Advanced Calculus, Supplement and Solutions
Horst R. Thieme, Arizona State University, Fall 2007. updated December 4, 2007
The real numbers
1.1 Ordered Fields [1, Sec.11]
We use the following symbols: N set of natural numbers (without
x E S) then cfw_Sn:1 converges to x and we invoke
1.17. The proof for x
inf S can be handled in a
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
x . -1
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
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
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
1. Limit of a function:
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
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
are necessarily true?
(a) cfw_an is bounded.
(b) cfw_an is monotone.
(c) cfw_an is Cauchy.
(d) cfw_a2k conve
44$; 329 anmerw m-l
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
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MAT 371: EXAM 1 REVIEW HINTS/SOLUTIONS
(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.
(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.
MAT 371: EXAM 2 REVIEW SOLUTIONS
(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
f (x) dx = f (b) f (a).
(c) If f is discontinuous at some points i
MAT 371: EXAM 3 REVIEW SOLUTIONS
(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.
(c) If the range S of cfw_an
MAT 371: SOLUTIONS TO FINAL REVIEW PROBLEMS
(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