$t$ (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.

18. If three equal subdivisions of $[ - 4,2 ]$ are used, what is the trapezoidal approximation of $\int _ { - 4 } ^ { 2 } \frac { e ^ { - x } } { 2 } d x ?$
(A) $e ^ { 2 } + e ^ { 0 } + e ^ { - 2 }$
(B) $e ^ { 4 } + e ^ { 2 } + e ^ { 0 }$
(C) $e ^ { 4 } + 2 e ^ { 2 } + 2 e ^ { 0 } + e ^ { - 2 }$
(D) $\frac { 1 } { 2 } \left( e ^ { 4 } + e ^ { 2 } + e ^ { 0 } + e ^ { - 2 } \right)$
(E) $\frac { 1 } { 2 } \left( e ^ { 4 } + 2 e ^ { 2 } + 2 e ^ { 0 } + e ^ { - 2 } \right)$

42. Calculate the approximate area of the shaded region in the figure by the trapezoidal rule, using divisions at $x = \frac { 4 } { 3 }$ and $x = \frac { 5 } { 3 }$.
(A) $\frac { 50 } { 27 }$
(B) $\frac { 251 } { 108 }$
(C) $\frac { 7 } { 3 }$
(D) $\frac { 127 } { 54 }$
(E) $\frac { 77 } { 27 }$

The graph of the velocity $v(t)$, in $\mathrm{ft/sec}$, of a car traveling on a straight road, for $0 \leq t \leq 50$, is shown above. A table of values for $v(t)$, at 5 second intervals of time $t$, is shown to the right of the graph.

$t$ (seconds)	$v(t)$ (feet per second)
0	0
5	12
10	20
15	30
20	55
25	70
30	78
35	81
40	75
45	60
50	72

(a) During what intervals of time is the acceleration of the car positive? Give a reason for your answer.
(b) Find the average acceleration of the car, in $\mathrm{ft/sec^2}$, over the interval $0 \leq t \leq 50$.
(c) Find one approximation for the acceleration of the car, in $\mathrm{ft/sec^2}$, at $t = 40$. Show the computations you used to arrive at your answer.
(d) Approximate $\int_{0}^{50} v(t)\, dt$ with a Riemann sum, using the midpoints of five subintervals of equal length. Using correct units, explain the meaning of this integral.

The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function $R$ of time $t$. The table below shows the rate as measured every 3 hours for a 24-hour period.

\begin{tabular}{ c } $t$

(hours)

&

$R ( t )$

(gallons per hour)

\hline 0 & 9.6
3 & 10.4 6 & 10.8 9 & 11.2 12 & 11.4 15 & 11.3 18 & 10.7 21 & 10.2 24 & 9.6 \end{tabular}
(a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate $\int _ { 0 } ^ { 24 } R ( t ) d t$. Using correct units, explain the meaning of your answer in terms of water flow.
(b) Is there some time $t , 0 < t < 24$, such that $R ^ { \prime } ( t ) = 0$ ? Justify your answer.
(c) The rate of water flow $R ( t )$ can be approximated by $Q ( t ) = \frac { 1 } { 79 } \left( 768 + 23 t - t ^ { 2 } \right)$. Use $Q ( t )$ to approximate the average rate of water flow during the 24-hour time period. Indicate units of measure.

The temperature, in degrees Celsius (${}^{\circ}\mathrm{C}$), of the water in a pond is a differentiable function $W$ of time $t$. The table below shows the water temperature as recorded every 3 days over a 15-day period.

\begin{tabular}{ c } $t$

(days)

&

$W(t)$

$\left({}^{\circ}\mathrm{C}\right)$

\hline\hline 0 & 20
3 & 31 6 & 28 9 & 24 12 & 22 15 & 21 \hline \end{tabular}
(a) Use data from the table to find an approximation for $W^{\prime}(12)$. Show the computations that lead to your answer. Indicate units of measure.
(b) Approximate the average temperature, in degrees Celsius, of the water over the time interval $0 \leq t \leq 15$ days by using a trapezoidal approximation with subintervals of length $\Delta t = 3$ days.
(c) A student proposes the function $P$, given by $P(t) = 20 + 10te^{(-t/3)}$, as a model for the temperature of the water in the pond at time $t$, where $t$ is measured in days and $P(t)$ is measured in degrees Celsius. Find $P^{\prime}(12)$. Using appropriate units, explain the meaning of your answer in terms of water temperature.
(d) Use the function $P$ defined in part (c) to find the average value, in degrees Celsius, of $P(t)$ over the time interval $0 \leq t \leq 15$ days.

The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function $R$ of time $t$. The graph of $R$ and a table of selected values of $R(t)$, for the time interval $0 \leq t \leq 90$ minutes, are shown above.
(a) Use data from the table to find an approximation for $R'(45)$. Show the computations that lead to your answer. Indicate units of measure.
(b) The rate of fuel consumption is increasing fastest at time $t = 45$ minutes. What is the value of $R''(45)$? Explain your reasoning.
(c) Approximate the value of $\int_{0}^{90} R(t)\,dt$ using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation less than the value of $\int_{0}^{90} R(t)\,dt$? Explain your reasoning.
(d) For $0 < b \leq 90$ minutes, explain the meaning of $\int_{0}^{b} R(t)\,dt$ in terms of fuel consumption for the plane. Explain the meaning of $\frac{1}{b}\int_{0}^{b} R(t)\,dt$ in terms of fuel consumption for the plane. Indicate units of measure in both answers.

A metal wire of length 8 centimeters (cm) is heated at one end. The table below gives selected values of the temperature $T ( x )$, in degrees Celsius $\left( { } ^ { \circ } \mathrm { C } \right)$, of the wire $x$ cm from the heated end. The function $T$ is decreasing and twice differentiable.

\begin{tabular}{ c } Distance

$x ( \mathrm {~cm} )$

& 0 & 1 & 5 & 6 & 8 \hline

Temperature

$T ( x ) \left( { } ^ { \circ } \mathrm { C } \right)$

& 100 & 93 & 70 & 62 & 55 \hline \end{tabular}
(a) Estimate $T ^ { \prime } ( 7 )$. Show the work that leads to your answer. Indicate units of measure.
(b) Write an integral expression in terms of $T ( x )$ for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure.
(c) Find $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) d x$, and indicate units of measure. Explain the meaning of $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) d x$ in terms of the temperature of the wire.
(d) Are the data in the table consistent with the assertion that $T ^ { \prime \prime } ( x ) > 0$ for every $x$ in the interval $0 < x < 8$ ? Explain 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?

A scientist measures the depth of the Doe River at Picnic Point. The river is 24 feet wide at this location. The measurements are taken in a straight line perpendicular to the edge of the river. The data are shown in the table above. The velocity of the water at Picnic Point, in feet per minute, is modeled by $v ( t ) = 16 + 2 \sin ( \sqrt { t + 10 } )$ for $0 \leq t \leq 120$ minutes. (a) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the area of the cross section of the river at Picnic Point, in square feet. Show the computations that lead to your answer. (b) The volumetric flow at a location along the river is the product of the cross-sectional area and the velocity of the water at that location. Use your approximation from part (a) to estimate the average value of the volumetric flow at Picnic Point, in cubic feet per minute, from $t = 0$ to $t = 120$ minutes. (c) The scientist proposes the function $f$, given by $f ( x ) = 8 \sin \left( \frac { \pi x } { 24 } \right)$, as a model for the depth of the water, in feet, at Picnic Point $x$ feet from the river's edge. Find the area of the cross section of the river at Picnic Point based on this model. (d) Recall that the volumetric flow is the product of the cross-sectional area and the velocity of the water at a location. To prevent flooding, water must be diverted if the average value of the volumetric flow at Picnic Point exceeds 2100 cubic feet per minute for a 20 -minute period. Using your answer from part (c), find the average value of the volumetric flow during the time interval $40 \leq t \leq 60$ minutes. Does this value indicate that the water must be diverted?

3. The figure above shows an aboveground swimming pool in the shape of a cylinder with a radius of 12 feet and a height of 4 feet. The pool contains 1000 cubic feet of water at time $t = 0$. During the time interval $0 \leq t \leq 12$ hours, water is pumped into the pool at the rate $P ( t )$ cubic feet per hour. The table above gives values of $P ( t )$ for selected values of $t$. During the same time interval, water is leaking from the pool at the rate $R ( t )$ cubic feet per hour, where $R ( t ) = 25 e ^ { - 0.05 t }$. (Note: The volume $V$ of a cylinder with radius $r$ and height $h$ is given by $V = \pi r ^ { 2 } h$.)
(a) Use a midpoint Riemann sum with three subintervals of equal length to approximate the total amount of water that was pumped into the pool during the time interval $0 \leq t \leq 12$ hours. Show the computations that lead to your answer.
(b) Calculate the total amount of water that leaked out of the pool during the time interval $0 \leq t \leq 12$ hours.
(c) Use the results from parts (a) and (b) to approximate the volume of water in the pool at time $t = 12$ hours. Round your answer to the nearest cubic foot.
(d) Find the rate at which the volume of water in the pool is increasing at time $t = 8$ hours. How fast is the water level in the pool rising at $t = 8$ hours? Indicate units of measure in both answers.

WRITE ALL WORK IN THE EXAM BOOKLET.

END OF PART A OF SECTION II

No calculator is allowed for these problems. [Figure]

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.

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

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

Train $A$ runs back and forth on an east-west section of railroad track. Train A's velocity, measured in meters per minute, is given by a differentiable function $v _ { A } ( t )$, where time $t$ is measured in minutes. Selected values for $v _ { A } ( t )$ are given in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 2 & 5 & 8 & 12 \hline

$v _ { A } ( t )$

(meters/minute)

& 0 & 100 & 40 & - 120 & - 150 \hline \end{tabular}
(a) Find the average acceleration of train $A$ over the interval $2 \leq t \leq 8$.
(b) Do the data in the table support the conclusion that train $A$'s velocity is $-100$ meters per minute at some time $t$ with $5 < t < 8$? Give a reason for your answer.
(c) At time $t = 2$, train $A$'s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train $A$, in meters from the Origin Station, at time $t = 12$. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time $t = 12$.
(d) A second train, train $B$, travels north from the Origin Station. At time $t$ the velocity of train $B$ is given by $v _ { B } ( t ) = - 5 t ^ { 2 } + 60 t + 25$, and at time $t = 2$ the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train $A$ and train $B$ is changing at time $t = 2$.

Johanna jogs along a straight path. For $0 \leq t \leq 40$, Johanna's velocity is given by a differentiable function $v$. Selected values of $v(t)$, where $t$ is measured in minutes and $v(t)$ is measured in meters per minute, are given in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 12 & 20 & 24 & 40 \hline $v(t)$ & 0 & 200 & 240 & -220 & 150 \hline

(meters per minute)

& & & & & \hline \end{tabular}
(a) Use the data in the table to estimate the value of $v'(16)$.
(b) Using correct units, explain the meaning of the definite integral $\int_0^{40} |v(t)|\, dt$ in the context of the problem. Approximate the value of $\int_0^{40} |v(t)|\, dt$ using a right Riemann sum with the four subintervals indicated in the table.
(c) Bob is riding his bicycle along the same path. For $0 \leq t \leq 10$, Bob's velocity is modeled by $B(t) = t^3 - 6t^2 + 300$, where $t$ is measured in minutes and $B(t)$ is measured in meters per minute. Find Bob's acceleration at time $t = 5$.
(d) Based on the model $B$ from part (c), find Bob's average velocity during the interval $0 \leq t \leq 10$.

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.

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.

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.

The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function $R$ of time $t$. The table below shows the rate as measured every 3 hours for a 24-hour period.

\begin{tabular}{ c } $t$

(hours)

&

$R(t)$

(gallons per hour)

\hline 0 & 9.6
3 & 10.4 6 & 10.8 9 & 11.2 12 & 11.4 15 & 11.3 18 & 10.7 21 & 10.2 24 & 9.6 \end{tabular}
(a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate $\int_{0}^{24} R(t)\, dt$. Using correct units, explain the meaning of your answer in terms of water flow.
(b) Is there some time $t$, $0 < t < 24$, such that $R'(t) = 0$? Justify your answer.
(c) The rate of water flow $R(t)$ can be approximated by $Q(t) = \frac{1}{79}\left(768 + 23t - t^2\right)$. Use $Q(t)$ to approximate the average rate of water flow during the 24-hour time period. Indicate units of measure.

The temperature, in degrees Celsius (${}^{\circ}\mathrm{C}$), of the water in a pond is a differentiable function $W$ of time $t$. The table below shows the water temperature as recorded every 3 days over a 15-day period.

\begin{tabular}{ c } $t$

(days)

&

$W(t)$

$\left({}^{\circ}\mathrm{C}\right)$

\hline\hline 0 & 20
3 & 31 6 & 28 9 & 24 12 & 22 15 & 21 \hline \end{tabular}
(a) Use data from the table to find an approximation for $W'(12)$. Show the computations that lead to your answer. Indicate units of measure.
(b) Approximate the average temperature, in degrees Celsius, of the water over the time interval $0 \leq t \leq 15$ days by using a trapezoidal approximation with subintervals of length $\Delta t = 3$ days.
(c) A student proposes the function $P$, given by $P(t) = 20 + 10te^{(-t/3)}$, as a model for the temperature of the water in the pond at time $t$, where $t$ is measured in days and $P(t)$ is measured in degrees Celsius. Find $P'(12)$. Using appropriate units, explain the meaning of your answer in terms of water temperature.
(d) Use the function $P$ defined in part (c) to find the average value, in degrees Celsius, of $P(t)$ over the time interval $0 \leq t \leq 15$ days.

A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. The table below gives the measurements of the diameter of the blood vessel at selected points along the length of the blood vessel, where $x$ represents the distance from one end of the blood vessel and $B(x)$ is a twice-differentiable function that represents the diameter at that point.

\begin{tabular}{ c } Distance

$x$

$(\mathrm{~mm})$

& 0 & 60 & 120 & 180 & 240 & 300 & 360 \hline

Diameter

$B(x)$

$(\mathrm{mm})$

& 24 & 30 & 28 & 30 & 26 & 24 & 26 \hline \end{tabular}
(a) Write an integral expression in terms of $B(x)$ that represents the average radius, in mm, of the blood vessel between $x = 0$ and $x = 360$.
(b) Approximate the value of your answer from part (a) using the data from the table and a midpoint Riemann sum with three subintervals of equal length. Show the computations that lead to your answer.
(c) Using correct units, explain the meaning of $\pi \int_{125}^{275} \left(\frac{B(x)}{2}\right)^2 dx$ in terms of the blood vessel.
(d) Explain why there must be at least one value $x$, for $0 < x < 360$, such that $B''(x) = 0$.

3. The figure above shows an aboveground swimming pool in the shape of a cylinder with a radius of 12 feet and a height of 4 feet. The pool contains 1000 cubic feet of water at time $t = 0$. During the time interval $0 \leq t \leq 12$ hours, water is pumped into the pool at the rate $P ( t )$ cubic feet per hour. The table above gives values of $P ( t )$ for selected values of $t$. During the same time interval, water is leaking from the pool at the rate $R ( t )$ cubic feet per hour, where $R ( t ) = 25 e ^ { - 0.05 t }$. (Note: The volume $V$ of a cylinder with radius $r$ and height $h$ is given by $V = \pi r ^ { 2 } h$.)
(a) Use a midpoint Riemann sum with three subintervals of equal length to approximate the total amount of water that was pumped into the pool during the time interval $0 \leq t \leq 12$ hours. Show the computations that lead to your answer.
(b) Calculate the total amount of water that leaked out of the pool during the time interval $0 \leq t \leq 12$ hours.
(c) Use the results from parts (a) and (b) to approximate the volume of water in the pool at time $t = 12$ hours. Round your answer to the nearest cubic foot.
(d) Find the rate at which the volume of water in the pool is increasing at time $t = 8$ hours. How fast is the water level in the pool rising at $t = 8$ hours? Indicate units of measure in both answers.

WRITE ALL WORK IN THE EXAM BOOKLET.

END OF PART A OF SECTION II

© 2010 The College Board. Visit the College Board on the Web: \href{http://www.collegeboard.com}{www.collegeboard.com}.
No calculator is allowed for these problems. [Figure]

The function $f$ is twice differentiable for $x > 0$ with $f(1) = 15$ and $f''(1) = 20$. Values of $f'$, the derivative of $f$, are given for selected values of $x$ in the table below.

$x$	1	1.1	1.2	1.3	1.4
$f'(x)$	8	10	12	13	14.5

(a) Write an equation for the line tangent to the graph of $f$ at $x = 1$. Use this line to approximate $f(1.4)$.
(b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate $\int_{1}^{1.4} f'(x)\, dx$. Use the approximation for $\int_{1}^{1.4} f'(x)\, dx$ to estimate the value of $f(1.4)$. Show the computations that lead to your answer.
(c) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f(1.4)$. Show the computations that lead to your answer.
(d) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f(1.4)$.

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

The height of a tree at time $t$ is given by a twice-differentiable function $H$, where $H ( t )$ is measured in meters and $t$ is measured in years. Selected values of $H ( t )$ are given in the table below.

\begin{tabular}{ c } $t$

(years)

& 2 & 3 & 5 & 7 & 10 \hline

$H ( t )$

(meters)

& 1.5 & 2 & 6 & 11 & 15 \hline \end{tabular}
(a) Use the data in the table to estimate $H ^ { \prime } ( 6 )$. Using correct units, interpret the meaning of $H ^ { \prime } ( 6 )$ in the context of the problem.
(b) Explain why there must be at least one time $t$, for $2 < t < 10$, such that $H ^ { \prime } ( t ) = 2$.
(c) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the average height of the tree over the time interval $2 \leq t \leq 10$.
(d) The height of the tree, in meters, can also be modeled by the function $G$, given by $G ( x ) = \frac { 100 x } { 1 + x }$, where $x$ is the diameter of the base of the tree, in meters. When the tree is 50 meters tall, the diameter of the base of the tree is increasing at a rate of 0.03 meter per year. According to this model, what is the rate of change of the height of the tree with respect to time, in meters per year, at the time when the tree is 50 meters tall?

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 $2 \pi \int _ { 0 } ^ { 4 } r f ( r ) d r$. Approximate the value of $2 \pi \int _ { 0 } ^ { 4 } r f ( r ) d r$ 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 $1 \leq r \leq 4$ ?