To find the linear approximation [tex]\( A + Bx \)[/tex] of the function [tex]\( \frac{1}{\sqrt{2-x}} \)[/tex] at [tex]\( x=0 \)[/tex], we follow these steps:
1. Identify the function and the point of approximation:
Given function: [tex]\( f(x) = \frac{1}{\sqrt{2-x}} \)[/tex]
Point of approximation: [tex]\( x = 0 \)[/tex]
2. Calculate [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = \frac{1}{\sqrt{2-0}} = \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} = 0.7071067811865475
\][/tex]
This is the value of [tex]\( A \)[/tex], as [tex]\( A = f(0) \)[/tex].
So, the value of [tex]\( A \)[/tex] in the linear approximation at [tex]\( x = 0 \)[/tex] is [tex]\( 0.7071067811865475 \)[/tex].
