00fall-final

# 00fall-final - x ) 6 = 0 for all x ∈ IR. b . If a...

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MAT 371 Advanced Calculus / 140 Dec 7, 2000 Final examination name 1 . Complete the “standard” deﬁnitions for: a. An inﬁnite sequence ( a n ) n ZZ + is a Cauchy sequence if . . . b. Suppose E IR, and x 0 E . A function f IR is diﬀerentiable at x 0 if . . . c. Suppose a, b IR. A function f : [ a, b ] 7→ IR is integrable over the interval [ a, b ] if . . . Include a precise deﬁnition of all technical terms used. d. A sequence { f n } n =1 of functions (with common domain E ) converges uniformly if . . . 2. a,b . State both forms of the fundamental theorem of calculus. c . State the mean value theorem (of diﬀerential calculus). d . State two theorems which guarantee that a function is integrable (the stronger, the better!) 3. Using only the deﬁnitions of diﬀerentiability and integrability show that f : x 7→ x a. is diﬀerentiable on (0 , 1), and a. is integrable on [0 , 1]. 4. Decide whether true. If true, give a brief argument why, else provide a counterexample. a . If a function f is one-to-one and diﬀerentiable on IR then f 0 (
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Unformatted text preview: x ) 6 = 0 for all x ∈ IR. b . If a function f is diﬀerentiable, then its derivative f is continuous. c . If a sequence of functions converges pointwise, then it converges uniformly. d . If a function f is integrable then it has at most a ﬁnite number of discontinuities. e . Suppose E ⊆ IR is closed, and z is an accumulation point of a sequence { x k } ∞ k =1 ⊆ E . If a function f : E 7→ IR is continuous at each x k , then f is continuous at z . 5. Suppose f : [ a, b ] 7→ IR is diﬀerentiable at x ∈ ( a, b ). Prove that f is continuous at x . 6. Suppose and f and g are integrable over [ a, b ] and c ∈ IR. Prove that ( f + cg ) is integrable over [ a, b ]. 7. Suppose a sequence { f n } ∞ n =1 of continuous functions deﬁned on a common interval [ a, b ] converges uniformly to a function f : [ a, b ] 7→ IR. Prove that f is continuous on [ a, b ]....
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## This note was uploaded on 04/02/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.

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