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04fall-test3

04fall-test3 - Suppose a,b,c,k,m ∈ R and a< c< b...

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MAT 371 Advanced Calculus /100 November 22, 2004 Test 3 name 1 2 3 4 5 6 Bonus 15 10 10 25 20 20 1. State precise definitions for the following a. A function f is increasing on an interval ( a, b ) if . . . b. A function f is differentiable on an interval ( a, b ) if . . . c. A function f is integrable over an interval [ a, b ] if . . . 2. a. State the Extreme Value Theorem. b. State the Mean Value Theorem. 3. (Without proofs) give examples of functions that have the following properties (or briefly explain why no such function exists). a. f : R R is differentiable, but f is not continuous. b. f : [0 , 1] R f has infinitely many discontinuities but is integrable over [0 , 1]. 4. Working directly from the definition a. show that every differentiable function is continuous, and b. show that if f is differentiable and f 0 on ( a, b ) then f is increasing on interval ( a, b ) 5. Suppose a, b, c, k, m
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Unformatted text preview: Suppose a,b,c,k,m ∈ R and a < c < b . Working directly from the definition show that the piecewise constant function f : [ a,b ] 7→ R defined by f ( x ) = k if a ≤ x ≤ c and f ( x ) = m if c < x ≤ b is integrable. 6. a. (Without proof), give an example of an integrable function f : [ a,b ] 7→ R such that f ≥ 0 and R b a f = 0, but f 6≡ 0. b. Suppose that a < b are real numbers. Working directly from the definition show that if f : [ a,b ] 7→ [0 , ∞ ) is continuous (and hence integrable), and f 6≡ 0 then R b a f > 0. Bonus: Sketch a proof that every monotone function f : [ a,b ] 7→ R is integrable. Hint: How many jump-discontinuities of height larger than | f ( b )-f ( a ) | / N for N ∈ Z + can f have?...
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