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. a (b) Perform an integration by parts to show that f(x) = f(a) + f'(a)(x – a) + | (x – t) f" (t) dt. 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 1 1 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). To get to the formula in (c), we have performed integrated by parts (d) 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) = e with a = 0 and n = 3.
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. a (b) Perform an integration by parts to show that f(x) = f(a) + f'(a)(x – a) + | (x – t) f" (t) dt. 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 1 1 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). To get to the formula in (c), we have performed integrated by parts (d) 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) = e with a = 0 and n = 3.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Just part d and e, thank you!
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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