Additional Problem 1: Let F(x) = R x
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
(x) = f(x)φ
(x) for every
x ∈ [a, b]. What would you call this result? (You may assume all additivity properties of
Additional Problem 2: Calculate Z b
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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