Exercise 8

Let $f$ and $g$ be differentiable functions for which the following information is known: $f(2) = 5$, $g(2) = -3$, $f'(2) = -1/2$, $g'(2) = 2$.

1. Let $h$ be the new function defined by the rule $h(x) = 3f(x) - 4g(x)$. Determine $h(2)$ and $h'(2)$.

2. Find an equation for the tangent line to $y = h(x)$ at the point $(2,h(2))$.

3. Let $p$ be the function defined by the rule $p(x) = -2f(x) + \frac{1}{2}g(x)$. Is $p$ increasing, decreasing, or neither at $a = 2$? Why?

4. Estimate the value of $p(2.03)$ by using the local linearization of $p$ at the point $(2,p(2))$.