Let $f$ be a function defined on the closed interval $- 5 \leq x \leq 5$ with $f ( 1 ) = 3$. The graph of $f ^ { \prime }$, the derivative of $f$, consists of two semicircles and two line segments, as shown above. (a) For $- 5 < x < 5$, find all values $x$ at which $f$ has a relative maximum. Justify your answer. (b) For $- 5 < x < 5$, find all values $x$ at which the graph of $f$ has a point of inflection. Justify your answer. (c) Find all intervals on which the graph of $f$ is concave up and also has positive slope. Explain your reasoning. (d) Find the absolute minimum value of $f ( x )$ over the closed interval $- 5 \leq x \leq 5$. Explain your reasoning.

The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function $r$ of time $t$, where $t$ is measured in minutes. For $0 < t < 12$, the graph of $r$ is concave down. The table below gives selected values of the rate of change, $r^{\prime}(t)$, of the radius of the balloon over the time interval $0 \leq t \leq 12$.

\begin{tabular}{c} $t$

(minutes)

& 0 & 2 & 5 & 7 & 11 & 12 \hline

$r^{\prime}(t)$

(feet per minute)

& 5.7 & 4.0 & 2.0 & 1.2 & 0.6 & 0.5 \hline \end{tabular}
The radius of the balloon is 30 feet when $t = 5$. (Note: The volume of a sphere of radius $r$ is given by $V = \frac{4}{3}\pi r^{3}$.)
(a) Estimate the radius of the balloon when $t = 5.4$ using the tangent line approximation at $t = 5$. Is your estimate greater than or less than the true value? Give a reason for your answer.
(b) Find the rate of change of the volume of the balloon with respect to time when $t = 5$. Indicate units of measure.
(c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate $\int_{0}^{12} r^{\prime}(t)\, dt$. Using correct units, explain the meaning of $\int_{0}^{12} r^{\prime}(t)\, dt$ in terms of the radius of the balloon.
(d) Is your approximation in part (c) greater than or less than $\int_{0}^{12} r^{\prime}(t)\, dt$? Give a reason for your answer.

Let $f$ be the function defined by $f(x) = k\sqrt{x} - \ln x$ for $x > 0$, where $k$ is a positive constant.
(a) Find $f^{\prime}(x)$ and $f^{\prime\prime}(x)$.
(b) For what value of the constant $k$ does $f$ have a critical point at $x = 1$? For this value of $k$, determine whether $f$ has a relative minimum, relative maximum, or neither at $x = 1$. Justify your answer.
(c) For a certain value of the constant $k$, the graph of $f$ has a point of inflection on the $x$-axis. Find this value of $k$.

Let $g$ be a continuous function with $g ( 2 ) = 5$. The graph of the piecewise-linear function $g ^ { \prime }$, the derivative of $g$, is shown above for $- 3 \leq x \leq 7$. (a) Find the $x$-coordinate of all points of inflection of the graph of $y = g ( x )$ for $- 3 < x < 7$. Justify your answer. (b) Find the absolute maximum value of $g$ on the interval $- 3 \leq x \leq 7$. Justify your answer. (c) Find the average rate of change of $g ( x )$ on the interval $- 3 \leq x \leq 7$. (d) Find the average rate of change of $g ^ { \prime } ( x )$ on the interval $- 3 \leq x \leq 7$. Does the Mean Value Theorem applied on the interval $- 3 \leq x \leq 7$ guarantee a value of $c$, for $- 3 < c < 7$, such that $g ^ { \prime \prime } ( c )$ is equal to this average rate of change? Why or why not?

Let $f$ be a function that is continuous on the interval $[ 0,4 )$. The function $f$ is twice differentiable except at $x = 2$. The function $f$ and its derivatives have the properties indicated in the table above, where DNE indicates that the derivatives of $f$ do not exist at $x = 2$.
(a) For $0 < x < 4$, find all values of $x$ at which $f$ has a relative extremum. Determine whether $f$ has a relative maximum or a relative minimum at each of these values. Justify your answer.
(b) On the axes provided, sketch the graph of a function that has all the characteristics of $f$.
(Note: Use the axes provided in the pink test booklet.)
(c) Let $g$ be the function defined by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t$ on the open interval $( 0,4 )$. For $0 < x < 4$, find all values of $x$ at which $g$ has a relative extremum. Determine whether $g$ has a relative maximum or a relative minimum at each of these values. Justify your answer.
(d) For the function $g$ defined in part (c), find all values of $x$, for $0 < x < 4$, at which the graph of $g$ has a point of inflection. Justify your answer.

The derivative of a function $f$ is defined by $$f'(x) = \begin{cases} g(x) & \text{for } -4 \leq x \leq 0 \\ 5e^{-x/3} - 3 & \text{for } 0 < x \leq 4 \end{cases}.$$ The graph of the continuous function $f'$, shown in the figure above, has $x$-intercepts at $x = -2$ and $x = 3\ln\left(\frac{5}{3}\right)$. The graph of $g$ on $-4 \leq x \leq 0$ is a semicircle, and $f(0) = 5$.
(a) For $-4 < x < 4$, find all values of $x$ at which the graph of $f$ has a point of inflection. Justify your answer.
(b) Find $f(-4)$ and $f(4)$.
(c) For $-4 \leq x \leq 4$, find the value of $x$ at which $f$ has an absolute maximum. Justify your answer.

2. The function $g$ is defined for $x > 0$ with $g ( 1 ) = 2 , g ^ { \prime } ( x ) = \sin \left( x + \frac { 1 } { x } \right)$, and $g ^ { \prime \prime } ( x ) = \left( 1 - \frac { 1 } { x ^ { 2 } } \right) \cos \left( x + \frac { 1 } { x } \right)$.
(a) Find all values of $x$ in the interval $0.12 \leq x \leq 1$ at which the graph of $g$ has a horizontal tangent line.
(b) On what subintervals of $( 0.12,1 )$, if any, is the graph of $g$ concave down? Justify your answer.
(c) Write an equation for the line tangent to the graph of $g$ at $x = 0.3$.
(d) Does the line tangent to the graph of $g$ at $x = 0.3$ lie above or below the graph of $g$ for $0.3 < x < 1$ ? Why?

$t$	0	2	4	6	8	10	12
$P ( t )$	0	46	53	57	60	62	63

[Figure]

Consider a differentiable function $f$ having domain all positive real numbers, and for which it is known that $f^{\prime}(x) = (4 - x)x^{-3}$ for $x > 0$.
(a) Find the $x$-coordinate of the critical point of $f$. Determine whether the point is a relative maximum, a relative minimum, or neither for the function $f$. Justify your answer.
(b) Find all intervals on which the graph of $f$ is concave down. Justify your answer.
(c) Given that $f(1) = 2$, determine the function $f$.

Let $f$ be the function given by $f ( x ) = 300 x - x ^ { 3 }$. On which of the following intervals is the function $f$ increasing?
(A) $( - \infty , - 10 ]$ and $[ 10 , \infty )$
(B) $[ - 10,10 ]$
(C) $[ 0,10 ]$ only
(D) $[ 0,10 \sqrt { 3 } ]$ only
(E) $[ 0 , \infty )$

A particle moves along the $x$-axis with its position at time $t$ given by $x ( t ) = ( t - a ) ( t - b )$, where $a$ and $b$ are constants and $a \neq b$. For which of the following values of $t$ is the particle at rest?
(A) $t = a b$
(B) $t = \frac { a + b } { 2 }$
(C) $t = a + b$
(D) $t = 2 ( a + b )$
(E) $t = a$ and $t = b$

Let $f$ be the function defined by $f ( x ) = \frac { \ln x } { x }$. What is the absolute maximum value of $f$ ?
(A) 1
(B) $\frac { 1 } { e }$
(C) 0
(D) $- e$
(E) $f$ does not have an absolute maximum value.

Let $g$ be the function given by $g ( x ) = x ^ { 2 } e ^ { k x }$, where $k$ is a constant. For what value of $k$ does $g$ have a critical point at $x = \frac { 2 } { 3 }$ ?
(A) $-3$
(B) $- \frac { 3 } { 2 }$
(C) $- \frac { 1 } { 3 }$
(D) 0
(E) There is no such $k$.

Let $g$ be a function with first derivative given by $g ^ { \prime } ( x ) = \int _ { 0 } ^ { x } e ^ { - t ^ { 2 } } d t$. Which of the following must be true on the interval $0 < x < 2$ ?
(A) $g$ is increasing, and the graph of $g$ is concave up.
(B) $g$ is increasing, and the graph of $g$ is concave down.
(C) $g$ is decreasing, and the graph of $g$ is concave up.
(D) $g$ is decreasing, and the graph of $g$ is concave down.
(E) $g$ is decreasing, and the graph of $g$ has a point of inflection on $0 < x < 2$.

If $( x + 2 y ) \cdot \frac { d y } { d x } = 2 x - y$, what is the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 3,0 )$ ?
(A) $- \frac { 10 } { 3 }$
(B) 0
(C) 2
(D) $\frac { 10 } { 3 }$
(E) Undefined

The graph of $f ^ { \prime }$, the derivative of the function $f$, is shown above. Which of the following statements must be true?
I. $f$ has a relative minimum at $x = - 3$.
II. The graph of $f$ has a point of inflection at $x = - 2$.
III. The graph of $f$ is concave down for $0 < x < 4$.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

If $f ^ { \prime } ( x ) = \sqrt { x ^ { 4 } + 1 } + x ^ { 3 } - 3 x$, then $f$ has a local maximum at $x =$
(A) $-2.314$
(B) $-1.332$
(C) $0.350$
(D) $0.829$
(E) $1.234$

For $- 1.5 < x < 1.5$, let $f$ be a function with first derivative given by $f ^ { \prime } ( x ) = e ^ { \left( x ^ { 4 } - 2 x ^ { 2 } + 1 \right) } - 2$. Which of the following are all intervals on which the graph of $f$ is concave down?
(A) $(-0.418, 0.418)$ only
(B) $( - 1,1 )$
(C) $( - 1.354 , - 0.409 )$ and $( 0.409,1.354 )$
(D) $( - 1.5 , - 1 )$ and $( 0,1 )$
(E) $( - 1.5 , - 1.354 ) , ( - 0.409,0 )$, and $( 1.354,1.5 )$

The graph of $f ^ { \prime }$, the derivative of $f$, is shown in the figure above. The function $f$ has a local maximum at $x =$
(A) $-3$
(B) $-1$
(C) 1
(D) 3
(E) 4

If $f ^ { \prime } ( x ) > 0$ for all real numbers $x$ and $\int _ { 4 } ^ { 7 } f ( t ) d t = 0$, which of the following could be a table of values for the function $f$ ?
(A)

$x$	$f ( x )$
4	$-4$
5	$-3$
7	0

(B)

$x$	$f ( x )$
4	$-4$
5	$-2$
7	5

(C)

$x$	$f ( x )$
4	$-4$
5	6
7	3

(D)

$x$	$f ( x )$
4	0
5	0
7	0

(E)

$x$	$f ( x )$
4	0
5	4
7	6

The graph of $f ^ { \prime \prime }$, the second derivative of $f$, is shown above for $- 2 \leq x \leq 4$. What are all intervals on which the graph of the function $f$ is concave down?
(A) $-1 < x < 1$
(B) $0 < x < 2$
(C) $1 < x < 3$ only
(D) $-2 < x < -1$ only
(E) $-2 < x < -1$ and $1 < x < 3$

Let $f$ be a polynomial function with values of $f ^ { \prime } ( x )$ at selected values of $x$ given in the table above.

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

Which of the following must be true for $-2 < x < 6$ ?
(A) The graph of $f$ is concave up.
(B) The graph of $f$ has at least two points of inflection.
(C) $f$ is increasing.
(D) $f$ has no critical points.
(E) $f$ has at least two relative extrema.

The figure above shows the graph of $f ^ { \prime }$, the derivative of a twice-differentiable function $f$, on the closed interval $0 \leq x \leq 8$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = 1$, $x = 3$, and $x = 5$. The areas of the regions between the graph of $f ^ { \prime }$ and the $x$-axis are labeled in the figure. The function $f$ is defined for all real numbers and satisfies $f ( 8 ) = 4$.
(a) Find all values of $x$ on the open interval $0 < x < 8$ for which the function $f$ has a local minimum. Justify your answer.
(b) Determine the absolute minimum value of $f$ on the closed interval $0 \leq x \leq 8$. Justify your answer.
(c) On what open intervals contained in $0 < x < 8$ is the graph of $f$ both concave down and increasing? Explain your reasoning.
(d) The function $g$ is defined by $g ( x ) = ( f ( x ) ) ^ { 3 }$. If $f ( 3 ) = - \frac { 5 } { 2 }$, find the slope of the line tangent to the graph of $g$ at $x = 3$.

The twice-differentiable functions $f$ and $g$ are defined for all real numbers $x$. Values of $f$, $f ^ { \prime }$, $g$, and $g ^ { \prime }$ for various values of $x$ are given in the table below.

$x$	-2	$- 2 < x < - 1$	-1	$- 1 < x < 1$	1	$1 < x < 3$	3
$f ( x )$	12	Positive	8	Positive	2	Positive	7
$f ^ { \prime } ( x )$	-5	Negative	0	Negative	0	Positive	$\frac { 1 } { 2 }$
$g ( x )$	-1	Negative	0	Positive	3	Positive	1
$g ^ { \prime } ( x )$	2	Positive	$\frac { 3 } { 2 }$	Positive	0	Negative	-2

(a) Find the $x$-coordinate of each relative minimum of $f$ on the interval $[ - 2, 3 ]$. Justify your answers.
(b) Explain why there must be a value $c$, for $- 1 < c < 1$, such that $f ^ { \prime \prime } ( c ) = 0$.
(c) The function $h$ is defined by $h ( x ) = \ln ( f ( x ) )$. Find $h ^ { \prime } ( 3 )$. Show the computations that lead to your answer.
(d) Evaluate $\displaystyle\int _ { - 2 } ^ { 3 } f ^ { \prime } ( g ( x ) ) g ^ { \prime } ( x ) \, dx$.

The figure above shows the graph of $f'$, the derivative of a twice-differentiable function $f$, on the interval $[-3, 4]$. The graph of $f'$ has horizontal tangents at $x = -1$, $x = 1$, and $x = 3$. The areas of the regions bounded by the $x$-axis and the graph of $f'$ on the intervals $[-2, 1]$ and $[1, 4]$ are 9 and 12, respectively.
(a) Find all $x$-coordinates at which $f$ has a relative maximum. Give a reason for your answer.
(b) On what open intervals contained in $-3 < x < 4$ is the graph of $f$ both concave down and decreasing? Give a reason for your answer.
(c) Find the $x$-coordinates of all points of inflection for the graph of $f$. Give a reason for your answer.
(d) Given that $f(1) = 3$, write an expression for $f(x)$ that involves an integral. Find $f(4)$ and $f(-2)$.

The figure above shows the graph of the piecewise-linear function $f$. For $- 4 \leq x \leq 12$, the function $g$ is defined by $g ( x ) = \int _ { 2 } ^ { x } f ( t ) \, dt$.
(a) Does $g$ have a relative minimum, a relative maximum, or neither at $x = 10$? Justify your answer.
(b) Does the graph of $g$ have a point of inflection at $x = 4$? Justify your answer.
(c) Find the absolute minimum value and the absolute maximum value of $g$ on the interval $- 4 \leq x \leq 12$. Justify your answers.
(d) For $- 4 \leq x \leq 12$, find all intervals for which $g ( x ) \leq 0$.