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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. -

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. -

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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 FTOC-2 means that f(x) = f(a) + / f'(t) dt. (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) + ¿f"(a)(x – a)² + ; / (x – t)² f(3® (t) dt. a Recall that f(k) (t) means the k-th derivative of f (with respect to t). (d) To get to the formula in (c), we have performed integrated by parts two times. Do this a third time and then write down what the general pattern after n times will be (you do not need to prove this though). (e) Apply your answer to (d) to f(x) = eª with a = O and n = = 3.
Transcribed Image Text: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 FTOC-2 means that f(x) = f(a) + / f'(t) dt. (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) + ¿f"(a)(x – a)² + ; / (x – t)² f(3® (t) dt. a Recall that f(k) (t) means the k-th derivative of f (with respect to t). (d) To get to the formula in (c), we have performed integrated by parts two times. Do this a third time and then write down what the general pattern after n times will be (you do not need to prove this though). (e) Apply your answer to (d) to f(x) = eª with a = O and n = = 3.
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