Chapter 4 - Applications of the Derivative - 4.1 Linear Approximation and Applications - Exercises - Page 174: 51

$$L(x)= 1$$

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