1. Suppose that f is an infinitely differentiable function (that is, all its derivatives exist and are continuous). Let a be some point in the domain of f. In this exercise we will investigate Taylor's theorem with integral remainder' (feel free to Google this if you want). (a) Explain why FT0C-2 means that f (x) = f(a)+ | f'(e) f'(t) dt. %3D (b) Perform an integration by parts to show that f(x) = f(a) + f'(a)(x – a) + | (x – t) f" (t) dt. a Hint: Try using u = (x – t) and v' = f"(t). (c) Perform another integration by parts (being careful to remember that we're integrating with respect to t, not x) to show that f(x) = f(a) + f'(a)(x – a) + "(a)(x – a)² +5 / (2 – t)² f() (t) dt. -

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