$\lim _ { x \rightarrow 0 } \frac { 1 - \cos ^ { 2 } ( 2 x ) } { ( 2 x ) ^ { 2 } } =$
(A) 0
(B) $\frac { 1 } { 4 }$
(C) $\frac { 1 } { 2 }$
(D) 1

If $f ( x ) = \int _ { 1 } ^ { x ^ { 3 } } \frac { 1 } { 1 + \ln t } d t$ for $x \geq 1$, then $f ^ { \prime } ( 2 ) =$
(A) $\frac { 1 } { 1 + \ln 2 }$
(B) $\frac { 12 } { 1 + \ln 2 }$
(C) $\frac { 1 } { 1 + \ln 8 }$
(D) $\frac { 12 } { 1 + \ln 8 }$

10. $\lim _ { h \rightarrow 0 } \frac { \sin ( x + h ) - \sin x } { h }$ is
(A) 0
(B) 1
(C) $\quad \sin x$
(D) $\cos x$
(E) nonexistent

The velocity of a particle moving along a straight line is given by $v ( t ) = 1.3 t \ln ( 0.2 t + 0.4 )$ for time $t \geq 0$. What is the acceleration of the particle at time $t = 1.2$?
(A) - 0.580
(B) - 0.548
(C) - 0.093
(D) 0.660

14. If $F ( x ) = \int _ { 1 } ^ { x ^ { 2 } } \sqrt { 1 + t ^ { 3 } } d t$, then $F ^ { \prime } ( x ) =$
(A) $2 x \sqrt { 1 + x ^ { 6 } }$
(B) $2 x \sqrt { 1 + x ^ { 3 } }$
(C) $\sqrt { 1 + x ^ { 6 } }$
(D) $\sqrt { 1 + x ^ { 3 } }$
(E) $\int _ { 1 } ^ { x ^ { 2 } } \frac { 3 t ^ { 2 } } { 2 \sqrt { 1 + t ^ { 3 } } } d t$

10. What is the instantaneous rate of change at $x = 2$ of the function $f$ given by $f ( x ) = \frac { x ^ { 2 } - 2 } { x - 1 }$ ?
(A) - 2
(B) $\frac { 1 } { 6 }$
(C) $\frac { 1 } { 2 }$
(D) 2
(E) 6

16. If $f ( x ) = \sin \left( e ^ { - x } \right)$, then $f ^ { \prime } ( x ) =$
(A) $- \cos \left( e ^ { - x } \right)$
(B) $\cos \left( e ^ { - x } \right) + e ^ { - x }$
(C) $\cos \left( e ^ { - x } \right) - e ^ { - x }$
(D) $e ^ { - x } \cos \left( e ^ { - x } \right)$
(E) $- e ^ { - x } \cos \left( e ^ { - x } \right)$ [Figure]

Suppose that the function $f$ has a continuous second derivative for all $x$, and that $f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 3$, and $f ^ { \prime \prime } ( 0 ) = 0$. Let $g$ be a function whose derivative is given by $g ^ { \prime } ( x ) = e ^ { - 2 x } \left( 3 f ( x ) + 2 f ^ { \prime } ( x ) \right)$ for all $x$.
(a) Write an equation of the line tangent to the graph of $f$ at the point where $x = 0$.
(b) Is there sufficient information to determine whether or not the graph of $f$ has a point of inflection when $x = 0$ ? Explain your answer.
(c) Given that $g ( 0 ) = 4$, write an equation of the line tangent to the graph of $g$ at the point where $x = 0$.
(d) Show that $g ^ { \prime \prime } ( x ) = e ^ { - 2 x } \left( - 6 f ( x ) - f ^ { \prime } ( x ) + 2 f ^ { \prime \prime } ( x ) \right)$. Does $g$ have a local maximum at $x = 0$ ? Justify your answer.

$f ( x ) = \frac { 2 x } { \sqrt { x ^ { 2 } + x + 1 } }$
a) $x ^ { 2 } + x + 1 > 0$
$x ^ { 2 } + x + ( 1 / 2 ) ^ { 2 } + 1 > 0$
$( x + 1 / 2 ) ^ { 2 } + 1 - 1 / 4 > 0$
$( x + 1 / 2 ) ^ { 2 } + 3 / 4 > 0$ for oll real $x$
Domain: All real $x$
b) see graph
c) $\lim _ { x \rightarrow \infty } \frac { 2 x } { \sqrt { x ^ { 2 } + x + 1 } } = \lim _ { x \rightarrow \infty } \frac { 2 x } { \sqrt { \frac { x } { x ^ { 2 } } + x + 1 } } = \lim _ { x \rightarrow \infty } \frac { 2 } { \sqrt { 1 + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } } } = \frac { 2 } { \sqrt { 1 + 0 + 0 } } = 2$
$\lim _ { x \rightarrow - \infty } \frac { 2 x } { \sqrt { x } ^ { 2 } + x + 1 } = \lim _ { x \rightarrow - \infty } \frac { 2 } { \sqrt { 1 + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } } } = - \frac { 2 } { \sqrt { 1 + 0 + 0 } } = - 2$
$y = 2 , y = - 2$
d) $f ^ { \prime } ( x ) = \frac { x + 2 } { \left( x ^ { 2 } + x + 1 \right) ^ { 3 / 2 } }$
$f ^ { \prime } ( x ) = 0$
$x + 2 = 0$
$x = - 2$
$f ( - 2 ) = - \frac { 4 } { \sqrt { 3 } } = - \frac { 4 \sqrt { 3 } } { 3 }$
Range: $\{ 4 : - 4 \sqrt { 3 } / 3 \leq 4 < 2 \}$

$V ( t ) = t \cos t , t \geq 0 , y ( 0 ) = 3$
a) $t \cos t > 0,0 \leq t \leq 5$
$t \cos t = 0$
$t = 0$
$$\begin{gathered} \cos t = 0 \quad t \quad \cdots \quad t \\ t = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \quad a _ { - } \frac { \pi } { 2 } \quad \frac { \pi } { 2 } \quad E \end{gathered}$$
$( 0 , \pi / 2 ) , ( 3 \pi / 2,5 )$
b) $$\begin{aligned} \Lambda ( t ) & = V ( t ) \\ \Lambda ( t ) & = \cos t + t - \sin t ) \\ & = \cos t - t \sin t \end{aligned}$$
() $Y ( t ) = \int Y ( t ) d t$
$$\begin{aligned} & y ( t ) = \int t \cos t d t \text { let } \quad \frac { d } { d } = t \quad \frac { d v } { d } \\ & = \cos t \\ & = t \sin t - \int \sin t d t \quad \frac { d t } { d t } \quad v \\ & \end{aligned}$$
$Y ( t ) = + \sin t - ( - \cos t ) + c$
$= t \sin t + \cos t + c$
$y | 0 | = 3$
$j = \cos 0 + c$
$c = 2$
$Y ( t ) = t \sin t + \cos t + 2$
d) $V ( t ) = t \cos t$
$t \cos t = 0$
$t \neq 0 \quad \cos t = 0$
$$t = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \ldots$$
$Y \left( \frac { \pi } { 2 } \right) = \frac { \pi } { 2 } \sin \frac { \pi } { 2 } + \cos \frac { \pi } { 2 } + 2 = \frac { \pi } { 2 } + 0 + 2 = \frac { \pi } { 2 } + 2$

Let $f$ be the function given by $f ( x ) = \sqrt { x ^ { 4 } - 16 x ^ { 2 } }$. (a) Find the domain of $f$. (b) Describe the symmetry, if any, of the graph of $f$. (c) Find $f ^ { \prime } ( x )$. (d) Find the slope of the line normal to the graph of $f$ at $x = 5$.

The twice-differentiable function $f$ is defined for all real numbers and satisfies the following conditions: $$f(0) = 2,\quad f'(0) = -4,\quad \text{and}\quad f''(0) = 3.$$
(a) The function $g$ is given by $g(x) = e^{ax} + f(x)$ for all real numbers, where $a$ is a constant. Find $g'(0)$ and $g''(0)$ in terms of $a$. Show the work that leads to your answers.
(b) The function $h$ is given by $h(x) = \cos(kx)f(x)$ for all real numbers, where $k$ is a constant. Find $h'(x)$ and write an equation for the line tangent to the graph of $h$ at $x = 0$.

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 $x = 2$.

Let $f$ be a twice-differentiable function such that $f ( 2 ) = 5$ and $f ( 5 ) = 2$. Let $g$ be the function given by $g ( x ) = f ( f ( x ) )$. (a) Explain why there must be a value $c$ for $2 < c < 5$ such that $f ^ { \prime } ( c ) = - 1$. (b) Show that $g ^ { \prime } ( 2 ) = g ^ { \prime } ( 5 )$. Use this result to explain why there must be a value $k$ for $2 < k < 5$ such that $g ^ { \prime \prime } ( k ) = 0$. (c) Show that if $f ^ { \prime \prime } ( x ) = 0$ for all $x$, then the graph of $g$ does not have a point of inflection. (d) Let $h ( x ) = f ( x ) - x$. Explain why there must be a value $r$ for $2 < r < 5$ such that $h ( r ) = 0$.

The functions $f$ and $g$ are given by $f ( x ) = \int _ { 0 } ^ { 3 x } \sqrt { 4 + t ^ { 2 } } d t$ and $g ( x ) = f ( \sin x )$. (a) Find $f ^ { \prime } ( x )$ and $g ^ { \prime } ( x )$. (b) Write an equation for the line tangent to the graph of $y = g ( x )$ at $x = \pi$. (c) Write, but do not evaluate, an integral expression that represents the maximum value of $g$ on the interval $0 \leq x \leq \pi$. Justify your answer.

The function $f$ is defined by $f ( x ) = \sqrt { 25 - x ^ { 2 } }$ for $- 5 \leq x \leq 5$.
(a) Find $f ^ { \prime } ( x )$.
(b) Write an equation for the line tangent to the graph of $f$ at $x = - 3$.
(c) Let $g$ be the function defined by $g ( x ) = \left\{ \begin{array} { l } f ( x ) \text { for } - 5 \leq x \leq - 3 \\ x + 7 \text { for } - 3 < x \leq 5 . \end{array} \right.$
Is $g$ continuous at $x = - 3$ ? Use the definition of continuity to explain your answer.
(d) Find the value of $\int _ { 0 } ^ { 5 } x \sqrt { 25 - x ^ { 2 } } d x$.

The function $f$ is defined by $f ( x ) = \sqrt { 25 - x ^ { 2 } }$ for $- 5 \leq x \leq 5$.
(a) Find $f ^ { \prime } ( x )$.
(b) Write an equation for the line tangent to the graph of $f$ at $x = - 3$.
(c) Let $g$ be the function defined by $g ( x ) = \left\{ \begin{array} { l } f ( x ) \text { for } - 5 \leq x \leq - 3 \\ x + 7 \text { for } - 3 < x \leq 5 . \end{array} \right.$
Is $g$ continuous at $x = - 3$ ? Use the definition of continuity to explain your answer.
(d) Find the value of $\int _ { 0 } ^ { 5 } x \sqrt { 25 - x ^ { 2 } } d x$.

If $y = \left( x ^ { 3 } - \cos x \right) ^ { 5 }$, then $y ^ { \prime } =$
(A) $5 \left( x ^ { 3 } - \cos x \right) ^ { 4 }$
(B) $5 \left( 3 x ^ { 2 } + \sin x \right) ^ { 4 }$
(C) $5 \left( 3 x ^ { 2 } + \sin x \right)$
(D) $5 \left( 3 x ^ { 2 } + \sin x \right) ^ { 4 } \cdot ( 6 x + \cos x )$
(E) $5 \left( x ^ { 3 } - \cos x \right) ^ { 4 } \cdot \left( 3 x ^ { 2 } + \sin x \right)$

If $f ( x ) = \sqrt { x ^ { 2 } - 4 }$ and $g ( x ) = 3 x - 2$, then the derivative of $f ( g ( x ) )$ at $x = 3$ is
(A) $\frac { 7 } { \sqrt { 5 } }$
(B) $\frac { 14 } { \sqrt { 5 } }$
(C) $\frac { 18 } { \sqrt { 5 } }$
(D) $\frac { 15 } { \sqrt { 21 } }$
(E) $\frac { 30 } { \sqrt { 21 } }$

$\lim _ { h \rightarrow 0 } \frac { \ln ( 4 + h ) - \ln ( 4 ) } { h }$ is
(A) 0
(B) $\frac { 1 } { 4 }$
(C) 1
(D) $e$
(E) nonexistent

The graph of $y = e ^ { \tan x } - 2$ crosses the $x$-axis at one point in the interval $[ 0,1 ]$. What is the slope of the graph at this point?
(A) 0.606
(B) 2
(C) 2.242
(D) 2.961
(E) 3.747

On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by $G ( t ) = 90 + 45 \cos \left( \frac { t ^ { 2 } } { 18 } \right)$, where $t$ is measured in hours and $0 \leq t \leq 8$. At the beginning of the workday $( t = 0 )$, the plant has 500 tons of unprocessed gravel. During the hours of operation, $0 \leq t \leq 8$, the plant processes gravel at a constant rate of 100 tons per hour.
(a) Find $G ^ { \prime } ( 5 )$. Using correct units, interpret your answer in the context of the problem.
(b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday.
(c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time $t = 5$ hours? Show the work that leads to your answer.
(d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer.

Grass clippings are placed in a bin, where they decompose. For $0 \leq t \leq 30$, the amount of grass clippings remaining in the bin is modeled by $A ( t ) = 6.687 ( 0.931 ) ^ { t }$, where $A ( t )$ is measured in pounds and $t$ is measured in days.
(a) Find the average rate of change of $A ( t )$ over the interval $0 \leq t \leq 30$. Indicate units of measure.
(b) Find the value of $A ^ { \prime } ( 15 )$. Using correct units, interpret the meaning of the value in the context of the problem.
(c) Find the time $t$ for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval $0 \leq t \leq 30$.
(d) For $t > 30$, $L ( t )$, the linear approximation to $A$ at $t = 30$, is a better model for the amount of grass clippings remaining in the bin. Use $L ( t )$ to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer.

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.