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 following symbols: N set of natural numbers (without
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 in the proof. You may want to give the
students some hi
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. > 0, = > 0, 0 < |x + 2| < , we have |f (x) + 4| = |x 2
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 . -1
Solution:
x0 ex
lim
x x = lim x x0 (e - 1) -1 1 = l
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 of the Mean Value Theorem on [r1 , r2 ] and hence ther
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 inf E is similar. Let Q be any neighborhood of sup E,
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 Work: | |f (x)| - |L| | |f (x) - L| for all x D by the t
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 . [1.35] (5 points) Let E = cfw_an : n N with accumulation p
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: (a) The limit L := lim exists. (b) The limit L := lim ex
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 accumulation point of E. Also, a function is automatically contin
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 accumulation point of the domain. 2. Decreasing function, i
Chapter 5
1. Riemann Integrable, Riemann Integral. This includes defining partition, upper Riemann sum, lower Riemann sum, upper Riemann integral, lower Riemann integral. Let f : [a, b] R be a bounded function. M := supcfw_f (x) | a x b, m := infcfw_f (x)
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: Since f is continuous at x0 , > 0, > 0, for x D with |x - x
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 monotone.
n=1
(c) cfw_an is Cauchy.
n=1
(d) cfw_a2k conve
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 chapters. A strong
foundation laid now will make life mu
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
Company
Renovations on a portion of existing museum
Sc
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 perfect dream home for the modern Indian family. Its a
home
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. Suitable assumptions are made and various
facts are stated. T
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 Justification
A3. Business Plan
A4. Economic Analysis
A5.
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 procedures
that should be included for this project:
Detaile
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 fact, a sequence is convergent if and only if it is Cauc
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 nor closed, than S c is neither open nor closed.
(d) If
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) False. If S is open, then S c is closed, but a set may be
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 = f (b) f (a).
(c) If f is discontinuous at some points i
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) False. For example, monotone functions are integrable b
MAT 371: REVIEW PROBLEMS
1. True or false?
(a) If = sup S for some S R and is not an accumulation point of S , then S .
(b) If the range S of cfw_an contains no accumulation points, then cfw_an cannot be convergent.
n=1
n=1
(c) If the range S of cfw_an
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
and therefore has no accumulation points.
(c) True. If