The graph of a differentiable function $f$ on the closed interval $[-3, 15]$ is shown in the figure above. The graph of $f$ has a horizontal tangent line at $x = 6$. Let $g(x) = 5 + \int_{6}^{x} f(t)\, dt$ for $-3 \leq x \leq 15$.
(a) Find $g(6)$, $g^{\prime}(6)$, and $g^{\prime\prime}(6)$.
(b) On what intervals is $g$ decreasing? Justify your answer.
(c) On what intervals is the graph of $g$ concave down? Justify your answer.
(d) Find a trapezoidal approximation of $\int_{-3}^{15} f(t)\, dt$ using six subintervals of length $\Delta t = 3$.

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 $m$?

Let $\ell$ be the line tangent to the graph of $y = x^{n}$ at the point $(1,1)$, where $n > 1$.
(a) Find $\displaystyle\int_{0}^{1} x^{n}\,dx$ in terms of $n$.
(b) Let $T$ be the triangular region bounded by $\ell$, the $x$-axis, and the line $x = 1$. Show that the area of $T$ is $\dfrac{1}{2n}$.
(c) Let $S$ be the region bounded by the graph of $y = x^{n}$, the line $\ell$, and the $x$-axis. Express the area of $S$ in terms of $n$ and determine the value of $n$ that maximizes the area of $S$.

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.

For time $t \geq 0$ hours, let $r ( t ) = 120 \left( 1 - e ^ { - 10 t ^ { 2 } } \right)$ represent the speed, in kilometers per hour, at which a car travels along a straight road. The number of liters of gasoline used by the car to travel $x$ kilometers is modeled by $g ( x ) = 0.05 x \left( 1 - e ^ { - x / 2 } \right)$. (a) How many kilometers does the car travel during the first 2 hours? (b) Find the rate of change with respect to time of the number of liters of gasoline used by the car when $t = 2$ hours. Indicate units of measure. (c) How many liters of gasoline have been used by the car when it reaches a speed of 80 kilometers per hour?

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).

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.

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.

There are 700 people in line for a popular amusement-park ride when the ride begins operation in the morning. Once it begins operation, the ride accepts passengers until the park closes 8 hours later. While there is a line, people move onto the ride at a rate of 800 people per hour. The graph above shows the rate, $r(t)$, at which people arrive at the ride throughout the day. Time $t$ is measured in hours from the time the ride begins operation.
(a) How many people arrive at the ride between $t = 0$ and $t = 3$? Show the computations that lead to your answer.
(b) Is the number of people waiting in line to get on the ride increasing or decreasing between $t = 2$ and $t = 3$? Justify your answer.
(c) At what time $t$ is the line for the ride the longest? How many people are in line at that time? Justify your answers.
(d) Write, but do not solve, an equation involving an integral expression of $r$ whose solution gives the earliest time $t$ at which there is no longer a line for the ride.

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.

A cylindrical can of radius 10 millimeters is used to measure rainfall in Stormville. The can is initially empty, and rain enters the can during a 60-day period. The height of water in the can is modeled by the function $S$, where $S(t)$ is measured in millimeters and $t$ is measured in days for $0 \leq t \leq 60$. The rate at which the height of the water is rising in the can is given by $S^{\prime}(t) = 2\sin(0.03t) + 1.5$.
(a) According to the model, what is the height of the water in the can at the end of the 60-day period?
(b) According to the model, what is the average rate of change in the height of water in the can over the 60-day period? Show the computations that lead to your answer. Indicate units of measure.
(c) Assuming no evaporation occurs, at what rate is the volume of water in the can changing at time $t = 7$? Indicate units of measure.
(d) During the same 60-day period, rain on Monsoon Mountain accumulates in a can identical to the one in Stormville. The height of the water in the can on Monsoon Mountain is modeled by the function $M$, where $M(t) = \frac{1}{400}\left(3t^3 - 30t^2 + 330t\right)$. The height $M(t)$ is measured in millimeters, and $t$ is measured in days for $0 \leq t \leq 60$. Let $D(t) = M^{\prime}(t) - S^{\prime}(t)$. Apply the Intermediate Value Theorem to the function $D$ on the interval $0 \leq t \leq 60$ to justify that there exists a time $t$, $0 < t < 60$, at which the heights of water in the two cans are changing at the same rate.

A 12,000-liter tank of water is filled to capacity. At time $t = 0$, water begins to drain out of the tank at a rate modeled by $r(t)$, measured in liters per hour, where $r$ is given by the piecewise-defined function $$r(t) = \begin{cases} \dfrac{600t}{t+3} & \text{for } 0 \leq t \leq 5 \\ 1000e^{-0.2t} & \text{for } t > 5 \end{cases}$$
(a) Is $r$ continuous at $t = 5$? Show the work that leads to your answer.
(b) Find the average rate at which water is draining from the tank between time $t = 0$ and time $t = 8$ hours.
(c) Find $r^{\prime}(3)$. Using correct units, explain the meaning of that value in the context of this problem.
(d) Write, but do not solve, an equation involving an integral to find the time $A$ when the amount of water in the tank is 9000 liters.

The continuous function $f$ is defined on the interval $-4 \leq x \leq 3$. The graph of $f$ consists of two quarter circles and one line segment, as shown in the figure. Let $g(x) = 2x + \int_{0}^{x} f(t)\,dt$.
(a) Find $g(-3)$. Find $g'(x)$ and evaluate $g'(-3)$.
(b) Determine the $x$-coordinate of the point at which $g$ has an absolute maximum on the interval $-4 \leq x \leq 3$. Justify your answer.
(c) Find all values of $x$ on the interval $-4 < x < 3$ for which the graph of $g$ has a point of inflection. Give a reason for your answer.
(d) Find the average rate of change of $f$ on the interval $-4 \leq x \leq 3$. There is no point $c$, $-4 < c < 3$, for which $f'(c)$ is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem.

Let $g$ be the piecewise-linear function defined on $[-2\pi, 4\pi]$ whose graph is given above, and let $f(x) = g(x) - \cos\left(\dfrac{x}{2}\right)$.
(a) Find $\int_{-2\pi}^{4\pi} f(x)\, dx$. Show the computations that lead to your answer.
(b) Find all $x$-values in the open interval $(-2\pi, 4\pi)$ for which $f$ has a critical point.
(c) Let $h(x) = \int_{0}^{3x} g(t)\, dt$. Find $h^{\prime}\!\left(-\dfrac{\pi}{3}\right)$.

The temperature of water in a tub at time $t$ is modeled by a strictly increasing, twice-differentiable function $W$, where $W ( t )$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t = 0$, the temperature of the water is $55 ^ { \circ } \mathrm { F }$. The water is heated for 30 minutes, beginning at time $t = 0$. Values of $W ( t )$ at selected times $t$ for the first 20 minutes are given in the table above.

$t$ (minutes)	0	4	9	15	20
$W ( t )$ (degrees Fahrenheit)	55.0	57.1	61.8	67.9	71.0

(a) Use the data in the table to estimate $W ^ { \prime } ( 12 )$. Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.
(b) Use the data in the table to evaluate $\int _ { 0 } ^ { 20 } W ^ { \prime } ( t ) d t$. Using correct units, interpret the meaning of $\int _ { 0 } ^ { 20 } W ^ { \prime } ( t ) d t$ in the context of this problem.
(c) For $0 \leq t \leq 20$, the average temperature of the water in the tub is $\frac { 1 } { 20 } \int _ { 0 } ^ { 20 } W ( t ) d t$. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate $\frac { 1 } { 20 } \int _ { 0 } ^ { 20 } W ( t ) d t$. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning.
(d) For $20 \leq t \leq 25$, the function $W$ that models the water temperature has first derivative given by $W ^ { \prime } ( t ) = 0.4 \sqrt { t } \cos ( 0.06 t )$. Based on the model, what is the temperature of the water at time $t = 25$ ?

Let $f$ be the continuous function defined on $[ - 4,3 ]$ whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let $g$ be the function given by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t$.
(a) Find the values of $g ( 2 )$ and $g ( - 2 )$.
(b) For each of $g ^ { \prime } ( - 3 )$ and $g ^ { \prime \prime } ( - 3 )$, find the value or state that it does not exist.
(c) Find the $x$-coordinate of each point at which the graph of $g$ has a horizontal tangent line. For each of these points, determine whether $g$ has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers.
(d) For $- 4 < x < 3$, find all values of $x$ for which the graph of $g$ has a point of inflection. Explain your reasoning.

Let $f$ be the continuous function defined on $[ - 4,3 ]$ whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let $g$ be the function given by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t$.
(a) Find the values of $g ( 2 )$ and $g ( - 2 )$.
(b) For each of $g ^ { \prime } ( - 3 )$ and $g ^ { \prime \prime } ( - 3 )$, find the value or state that it does not exist.
(c) Find the $x$-coordinate of each point at which the graph of $g$ has a horizontal tangent line. For each of these points, determine whether $g$ has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers.
(d) For $- 4 < x < 3$, find all values of $x$ for which the graph of $g$ has a point of inflection. Explain your reasoning.