$t$ (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.

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.

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 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 $t = 25$ ?

Hot water is dripping through a coffeemaker, filling a large cup with coffee. The amount of coffee in the cup at time $t$, $0 \leq t \leq 6$, is given by a differentiable function $C$, where $t$ is measured in minutes. Selected values of $C ( t )$, measured in ounces, are given in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 1 & 2 & 3 & 4 & 5 & 6 \hline

$C ( t )$

(ounces)

& 0 & 5.3 & 8.8 & 11.2 & 12.8 & 13.8 & 14.5 \hline \end{tabular}
(a) Use the data in the table to approximate $C ^ { \prime } ( 3.5 )$. Show the computations that lead to your answer, and indicate units of measure.
(b) Is there a time $t$, $2 \leq t \leq 4$, at which $C ^ { \prime } ( t ) = 2$? Justify your answer.
(c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of $\frac { 1 } { 6 } \int _ { 0 } ^ { 6 } C ( t ) \, dt$. Using correct units, explain the meaning of $\frac { 1 } { 6 } \int _ { 0 } ^ { 6 } C ( t ) \, dt$ in the context of the problem.
(d) The amount of coffee in the cup, in ounces, is modeled by $B ( t ) = 16 - 16 e ^ { - 0.4 t }$. Using this model, find the rate at which the amount of coffee in the cup is changing when $t = 5$.

Water is pumped into a tank at a rate modeled by $W ( t ) = 2000 e ^ { - t ^ { 2 } / 20 }$ liters per hour for $0 \leq t \leq 8$, where $t$ is measured in hours. Water is removed from the tank at a rate modeled by $R ( t )$ liters per hour, where $R$ is differentiable and decreasing on $0 \leq t \leq 8$. Selected values of $R ( t )$ are shown in the table below. At time $t = 0$, there are 50,000 liters of water in the tank.

\begin{tabular}{ c } $t$

(hours)

& 0 & 1 & 3 & 6 & 8 \hline

$R ( t )$

(liters / hour)

& 1340 & 1190 & 950 & 740 & 700 \hline \end{tabular}
(a) Estimate $R ^ { \prime } ( 2 )$. Show the work that leads to your answer. Indicate units of measure.
(b) Use a left Riemann sum with the four subintervals indicated by the table to estimate the total amount of water removed from the tank during the 8 hours. Is this an overestimate or an underestimate of the total amount of water removed? Give a reason for your answer.
(c) Use your answer from part (b) to find an estimate of the total amount of water in the tank, to the nearest liter, at the end of 8 hours.
(d) For $0 \leq t \leq 8$, is there a time $t$ when the rate at which water is pumped into the tank is the same as the rate at which water is removed from the tank? Explain why or why not.

The velocity of a particle, $P$, moving along the $x$-axis is given by the differentiable function $v_P$, where $v_P(t)$ is measured in meters per hour and $t$ is measured in hours. Selected values of $v_P(t)$ are shown in the table below. Particle $P$ is at the origin at time $t = 0$.

\begin{tabular}{ c } $t$

(hours)

& 0 & 0.3 & 1.7 & 2.8 & 4 \hline

$v_P(t)$

(meters per hour)

& 0 & 55 & -29 & 55 & 48 \hline \end{tabular}
(a) Justify why there must be at least one time $t$, for $0.3 \leq t \leq 2.8$, at which $v_P'(t)$, the acceleration of particle $P$, equals 0 meters per hour per hour.
(b) Use a trapezoidal sum with the three subintervals $[0, 0.3]$, $[0.3, 1.7]$, and $[1.7, 2.8]$ to approximate the value of $\int_0^{2.8} v_P(t)\, dt$.
(c) A second particle, $Q$, also moves along the $x$-axis so that its velocity for $0 \leq t \leq 4$ is given by $v_Q(t) = 45\sqrt{t}\cos\left(0.063t^2\right)$ meters per hour. Find the time interval during which the velocity of particle $Q$ is at least 60 meters per hour. Find the distance traveled by particle $Q$ during the interval when the velocity of particle $Q$ is at least 60 meters per hour.
(d) At time $t = 0$, particle $Q$ is at position $x = -90$. Using the result from part (b) and the function $v_Q$ from part (c), approximate the distance between particles $P$ and $Q$ at time $t = 2.8$.

The density of a bacteria population in a circular petri dish at a distance $r$ centimeters from the center of the dish is given by an increasing, differentiable function $f$, where $f(r)$ is measured in milligrams per square centimeter. Values of $f(r)$ for selected values of $r$ are given in the table below.

\begin{tabular}{ c } $r$

(centimeters)

& 0 & 1 & 2 & 2.5 & 4 \hline

$f ( r )$

(milligrams per square centimeter)

& 1 & 2 & 6 & 10 & 18 \hline \end{tabular}
(a) Use the data in the table to estimate $f^{\prime}(2.25)$. Using correct units, interpret the meaning of your answer in the context of this problem.
(b) The total mass, in milligrams, of bacteria in the petri dish is given by the integral expression $2\pi \int_{0}^{4} r f(r)\, dr$. Approximate the value of $2\pi \int_{0}^{4} r f(r)\, dr$ using a right Riemann sum with the four subintervals indicated by the data in the table.
(c) Is the approximation found in part (b) an overestimate or underestimate of the total mass of bacteria in the petri dish? Explain your reasoning.
(d) The density of bacteria in the petri dish, for $1 \leq r \leq 4$, is modeled by the function $g$ defined by $g(r) = 2 - 16(\cos(1.57\sqrt{r}))^{3}$. For what value of $k$, $1 < k < 4$, is $g(k)$ equal to the average value of $g(r)$ on the interval $1 \leq r \leq 4$?

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.

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?

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 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 $t = 25$ ?

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.

A student starts reading a book at time $t = 0$ minutes and continues reading for the next 10 minutes. The rate at which the student reads is modeled by the differentiable function $R$, where $R ( t )$ is measured in words per minute. Selected values of $R ( t )$ are given in the table shown.

$t$ (minutes)	0	2	8	10
$R ( t )$ (words per minute)	90	100	150	162

A. Approximate $R ^ { \prime } ( 1 )$ using the average rate of change of $R$ over the interval $0 \leq t \leq 2$. Show the work that leads to your answer. Indicate units of measure.
B. Must there be a value $c$, for $0 < c < 10$, such that $R ( c ) = 155$ ? Justify your answer.
C. Use a trapezoidal sum with the three subintervals indicated by the data in the table to approximate the value of $\int _ { 0 } ^ { 10 } R ( t ) d t$. Show the work that leads to your answer.
D. A teacher also starts reading at time $t = 0$ minutes and continues reading for the next 10 minutes. The rate at which the teacher reads is modeled by the function $W$ defined by $W ( t ) = - \frac { 3 } { 10 } t ^ { 2 } + 8 t + 100$, where $W ( t )$ is measured in words per minute. Based on the model, how many words has the teacher read by the end of the 10 minutes? Show the work that leads to your answer.