Find the linear approximation of f(x)=√x when x = 81.1108O L(x)=3+ -(x-81)OL(x)=81+1108-(x-81)27○ L(x)=3-(x-81)27○ L(x)=81+ ²(x-81)4

We have given function fx=x4 Put x=81 f81=814=344=3 Linear approximation at x=a is given as…

Answered: Find the linear approximation of…

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

Find the linear approximation of f(x)=√x when x = 81.
1
108
O L(x)=3+ -(x-81)
OL(x)=81+
1
108
-(x-81)
27
○ L(x)=3-(x-81)
27
○ L(x)=81+ ²(x-81)
4

Expert Solution

Step 1

We have given function

$f (x) = \sqrt[4]{x}$

Put $x = 81$

$f (81) =$ $\sqrt[4]{81} =$ $\sqrt[4]{3^{4}}$ $= 3$

Linear approximation at x=a is given as

$L (x) =$ $f (a)$ $+ f^{'} (a)$ $(x - a)$ ..........(i)

$f^{'} (x) =$ $\frac{d}{d x} (\sqrt[4]{x})$ $= \frac{d}{d x} (x^{\frac{1}{4}})$ $= \frac{1}{4} x^{\frac{1}{4} - 1}$

$f^{'} (x) =$ $\frac{1}{4} x^{- \frac{3}{4}}$ $= \frac{1}{4}$ $(\frac{1}{x^{\frac{3}{4}}})$

Put $x = 81$

$f^{'} (81) =$ $\frac{1}{4} \cdot \frac{1}{{(81)}^{\frac{3}{4}}}$ $= \frac{1}{4} \cdot \frac{1}{3^{3}}$ $= \frac{1}{4 x 27}$ $= \frac{1}{108}$

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Find the linear approximation of f(x)=√x when x = 81. 1 108 O L(x)=3+ -(x-81) OL(x)=81+ 1 108 -(x-81) 27 ○ L(x)=3-(x-81) 27 ○ L(x)=81+ ²(x-81) 4