CORNELL UNIVERSITY
STSCI/ILRST 4550 & ORIE 5550
Applied Time Series Analysis
Spring 2014: Prof. David S. Matteson
Assignment #10: Extra Credit
This assignment is extra credit and due at 3:45PM on Friday, May 9, 2014. Late homework
is not accepted, and ho
INTRODUCTION
TO REAL ANALYSIS
William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus
Department of Mathematics
Trinity University
San Antonio, Texas, USA
wtrench@trinity.edu
This book has been judged to meet the evaluation criteria set by
the
Chapter 3 Continuous Functions
Continuity is a very important concept in analysis. The tool that we shall use to study
continuity will be sequences. There are important results concerning the subsets of
the real numbers and the continuity of the function:
4/19/2014
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Science & Mathematics > Mathematics
Math 3110 Homework 14 Solutions
Exercise 15.1.3
If f (x) changes sign from negative to positive on [a, b], then there are some h, k 2 [a, b] with
h < k, f (h) < 0, and f (k) > 0. Since f (x) is dierentiable on [a, b] and [h, k] [a, b],
f (x) is dierentiab
HW 5 Hints and Solutions
13.1.1 (a) Suppose S = [a1 , b1 ] [a2 , b2 ] [an , bn ]. If xn is an innite sequence of points in S, then one of the intervals [ai , bi ] must contain innitely
many of the xi . Let yj be an innite sub-sequence of xj such that yj i
Math 101: Selected solutions, Homeworks 6-8
Hw 6: Problem 1b
Suppose g : R ! R equals 1 on all rational points and g is continuous. Towards contradiction, suppose g(x) 6= 1 for some x 2 R. Then we may let = |g(x) 1| > 0 and nd such
that for any x0 2 (x
,
HW 8 SOLUTIONS
The Way of Analysis
p. 176:
1.) Dene x+ = mincfw_x, 0. Show that f (x) = xk is C 1 if k is an integer greater than one:
+
The only issue is what happens at 0. (For x > 0, the function is just x, which we know is
C 1 . For x < 0, the functio
The Unit Step Functions and Laplace Transforms
The Unit Step Function, U, is like a switch that turns a function on or off over a specified
interval. By definition,
U =
0 0<
1 0
For example, consider the product U. The function is turned off when 0 < and
Math 3110 Homework 11 Solutions
Exercise 12.1.1
Let f be a continuous function whose values are always rational numbers. Assume for
contradiction that f is not a constant function. Then there exist x1 , x2 with x1 < x2 such
that f (x1 ) 6= f (x2 ). Withou
Mathematics 3100 - Homework V
Wednesday, March 3, 2004
1. Prove the Product theorem: an ! L, an ! M =) an bn ! LM by following through
the ideas given in the book.
Proof.
|an bn
LM | = |(an
L)bn + L(bn
M )| |an
L|bn | + |L|bn
M |.
In Problem 3.4, we have
Taylor series Homework set solutions.
17.1.1
Let f (x) = (1 + x)r . Then, by the usual dierentiation formulas, we have that
f (k) (x) = r(r 1) (r k + 1)(1 + x)rk .
Therefore,
n
Tn (x) =
k=0
Observation 1
f (k) (0) k
x =
k!
n
k=0
r(r 1) (r k + 1) k
x .
k!
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Chapter
3
TRIGONOMETRIC FUNCTIONS
A mathematician knows how to solve a problem,
he can not solve it. MILNE
3.1 Introduction
The word trigonometry is derived from the Greek words
trigon and metron and it means measuring the sides of
a triangle. The subjec
Math 3110 Homework 13 Solutions
Exercise 14.1.5
= . If |x| < , then
1
|f (x) f (0)| = x sin
|x| <
x
Let > 0 be given, and let
= .
This shows that f is continuous at 0.
In order for f (x) to be dierentiable at 0, we would need for
f (x)
f (0) = lim
x!0
Solutions to homework 12
14.1.6
By construction
f (x) f (a)
e(x)
=
f (a).
xa
xa
By the limit theorems (addition)
e(x)
=
xa x a
lim
f (x) f (a)
xa
xa
lim
f (a) = f (a) f (a) = 0.
L(x) is the tangent line to f (x) at x = a, e(x) is the dierence between f
Solutions to homework 12
14.1.6
By construction
f (x) f (a)
e(x)
=
f (a).
xa
xa
By the limit theorems (addition)
e(x)
=
xa x a
lim
f (x) f (a)
xa
xa
lim
f (a) = f (a) f (a) = 0.
L(x) is the tangent line to f (x) at x = a, e(x) is the dierence between f
HW 6 Hints and Solutions
15.2.1 Let a be any point in the interval I. Let b be any other. Then by
using the Mean Value Theorem, f (b)f (a) = f (c) for some c between a and
ba
b. Since |f (c)| M this means |f (b) f (a)| M|b a| for all b in the
interval I.
Math 3110
Spring 09
Homework 9:
Selected Solutions
[12-3]: Claim: Let f : [a, a] ! R be continuous, with f (0) > f (a), f (a). In this case there
exist x1 , x2 2 [a, a] such that a horizontal chord of length exactly a joins (x1 , f (x1 ) to (x2 , f (x2 ).
Math 3110 Homework 12 Solutions
Exercise 13.1.1
(a) At least one of the intervals must contain innitely many terms of the sequence, for
otherwise, the sequence would only have nitely many terms in each of nitely many intervals,
and hence only nitely many
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Functions: Even and Odd Functions and Composition
Definitions
Even Function A function is even if
for all .
The graph of an even function is symmetric about -axis. This means that the graph for negative
is the mirror image of that for positive .
Example
CHAPTER II
THE LIMIT OF A SEQUENCE OF NUMBERS
DEFINITION OF THE NUMBER e.
This chapter contains the beginnings of the most important, and probably the
most subtle, notion in mathematical analysis, i.e., the concept of a limit. Though
Newton and Leibniz di
CHAPTER II
THE LIMIT OF A SEQUENCE OF NUMBERS
DEFINITION OF THE NUMBER e.
This chapter contains the beginnings of the most important, and probably the
most subtle, notion in mathematical analysis, i.e., the concept of a limit. Though
Newton and Leibniz di
Mathematics 3100 - Homework VI Wednesday, March 26, 2004 1. Given a set of nested intervals [an , bn ], n = 0, 1, 2, , for which bn an 0, prove the sequence a0 , b0 , a1 , b1 , a2 , b2 , , an , bn , converges. Proof. Let the sequence be cfw_cn . Then we c