Find the linear approximation of $LaTeX: \displaystyle f(x) = \sqrt{x + 225}$ at $LaTeX: \displaystyle a = 0$ and use it to approximate $LaTeX: \displaystyle \sqrt{224.16}$

Using the formula for the linearization $LaTeX: \displaystyle L(x) = f'(a)(x-a)+f(a)$ gives: $LaTeX: f(x) = \sqrt{x + 225} \approx L(x) = \frac{x}{30} + 15$ To approximate $LaTeX: \displaystyle \sqrt{224.16}$ use $LaTeX: \displaystyle x = 224.16-225 = -0.84$ . $LaTeX: \sqrt{224.16} = f(-0.84) \approx L(-0.84) = 14.972000$