrable function on the interval [a, b].
(a) Consider the function g(x) = sin(f (x). I claim that g is Riemann integrable on
[a, b]
Some Hw Solutions 7
Matthew Reed
Problem 12.6.1 Show that the equation f (x, y) = x2 y 2 + 2exy 4 2e2 = 0 can be
solved for y in terms of x in a neighborhood of the point x = 1 with y(1) = 2. Calculat
Solutions to some Hw Problems 1
Matthew Reed
January 23, 2013
Problem 8.2.4 Consider some continuous function f 0 on the interval [a, b]. Since
f is continuous we know that its Riemann integral over [
Solutions to Some Homework Problems 5
Matthew Reed
Problem 12.3.2 Calculate F 0 (y) for each of the following functions F :
R1
2 2
(a) F (y) = 0 ex y dx
Solution: The first thing to notice is that the
Some Hw Solutions 2
Matthew Reed
Problem 8.4.4
Let f and g be continuous on (a, b] and such that |f (x)| |g(x)| for all a < x b. If
Rb
Rb
the integral a g(x)dx is absolutely convergent, show that so a
Some Hw Solutions 4
Matthew Reed
Problem 11.2.8 The identity
|x + y|2 + |x y|2 = 2(|x|2 + |y|2 )
is known as the parallelogram law.
(a) Prove the identity is valid for all x, y Rn .
Solution: Recall t
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 hyperbo
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
A
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 an
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 Chan
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,
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
MAT 125B Final Exam
Last Name (PRINT):
First Name (PRINT):
Student ID #:
Instructions:
1. Do not open your test until you are told to begin.
2. Use a pen to print your name in the spaces above.
3. No