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Math 3210-1 HW 16 Due Wednesday, July 25 Properties of the Riemann Integral 1. A function f on [a, b] is called a step function if there exists a partition P = {a = u0 < u1 < · · · < um = b} of [a, b] such that f is constant on each interval (uj−1 , uj ), say f (x) = cj for x ∈ (uj−1 , uj ). Rb (a) Show that a step function f is integrable and evaluate a f . R4 15 if 0 ≤ x < 1 (b) Given P (x) = , evaluate 0 P (x) dx. Note: P (x) is called the 15 + 13n if n ≤ x < n + 1 postage-stamp function. Do you see why? 2. Find an example of a function f : [0, 1] → R such that f is not integrable on [0, 1] by |f | is integrable on [0, 1]. Rb Rb 3. Suppose that f and g are continuous function on [a, b] such that a f = a g. Prove that there exists x ∈ [a, b] such that f (x) = g(x). The Fundamental Theorem of Calculus 4. Calculate Z x 2 1 (a) lim et dt x→0 x 0 Z 3+h 2 1 et dt (b) lim h→0 h 3 5. Let f be defined as follows: f (t) = 0 for t < 0; f (t) = t for 0 ≤ t ≤ 1; f (t) = 4 for t > 1. Rx (a) Determine the function F (x) = 0 f (t) dt. (b) Sketch F . Where is F continuous? (c) Where is F differentiable? Calculate F ′ at the points of differentiability. 6. Let f : [0, 1] → R be a continuous function with continuous second derivative f ′′ , and f (0) = f ′ (1) = 0. Z 1 f (x)f ′′ (x) dx = 0, then f ≡ 0. Prove that if 0

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