Find the linear approximation of $LaTeX: \displaystyle f(x) = \sqrt{x + 169}$ at $LaTeX: \displaystyle a = 0$ and use it to approximate $LaTeX: \displaystyle \sqrt{169.08}$

Using the formula for the linearization $LaTeX: \displaystyle L(x) = f'(a)(x-a)+f(a)$ gives: $LaTeX: f(x) = \sqrt{x + 169} \approx L(x) = \frac{x}{26} + 13$ To approximate $LaTeX: \displaystyle \sqrt{169.08}$ use $LaTeX: \displaystyle x = 169.08-169 = 0.08$ . $LaTeX: \sqrt{169.08} = f(0.08) \approx L(0.08) = 13.003077$