Two functions from a table: product…

Name: ______________________ Date: ____________ Two functions from a table: product, quotient, tangent line and a limit (9 points)

Table

Values of f, g and their derivatives

The functions $f$ and $g$ are differentiable for all real numbers. The table gives values of the functions and their derivatives at selected values of $x$ . No calculator is allowed.

Values of $f(x)$ , $f'(x)$ , $g(x)$ and $g'(x)$
$x$	1	2	3	4
$f(x)$	4	1	3	6
$f'(x)$	$-2$	3	5	1
$g(x)$	1	3	2	$-1$
$g'(x)$	4	$-2$	3	$-4$

(a) Let $h(x) = f(x)g(x)$ . Find $h'(2)$ . Show the work that leads to your answer. [2 points]
(b) Let $k(x) = \frac{f(x)}{g(x)}$ . Find $k'(3)$ . Show the work that leads to your answer. [2 points]
(c) Let $m(x) = 4f(x) - \ln x$ for $x > 0$ . Write an equation for the line tangent to the graph of $m$ at $x = 1$ . [2 points]
(d) (i) Find $\lim_{x \to 4} \frac{f(x) - 6}{x - 4}$ . Explain your reasoning.
(ii) Explain why there must be a value $c$ , with $1 < c < 4$ , such that $g(c) = 0$ . [3 points]

The question

Values of f, g and their derivatives