Answered: Find the linear approximation to…

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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Find the linear approximation to y=arcsinh(2x) at x=6.

Expert Solution

Step 1

$Given : y = arcsinh (2 x) and x_{0} = 6 To find : The linear approximation to y = arcsinh (2 x) at x_{0} = 6$

Step 2

$A l i n e a r a p p r o x i m a t i o n i s g i v e n b y L (x) \approx f (x_{0}) + f' (x_{0}) (x - x_{0}) . . . (1) G i v e n t h a t y = f (x) = a r c \sin h (2 x) a n d x_{0} = 6 N o w, y_{0} = f (x_{0}) = a r c \sin h (2 x_{0}) y_{0} = f (x_{0}) = a r c \sin h [(2) (6)] f (x_{0}) = a r c \sin h (12) f i n d t h e d e i v a t i v e o f f u n c t i o n f (x) = a r c \sin h (2 x) f' (x) = \frac{2}{\sqrt{4 x^{2} + 1}} f' (x_{0}) = \frac{2}{\sqrt{4 {x_{0}}^{2} + 1}} f' (x_{0}) = \frac{2}{\sqrt{145}} . . . (x_{0} = 6)$

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