1.
Suppose you know that the function \(f\) passes through the point \((4,3)\) and has first derivative
\begin{equation*}
f'(x) = \sqrt{x}+ 5.
\end{equation*}
(a)
Find the equation of the tangent line to the the function \(f(x)\) at the point \((4,3)\text{.}\)
(b)
Use the tangent line (or the equivalent local linearization) to approximate \(f(4.1)\text{.}\) Show your work.
(c)
Suppose you found out that \(f''(4) = 0.25\text{.}\) What does this tell you about the shape of \(f\) near \(x = 4\text{?}\) Does this mean your approximation for \(f(4.1)\) is an over estimate or under estimate? Briefly explain.
