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. -
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Question is in the picture, thanks for the help!
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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