Find the linearization at x = a.ITf(x) = sin (x), a =4(Use symbolic notation and fractions where needed.)

Given, The linearization at x=a. fx=sin3x, a=π4We know that, The…

Answered: Find the linearization at x = a. IT…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter3: The Derivative

Section3.CR: Chapter 3 Review

Problem 59CR

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Question

Find the linearization at x = a.
IT
f(x) = sin (x), a =
4
(Use symbolic notation and fractions where needed.)

Expert Solution

Step 1

$G i v e n, T h e l i n e a r i z a t i o n a t x = a . f (x) = \sin^{3} (x), a = \frac{π}{4} W e k n o w t h a t, T h e l i n e a r i z a t i o n o f a f u n c t i o n f (x) a t x = a i s g i v e n a s L (x) = f (x) + f' (x) (x - a)$

Step 2

$N o w, ∴ f (x) = \sin^{3} (x) a t x = \frac{π}{4} \Rightarrow f (\frac{π}{4}) = {(\sin (\frac{π}{4}))}^{3} \Rightarrow f (\frac{π}{4}) = {(\frac{1}{\sqrt{2}})}^{3} \Rightarrow f (\frac{π}{4}) = \frac{1}{2 \sqrt{2}} ∴ f (x) = \sin^{3} (x) D i f f e r e n t i a t i n g w . r . t x, w e h a v e \Rightarrow f' (x) = \frac{d}{d x} (\sin^{3} (x)) \Rightarrow f' (x) = 3 \sin^{2} (x) \frac{d}{d x} (\sin (x)) \Rightarrow f' (x) = 3 \sin^{2} (x) \cos (x) a t x = \frac{π}{4} \Rightarrow f' (\frac{π}{4}) = 3 \sin^{2} (\frac{π}{4}) \cos (\frac{π}{4}) \Rightarrow f' (\frac{π}{4}) = 3 {(\frac{1}{\sqrt{2}})}^{2} (\frac{1}{\sqrt{2}}) \Rightarrow f' (\frac{π}{4}) = 3 \times \frac{1}{2} \times \frac{1}{\sqrt{2}} \Rightarrow f' (\frac{π}{4}) = \frac{3}{2 \sqrt{2}}$

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