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.

Let $R$ be the region in the first quadrant bounded by the $x$-axis and the graphs of $y = \ln x$ and $y = 5 - x$, as shown in the figure above.
(a) Find the area of $R$.
(b) Region $R$ is the base of a solid. For the solid, each cross section perpendicular to the $x$-axis is a square. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid.
(c) The horizontal line $y = k$ divides $R$ into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $k$.

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 point, determine whether it is a relative maximum, a relative minimum, or neither.
(d) For $- 4 < x < 3$, find all values of $x$ for which the graph of $g$ has a point of inflection. Explain your reasoning.

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.?

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 $S ^ { \prime } ( t ) = 2 \sin ( 0.03 t ) + 1.5$.
(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 $M ( t ) = \frac { 1 } { 400 } \left( 3 t ^ { 3 } - 30 t ^ { 2 } + 330 t \right)$. 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.

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.

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.

Which of the following integrals gives the length of the curve $y = \ln x$ from $x = 1$ to $x = 2$ ?
(A) $\int _ { 1 } ^ { 2 } \sqrt { 1 + \frac { 1 } { x ^ { 2 } } } d x$
(B) $\int _ { 1 } ^ { 2 } \left( 1 + \frac { 1 } { x ^ { 2 } } \right) d x$
(C) $\int _ { 1 } ^ { 2 } \sqrt { 1 + e ^ { 2 x } } d x$
(D) $\int _ { 1 } ^ { 2 } \sqrt { 1 + ( \ln x ) ^ { 2 } } d x$
(E) $\int _ { 1 } ^ { 2 } \left( 1 + ( \ln x ) ^ { 2 } \right) d x$

$\int _ { 1 } ^ { 4 } t ^ { - 3 / 2 } d t =$
(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) $h ^ { \prime \prime } ( 6 ) < h ^ { \prime } ( 6 ) < h ( 6 )$

$\int _ { 1 } ^ { \infty } x e ^ { - x ^ { 2 } } d x$ is
(A) $- \frac { 1 } { e }$
(B) $\frac { 1 } { 2 e }$
(C) $\frac { 1 } { e }$
(D) $\frac { 2 } { e }$
(E) divergent

The graph of the piecewise linear function $f$ is shown above. What is the value of $\int _ { - 1 } ^ { 9 } ( 3 f ( x ) + 2 ) d x$ ?
(A) 7.5
(B) 9.5
(C) 27.5
(D) 47
(E) 48.5

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 function $f$ is defined on the closed interval $[ - 5, 4 ]$. The graph of $f$ consists of three line segments and is shown in the figure above. Let $g$ be the function defined by $g ( x ) = \int _ { - 3 } ^ { x } f ( t ) \, dt$.
(a) Find $g ( 3 )$.
(b) On what open intervals contained in $- 5 < x < 4$ is the graph of $g$ both increasing and concave down? Give a reason for your answer.
(c) The function $h$ is defined by $h ( x ) = \frac { g ( x ) } { 5 x }$. Find $h ^ { \prime } ( 3 )$.
(d) The function $p$ is defined by $p ( x ) = f \left( x ^ { 2 } - x \right)$. Find the slope of the line tangent to the graph of $p$ at the point where $x = - 1$.

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.

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

$v ( t )$

(meters per minute)

& 0 & 200 & 240 & - 220 & 150 \hline \end{tabular}
(a) Use the data in the table to estimate the value of $v ^ { \prime } ( 16 )$.
(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 $B ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 300$, 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 $0 \leq t \leq 10$.

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?

A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height $h$ feet is given by the function $A$, where $A(h)$ is measured in square feet. The function $A$ is continuous and decreases as $h$ increases. Selected values for $A(h)$ are given in the table below.

\begin{tabular}{ c } $h$

(feet)

& 0 & 2 & 5 & 10 \hline

$A ( h )$

(square feet)

& 50.3 & 14.4 & 6.5 & 2.9 \hline \end{tabular}
(a) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the volume of the tank. Indicate units of measure.
(b) Does the approximation in part (a) overestimate or underestimate the volume of the tank? Explain your reasoning.
(c) The area, in square feet, of the horizontal cross section at height $h$ feet is modeled by the function $f$ given by $f(h) = \frac{50.3}{e^{0.2h} + h}$. Based on this model, find the volume of the tank. Indicate units of measure.
(d) Water is pumped into the tank. When the height of the water is 5 feet, the height is increasing at the rate of 0.26 foot per minute. Using the model from part (c), find the rate at which the volume of water is changing with respect to time when the height of the water is 5 feet. Indicate units of measure.

People enter a line for an escalator at a rate modeled by the function $r$ given by
$$r ( t ) = \begin{cases} 44 \left( \frac { t } { 100 } \right) ^ { 3 } \left( 1 - \frac { t } { 300 } \right) ^ { 7 } & \text { for } 0 \leq t \leq 300 \\ 0 & \text { for } t > 300 \end{cases}$$
where $r ( t )$ is measured in people per second and $t$ is measured in seconds. As people get on the escalator, they exit the line at a constant rate of 0.7 person per second. There are 20 people in line at time $t = 0$.
(a) How many people enter the line for the escalator during the time interval $0 \leq t \leq 300$ ?
(b) During the time interval $0 \leq t \leq 300$, there are always people in line for the escalator. How many people are in line at time $t = 300$ ?
(c) For $t > 300$, what is the first time $t$ that there are no people in line for the escalator?
(d) For $0 \leq t \leq 300$, at what time $t$ is the number of people in line a minimum? To the nearest whole number, find the number of people in line at this time. Justify your answer.

The graph of the continuous function $g$, the derivative of the function $f$, is shown above. The function $g$ is piecewise linear for $- 5 \leq x < 3$, and $g ( x ) = 2 ( x - 4 ) ^ { 2 }$ for $3 \leq x \leq 6$.
(a) If $f ( 1 ) = 3$, what is the value of $f ( - 5 )$ ?
(b) Evaluate $\int _ { 1 } ^ { 6 } g ( x ) d x$.
(c) For $- 5 < x < 6$, on what open intervals, if any, is the graph of $f$ both increasing and concave up? Give a reason for your answer.
(d) Find the $x$-coordinate of each point of inflection of the graph of $f$. Give a reason for your answer.

Fish enter a lake at a rate modeled by the function $E$ given by $E ( t ) = 20 + 15 \sin \left( \frac { \pi t } { 6 } \right)$. Fish leave the lake at a rate modeled by the function $L$ given by $L ( t ) = 4 + 2 ^ { 0.1 t ^ { 2 } }$. Both $E ( t )$ and $L ( t )$ are measured in fish per hour, and $t$ is measured in hours since midnight $( t = 0 )$.
(a) How many fish enter the lake over the 5-hour period from midnight $( t = 0 )$ to 5 A.M. $( t = 5 )$? Give your answer to the nearest whole number.
(b) What is the average number of fish that leave the lake per hour over the 5-hour period from midnight $( t = 0 )$ to 5 A.M. $( t = 5 )$?
(c) At what time $t$, for $0 \leq t \leq 8$, is the greatest number of fish in the lake? Justify your answer.
(d) Is the rate of change in the number of fish in the lake increasing or decreasing at 5 A.M. ($t = 5$)? Explain your reasoning.

The continuous function $f$ is defined on the closed interval $- 6 \leq x \leq 5$. The figure above shows a portion of the graph of $f$, consisting of two line segments and a quarter of a circle centered at the point $( 5, 3 )$. It is known that the point $( 3, 3 - \sqrt { 5 } )$ is on the graph of $f$.
(a) If $\int _ { - 6 } ^ { 5 } f ( x ) \, dx = 7$, find the value of $\int _ { - 6 } ^ { - 2 } f ( x ) \, dx$. Show the work that leads to your answer.
(b) Evaluate $\int _ { 3 } ^ { 5 } \left( 2 f ^ { \prime } ( x ) + 4 \right) dx$.
(c) The function $g$ is given by $g ( x ) = \int _ { - 2 } ^ { x } f ( t ) \, dt$. Find the absolute maximum value of $g$ on the interval $- 2 \leq x \leq 5$. Justify your answer.
(d) Find $\lim _ { x \rightarrow 1 } \frac { 10 ^ { x } - 3 f ^ { \prime } ( x ) } { f ( x ) - \arctan x }$.