The height of the water in a conical storage tank is modeled by a differentiable function $h$, where $h ( t )$ is measured in meters and $t$ is measured in hours. At time $t = 0$, the height of the water in the tank is 25 meters. The height is changing at the rate $h ^ { \prime } ( t ) = 2 - \frac { 24 e ^ { - 0.025 t } } { t + 4 }$ meters per hour for $0 \leq t \leq 24$.
(a) When the height of the water in the tank is $h$ meters, the volume of water is $V = \frac { 1 } { 3 } \pi h ^ { 3 }$. At what rate is the volume of water changing at time $t = 0$ ? Indicate units of measure.
(b) What is the minimum height of the water during the time period $0 \leq t \leq 24$ ? Justify your answer.
(c) The line tangent to the graph of $h$ at $t = 16$ is used to approximate the height of the water in the tank. Using the tangent line approximation, at what time $t$ does the height of the water return to 25 meters?

An ice sculpture in the form of a sphere melts in such a way that it maintains its spherical shape. The volume of the sphere is decreasing at a constant rate of $2 \pi$ cubic meters per hour. At what rate, in square meters per hour, is the surface area of the sphere decreasing at the moment when the radius is 5 meters? (Note: For a sphere of radius $r$, the surface area is $4 \pi r ^ { 2 }$ and the volume is $\frac { 4 } { 3 } \pi r ^ { 3 }$.)
(A) $\frac { 4 \pi } { 5 }$
(B) $40 \pi$
(C) $80 \pi ^ { 2 }$
(D) $100 \pi$

The radius of a right circular cylinder is increasing at a rate of 2 units per second. The height of the cylinder is decreasing at a rate of 5 units per second. Which of the following expressions gives the rate at which the volume of the cylinder is changing with respect to time in terms of the radius $r$ and height $h$ of the cylinder? (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r ^ { 2 } h$.)
(A) $- 20 \pi r$
(B) $- 2 \pi r h$
(C) $4 \pi r h - 5 \pi r ^ { 2 }$
(D) $4 \pi r h + 5 \pi r ^ { 2 }$

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.

In the figure above, line $\ell$ is tangent to the graph of $y = \frac { 1 } { x ^ { 2 } }$ at point $P$, with coordinates $\left( w , \frac { 1 } { w ^ { 2 } } \right)$, where $w > 0$. Point $Q$ has coordinates $( w , 0 )$. Line $\ell$ crosses the $x$-axis at point $R$, with coordinates $( k , 0 )$.
(a) Find the value of $k$ when $w = 3$.
(b) For all $w > 0$, find $k$ in terms of $w$.
(c) Suppose that $w$ is increasing at the constant rate of 7 units per second. When $w = 5$, what is the rate of change of $k$ with respect to time?
(d) Suppose that $w$ is increasing at the constant rate of 7 units per second. When $w = 5$, what is the rate of change of the area of $\triangle PQR$ with respect to time? Determine whether the area is increasing or decreasing at this instant.

A container has the shape of an open right circular cone. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth $h$ is changing at the constant rate of $\frac { - 3 } { 10 } \text{ cm/hr}$. (Note: The volume of a cone of height $h$ and radius $r$ is given by $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$.)
(a) Find the volume $V$ of water in the container when $h = 5 \text{ cm}$. Indicate units of measure.
(b) Find the rate of change of the volume of water in the container, with respect to time, when $h = 5 \text{ cm}$. Indicate units of measure.
(c) Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportionality?

Ship $A$ is traveling due west toward Lighthouse Rock at a speed of 15 kilometers per hour ($\mathrm{km/hr}$). Ship $B$ is traveling due north away from Lighthouse Rock at a speed of $10\mathrm{~km/hr}$. Let $x$ be the distance between Ship $A$ and Lighthouse Rock at time $t$, and let $y$ be the distance between Ship $B$ and Lighthouse Rock at time $t$, as shown in the figure above.
(a) Find the distance, in kilometers, between Ship $A$ and Ship $B$ when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.
(b) Find the rate of change, in $\mathrm{km/hr}$, of the distance between the two ships when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.
(c) Let $\theta$ be the angle shown in the figure. Find the rate of change of $\theta$, in radians per hour, when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.

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.

Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its radius and height changing with time. (Note: The volume $V$ of a right circular cylinder with radius $r$ and height $h$ is given by $V = \pi r ^ { 2 } h$.)
(a) At the instant when the radius of the oil slick is 100 centimeters and the height is 0.5 centimeter, the radius is increasing at the rate of 2.5 centimeters per minute. At this instant, what is the rate of change of the height of the oil slick with respect to time, in centimeters per minute?
(b) A recovery device arrives on the scene and begins removing oil. The rate at which oil is removed is $R ( t ) = 400 \sqrt { t }$ cubic centimeters per minute, where $t$ is the time in minutes since the device began working. Oil continues to leak at the rate of 2000 cubic centimeters per minute. Find the time $t$ when the oil slick reaches its maximum volume. Justify your answer.
(c) By the time the recovery device began removing oil, 60,000 cubic centimeters of oil had already leaked. Write, but do not evaluate, an expression involving an integral that gives the volume of oil at the time found in part (b).

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.

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 $t = 40$?

If $( x + 2 y ) \cdot \frac { d y } { d x } = 2 x - y$, what is the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 3,0 )$ ?
(A) $- \frac { 10 } { 3 }$
(B) 0
(C) 2
(D) $\frac { 10 } { 3 }$
(E) Undefined

A person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate of 4 feet per second. If the height of the streetlight is 15 feet, what is the rate at which the person's shadow is lengthening?
(A) $1.5 \text{ ft/sec}$
(B) $2.667 \text{ ft/sec}$
(C) $3.75 \text{ ft/sec}$
(D) $6 \text{ ft/sec}$
(E) $10 \text{ ft/sec}$

On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by $G ( t ) = 90 + 45 \cos \left( \frac { t ^ { 2 } } { 18 } \right)$, where $t$ is measured in hours and $0 \leq t \leq 8$. At the beginning of the workday $( t = 0 )$, the plant has 500 tons of unprocessed gravel. During the hours of operation, $0 \leq t \leq 8$, the plant processes gravel at a constant rate of 100 tons per hour.
(a) Find $G ^ { \prime } ( 5 )$. Using correct units, interpret your answer in the context of the problem.
(b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday.
(c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time $t = 5$ hours? Show the work that leads to your answer.
(d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer.

The inside of a funnel of height 10 inches has circular cross sections, as shown in the figure above. At height $h$, the radius of the funnel is given by $r = \frac { 1 } { 20 } \left( 3 + h ^ { 2 } \right)$, where $0 \leq h \leq 10$. The units of $r$ and $h$ are inches.
(a) Find the average value of the radius of the funnel.
(b) Find the volume of the funnel.
(c) The funnel contains liquid that is draining from the bottom. At the instant when the height of the liquid is $h = 3$ inches, the radius of the surface of the liquid is decreasing at a rate of $\frac { 1 } { 5 }$ inch per second. At this instant, what is the rate of change of the height of the liquid with respect to time?

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 $V = \frac{1}{3}\pi r^2 h$.)

A particle moves along the curve defined by the equation $y = x^{3} - 3x$. The $x$-coordinate of the particle, $x(t)$, satisfies the equation $\dfrac{dx}{dt} = \dfrac{1}{\sqrt{2t+1}}$, for $t \geq 0$ with initial condition $x(0) = -4$.
(a) Find $x(t)$ in terms of $t$.
(b) Find $\dfrac{dy}{dt}$ in terms of $t$.
(c) Find the location and speed of the particle at time $t = 4$.

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.

The fuel consumption of a car, in miles per gallon (mpg), is modeled by $F ( s ) = 6 e ^ { \left( \frac { s } { 20 } - \frac { s ^ { 2 } } { 2400 } \right) }$, where $s$ is the speed of the car, in miles per hour. If the car is traveling at 50 miles per hour and its speed is changing at the rate of 20 miles/hour$^{2}$, what is the rate at which its fuel consumption is changing?
(A) 0.215 mpg per hour
(B) 4.299 mpg per hour
(C) 19.793 mpg per hour
(D) 25.793 mpg per hour
(E) 515.855 mpg per hour

On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by $G ( t ) = 90 + 45 \cos \left( \frac { t ^ { 2 } } { 18 } \right)$, where $t$ is measured in hours and $0 \leq t \leq 8$. At the beginning of the workday $( t = 0 )$, the plant has 500 tons of unprocessed gravel. During the hours of operation, $0 \leq t \leq 8$, the plant processes gravel at a constant rate of 100 tons per hour.
(a) Find $G ^ { \prime } ( 5 )$. Using correct units, interpret your answer in the context of the problem.
(b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday.
(c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time $t = 5$ hours? Show the work that leads to your answer.
(d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify 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 $t = 5$.

The rate at which rainwater flows into a drainpipe is modeled by the function $R$, where $R ( t ) = 20 \sin \left( \frac { t ^ { 2 } } { 35 } \right)$ cubic feet per hour, $t$ is measured in hours, and $0 \leq t \leq 8$. The pipe is partially blocked, allowing water to drain out the other end of the pipe at a rate modeled by $D ( t ) = - 0.04 t ^ { 3 } + 0.4 t ^ { 2 } + 0.96 t$ cubic feet per hour, for $0 \leq t \leq 8$. There are 30 cubic feet of water in the pipe at time $t = 0$.
(a) How many cubic feet of rainwater flow into the pipe during the 8-hour time interval $0 \leq t \leq 8$?
(b) Is the amount of water in the pipe increasing or decreasing at time $t = 3$ hours? Give a reason for your answer.
(c) At what time $t$, $0 \leq t \leq 8$, is the amount of water in the pipe at a minimum? Justify your answer.
(d) The pipe can hold 50 cubic feet of water before overflowing. For $t > 8$, water continues to flow into and out of the pipe at the given rates until the pipe begins to overflow. Write, but do not solve, an equation involving one or more integrals that gives the time $w$ when the pipe will begin to overflow.

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 inside of a funnel of height 10 inches has circular cross sections, as shown in the figure above. At height $h$, the radius of the funnel is given by $r = \frac { 1 } { 20 } \left( 3 + h ^ { 2 } \right)$, where $0 \leq h \leq 10$. The units of $r$ and $h$ are inches.
(a) Find the average value of the radius of the funnel.
(b) Find the volume of the funnel.
(c) The funnel contains liquid that is draining from the bottom. At the instant when the height of the liquid is $h = 3$ inches, the radius of the surface of the liquid is decreasing at a rate of $\frac { 1 } { 5 }$ inch per second. At this instant, what is the rate of change of the height of the liquid with respect to time?