Chapter 4 - Applications of the Derivative - 4.1 Linear Approximation and Applications - Exercises - Page 174: 49

$$L(x) \approx \frac{-1}{2}x +1 $$

Given $$y=(1+x)^{-1 / 2}, \quad a=0 $$ Since $y(0)=1$ and \begin{align*} y'(x)&=\frac{-1}{2}(1+x)^{-3/ 2}\\ y'(0)&=\frac{-1}{2} \end{align*} Then the linearization at $a=0$ is given by \begin{align*} L(x)&\approx y^{\prime}(a)(x-a)+y(a)\\ &=\frac{-1}{2}\left(x-0\right)+1 \\ &=\frac{-1}{2}x +1 \end{align*}