Homework 1 Solutions
Theorem
Suppose first that fn uniformly converge to f on A. The definition says that for any > 0, there is
some N 2 N so that if n > N and x 2 A, |f (x) fn (x)| < . Let us choose
Math 163
Spring 2017
Claudia Richoux
Homework 3
Problem 1: 26-8. Prove that if f is differentiable at z, then f is continuous at z.
Proof. Since f is differentiable at z, we know that f 0 (z) exists.
Math 163
Spring 2017
Claudia Richoux
Homework 4
Problem 1.
a. Prove that C([0, 1]), the space of continuous functions on [0, 1] is a vector space under the
usual function addition and scalar multiplic
Math 163
Spring 2017
Claudia Richoux
Homework 6
Problem 1. 2-4. Let g be a continuous real-valued function on the unit circle cfw_x R2 :
|x| = 1 such that g(0, 1) = g(1, 0) = 0 and g(x) = g(x). Define
Math 163
Spring 2017
Claudia Richoux
Homework 1
Problem 24-2. For each of the following sequences cfw_fn , determine the pointwise limit of
cfw_fn (if it exists) on the indicated interval, and decide
MATH 163, SHEET 13: THE EUCLIDEAN SPACE Rn
For the next three sheets we will be studying multivariable calculus, which is essentially
calculus on Rn . First we need to understand the space Rn .
Defini
SHEET 10: UNIFORM CONTINUITY AND INTEGRATION
We will now consider a notion of continuity that is stronger than ordinary continuity.
Definition 10.1. Let f : A R be a function. We say that f is uniform
SHEET 12: SERIES
In this sheet we use many results about limits of sequences, though we will not prove each
of them explicitly. We remark that most theorems about limits of functions have completely
a
SHEET 11: THE FUNDAMENTAL THEOREM OF CALCULUS AND
INVERSE FUNCTIONS
Lemma 11.1. Let f : [a, b] R be continuous at p (a, b). Define functions m and M
by:
(
infcfw_f (x) | p x p + h if h 0,
m(h) =
infcf
Math 163
Spring 2017
Claudia Richoux
Homework 7
Problem 1. 1-30.P
Let f : [a, b] R be an increasing function. If x1 , , xn [a, b] are
distinct, show that ni=1 o(f, xi ) < f (b) f (a).
Proof. We can su
Math 163
Spring 2017
Claudia Richoux
Homework 2
P
n
Problem 1: 24-18. Suppose that f (x) =
n=0 an x converges for some x0 , and that a0 6=
0; for simplicity, well assume that a0 = 1. Let bn be the se
Math 163
Spring 2017
Claudia Richoux
Homework 5
Problem 1: 1-16. Find the interior, exterior, and boundary of the following sets
cfw_x Rn : |x| 1.
Interior. cfw_x Rn : |x| < 1
Exterior. cfw_x Rn : |x|
Homework 2 Solutions
Extra Problem 1 In order to find the Taylor polynomial for sech, well need to find the derivatives of sech and
1
tanh in terms of each other. For example, since sech = cosh
and kn
Homework 7 Solutions
Problem 2 If you tried to do this question directly, I have very bad news for you. Unfortunately, unless you accept
some sort of conditions on your derivatives like continuity, yo
Homework 6 Solutions
Problem 2 Suppose we have A Rn compact and B A infinite. Suppose further that there is no point of A
that is a limit point of B. What this means is that for each point x of A, sin
Homework 4 Solutions
Problem 4 When I read this question, I assumed that Dr. Boller was not allowing use of the Cauchy-Schwarz
Inequality, because whats the use of doing the case of R2 by hand if your
Spring Quarter 2017
Math 16300/31: Honors Calculus III
John Boller
Homework 7, Version 2
Due: Wednesday, May 31, 2017
1. Read Spivaks Calculus on Manifolds, Chapters 2 and 3, especially pages 1533 and
Spring Quarter 2017
Math 16300/31: Honors Calculus III
John Boller
Homework 5, Version 3
Due: Friday, May 5, 2017
1. Read Spivaks Calculus on Manifolds, Chapter 1, especially pages 513.
2. (*) Let cfw
Spring Quarter 2017
Math 16300/31: Honors Calculus III
John Boller
Homework 6, Version 2
Due: Friday, May 12, 2017
1. Read Spivaks Calculus on Manifolds, Chapter 1 and 2, especially pages 1118.
2. Let
EXAM 1 OUTLINE
1. General information
Format.
1) True/False or short answer problems. Questions about general theory, examples and counterexamples, and definitions. Review examples/counter-examples fr
University of Chicago
Math 163-41
Spring 2016
EXAM 1 SOLUTIONS
1. (9 points) True or False. Determine if the statement is true or false. Place answers in the grid on
the front page.
(a) If cfw_an is
University of Chicago
Math 163-41
Spring 2016
Basic linear algebra concepts.
1. Rn ; vectors in Rn ; vector arithmetic
2. Linear independence, spanning, standard basis for Rn
3. Definition of a linear
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SHEET 9: SEQUENCES and LIMITS
We will now work with the real numbers R instead of an arbitrary continuum C. Accordingly, let us now use the standard notation (a, b) for the region ab = cfw_x R : a < x
MATH 162, SHEET 8: THE REAL NUMBERS
This sheet is concerned with proving that the continuum R is an ordered eld. Addition
and multiplication on R are dened in terms of addition and multiplication on Q
MATH 162, SHEET 6: THE FIELD AXIOMS
We will formalize the notions of addition and multiplication in structures called elds. A
eld with a compatible order is called an ordered eld. We will see that Q a
MATH 161, SHEET 5: CONTINUOUS FUNCTIONS
Denition 5.1. Let A X C. We say that A is open in X if it is the intersection of X
with an open set, and closed in X if it is the intersection of X with a close
MATH 161, SHEET 4: CONNECTEDNESS, BOUNDEDNESS,
COMPACTNESS
At the end of script 3, we dened what it means for a topological space to be connected,
using separated sets (Denition 3.25). The following d
MATH 161, SHEET 2: INTRODUCING THE CONTINUUM
This sheet introduces the continuum C through a series of axioms.
Axiom 1. The continuum is a nonempty set C.
We often refer to elements of C as points.
De