The temperature of a room, in degrees Fahrenheit, is modeled by $H$, a differentiable function of the number of minutes after the thermostat is adjusted. Of the following, which is the best interpretation of $H ^ { \prime } ( 5 ) = 2$ ?
(A) The temperature of the room is 2 degrees Fahrenheit, 5 minutes after the thermostat is adjusted.
(B) The temperature of the room increases by 2 degrees Fahrenheit during the first 5 minutes after the thermostat is adjusted.
(C) The temperature of the room is increasing at a constant rate of $\frac { 2 } { 5 }$ degree Fahrenheit per minute.
(D) The temperature of the room is increasing at a rate of 2 degrees Fahrenheit per minute, 5 minutes after the thermostat is adjusted.

The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function $r$ of time $t$, where $t$ is measured in minutes. For $0 < t < 12$, the graph of $r$ is concave down. The table below gives selected values of the rate of change, $r^{\prime}(t)$, of the radius of the balloon over the time interval $0 \leq t \leq 12$.

\begin{tabular}{c} $t$

(minutes)

& 0 & 2 & 5 & 7 & 11 & 12 \hline

$r^{\prime}(t)$

(feet per minute)

& 5.7 & 4.0 & 2.0 & 1.2 & 0.6 & 0.5 \hline \end{tabular}
The radius of the balloon is 30 feet when $t = 5$. (Note: The volume of a sphere of radius $r$ is given by $V = \frac{4}{3}\pi r^{3}$.)
(a) Estimate the radius of the balloon when $t = 5.4$ using the tangent line approximation at $t = 5$. Is your estimate greater than or less than the true value? Give a reason for your answer.
(b) Find the rate of change of the volume of the balloon with respect to time when $t = 5$. Indicate units of measure.
(c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate $\int_{0}^{12} r^{\prime}(t)\, dt$. Using correct units, explain the meaning of $\int_{0}^{12} r^{\prime}(t)\, dt$ in terms of the radius of the balloon.
(d) Is your approximation in part (c) greater than or less than $\int_{0}^{12} r^{\prime}(t)\, dt$? Give a reason for your answer.

Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiable function $B$ models Ben's position on the track, measured in meters from the western end of the track, at time $t$, measured in seconds from the start of the ride. The table below gives values for $B(t)$ and Ben's velocity, $v(t)$, measured in meters per second, at selected times $t$.

\begin{tabular}{ c } $t$

(seconds)

& 0 & 10 & 40 & 60 \hline

$B(t)$

(meters)

& 100 & 136 & 9 & 49 \hline

$v(t)$

(meters per second)

& 2.0 & 2.3 & 2.5 & 4.6 \hline \end{tabular}
(a) Use the data in the table to approximate Ben's acceleration at time $t = 5$ seconds. Indicate units of measure.
(b) Using correct units, interpret the meaning of $\int_{0}^{60} |v(t)|\, dt$ in the context of this problem. Approximate $\int_{0}^{60} |v(t)|\, dt$ using a left Riemann sum with the subintervals indicated by the data in the table.
(c) For $40 \leq t \leq 60$, must there be a time $t$ when Ben's velocity is 2 meters per second? Justify your answer.
(d) A light is directly above the western end of the track. Ben rides so that at time $t$, the distance $L(t)$ between Ben and the light satisfies $(L(t))^2 = 12^2 + (B(t))^2$. At what rate is the distance between Ben and the light changing at time $t = 40$?

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?

An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function $r$, where $r(t)$ is measured in centimeters and $t$ is measured in days. The table below gives selected values of $r'(t)$, the rate of change of the radius, over the time interval $0 \leq t \leq 12$.

$t$ (days)	0	3	7	10	12
$r'(t)$ (centimeters per day)	$-6.1$	$-5.0$	$-4.4$	$-3.8$	$-3.5$

(a) Approximate $r''(8.5)$ using the average rate of change of $r'$ over the interval $7 \leq t \leq 10$. Show the computations that lead to your answer, and indicate units of measure.
(b) Is there a time $t$, $0 \leq t \leq 3$, for which $r'(t) = -6$? Justify your answer.
(c) Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of $\int_{0}^{12} r'(t)\,dt$.
(d) The height of the cone decreases at a rate of 2 centimeters per day. At time $t = 3$ days, the radius is 100 centimeters and the height is 50 centimeters. Find the rate of change of the volume of the cone with respect to time, in cubic centimeters per day, at time $t = 3$ days. (The volume $V$ of a cone with radius $r$ and height $h$ is $V = \frac{1}{3}\pi r^2 h$.)

The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function $r$ of time $t$, where $t$ is measured in minutes. For $0 < t < 12$, the graph of $r$ is concave down. The table below gives selected values of the rate of change, $r'(t)$, of the radius of the balloon over the time interval $0 \leq t \leq 12$. The radius of the balloon is 30 feet when $t = 5$.

$t$ (minutes)	0	2	5	7	11	12
$r'(t)$ (feet per minute)	5.7	4.0	2.0	1.2	0.6	0.5

(Note: The volume of a sphere of radius $r$ is given by $V = \frac{4}{3}\pi r^3$.)
(a) Estimate the radius of the balloon when $t = 5.4$ using the tangent line approximation at $t = 5$. Is your estimate greater than or less than the true value? Give a reason for your answer.
(b) Find the rate of change of the volume of the balloon with respect to time when $t = 5$. Indicate units of measure.
(c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate $\int_{0}^{12} r'(t)\, dt$. Using correct units, explain the meaning of $\int_{0}^{12} r'(t)\, dt$ in terms of the radius of the balloon.
(d) Is your approximation in part (c) greater than or less than $\int_{0}^{12} r'(t)\, dt$? Give a reason for your answer.

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 $t = 5$.

Researchers on a boat are investigating plankton cells in a sea. At a depth of $h$ meters, the density of plankton cells, in millions of cells per cubic meter, is modeled by $p ( h ) = 0.2 h ^ { 2 } e ^ { - 0.0025 h ^ { 2 } }$ for $0 \leq h \leq 30$ and is modeled by $f ( h )$ for $h \geq 30$. The continuous function $f$ is not explicitly given.
(a) Find $p ^ { \prime } ( 25 )$. Using correct units, interpret the meaning of $p ^ { \prime } ( 25 )$ in the context of the problem.
(b) Consider a vertical column of water in this sea with horizontal cross sections of constant area 3 square meters. To the nearest million, how many plankton cells are in this column of water between $h = 0$ and $h = 30$ meters?
(c) There is a function $u$ such that $0 \leq f ( h ) \leq u ( h )$ for all $h \geq 30$ and $\int _ { 30 } ^ { \infty } u ( h ) d h = 105$. The column of water in part (b) is $K$ meters deep, where $K > 30$. Write an expression involving one or more integrals that gives the number of plankton cells, in millions, in the entire column. Explain why the number of plankton cells in the column is less than or equal to 2000 million.
(d) The boat is moving on the surface of the sea. At time $t \geq 0$, the position of the boat is $( x ( t ) , y ( t ) )$, where $x ^ { \prime } ( t ) = 662 \sin ( 5 t )$ and $y ^ { \prime } ( t ) = 880 \cos ( 6 t )$. Time $t$ is measured in hours, and $x ( t )$ and $y ( t )$ are measured in meters. Find the total distance traveled by the boat over the time interval $0 \leq t \leq 1$.

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?

An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function $r$, where $r ( t )$ is measured in centimeters and $t$ is measured in days. The table below gives selected values of $r ^ { \prime } ( t )$, the rate of change of the radius, over the time interval $0 \leq t \leq 12$.

\begin{tabular}{ c } $t$

(days)

& 0 & 3 & 7 & 10 & 12 \hline

$r ^ { \prime } ( t )$

(centimeters per day)

& - 6.1 & - 5.0 & - 4.4 & - 3.8 & - 3.5 \hline \end{tabular}
(a) Approximate $r ^ { \prime \prime } ( 8.5 )$ using the average rate of change of $r ^ { \prime }$ over the interval $7 \leq t \leq 10$. Show the computations that lead to your answer, and indicate units of measure.
(b) Is there a time $t , 0 \leq t \leq 3$, for which $r ^ { \prime } ( t ) = - 6$ ? Justify your answer.
(c) Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of $\int _ { 0 } ^ { 12 } r ^ { \prime } ( t ) d t$.
(d) The height of the cone decreases at a rate of 2 centimeters per day. At time $t = 3$ days, the radius is 100 centimeters and the height is 50 centimeters. Find the rate of change of the volume of the cone with respect to time, in cubic centimeters per day, at time $t = 3$ days. (The volume $V$ of a cone with radius $r$ and height $h$ is $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$.)

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.