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Homework 11
You may work together, but the solutions must be written up in your own words. Show all work and justify all answers.
1. Ross 32.8
2. Let a, b ∈ R, a < b. Let f : [a, b] → R be a function such that f(x) = 0 for all x ∈ [a, b]\S, where S = {s1, ..., sk} is a finite subset of [a, b]. Prove that f is integrable and find R b a f.
3. Ross 33.7 and 33.8 (a).
4. Let f : [a, b] → R be a continuous function such that f(x) ≥ 0 for all x ∈ [a, b] and R b a f = 0. Prove that f(x) = 0 for all x ∈ R.
5. Ross 34.2
6. Define f : R → R by f(x) = 0 for x < 0, f(x) = x for x ∈ [0, 1] and f(x) = 4 for x > 1. Define F : R → R by F(x) = R x 0 f. a) Determine F(x) for each x ∈ R. b) At which x ∈ R is F continuous? c) At which x ∈ R is F differentiable? For those x, what is F 0 (x)?
7. Ross 34.5
8. Ross 34.7. Indicate precisely how you use change of variables, and check that all conditions of the theorem are met. 1