05sprg-test3

05sprg-test3 - f : S 7 R such that f does not have a...

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MAT 371 Advanced Calculus /100 April 13, 2005 Test 3 name 1 2 3 4 5 6 20 15 15 15 15 20 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 1. a. Define Cauchy sequence . b. Define complete . c. Define compact . d. Define differentiable e. State the Mean Value Theorem . 2. a. Show that every uniformly continuous function f : (0 , 1] 7→ R is bounded. b. Give an example of a continuous function f : (0 , 1] 7→ R that is not bounded. Bonus. Suppose that f : (0 , 1] 7→ R is uniformly continuous. Show that there exists a con- tinuous extension f : [0 , 1] 7→ R of f , i.e. f is continuous and for all x (0 , 1], f ( x ) = f ( x ). 3. a. Find an open cover of the set S = { 1 n : n Z + } ⊆ R that does not have a finite subcover. b. Working directly from the definition, prove that the set K = S ∪ { 0 } ⊆ R is compact. 4. a. Prove that if K R is compact and f : K 7→ R is continuous then f ( K ) is compact. b. Explain why this implies that for every continuous function f : K 7→ R there exists c [ a, b ] such that for all x [ a, b ], f ( c ) f ( x ). c. Give an examples of a bounded set S R and a continuous function
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Unformatted text preview: f : S 7 R such that f does not have a maximum value on S . 5. Suppose that f : R 7 R and g : R 7 R are dierentiable functions. Prove that the dierence f-g is dierentiable. Bonus. Prove that the product f g is dierentiable. 6. a. Suppose that f : ( a, b ) 7 R is a decreasing function. Prove that for every c ( a, b ) either f is not dierentiable at c , or f ( c ) 0. b. Suppose that f : ( a, b ) 7 R is a dierentiable function and and for all c ( a, b ), f ( c ) < 0. Prove that f is decreasing. Bonus. Give an example of a continuous function f : R 7 R that is dierentiable everywhere, and such that f (0) < 0, but f is not decreasing on any open interval containing 0. Prove that your example has indeed all the required properties....
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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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