Advanced Calculus, Supplement and Solutions
Horst R. Thieme, Arizona State University, Fall 2007. updated December 4, 2007
2
Chapter 1
The real numbers
1.1 Ordered Fields [1, Sec.11]
We use the fo
Abstract
This paper presents the FEP analysis of The Petronas Towers, Malaysia and a
PDRI analysis for the project. Various problems, their causes and implications
on the project are discussed. Suitab
23.
24.
27.
x E S) then cfw_Sn:1 converges to x and we invoke
1.17. The proof for x
similar fashion.
inf S can be handled in a
El
This is a difficult problem, simply because it involves
several steps
Math 341 Homework # 8 P104. 13, 14, 17. P105. 19, 20, 21, 22. 13. Let f : D R be continuous at x0 D. Prove that there is M > 0 and a neighborhood Q of x0 such that |f (x)| M for all x Q D. Proof: Sinc
Math 341 Homework # 5 P79. 1, 3, 5, 8. 2 1. Dene f : (2, 0) R by f (x) = x +24 . Prove that f has a limit at 2, x and nd it. Proof: Note that (x + 2)(x 2) = x 2. f (x) = x+2 Guess the limit will be 4.
Math 341 Homework # 11 P131. 32(b), 33, 35, 37. P165. 3, 5. P166. 8, 9. 32(b). Assume the rules for differentiating the elementary functions, and use L'Hospital's rule and find the limit
x0 ex
lim
x .
MAA 4200: Homework 7 Solutions
July 25, 2009
[4.18] Proof: Let f (x) = x3 - 3x + b = 0. Then f (x) = 3x2 - 3 < 0 for x (-1, 1). Suppose f has two roots r1 , r2 [-1, 1]. Then f satisfies the hypotheses
MAA 4200: Homework 6 Solutions
July 13, 2009
[3.35] Proof: Suppose E compact and nonempty. By definition of compact, E is bounded and so sup E and inf E exist. Suppose sup E is not in E. The proof for
MAA 4200: Homework 5 Solutions
[3.2] We must show that limx-3 f (x) = -12.
June 29, 2009
Preliminary Work: For x = -3, |f (x) + 12| = |2(x - 3) + 12| = |2x + 6| = 2|x + 3|. Proof: Let > 0. Choose = /2
MAA 4200: Homework Solutions
Sequential Limit Theorem = Theorem 2.1 Algebra of Limits Theorem = Theorem 2.4
June 19, 2009
[2.12] Prove that if limxx0 f (x) = L then limxx0 |f (x)| = |L|. Preliminary W
MAA 4200: Homework 3 Solutions
June 17, 2009
[1.34] (5 points) If an = (-1)n (1 - 1/n), then the subsequence a2k = (1 - 1/2k) 1 as k by the Algebra of Limits Theorem and the fact that 1/k 0 as k . [1.
Chapter 4.
1. Three equivalent definitions of differentiable and derivative: Let f : D R with x0 an accumulation of D, and x0 D. The following are three are equivalent definitions of differentiable: (
Chapter 3.
1. f : E R is continuous at x0 if . 2. x E is said to be an isolated point in E if there is an > 0 such that (x - , x + ) E = cfw_x. (Note that an isolated point of E is never an accumulati
Chapter 2.
1. Limit of a function:
xx0
lim f (x) = L
See page 64. Make sure to include the hypotheses and don't forget the "zero less than" in 0 < |x - x0 | and the fact that x0 must be an accumulat
MAT 371: EXAM 1 REVIEW
1. Let cfw_an be a convergent sequence of real numbers with limit A. Which of the following statements
n=1
are necessarily true?
(a) cfw_an is bounded.
n=1
(b) cfw_an is mono
MAT 371: SOLUTIONS TO FINAL REVIEW PROBLEMS
1.
(a) True. This was discussed in the Exam I review.
(b) False. For example, the constant sequence an = 0 is convergent but has range cfw_0, which is nite
44$; 329 anmerw m-l
41 SEQUENGES
This chapter can be frustrating as it is usually necessary to
cover the material in painstaking detail. But dont despair,
the extra time and effort pays off in later c
[IFT 102]
Introduction to Java Technologies
Lab 6: Exceptions
Score: 50 pts
I.
Exceptions Arent Always Errors
File CountLetters.java contains a program that reads a word from the user and prints the n
[IFT 102]
Introduction to Java Technologies
hwk 5
Score: 30 pts
I.
The Exercises below are from the required Textbook, page 481, 533,
and 534 (3 pts * 6)
Ex 9.3
Ex 9.7
Ex. 9.8
Ex 10.1
Ex 10.2
Ex. 10.3
CHEMICAL PLANT PDRI EVALUATION
FEP
By,
Group 2
Tyler Johnson
Kotresha M M J
Venkata Vamsi
Emani
Mitch Shaw
Hossein Vashani
Hariharan
CHEMICAL PLANT PROJECT
New chemical plant for Mythology Chemical
C
Homes with expensive features.
Yet, inexpensively priced.
Near Rajarajeshwari Medical College
Mysore Road-NICE Junction, Bangalore
Welcome to Provident Sunworth.
Rich in features, Sunworth is the perf
PDRI Building Projects
Project Score Sheet (Unweighted)
SECTION I - BASIS OF PROJECT DECISION
Definition Level
CATEGORY
Element
0
1
2
3
4
5
Score
A. BUSINESS STRATEGY
A1. Building Use
A2. Business Jus
Recommendations
Based on the limited information provided about the project, the following items should be
addressed for more clarification. The results will clarify the next necessary steps and proce
MAT 371: EXAM 1 REVIEW HINTS/SOLUTIONS
1.
(a) True. A convergent sequence must be bounded.
(b) False. For example, if an = (1)n /n, then cfw_an converges to 0, but is not monotone.
n=1
(c) True. In f
MAT 371: EXAM 2 REVIEW
1. Determine whether the statements below are true or false.
(a) Any function f : Z R is continuous.
(b) If S R is open, then S c is not open.
(c) If a set S R is neither open n
MAT 371: EXAM 2 REVIEW SOLUTIONS
1.
(a) True. Note that Z has no accumulation points if x0 D is not an accumulation point of D, then
a function f : D R will be automatically continuous at x0 .
(b) Fal
MAT 371: EXAM 3 REVIEW
1. Determine whether the statements below are true or false.
(a) If f : [a, b] R is dierentiable, then f R(x) on [a, b].
(b) If f is dierentiable on [a, b], then
b
a
f (x) dx =
MAT 371: EXAM 3 REVIEW SOLUTIONS
1.
(a) True. If f is dierentiable on [a, b], it is necessarily continuous, and so also integrable, there.
(b) False. The derivative f may not be integrable there.
(c)