The figure above shows the graph of $f ^ { \prime }$, the derivative of a twice-differentiable function $f$, on the closed interval $[ 0,4 ]$. The areas of the regions bounded by the graph of $f ^ { \prime }$ and the $x$-axis on the intervals $[ 0,1 ] , [ 1,2 ] , [ 2,3 ]$, and $[ 3,4 ]$ are $2, 6, 10$, and $14$, respectively. The graph of $f ^ { \prime }$ has horizontal tangents at $x = 0.6 , x = 1.6$, $x = 2.5$, and $x = 3.5$. It is known that $f ( 2 ) = 5$.
(a) On what open intervals contained in $( 0,4 )$ is the graph of $f$ both decreasing and concave down? Give a reason for your answer.
(b) Find the absolute minimum value of $f$ on the interval $[ 0,4 ]$. Justify your answer.
(c) Evaluate $\int _ { 0 } ^ { 4 } f ( x ) f ^ { \prime } ( x ) \, d x$.
(d) The function $g$ is defined by $g ( x ) = x ^ { 3 } f ( x )$. Find $g ^ { \prime } ( 2 )$. Show the work that leads to your answer.

The graph of $f ^ { \prime }$, the derivative of the function $f$, is shown above. Which of the following could be the graph of $f$?
(A), (B), (C), (D) [graphs as shown in the exam]

The derivative of a function $f$ is given by $f ^ { \prime } ( x ) = e ^ { \sin x } - \cos x - 1$ for $0 < x < 9$. On what intervals is $f$ decreasing?
(A) $0 < x < 0.633$ and $4.115 < x < 6.916$
(B) $0 < x < 1.947$ and $5.744 < x < 8.230$
(C) $0.633 < x < 4.115$ and $6.916 < x < 9$
(D) $1.947 < x < 5.744$ and $8.230 < x < 9$

$x$	- 1	0	2	4	5
$f ^ { \prime } ( x )$	11	9	8	5	2

Let $f$ be a twice-differentiable function. Values of $f ^ { \prime }$, the derivative of $f$, at selected values of $x$ are given in the table above. Which of the following statements must be true?
(A) $f$ is increasing for $- 1 \leq x \leq 5$.
(B) The graph of $f$ is concave down for $- 1 < x < 5$.
(C) There exists $c$, where $- 1 < c < 5$, such that $f ^ { \prime } ( c ) = - \frac { 3 } { 2 }$.
(D) There exists $c$, where $- 1 < c < 5$, such that $f ^ { \prime \prime } ( c ) = - \frac { 3 } { 2 }$.

Let $f$ be the function with derivative defined by $f ^ { \prime } ( x ) = 2 + ( 2 x - 8 ) \sin ( x + 3 )$.
How many points of inflection does the graph of $f$ have on the interval $0 < x < 9$?
(A) One
(B) Two
(C) Three
(D) Four

Let $f$ be the function given by $f(x) = 2xe^{2x}$.
(a) Find $\lim_{x \rightarrow -\infty} f(x)$ and $\lim_{x \rightarrow \infty} f(x)$.
(b) Find the absolute minimum value of $f$. Justify that your answer is an absolute minimum.
(c) What is the range of $f$?
(d) Consider the family of functions defined by $y = bxe^{bx}$, where $b$ is a nonzero constant. Show that the absolute minimum value of $bxe^{bx}$ is the same for all nonzero values of $b$.

Let $h$ be a function defined for all $x \neq 0$ such that $h(4) = -3$ and the derivative of $h$ is given by $h^{\prime}(x) = \dfrac{x^{2} - 2}{x}$ for all $x \neq 0$.
(a) Find all values of $x$ for which the graph of $h$ has a horizontal tangent, and determine whether $h$ has a local maximum, a local minimum, or neither at each of these values. Justify your answers.
(b) On what intervals, if any, is the graph of $h$ concave up? Justify your answer.
(c) Write an equation for the line tangent to the graph of $h$ at $x = 4$.
(d) Does the line tangent to the graph of $h$ at $x = 4$ lie above or below the graph of $h$ for $x > 4$? Why?

A cubic polynomial function $f$ is defined by $$f(x) = 4x^{3} + ax^{2} + bx + k$$ where $a$, $b$, and $k$ are constants. The function $f$ has a local minimum at $x = -1$, and the graph of $f$ has a point of inflection at $x = -2$.
(a) Find the values of $a$ and $b$.
(b) If $\displaystyle\int_{0}^{1} f(x)\,dx = 32$, what is the value of $k$?

The number of gallons, $P(t)$, of a pollutant in a lake changes at the rate $P^{\prime}(t) = 1 - 3e^{-0.2\sqrt{t}}$ gallons per day, where $t$ is measured in days. There are 50 gallons of the pollutant in the lake at time $t = 0$. The lake is considered to be safe when it contains 40 gallons or less of pollutant.
(a) Is the amount of pollutant increasing at time $t = 9$? Why or why not?
(b) For what value of $t$ will the number of gallons of pollutant be at its minimum? Justify your answer.
(c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your answer.
(d) An investigator uses the tangent line approximation to $P(t)$ at $t = 0$ as a model for the amount of pollutant in the lake. At what time $t$ does this model predict that the lake becomes safe?

Let $f$ be a function defined on the closed interval $-3 \leq x \leq 4$ with $f(0) = 3$. The graph of $f'$, the derivative of $f$, consists of one line segment and a semicircle, as shown above.
(a) On what intervals, if any, is $f$ increasing? Justify your answer.
(b) Find the $x$-coordinate of each point of inflection of the graph of $f$ on the open interval $-3 < x < 4$. Justify your answer.
(c) Find an equation for the line tangent to the graph of $f$ at the point $(0, 3)$.
(d) Find $f(-3)$ and $f(4)$. Show the work that leads to your answers.

The figure above shows the graph of $f'$, the derivative of the function $f$, on the closed interval $-1 \leq x \leq 5$. The graph of $f'$ has horizontal tangent lines at $x = 1$ and $x = 3$. The function $f$ is twice differentiable with $f(2) = 6$.
(a) Find the $x$-coordinate of each of the points of inflection of the graph of $f$. Give a reason for your answer.
(b) At what value of $x$ does $f$ attain its absolute minimum value on the closed interval $-1 \leq x \leq 5$? At what value of $x$ does $f$ attain its absolute maximum value on the closed interval $-1 \leq x \leq 5$? Show the analysis that leads to your answers.
(c) Let $g$ be the function defined by $g(x) = x f(x)$. Find an equation for the line tangent to the graph of $g$ at $x = 2$.

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 below, where DNE indicates that the derivatives of $f$ do not exist at $x = 2$.

$x$	0	$0 < x < 1$	1	$1 < x < 2$	2	$2 < x < 3$	3	$3 < x < 4$
$f ( x )$	- 1	Negative	0	Positive	2	Positive	0	Negative
$f ^ { \prime } ( x )$	4	Positive	0	Positive	DNE	Negative	- 3	Negative
$f ^ { \prime \prime } ( x )$	- 2	Negative	0	Positive	DNE	Negative	0	Positive

(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 $f$.
(c) Let $g$ be the function defined by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) \, dt$ 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.

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 $f$ be the function given by $f ( x ) = \frac { \ln x } { x }$ for all $x > 0$. The derivative of $f$ is given by $f ^ { \prime } ( x ) = \frac { 1 - \ln x } { x ^ { 2 } }$.
(a) Write an equation for the line tangent to the graph of $f$ at $x = e ^ { 2 }$.
(b) Find the $x$-coordinate of the critical point of $f$. Determine whether this point is a relative minimum, a relative maximum, or neither for the function $f$. Justify your answer.
(c) The graph of the function $f$ has exactly one point of inflection. Find the $x$-coordinate of this point.
(d) Find $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$.

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.

The function $g$ is defined and differentiable on the closed interval $[-7, 5]$ and satisfies $g(0) = 5$. The graph of $y = g'(x)$, the derivative of $g$, consists of a semicircle and three line segments, as shown in the figure above.
(a) Find $g(3)$ and $g(-2)$.
(b) Find the $x$-coordinate of each point of inflection of the graph of $y = g(x)$ on the interval $-7 < x < 5$. Explain your reasoning.
(c) The function $h$ is defined by $h(x) = g(x) - \frac{1}{2}x^2$. Find the $x$-coordinate of each critical point of $h$, where $-7 < x < 5$, and classify each critical point as the location of a relative minimum, relative maximum, or neither a minimum nor a maximum. Explain your reasoning.

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) $[ 0 , \infty )$

The graph of a differentiable function $f$ is shown above. If $h ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$, which of the following is true?
(A) $h ( 6 ) < h ^ { \prime } ( 6 ) < h ^ { \prime \prime } ( 6 )$
(B) $h ( 6 ) < h ^ { \prime \prime } ( 6 ) < h ^ { \prime } ( 6 )$
(C) $h ^ { \prime } ( 6 ) < h ( 6 ) < h ^ { \prime \prime } ( 6 )$
(D) $h ^ { \prime \prime } ( 6 ) < h ( 6 ) < h ^ { \prime } ( 6 )$
(E) $h ^ { \prime \prime } ( 6 ) < h ^ { \prime } ( 6 ) < h ( 6 )$

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 $0 < x < 2$.

The graph of the function $f$ is shown in the figure above. For which of the following values of $x$ is $f ^ { \prime } ( x )$ positive and increasing?
(A) $a$
(B) $b$
(C) $c$
(D) $d$
(E) $e$

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) $1.234$