Answered: Find the linear approximation to…

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.2: Introduction To Conics: parabolas

Problem 4ECP: Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.

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Question

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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