The function $f$ is defined by $f ( x ) = \left\{ \begin{array} { l l } 2 & \text { for } x < 3 \\ x - 1 & \text { for } x \geq 3 \end{array} \right.$. What is the value of $\int _ { 1 } ^ { 5 } f ( x ) d x$ ?
(A) 2
(B) 6
(C) 8
(D) 10
(E) 12

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) $h ^ { \prime \prime } ( 6 ) < h ^ { \prime } ( 6 ) < h ( 6 )$

A particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v ( t )$, and the acceleration of the particle at time $t$ is given by $a ( t )$. Which of the following gives the average velocity of the particle from time $t = 0$ to time $t = 8$ ?
(A) $\frac { a ( 8 ) - a ( 0 ) } { 8 }$
(B) $\frac { 1 } { 8 } \int _ { 0 } ^ { 8 } v ( t ) d t$
(C) $\frac { 1 } { 8 } \int _ { 0 } ^ { 8 } | v ( t ) | d t$
(D) $\frac { 1 } { 2 } \int _ { 0 } ^ { 8 } v ( t ) d t$
(E) $\frac { v ( 0 ) + v ( 8 ) } { 2 }$

Water is pumped into a tank at a rate of $r ( t ) = 30 \left( 1 - e ^ { - 0.16 t } \right)$ gallons per minute, where $t$ is the number of minutes since the pump was turned on. If the tank contained 800 gallons of water when the pump was turned on, how much water, to the nearest gallon, is in the tank after 20 minutes?
(A) 380 gallons
(B) 420 gallons
(C) 829 gallons
(D) 1220 gallons
(E) 1376 gallons

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 $h ( x ) = \dfrac { g ( x ) } { 5 x }$. 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 $D(t) = -0.04t^3 + 0.4t^2 + 0.96t$ 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.

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.

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 $L(t) = 4 + 2^{0.1t^2}$. 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 $\int_{3}^{5} \left(2f'(x) + 4\right) dx$.
(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 $\lim_{x \to 1} \dfrac{10^x - 3f'(x)}{f(x) - \arctan x}$.

Let $f$ be a continuous function defined on the closed interval $-4 \leq x \leq 6$. The graph of $f$, consisting of four line segments, is shown above. Let $G$ be the function defined by $G(x) = \int_{0}^{x} f(t)\, dt$.
(a) On what open intervals is the graph of $G$ concave up? Give a reason for your answer.
(b) Let $P$ be the function defined by $P(x) = G(x) \cdot f(x)$. Find $P^{\prime}(3)$.
(c) Find $\lim_{x \rightarrow 2} \frac{G(x)}{x^{2} - 2x}$.
(d) Find the average rate of change of $G$ on the interval $[-4, 2]$. Does the Mean Value Theorem guarantee a value $c$, $-4 < c < 2$, for which $G^{\prime}(c)$ is equal to this average rate of change? Justify your answer.

From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by $A(t) = 450\sqrt{\sin(0.62t)}$, where $t$ is the number of hours after 5 A.M. and $A(t)$ is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line.
(a) Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at the toll plaza from 6 A.M. $(t=1)$ to 10 A.M. $(t=5)$.
(b) Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from 6 A.M. $(t=1)$ to 10 A.M. $(t=5)$.
(c) Is the rate at which vehicles arrive at the toll plaza at 6 A.M. ($t=1$) increasing or decreasing? Give a reason for your answer.
(d) A line forms whenever $A(t) \geq 400$. The number of vehicles in line at time $t$, for $a \leq t \leq 4$, is given by $N(t) = \int_{a}^{t}(A(x) - 400)\,dx$, where $a$ is the time when a line first begins to form. To the nearest whole number, find the greatest number of vehicles in line at the toll plaza in the time interval $a \leq t \leq 4$. Justify your answer.

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 graph of the function $f$, consisting of three line segments, is given above. Let $g(x) = \int_{1}^{x} f(t)\, dt$.
(a) Compute $g(4)$ and $g(-2)$.
(b) Find the instantaneous rate of change of $g$, with respect to $x$, at $x = 1$.
(c) Find the absolute minimum value of $g$ on the closed interval $[-2, 4]$. Justify your answer.
(d) The second derivative of $g$ is not defined at $x = 1$ and $x = 2$. How many of these values are $x$-coordinates of points of inflection of the graph of $g$? Justify your answer.

The rate at which people enter an amusement park on a given day is modeled by the function $E$ defined by $$E ( t ) = \frac { 15600 } { \left( t ^ { 2 } - 24 t + 160 \right) }$$ The rate at which people leave the same amusement park on the same day is modeled by the function $L$ defined by $$L ( t ) = \frac { 9890 } { \left( t ^ { 2 } - 38 t + 370 \right) }$$ Both $E ( t )$ and $L ( t )$ are measured in people per hour and time $t$ is measured in hours after midnight. These functions are valid for $9 \leq t \leq 23$, the hours during which the park is open. At time $t = 9$, there are no people in the park.
(a) How many people have entered the park by 5:00 P.M. ( $t = 17$ )? Round your answer to the nearest whole number.
(b) The price of admission to the park is $\$15$ until 5:00 P.M. ( $t = 17$ ). After 5:00 P.M., the price of admission to the park is $\$11$. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number.
(c) Let $H ( t ) = \int _ { 9 } ^ { t } ( E ( x ) - L ( x ) ) d x$ for $9 \leq t \leq 23$. The value of $H ( 17 )$ to the nearest whole number is 3725. Find the value of $H ^ { \prime } ( 17 )$, and explain the meaning of $H ( 17 )$ and $H ^ { \prime } ( 17 )$ in the context of the amusement park.
(d) At what time $t$, for $9 \leq t \leq 23$, does the model predict that the number of people in the park is a maximum?

The graph of the function $f$ shown above consists of two line segments. Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$.
(a) Find $g ( - 1 ) , g ^ { \prime } ( - 1 )$, and $g ^ { \prime \prime } ( - 1 )$.
(b) For what values of $x$ in the open interval $( - 2, 2 )$ is $g$ increasing? Explain your reasoning.
(c) For what values of $x$ in the open interval $( - 2, 2 )$ is the graph of $g$ concave down? Explain your reasoning.
(d) On the axes provided, sketch the graph of $g$ on the closed interval $[ - 2, 2 ]$.

The graph of a differentiable function $f$ on the closed interval $[-3, 15]$ is shown in the figure above. The graph of $f$ has a horizontal tangent line at $x = 6$. Let $g(x) = 5 + \int_{6}^{x} f(t)\, dt$ for $-3 \leq x \leq 15$.
(a) Find $g(6)$, $g'(6)$, and $g''(6)$.
(b) On what intervals is $g$ decreasing? Justify your answer.
(c) On what intervals is the graph of $g$ concave down? Justify your answer.
(d) Find a trapezoidal approximation of $\int_{-3}^{15} f(t)\, dt$ using six subintervals of length $\Delta t = 3$.

Let $f$ be a function defined on the closed interval $[0,7]$. The graph of $f$, consisting of four line segments, is shown above. Let $g$ be the function given by $g(x) = \int_{2}^{x} f(t)\, dt$.
(a) Find $g(3)$, $g'(3)$, and $g''(3)$.
(b) Find the average rate of change of $g$ on the interval $0 \leq x \leq 3$.
(c) For how many values $c$, where $0 < c < 3$, is $g'(c)$ equal to the average rate found in part (b)? Explain your reasoning.
(d) Find the $x$-coordinate of each point of inflection of the graph of $g$ on the interval $0 < x < 7$. Justify your answer.

Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function $F$ defined by
$$F ( t ) = 82 + 4 \sin \left( \frac { t } { 2 } \right) \text { for } 0 \leq t \leq 30$$
where $F ( t )$ is measured in cars per minute and $t$ is measured in minutes.
(a) To the nearest whole number, how many cars pass through the intersection over the 30 -minute period?
(b) Is the traffic flow increasing or decreasing at $t = 7$ ? Give a reason for your answer.
(c) What is the average value of the traffic flow over the time interval $10 \leq t \leq 15$ ? Indicate units of measure.
(d) What is the average rate of change of the traffic flow over the time interval $10 \leq t \leq 15$ ? Indicate units of measure.

A test plane flies in a straight line with positive velocity $v ( t )$, in miles per minute at time $t$ minutes, where $v$ is a differentiable function of $t$. Selected values of $v ( t )$ for $0 \leq t \leq 40$ are shown in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 \hline

$v ( t )$

(miles per minute)

& 7.0 & 9.2 & 9.5 & 7.0 & 4.5 & 2.4 & 2.4 & 4.3 & 7.3 \hline \end{tabular}
(a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table to approximate $\int _ { 0 } ^ { 40 } v ( t ) d t$. Show the computations that lead to your answer. Using correct units, explain the meaning of $\int _ { 0 } ^ { 40 } v ( t ) d t$ in terms of the plane's flight.
(b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval $0 < t < 40$ ? Justify your answer.
(c) The function $f$, defined by $f ( t ) = 6 + \cos \left( \frac { t } { 10 } \right) + 3 \sin \left( \frac { 7 t } { 40 } \right)$, is used to model the velocity of the plane, in miles per minute, for $0 \leq t \leq 40$. According to this model, what is the acceleration of the plane at $t = 23$ ? Indicate units of measure.
(d) According to the model $f$, given in part (c), what is the average velocity of the plane, in miles per minute, over the time interval $0 \leq t \leq 40$ ?

An object moving along a curve in the $x y$-plane has position $( x ( t ) , y ( t ) )$ at time $t \geq 0$ with $\frac { d x } { d t } = 3 + \cos \left( t ^ { 2 } \right)$. The derivative $\frac { d y } { d t }$ is not explicitly given. At time $t = 2$, the object is at position $( 1,8 )$.
(a) Find the $x$-coordinate of the position of the object at time $t = 4$.
(b) At time $t = 2$, the value of $\frac { d y } { d t }$ is - 7 . Write an equation for the line tangent to the curve at the point $( x ( 2 ) , y ( 2 ) )$.
(c) Find the speed of the object at time $t = 2$.
(d) For $t \geq 3$, the line tangent to the curve at $( x ( t ) , y ( t ) )$ has a slope of $2 t + 1$. Find the acceleration vector of the object at time $t = 4$.

2. At an intersection in Thomasville, Oregon, cars turn left at the rate $L ( t ) = 60 \sqrt { t } \sin ^ { 2 } \left( \frac { t } { 3 } \right)$ cars per hour over the time interval $0 \leq t \leq 18$ hours. The graph of $y = L ( t )$ is shown above.
(a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval $0 \leq t \leq 18$ hours.
(b) Traffic engineers will consider turn restrictions when $L ( t ) \geq 150$ cars per hour. Find all values of $t$ for which $L ( t ) \geq 150$ and compute the average value of $L$ over this time interval. Indicate units of measure.
(c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.

WRITE ALL WORK IN THE PINK EXAM BOOKLET.

© 2006 The College Board. All rights reserved. Visit \href{http://apcentral.collegeboard.com}{apcentral.collegeboard.com} (for AP professionals) and \href{http://www.collegeboard.com/apstudents}{www.collegeboard.com/apstudents} (for students and parents). [Figure]

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.

3. The graph of the function $f$ shown above consists of six line segments. Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
(a) Find $g ( 4 ) , g ^ { \prime } ( 4 )$, and $g ^ { \prime \prime } ( 4 )$.
(b) Does $g$ have a relative minimum, a relative maximum, or neither at $x = 1$ ? Justify your answer.
(c) Suppose that $f$ is defined for all real numbers $x$ and is periodic with a period of length 5 . The graph above shows two periods of $f$. Given that $g ( 5 ) = 2$, find $g ( 10 )$ and write an equation for the line tangent to the graph of $g$ at $x = 108$.

WRITE ALL WORK IN THE PINK EXAM BOOKLET.

END OF PART A OF SECTION II

No calculator is allowed for these problems.

\begin{tabular}{ c } $t$

(seconds)

& 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \hline

$v ( t )$

(feet per second)

& 5 & 14 & 22 & 29 & 35 & 40 & 44 & 47 & 49 \hline \end{tabular}

1. Rocket $A$ has positive velocity $v ( t )$ after being launched upward from an initial height of 0 feet at time $t = 0$ seconds. The velocity of the rocket is recorded for selected values of $t$ over the interval $0 \leq t \leq 80$ seconds, as shown in the table above.
   (a) Find the average acceleration of rocket $A$ over the time interval $0 \leq t \leq 80$ seconds. Indicate units of measure.
   (b) Using correct units, explain the meaning of $\int _ { 10 } ^ { 70 } v ( t ) d t$ in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate $\int _ { 10 } ^ { 70 } v ( t ) d t$.
   (c) Rocket $B$ is launched upward with an acceleration of $a ( t ) = \frac { 3 } { \sqrt { t + 1 } }$ feet per second per second. At time $t = 0$ seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time $t = 80$ seconds? Explain your answer.

WRITE ALL WORK IN THE PINK EXAM BOOKLET.

© 2006 The College Board. All rights reserved. Visit \href{http://apcentral.collegeboard.com}{apcentral.collegeboard.com} (for AP professionals) and \href{http://www.collegeboard.com/apstudents}{www.collegeboard.com/apstudents} (for students and parents).

A car is traveling on a straight road. For $0 \leq t \leq 24$ seconds, the car's velocity $v ( t )$, in meters per second, is modeled by the piecewise-linear function defined by the graph above.
(a) Find $\int _ { 0 } ^ { 24 } v ( t ) \, d t$. Using correct units, explain the meaning of $\int _ { 0 } ^ { 24 } v ( t ) \, d t$.
(b) For each of $v ^ { \prime } ( 4 )$ and $v ^ { \prime } ( 20 )$, find the value or explain why it does not exist. Indicate units of measure.
(c) Let $a ( t )$ be the car's acceleration at time $t$, in meters per second per second. For $0 < t < 24$, write a piecewise-defined function for $a ( t )$.
(d) Find the average rate of change of $v$ over the interval $8 \leq t \leq 20$. Does the Mean Value Theorem guarantee a value of $c$, for $8 < c < 20$, such that $v ^ { \prime } ( c )$ is equal to this average rate of change? Why or why not?

At an intersection in Thomasville, Oregon, cars turn left at the rate $L(t) = 60\sqrt{t}\sin^{2}\left(\frac{t}{3}\right)$ cars per hour over the time interval $0 \leq t \leq 18$ hours.
(a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval $0 \leq t \leq 18$ hours.
(b) Traffic engineers will consider turn restrictions when $L(t) \geq 150$ cars per hour. Find all values of $t$ for which $L(t) \geq 150$ and compute the average value of $L$ over this time interval. Indicate units of measure.
(c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.