Exercise 11

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 $h(x) = g(x) \cdot f(x)$. Determine $h(2)$ and $h'(2)$.

2. Find an equation for the tangent line to $y = h(x)$ at the point $(2,h(2))$ (where $h$ is the function defined in (a)).

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

4. Estimate the value of $r(2.06)$ (where $r$ is the function defined in (c)) by using the local linearization of $r$ at the point $(2,r(2))$.