Question

Find a linear approximation for the following functions at the indicated points.

g(x) = ${\displaystyle \sqrt{x^2+9}}$,  x₀ = – 4

Answer 1

g(x)= ${\displaystyle \sqrt{x^2+9}}$, x₀ = – 4

g(x)= ${\displaystyle \frac{1}{2\sqrt{x^2+9}}\left(2x\right)=\frac{x}{\sqrt{x^2+9}}}$

g(x₀) = ${\displaystyle \text{g}(-4)}$

= ${\displaystyle \sqrt{16+9}}$

= ${\displaystyle \sqrt{25}}$

= 5

g'(x₀) = ${\displaystyle \frac{-4}{\sqrt{25}}}$

= ${\displaystyle \frac{-4}{5}}$

The required linear approximation L(x) = ${\displaystyle \text{g}\left(x_0\right)+\text{g'}\left(x_0\right)(x-x_0)}$

= ${\displaystyle 5-\frac{4}{5}(x+4)}$

= ${\displaystyle 5-\frac{4x}{5}-\frac{16}{5}}$

= ${\displaystyle \frac{9}{5}-\frac{4x}{5}}$

= ${\displaystyle \frac{9-4x}{5}}$