A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function $f$, where $f(t)$ is measured in gallons per second and $t$ is measured in seconds since pumping began. Selected values of $f(t)$ are given in the table.

\begin{tabular}{ c } $t$

(seconds)

& 0 & 60 & 90 & 120 & 135 & 150 \hline

$f(t)$

(gallons per second)

& 0 & 0.1 & 0.15 & 0.1 & 0.05 & 0 \hline \end{tabular}
(a) Using correct units, interpret the meaning of $\int_{60}^{135} f(t)\, dt$ in the context of the problem. Use a right Riemann sum with the three subintervals $[60,90]$, $[90,120]$, and $[120,135]$ to approximate the value of $\int_{60}^{135} f(t)\, dt$.
(b) Must there exist a value of $c$, for $60 < c < 120$, such that $f'(c) = 0$? Justify your answer.
(c) The rate of flow of gasoline, in gallons per second, can also be modeled by $g(t) = \left(\frac{t}{500}\right)\cos\left(\left(\frac{t}{120}\right)^{2}\right)$ for $0 \leq t \leq 150$. Using this model, find the average rate of flow of gasoline over the time interval $0 \leq t \leq 150$. Show the setup for your calculations.
(d) Using the model $g$ defined in part (c), find the value of $g'(140)$. Interpret the meaning of your answer in the context of the problem.

The temperature of coffee in a cup at time $t$ minutes is modeled by a decreasing differentiable function $C$, where $C(t)$ is measured in degrees Celsius. For $0 \leq t \leq 12$, selected values of $C(t)$ are given in the table shown.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 3 & 7 & 12 \hline

$C(t)$

(degrees Celsius)

& 100 & 85 & 69 & 55 \hline \end{tabular}
(a) Approximate $C'(5)$ using the average rate of change of $C$ over the interval $3 \leq t \leq 7$. Show the work that leads to your answer and include units of measure.
(b) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the value of $\int_{0}^{12} C(t)\, dt$. Interpret the meaning of $\frac{1}{12} \int_{0}^{12} C(t)\, dt$ in the context of the problem.
(c) For $12 \leq t \leq 20$, the rate of change of the temperature of the coffee is modeled by $C'(t) = \frac{-24.55 e^{0.01t}}{t}$, where $C'(t)$ is measured in degrees Celsius per minute. Find the temperature of the coffee at time $t = 20$. Show the setup for your calculations.
(d) For the model defined in part (c), it can be shown that $C''(t) = \frac{0.2455 e^{0.01t}(100 - t)}{t^2}$. For $12 < t < 20$, determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate. Give a reason for your answer.

The graph of the differentiable function $f$, shown for $-6 \leq x \leq 7$, has a horizontal tangent at $x = -2$ and is linear for $0 \leq x \leq 7$. Let $R$ be the region in the second quadrant bounded by the graph of $f$, the vertical line $x = -6$, and the $x$- and $y$-axes. Region $R$ has area 12.
(a) The function $g$ is defined by $g(x) = \int_{0}^{x} f(t)\, dt$. Find the values of $g(-6)$, $g(4)$, and $g(6)$.
(b) For the function $g$ defined in part (a), find all values of $x$ in the interval $0 \leq x \leq 6$ at which the graph of $g$ has a critical point. Give a reason for your answer.
(c) The function $h$ is defined by $h(x) = \int_{-6}^{x} f'(t)\, dt$. Find the values of $h(6)$, $h'(6)$, and $h''(6)$. Show the work that leads to your answers.

An invasive species of plant appears in a fruit grove at time $t = 0$ and begins to spread. The function $C$ defined by $C ( t ) = 7.6 \arctan ( 0.2 t )$ models the number of acres in the fruit grove affected by the species $t$ weeks after the species appears. It can be shown that $C ^ { \prime } ( t ) = \frac { 38 } { 25 + t ^ { 2 } }$.
(Note: Your calculator should be in radian mode.)
A. Find the average number of acres affected by the invasive species from time $t = 0$ to time $t = 4$ weeks. Show the setup for your calculations.
B. Find the time $t$ when the instantaneous rate of change of $C$ equals the average rate of change of $C$ over the time interval $0 \leq t \leq 4$. Show the setup for your calculations.
C. Assume that the invasive species continues to spread according to the given model for all times $t > 0$. Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression.
D. At time $t = 4$ weeks after the invasive species appears in the fruit grove, measures are taken to counter the spread of the species. The function $A$, defined by $A ( t ) = C ( t ) - \int _ { 4 } ^ { t } 0.1 \cdot \ln ( x ) d x$, models the number of acres affected by the species over the time interval $4 \leq t \leq 36$. At what time $t$, for $4 \leq t \leq 36$, does $A$ attain its maximum value? Justify your answer.

The continuous function $f$ is defined on the closed interval $- 6 \leq x \leq 12$. The graph of $f$, consisting of two semicircles and one line segment, is shown in the figure.
Let $g$ be the function defined by $g ( x ) = \int _ { 6 } ^ { x } f ( t ) d t$.
A. Find $g ^ { \prime } ( 8 )$. Give a reason for your answer.
B. Find all values of $x$ in the open interval $- 6 < x < 12$ at which the graph of $g$ has a point of inflection. Give a reason for your answer.
C. Find $g ( 12 )$ and $g ( 0 )$. Label your answers.
D. Find the value of $x$ at which $g$ attains an absolute minimum on the closed interval $- 6 \leq x \leq 12$. Justify your answer.

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 $[ - 2, 2 ]$.

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 $g(6)$, $g'(6)$, and $g''(6)$.
(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 a function defined on the closed interval $[0,7]$. The graph of $f$, consisting of four line segments, is shown above. Let $g$ be the function given by $g(x) = \int_{2}^{x} f(t)\, dt$.
(a) Find $g(3)$, $g'(3)$, and $g''(3)$.
(b) Find the average rate of change of $g$ on the interval $0 \leq x \leq 3$.
(c) For how many values $c$, where $0 < c < 3$, is $g'(c)$ equal to the average rate found in part (b)? Explain your reasoning.
(d) Find the $x$-coordinate of each point of inflection of the graph of $g$ on the interval $0 < x < 7$. Justify your answer.

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.
(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 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.

Concert tickets went on sale at noon $( t = 0 )$ and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time $t$ is modeled by a twice-differentiable function $L$ for $0 \leq t \leq 9$. Values of $L ( t )$ at various times $t$ are shown in the table below.

$t$ (hours)	0	1	3	4	7	8	9
$L ( t )$ (people)	120	156	176	126	150	80	0

(a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:30 P.M. $( t = 5.5 )$. Show the computations that lead to your answer. Indicate units of measure.
(b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale.
(c) For $0 \leq t \leq 9$, what is the fewest number of times at which $L ^ { \prime } ( t )$ must equal 0 ? Give a reason for your answer.
(d) The rate at which tickets were sold for $0 \leq t \leq 9$ is modeled by $r ( t ) = 550 t e ^ { - t / 2 }$ tickets per hour. Based on the model, how many tickets were sold by 3 P.M. ( $t = 3$ ), to the nearest whole number?

Let $f$ be a function that is twice differentiable for all real numbers. The table below 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.?

As a pot of tea cools, the temperature of the tea is modeled by a differentiable function $H$ for $0 \leq t \leq 10$, where time $t$ is measured in minutes and temperature $H(t)$ is measured in degrees Celsius. Values of $H(t)$ at selected values of time $t$ are shown in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 2 & 5 & 9 & 10 \hline

$H(t)$

(degrees Celsius)

& 66 & 60 & 52 & 44 & 43 \hline \end{tabular}
(a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time $t = 3.5$. Show the computations that lead to your answer.
(b) Using correct units, explain the meaning of $\frac{1}{10}\int_{0}^{10} H(t)\, dt$ in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate $\frac{1}{10}\int_{0}^{10} H(t)\, dt$.
(c) Evaluate $\int_{0}^{10} H'(t)\, dt$. Using correct units, explain the meaning of the expression in the context of this problem.
(d) At time $t = 0$, biscuits with temperature $100^\circ\mathrm{C}$ were removed from an oven. The temperature of the biscuits at time $t$ is modeled by a differentiable function $B$ for which it is known that $B'(t) = -13.84e^{-0.173t}$. Using the given models, at time $t = 10$, how much cooler are the biscuits than the tea?

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 above. 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.

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 $g(x) = \int_{1}^{x} f(t)\, dt$.
(a) Find the values of $g(2)$ and $g(-2)$.
(b) For each of $g'(-3)$ and $g''(-3)$, 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.

The graph of $f$ is shown above for $0 \leq x \leq 4$. What is the value of $\int _ { 0 } ^ { 4 } f ( x ) d x$ ?
(A) - 1
(B) 0
(C) 2
(D) 6
(E) 12

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.

A tank contains 50 liters of oil at time $t = 4$ hours. Oil is being pumped into the tank at a rate $R ( t )$, where $R ( t )$ is measured in liters per hour, and $t$ is measured in hours. Selected values of $R ( t )$ are given in the table above. Using a right Riemann sum with three subintervals and data from the table, what is the approximation of the number of liters of oil that are in the tank at time $t = 15$ hours?

$t$ (hours)	4	7	12	15
$R ( t )$ (liters/hour)	6.5	6.2	5.9	5.6

(A) 64.9
(B) 68.2
(C) 114.9
(D) 116.6
(E) 118.2

$\int _ { 1 } ^ { 4 } t ^ { - 3 / 2 } d t =$
(A) - 1
(B) $- \frac { 7 } { 8 }$
(C) $- \frac { 1 } { 2 }$
(D) $\frac { 1 } { 2 }$
(E) 1

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) $h ^ { \prime \prime } ( 6 ) < h ^ { \prime } ( 6 ) < h ( 6 )$