The graph of a differentiable function $f$ is shown above for $- 3 \leq x \leq 3$. The graph of $f$ has horizontal tangent lines at $x = - 1 , x = 1$, and $x = 2$. The areas of regions $A , B , C$, and $D$ are 5, 4, 5, and 3, respectively. Let $g$ be the antiderivative of $f$ such that $g ( 3 ) = 7$.
(a) Find all values of $x$ on the open interval $- 3 < x < 3$ for which the function $g$ has a relative maximum. Justify your answer.
(b) On what open intervals contained in $- 3 < x < 3$ is the graph of $g$ concave up? Give a reason for your answer.
(c) Find the value of $\lim _ { x \rightarrow 0 } \frac { g ( x ) + 1 } { 2 x }$, or state that it does not exist. Show the work that leads to your answer.
(d) Let $h$ be the function defined by $h ( x ) = 3 f ( 2 x + 1 ) + 4$. Find the value of $\int _ { - 2 } ^ { 1 } h ( x ) d x$.

$t$ (hours)	0	2	4	6	8	10	12
$R ( t )$ (vehicles per hour)	2935	3653	3442	3010	3604	1986	2201

On a certain weekday, the rate at which vehicles cross a bridge is modeled by the differentiable function $R$ for $0 \leq t \leq 12$, where $R ( t )$ is measured in vehicles per hour and $t$ is the number of hours since 7:00 A.M. $(t = 0)$. Values of $R ( t )$ for selected values of $t$ are given in the table above.
(a) Use the data in the table to approximate $R ^ { \prime } ( 5 )$. Show the computations that lead to your answer. Using correct units, explain the meaning of $R ^ { \prime } ( 5 )$ in the context of the problem.
(b) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of $\int _ { 0 } ^ { 12 } R ( t ) \, d t$. Indicate units of measure.
(c) On a certain weekend day, the rate at which vehicles cross the bridge is modeled by the function $H$ defined by $H ( t ) = - t ^ { 3 } - 3 t ^ { 2 } + 288 t + 1300$ for $0 \leq t \leq 17$, where $H ( t )$ is measured in vehicles per hour and $t$ is the number of hours since 7:00 A.M. $(t = 0)$. According to this model, what is the average number of vehicles crossing the bridge per hour on the weekend day for $0 \leq t \leq 12$?
(d) For $12 < t < 17$, $L ( t )$, the local linear approximation to the function $H$ given in part (c) at $t = 12$, is a better model for the rate at which vehicles cross the bridge on the weekend day. Use $L ( t )$ to find the time $t$, for $12 < t < 17$, at which the rate of vehicles crossing the bridge is 2000 vehicles per hour. Show the work that leads to your answer.

The figure above shows the graph of the continuous function $g$ on the interval $[ 0,8 ]$. Let $h$ be the function defined by $h ( x ) = \int _ { 3 } ^ { x } g ( t ) \, d t$. On what intervals is $h$ increasing?
(A) $[ 2,5 ]$ only
(B) $[ 1,7 ]$
(C) $[ 0,1 ]$ and $[ 3,7 ]$
(D) $[ 1,3 ]$ and $[ 7,8 ]$

Which of the following limits is equal to $\int _ { 3 } ^ { 5 } x ^ { 4 } d x$ ?
(A) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 3 + \frac { k } { n } \right) ^ { 4 } \frac { 1 } { n }$
(B) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 3 + \frac { k } { n } \right) ^ { 4 } \frac { 2 } { n }$
(C) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 3 + \frac { 2 k } { n } \right) ^ { 4 } \frac { 1 } { n }$
(D) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 3 + \frac { 2 k } { n } \right) ^ { 4 } \frac { 2 } { n }$

The function $f$ is continuous for $- 4 \leq x \leq 4$. The graph of $f$ shown above consists of five line segments. What is the average value of $f$ on the interval $- 4 \leq x \leq 4$ ?
(A) $\frac { 1 } { 8 }$
(B) $\frac { 3 } { 16 }$
(C) $\frac { 15 } { 16 }$
(D) $\frac { 3 } { 2 }$

A rain barrel collects water off the roof of a house during three hours of heavy rainfall. The height of the water in the barrel increases at the rate of $r ( t ) = 4 t ^ { 3 } e ^ { - 1.5 t }$ feet per hour, where $t$ is the time in hours since the rain began. At time $t = 1$ hour, the height of the water is 0.75 foot. What is the height of the water in the barrel at time $t = 2$ hours?
(A) 1.361 ft
(B) 1.500 ft
(C) 1.672 ft
(D) 2.111 ft

Honey is poured through a funnel at a rate of $r ( t ) = 4 e ^ { - 0.35 t }$ ounces per minute, where $t$ is measured in minutes. How many ounces of honey are poured through the funnel from time $t = 0$ to time $t = 3$?
(A) 0.910
(B) 1.400
(C) 2.600
(D) 7.429

The temperature outside a house during a 24-hour period is given by $$F(t) = 80 - 10\cos\left(\frac{\pi t}{12}\right), \quad 0 \leq t \leq 24,$$ where $F(t)$ is measured in degrees Fahrenheit and $t$ is measured in hours.
(a) Sketch the graph of $F$ on the grid provided.
(b) Find the average temperature, to the nearest degree Fahrenheit, between $t = 6$ and $t = 14$.
(c) An air conditioner cooled the house whenever the outside temperature was at or above 78 degrees Fahrenheit. For what values of $t$ was the air conditioner cooling the house?
(d) The cost of cooling the house accumulates at the rate of $\$0.05$ per hour for each degree the outside temperature exceeds 78 degrees Fahrenheit. What was the total cost, to the nearest cent, to cool the house for this 24-hour period?

The graph of the function $f$, consisting of three line segments, is given above. Let $g ( x ) = \int _ { 1 } ^ { x } f ( t ) \, dt$.
(a) Compute $g ( 4 )$ and $g ( - 2 )$.
(b) Find the instantaneous rate of change of $g$, with respect to $x$, at $x = 1$.
(c) Find the absolute minimum value of $g$ on the closed interval $[ - 2, 4 ]$. Justify your answer.
(d) The second derivative of $g$ is not defined at $x = 1$ and $x = 2$. How many of these values are $x$-coordinates of points of inflection of the graph of $g$ ? Justify your answer.

The rate at which people enter an amusement park on a given day is modeled by the function $E$ defined by $$E ( t ) = \frac { 15600 } { \left( t ^ { 2 } - 24 t + 160 \right) }$$ The rate at which people leave the same amusement park on the same day is modeled by the function $L$ defined by $$L ( t ) = \frac { 9890 } { \left( t ^ { 2 } - 38 t + 370 \right) }$$ Both $E ( t )$ and $L ( t )$ are measured in people per hour and time $t$ is measured in hours after midnight. These functions are valid for $9 \leq t \leq 23$, the hours during which the park is open. At time $t = 9$, there are no people in the park.
(a) How many people have entered the park by 5:00 P.M. ( $t = 17$ )? Round your answer to the nearest whole number.
(b) The price of admission to the park is $\$15$ until 5:00 P.M. ( $t = 17$ ). After 5:00 P.M., the price of admission to the park is $\$11$. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number.
(c) Let $H ( t ) = \int _ { 9 } ^ { t } ( E ( x ) - L ( x ) ) d x$ for $9 \leq t \leq 23$. The value of $H ( 17 )$ to the nearest whole number is 3725. Find the value of $H ^ { \prime } ( 17 )$, and explain the meaning of $H ( 17 )$ and $H ^ { \prime } ( 17 )$ in the context of the amusement park.
(d) At what time $t$, for $9 \leq t \leq 23$, does the model predict that the number of people in the park is a maximum?

The graph of the function $f$ shown above consists of two line segments. Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$.
(a) Find $g ( - 1 ) , g ^ { \prime } ( - 1 )$, and $g ^ { \prime \prime } ( - 1 )$.
(b) For what values of $x$ in the open interval $( - 2, 2 )$ is $g$ increasing? Explain your reasoning.
(c) For what values of $x$ in the open interval $( - 2, 2 )$ is the graph of $g$ concave down? Explain your reasoning.
(d) On the axes provided, sketch the graph of $g$ on the closed interval $[ - 2, 2 ]$.

Let $f$ be a function that is differentiable for all real numbers. The table below gives the values of $f$ and its derivative $f ^ { \prime }$ for selected points $x$ in the closed interval $- 1.5 \leq x \leq 1.5$. The second derivative of $f$ has the property that $f ^ { \prime \prime } ( x ) > 0$ for $- 1.5 \leq x \leq 1.5$.

$x$	- 1.5	- 1.0	- 0.5	0	0.5	1.0	1.5
$f ( x )$	- 1	- 4	- 6	- 7	- 6	- 4	- 1
$f ^ { \prime } ( x )$	- 7	- 5	- 3	0	3	5	7

(a) Evaluate $\int _ { 0 } ^ { 1.5 } \left( 3 f ^ { \prime } ( x ) + 4 \right) d x$. Show the work that leads to your answer.
(b) Write an equation of the line tangent to the graph of $f$ at the point where $x = 1$. Use this line to approximate the value of $f ( 1.2 )$. Is this approximation greater than or less than the actual value of $f ( 1.2 )$? Give a reason for your answer.
(c) Find a positive real number $r$ having the property that there must exist a value $c$ with $0 < c < 0.5$ and $f ^ { \prime \prime } ( c ) = r$. Give a reason for your answer.
(d) Let $g$ be the function given by $g ( x ) = \begin{cases} 2 x ^ { 2 } - x - 7 & \text { for } x < 0 \\ 2 x ^ { 2 } + x - 7 & \text { for } x \geq 0 . \end{cases}$ The graph of $g$ passes through each of the points $( x , f ( x ) )$ given in the table above. Is it possible that $f$ and $g$ are the same function? Give a reason for your answer.

Let $f$ be the function defined by $$f(x) = \begin{cases} \sqrt{x+1} & \text{for } 0 \leq x \leq 3 \\ 5 - x & \text{for } 3 < x \leq 5. \end{cases}$$ (a) Is $f$ continuous at $x = 3$? Explain why or why not.
(b) Find the average value of $f(x)$ on the closed interval $0 \leq x \leq 5$.
(c) Suppose the function $g$ is defined by $$g(x) = \begin{cases} k\sqrt{x+1} & \text{for } 0 \leq x \leq 3 \\ mx + 2 & \text{for } 3 < x \leq 5, \end{cases}$$ where $k$ and $m$ are constants. If $g$ is differentiable at $x = 3$, what are the values of $k$ and $m$?

For $0 \leq t \leq 31$, the rate of change of the number of mosquitoes on Tropical Island at time $t$ days is modeled by $R(t) = 5\sqrt{t}\cos\left(\frac{t}{5}\right)$ mosquitoes per day. There are 1000 mosquitoes on Tropical Island at time $t = 0$.
(a) Show that the number of mosquitoes is increasing at time $t = 6$.
(b) At time $t = 6$, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes increasing at a decreasing rate? Give a reason for your answer.
(c) According to the model, how many mosquitoes will be on the island at time $t = 31$? Round your answer to the nearest whole number.
(d) To the nearest whole number, what is the maximum number of mosquitoes for $0 \leq t \leq 31$? Show the analysis that leads to your conclusion.

The tide removes sand from Sandy Point Beach at a rate modeled by the function $R$, given by $$R ( t ) = 2 + 5 \sin \left( \frac { 4 \pi t } { 25 } \right)$$ A pumping station adds sand to the beach at a rate modeled by the function $S$, given by $$S ( t ) = \frac { 15 t } { 1 + 3 t }$$ Both $R ( t )$ and $S ( t )$ have units of cubic yards per hour and $t$ is measured in hours for $0 \leq t \leq 6$. At time $t = 0$, the beach contains 2500 cubic yards of sand.
(a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure.
(b) Write an expression for $Y ( t )$, the total number of cubic yards of sand on the beach at time $t$.
(c) Find the rate at which the total amount of sand on the beach is changing at time $t = 4$.
(d) For $0 \leq t \leq 6$, at what time $t$ is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.

A water tank at Camp Newton holds 1200 gallons of water at time $t = 0$. During the time interval $0 \leq t \leq 18$ hours, water is pumped into the tank at the rate $$W(t) = 95\sqrt{t}\sin^2\left(\frac{t}{6}\right) \text{ gallons per hour.}$$ During the same time interval, water is removed from the tank at the rate $$R(t) = 275\sin^2\left(\frac{t}{3}\right) \text{ gallons per hour.}$$
(a) Is the amount of water in the tank increasing at time $t = 15$? Why or why not?
(b) To the nearest whole number, how many gallons of water are in the tank at time $t = 18$?
(c) At what time $t$, for $0 \leq t \leq 18$, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion.
(d) For $t > 18$, no water is pumped into the tank, but water continues to be removed at the rate $R(t)$ until the tank becomes empty. Let $k$ be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of $k$.

The graph of the function $f$ consists of three line segments.
(a) Let $g$ be the function given by $g(x) = \int_{-4}^{x} f(t)\, dt$. For each of $g(-1)$, $g'(-1)$, and $g''(-1)$, find the value or state that it does not exist.
(b) For the function $g$ defined in part (a), find the $x$-coordinate of each point of inflection of the graph of $g$ on the open interval $-4 < x < 3$. Explain your reasoning.
(c) Let $h$ be the function given by $h(x) = \int_{x}^{3} f(t)\, dt$. Find all values of $x$ in the closed interval $-4 \leq x \leq 3$ for which $h(x) = 0$.
(d) For the function $h$ defined in part (c), find all intervals on which $h$ is decreasing. Explain your reasoning.

At an intersection in Thomasville, Oregon, cars turn left at the rate $L(t) = 60\sqrt{t}\sin^{2}\left(\frac{t}{3}\right)$ cars per hour over the time interval $0 \leq t \leq 18$ hours. The graph of $y = L(t)$ is shown above.
(a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval $0 \leq t \leq 18$ hours.
(b) Traffic engineers will consider turn restrictions when $L(t) \geq 150$ cars per hour. Find all values of $t$ for which $L(t) \geq 150$ and compute the average value of $L$ over this time interval. Indicate units of measure.
(c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.

The graph of the function $f$ shown above consists of six line segments. Let $g$ be the function given by $g(x) = \int_{0}^{x} f(t)\, dt$.
(a) Find $g(4)$, $g'(4)$, and $g''(4)$.
(b) Does $g$ have a relative minimum, a relative maximum, or neither at $x = 1$? Justify your answer.
(c) Suppose that $f$ is defined for all real numbers $x$ and is periodic with a period of length 5. The graph above shows two periods of $f$. Given that $g(5) = 2$, find $g(10)$ and write an equation for the line tangent to the graph of $g$ at $x = 108$.

The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval $0 \leq t \leq 7$, where $t$ is measured in hours. In this model, rates are given as follows:
(i) The rate at which water enters the tank is $f(t) = 100t^{2}\sin(\sqrt{t})$ gallons per hour for $0 \leq t \leq 7$.
(ii) The rate at which water leaves the tank is $$g(t) = \left\{ \begin{array}{r} 250 \text{ for } 0 \leq t < 3 \\ 2000 \text{ for } 3 < t \leq 7 \end{array} \right. \text{ gallons per hour.}$$ The graphs of $f$ and $g$, which intersect at $t = 1.617$ and $t = 5.076$, are shown in the figure above. At time $t = 0$, the amount of water in the tank is 5000 gallons.
(a) How many gallons of water enter the tank during the time interval $0 \leq t \leq 7$? Round your answer to the nearest gallon.
(b) For $0 \leq t \leq 7$, find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer.
(c) For $0 \leq t \leq 7$, at what time $t$ is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer.

The rate at which people enter an auditorium for a rock concert is modeled by the function $R$ given by $R(t) = 1380t^{2} - 675t^{3}$ for $0 \leq t \leq 2$ hours; $R(t)$ is measured in people per hour. No one is in the auditorium at time $t = 0$, when the doors open. The doors close and the concert begins at time $t = 2$.
(a) How many people are in the auditorium when the concert begins?
(b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer.
(c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function $w$ models the total wait time for all the people who enter the auditorium before time $t$. The derivative of $w$ is given by $w'(t) = (2 - t)R(t)$. Find $w(2) - w(1)$, the total wait time for those who enter the auditorium after time $t = 1$.
(d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c).

Mighty Cable Company manufactures cable that sells for $\$120$ per meter. For a cable of fixed length, the cost of producing a portion of the cable varies with its distance from the beginning of the cable. Mighty reports that the cost to produce a portion of a cable that is $x$ meters from the beginning of the cable is $6\sqrt{x}$ dollars per meter. (Note: Profit is defined to be the difference between the amount of money received by the company for selling the cable and the company's cost of producing the cable.)
(a) Find Mighty's profit on the sale of a 25-meter cable.
(b) Using correct units, explain the meaning of $\int_{25}^{30} 6\sqrt{x} \, dx$ in the context of this problem.
(c) Write an expression, involving an integral, that represents Mighty's profit on the sale of a cable that is $k$ meters long.
(d) Find the maximum profit that Mighty could earn on the sale of one cable. Justify your answer.

Let $f$ be a function that is twice differentiable for all real numbers. The table above gives values of $f$ for selected points in the closed interval $2 \leq x \leq 13$.

$x$	2	3	5	8	13
$f(x)$	1	4	$-2$	3	6

(a) Estimate $f'(4)$. Show the work that leads to your answer.
(b) Evaluate $\int_{2}^{13} \left(3 - 5f'(x)\right) dx$. Show the work that leads to your answer.
(c) Use a left Riemann sum with subintervals indicated by the data in the table to approximate $\int_{2}^{13} f(x) \, dx$. Show the work that leads to your answer.
(d) Suppose $f'(5) = 3$ and $f''(x) < 0$ for all $x$ in the closed interval $5 \leq x \leq 8$. Use the line tangent to the graph of $f$ at $x = 5$ to show that $f(7) \leq 4$. Use the secant line for the graph of $f$ on $5 \leq x \leq 8$ to show that $f(7) \geq \frac{4}{3}$.

There is no snow on Janet's driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow accumulates on the driveway at a rate modeled by $f(t) = 7te^{\cos t}$ cubic feet per hour, where $t$ is measured in hours since midnight. Janet starts removing snow at 6 A.M. ($t = 6$). The rate $g(t)$, in cubic feet per hour, at which Janet removes snow from the driveway at time $t$ hours after midnight is modeled by $$g(t) = \begin{cases} 0 & \text{for } 0 \leq t < 6 \\ 125 & \text{for } 6 \leq t < 7 \\ 108 & \text{for } 7 \leq t \leq 9. \end{cases}$$
(a) How many cubic feet of snow have accumulated on the driveway by 6 A.M.?
(b) Find the rate of change of the volume of snow on the driveway at 8 A.M.
(c) Let $h(t)$ represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time $t$ hours after midnight. Express $h$ as a piecewise-defined function with domain $0 \leq t \leq 9$.
(d) How many cubic feet of snow are on the driveway at 9 A.M.?

A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon ($t = 0$) and 8 P.M. ($t = 8$). The number of entries in the box $t$ hours after noon is modeled by a differentiable function $E$ for $0 \leq t \leq 8$. Values of $E(t)$, in hundreds of entries, at various times $t$ are shown in the table below.

\begin{tabular}{ c } $t$

(hours)

& 0 & 2 & 5 & 7 & 8 \hline

$E(t)$

(hundreds of

entries)

& 0 & 4 & 13 & 21 & 23 \hline \end{tabular}
(a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time $t = 6$. Show the computations that lead to your answer.
(b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of $\frac{1}{8}\int_{0}^{8} E(t)\, dt$. Using correct units, explain the meaning of $\frac{1}{8}\int_{0}^{8} E(t)\, dt$ in terms of the number of entries.
(c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function $P$, where $P(t) = t^3 - 30t^2 + 298t - 976$ hundreds of entries per hour for $8 \leq t \leq 12$. According to the model, how many entries had not yet been processed by midnight ($t = 12$)?
(d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer.