Determine f(44) (x) for f(x) = e-#sin(x). Hint: Check the first few derivatives and deduce the pattern.tangent?%24deleleR.Os lockH.Baltcontrolcommandcommandoption

Answered: Determine f(44) (x) for ƒ(x) =…

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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Question

Determine f(44) (x) for f(x) = e-#sin(x). Hint: Check the first few derivatives and deduce the pattern.
tangent?
%24
delele
R.
Os lock
H.
B
alt
control
command
command
option

Expert Solution

Step 1

The given function is:

$f (x) = e^{- x} \sin (x)$

Now differentiate the function,

$\begin{array}{rcl} f^{(1)} (x) & = & \frac{d}{d x} (e^{- x} \sin (x)) \\ = & - e^{- x} \sin (x) + e^{- x} \cos (x) \\ f^{(2)} (x) & = & e^{- x} \sin (x) - e^{- x} \cos (x) - e^{- x} \cos (x) - e^{- x} \sin (x) \\ = & - 2 e^{- x} \cos (x) \\ f^{(3)} (x) & = & \frac{d}{d x} (- 2 e^{- x} \cos (x)) \\ = & 2 (e^{- x} \cos (x) + e^{- x} \sin (x)) \\ f^{(4)} (x) & = & \frac{d}{d x} 2 (e^{- x} \cos (x) + e^{- x} \sin (x)) \\ = & - 4 e^{- x} \sin (x) \\ = & - 4 f (x) \end{array}$