The functions $f$ and $g$ are differentiable for all real numbers, and $g$ is strictly increasing. The table below gives values of the functions and their first derivatives at selected values of $x$.

$x$	$f(x)$	$f^{\prime}(x)$	$g(x)$	$g^{\prime}(x)$
1	6	4	2	5
2	9	2	3	1
3	10	-4	4	2
4	-1	3	6	7

The function $h$ is given by $h(x) = f(g(x)) - 6$.
(a) Explain why there must be a value $r$ for $1 < r < 3$ such that $h(r) = -5$.
(b) Explain why there must be a value $c$ for $1 < c < 3$ such that $h^{\prime}(c) = -5$.
(c) Let $w$ be the function given by $w(x) = \int_{1}^{g(x)} f(t)\, dt$. Find the value of $w^{\prime}(3)$.
(d) If $g^{-1}$ is the inverse function of $g$, write an equation for the line tangent to the graph of $y = g^{-1}(x)$ at $x = 2$.

The functions $f$ and $g$ have continuous second derivatives. The table below gives values of the functions and their derivatives at selected values of $x$.

$x$	$f ( x )$	$f ^ { \prime } ( x )$	$g ( x )$	$g ^ { \prime } ( x )$
1	$-6$	3	2	8
2	2	$-2$	$-3$	0
3	8	7	6	2
6	4	5	3	$-1$

(a) Let $k ( x ) = f ( g ( x ) )$. Write an equation for the line tangent to the graph of $k$ at $x = 3$.
(b) Let $h ( x ) = \frac { g ( x ) } { f ( x ) }$. Find $h ^ { \prime } ( 1 )$.
(c) Evaluate $\int _ { 1 } ^ { 3 } f ^ { \prime \prime } ( 2 x ) \, d x$.

Let $f$ be the function defined by $f(x) = \cos(2x) + e^{\sin x}$.
Let $g$ be a differentiable function. The table below gives values of $g$ and its derivative $g'$ at selected values of $x$.

\multicolumn{1}{|c|}{$x$}	$g(x)$	$g'(x)$
-5	10	-3
-4	5	-1
-3	2	4
-2	3	1
-1	1	-2
0	0	-3

Let $h$ be the function whose graph, consisting of five line segments, is shown in the figure above.
(a) Find the slope of the line tangent to the graph of $f$ at $x = \pi$.
(b) Let $k$ be the function defined by $k(x) = h(f(x))$. Find $k'(\pi)$.
(c) Let $m$ be the function defined by $m(x) = g(-2x) \cdot h(x)$. Find $m'(2)$.
(d) Is there a number $c$ in the closed interval $[-5, -3]$ such that $g'(c) = -4$? Justify your answer.

The functions $f$ and $g$ are twice differentiable. The table shown gives values of the functions and their first derivatives at selected values of $x$.

$x$	0	2	4	7
$f(x)$	10	7	4	5
$f'(x)$	$\frac{3}{2}$	$-8$	3	6
$g(x)$	1	2	$-3$	0
$g'(x)$	5	4	2	8

(a) Let $h$ be the function defined by $h(x) = f(g(x))$. Find $h'(7)$. Show the work that leads to your answer.
(b) Let $k$ be a differentiable function such that $k'(x) = (f(x))^{2} \cdot g(x)$. Is the graph of $k$ concave up or concave down at the point where $x = 4$? Give a reason for your answer.
(c) Let $m$ be the function defined by $m(x) = 5x^{3} + \int_{0}^{x} f'(t)\, dt$. Find $m(2)$. Show the work that leads to your answer.
(d) Is the function $m$ defined in part (c) increasing, decreasing, or neither at $x = 2$? Justify your answer.