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Additional Problem 1: Let F(x) = R x a f dφ for a ≤ x ≤ b where f is continuous on [a, b] and φ is differentiable and increasing on [a, b]. Show that F 0 (x) = f(x)φ 0 (x) for every x ∈ [a, b]. What would you call this result? (You may assume all additivity properties of the integral.) Additional Problem 2: Calculate Z b a α(x) dα(x) assuming the integral exists. Hints: For 11, mimic the proof of a sequence converging iff it is a Cauchy sequence.

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