A race car is traveling on a straight track at a velocity of 80 meters per second when the brakes are applied at time $t = 0$ seconds. From time $t = 0$ to the moment the race car stops, the acceleration of the race car is given by $a ( t ) = - 6 t ^ { 2 } - t$ meters per second per second. During this time period, how far does the race car travel?
(A) 188.229 m
(B) 198.766 m
(C) 260.042 m
(D) 267.089 m

A particle moves along the $y$-axis with velocity given by $v ( t ) = t \sin \left( t ^ { 2 } \right)$ for $t \geq 0$.
(a) In which direction (up or down) is the particle moving at time $t = 1.5$ ? Why?
(b) Find the acceleration of the particle at time $t = 1.5$. Is the velocity of the particle increasing at $t = 1.5$ ? Why or why not?
(c) Given that $y ( t )$ is the position of the particle at time $t$ and that $y ( 0 ) = 3$, find $y ( 2 )$.
(d) Find the total distance traveled by the particle from $t = 0$ to $t = 2$.

A car is traveling on a straight road with velocity $55\,\mathrm{ft/sec}$ at time $t = 0$. For $0 \leq t \leq 18$ seconds, the car's acceleration $a(t)$, in $\mathrm{ft/sec}^{2}$, is the piecewise linear function defined by the graph above.
(a) Is the velocity of the car increasing at $t = 2$ seconds? Why or why not?
(b) At what time in the interval $0 \leq t \leq 18$, other than $t = 0$, is the velocity of the car $55\,\mathrm{ft/sec}$? Why?
(c) On the time interval $0 \leq t \leq 18$, what is the car's absolute maximum velocity, in $\mathrm{ft/sec}$, and at what time does it occur? Justify your answer.
(d) At what times in the interval $0 \leq t \leq 18$, if any, is the car's velocity equal to zero? Justify your answer.

An object moves along the $x$-axis with initial position $x ( 0 ) = 2$. The velocity of the object at time $t \geq 0$ is given by $v ( t ) = \sin \left( \frac { \pi } { 3 } t \right)$.
(a) What is the acceleration of the object at time $t = 4$?
(b) Consider the following two statements. Statement I: For $3 < t < 4.5$, the velocity of the object is decreasing. Statement II: For $3 < t < 4.5$, the speed of the object is increasing. Are either or both of these statements correct? For each statement provide a reason why it is correct or not correct.
(c) What is the total distance traveled by the object over the time interval $0 \leq t \leq 4$?
(d) What is the position of the object at time $t = 4$?

A particle moves along the $x$-axis so that its velocity $v$ at any time $t$, for $0 \leq t \leq 16$, is given by $v(t) = e^{2\sin t} - 1$. At time $t = 0$, the particle is at the origin.
(a) On the axes provided, sketch the graph of $v(t)$ for $0 \leq t \leq 16$.
(b) During what intervals of time is the particle moving to the left? Give a reason for your answer.
(c) Find the total distance traveled by the particle from $t = 0$ to $t = 4$.
(d) Is there any time $t$, $0 < t \leq 16$, at which the particle returns to the origin? Justify your answer.

A particle moves along the $x$-axis so that its velocity at time $t$ is given by $$v(t) = -(t+1)\sin\left(\frac{t^2}{2}\right)$$ At time $t = 0$, the particle is at position $x = 1$.
(a) Find the acceleration of the particle at time $t = 2$. Is the speed of the particle increasing at $t = 2$? Why or why not?
(b) Find all times $t$ in the open interval $0 < t < 3$ when the particle changes direction. Justify your answer.
(c) Find the total distance traveled by the particle from time $t = 0$ until time $t = 3$.
(d) During the time interval $0 \leq t \leq 3$, what is the greatest distance between the particle and the origin? Show the work that leads to your answer.

A particle moves along the $x$-axis so that its velocity $v$ at time $t$, for $0 \leq t \leq 5$, is given by $v(t) = \ln\left(t^2 - 3t + 3\right)$. The particle is at position $x = 8$ at time $t = 0$.
(a) Find the acceleration of the particle at time $t = 4$.
(b) Find all times $t$ in the open interval $0 < t < 5$ at which the particle changes direction. During which time intervals, for $0 \leq t \leq 5$, does the particle travel to the left?
(c) Find the position of the particle at time $t = 2$.
(d) Find the average speed of the particle over the interval $0 \leq t \leq 2$.

A particle moves along the $x$-axis with position at time $t$ given by $x(t) = e^{-t}\sin t$ for $0 \leq t \leq 2\pi$.
(a) Find the time $t$ at which the particle is farthest to the left. Justify your answer.
(b) Find the value of the constant $A$ for which $x(t)$ satisfies the equation $Ax^{\prime\prime}(t) + x^{\prime}(t) + x(t) = 0$ for $0 < t < 2\pi$.

A particle moves along the $x$-axis so that its velocity at time $t$, for $0 \leq t \leq 6$, is given by a differentiable function $v$ whose graph is shown above. The velocity is 0 at $t = 0 , t = 3$, and $t = 5$, and the graph has horizontal tangents at $t = 1$ and $t = 4$. The areas of the regions bounded by the $t$-axis and the graph of $v$ on the intervals $[ 0,3 ] , [ 3,5 ]$, and $[ 5,6 ]$ are 8, 3, and 2, respectively. At time $t = 0$, the particle is at $x = - 2$.
(a) For $0 \leq t \leq 6$, find both the time and the position of the particle when the particle is farthest to the left. Justify your answer.
(b) For how many values of $t$, where $0 \leq t \leq 6$, is the particle at $x = - 8$ ? Explain your reasoning.
(c) On the interval $2 < t < 3$, is the speed of the particle increasing or decreasing? Give a reason for your answer.
(d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer.

Caren rides her bicycle along a straight road from home to school, starting at home at time $t = 0$ minutes and arriving at school at time $t = 12$ minutes. During the time interval $0 \leq t \leq 12$ minutes, her velocity $v(t)$, in miles per minute, is modeled by the piecewise-linear function whose graph is shown above.
(a) Find the acceleration of Caren's bicycle at time $t = 7.5$ minutes. Indicate units of measure.
(b) Using correct units, explain the meaning of $\int_{0}^{12} |v(t)| \, dt$ in terms of Caren's trip. Find the value of $\int_{0}^{12} |v(t)| \, dt$.
(c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer.
(d) Larry also rides his bicycle along a straight road from home to school in 12 minutes. His velocity is modeled by the function $w$ given by $w(t) = \frac{\pi}{15} \sin\left(\frac{\pi}{12} t\right)$, where $w(t)$ is in miles per minute for $0 \leq t \leq 12$ minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer.

For $0 \leq t \leq 6$, a particle is moving along the $x$-axis. The particle's position, $x(t)$, is not explicitly given. The velocity of the particle is given by $v(t) = 2\sin\left(e^{t/4}\right) + 1$. The acceleration of the particle is given by $a(t) = \frac{1}{2}e^{t/4}\cos\left(e^{t/4}\right)$ and $x(0) = 2$.
(a) Is the speed of the particle increasing or decreasing at time $t = 5.5$? Give a reason for your answer.
(b) Find the average velocity of the particle for the time period $0 \leq t \leq 6$.
(c) Find the total distance traveled by the particle from time $t = 0$ to $t = 6$.
(d) For $0 \leq t \leq 6$, the particle changes direction exactly once. Find the position of the particle at that time.

For $0 \leq t \leq 12$, a particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v ( t ) = \cos \left( \frac { \pi } { 6 } t \right)$. The particle is at position $x = - 2$ at time $t = 0$.
(a) For $0 \leq t \leq 12$, when is the particle moving to the left?
(b) Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time $t = 0$ to time $t = 6$.
(c) Find the acceleration of the particle at time $t$. Is the speed of the particle increasing, decreasing, or neither at time $t = 4$ ? Explain your reasoning.
(d) Find the position of the particle at time $t = 4$.

A particle moves along the $x$-axis. The velocity of the particle at time $t$ is $6 t - t ^ { 2 }$. What is the total distance traveled by the particle from time $t = 0$ to $t = 3$ ?
(A) 3
(B) 6
(C) 9
(D) 18
(E) 27

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) $\frac { v ( 0 ) + v ( 8 ) } { 2 }$

The graph above gives the velocity, $v$, in ft/sec, of a car for $0 \leq t \leq 8$, where $t$ is the time in seconds. Of the following, which is the best estimate of the distance traveled by the car from $t = 0$ until the car comes to a complete stop?
(A) 21 ft
(B) 26 ft
(C) 180 ft
(D) 210 ft
(E) 260 ft

A particle moves along a straight line. For $0 \leq t \leq 5$, the velocity of the particle is given by $v ( t ) = - 2 + \left( t ^ { 2 } + 3 t \right) ^ { 6 / 5 } - t ^ { 3 }$, and the position of the particle is given by $s ( t )$. It is known that $s ( 0 ) = 10$.
(a) Find all values of $t$ in the interval $2 \leq t \leq 4$ for which the speed of the particle is 2.
(b) Write an expression involving an integral that gives the position $s ( t )$. Use this expression to find the position of the particle at time $t = 5$.
(c) Find all times $t$ in the interval $0 \leq t \leq 5$ at which the particle changes direction. Justify your answer.
(d) Is the speed of the particle increasing or decreasing at time $t = 4$? Give a reason for your answer.

A particle, $P$, is moving along the $x$-axis. The velocity of particle $P$ at time $t$ is given by $v_{P}(t) = \sin\left(t^{1.5}\right)$ for $0 \leq t \leq \pi$. At time $t = 0$, particle $P$ is at position $x = 5$.
A second particle, $Q$, also moves along the $x$-axis. The velocity of particle $Q$ at time $t$ is given by $v_{Q}(t) = (t - 1.8) \cdot 1.25^{t}$ for $0 \leq t \leq \pi$. At time $t = 0$, particle $Q$ is at position $x = 10$.
(a) Find the positions of particles $P$ and $Q$ at time $t = 1$.
(b) Are particles $P$ and $Q$ moving toward each other or away from each other at time $t = 1$? Explain your reasoning.
(c) Find the acceleration of particle $Q$ at time $t = 1$. Is the speed of particle $Q$ increasing or decreasing at time $t = 1$? Explain your reasoning.
(d) Find the total distance traveled by particle $P$ over the time interval $0 \leq t \leq \pi$.

Particle $P$ moves along the $x$-axis such that, for time $t > 0$, its position is given by $x_P(t) = 6 - 4e^{-t}$. Particle $Q$ moves along the $y$-axis such that, for time $t > 0$, its velocity is given by $v_Q(t) = \dfrac{1}{t^2}$. At time $t = 1$, the position of particle $Q$ is $y_Q(1) = 2$.
(a) Find $v_P(t)$, the velocity of particle $P$ at time $t$.
(b) Find $a_Q(t)$, the acceleration of particle $Q$ at time $t$. Find all times $t$, for $t > 0$, when the speed of particle $Q$ is decreasing. Justify your answer.
(c) Find $y_Q(t)$, the position of particle $Q$ at time $t$.
(d) As $t \to \infty$, which particle will eventually be farther from the origin? Give a reason for your answer.

Stephen swims back and forth along a straight path in a 50-meter-long pool for 90 seconds. Stephen's velocity is modeled by $v(t) = 2.38e^{-0.02t}\sin\left(\frac{\pi}{56}t\right)$, where $t$ is measured in seconds and $v(t)$ is measured in meters per second.
(a) Find all times $t$ in the interval $0 < t < 90$ at which Stephen changes direction. Give a reason for your answer.
(b) Find Stephen's acceleration at time $t = 60$ seconds. Show the setup for your calculations, and indicate units of measure. Is Stephen speeding up or slowing down at time $t = 60$ seconds? Give a reason for your answer.
(c) Find the distance between Stephen's position at time $t = 20$ seconds and his position at time $t = 80$ seconds. Show the setup for your calculations.
(d) Find the total distance Stephen swims over the time interval $0 \leq t \leq 90$ seconds. Show the setup for your calculations.

A particle moves along the $x$-axis so that its velocity at time $t \geq 0$ is given by $v(t) = \ln\left(t^2 - 4t + 5\right) - 0.2t$.
(a) There is one time, $t = t_R$, in the interval $0 < t < 2$ when the particle is at rest (not moving). Find $t_R$. For $0 < t < t_R$, is the particle moving to the right or to the left? Give a reason for your answer.
(b) Find the acceleration of the particle at time $t = 1.5$. Show the setup for your calculations. Is the speed of the particle increasing or decreasing at time $t = 1.5$? Explain your reasoning.
(c) The position of the particle at time $t$ is $x(t)$, and its position at time $t = 1$ is $x(1) = -3$. Find the position of the particle at time $t = 4$. Show the setup for your calculations.
(d) Find the total distance traveled by the particle over the interval $1 \leq t \leq 4$. Show the setup for your calculations.

Two particles, $H$ and $J$, are moving along the $x$-axis. For $0 \leq t \leq 5$, the position of particle $H$ at time $t$ is given by $x _ { H } ( t ) = e ^ { t ^ { 2 } - 4 t }$ and the velocity of particle $J$ at time $t$ is given by $v _ { J } ( t ) = 2 t \left( t ^ { 2 } - 1 \right) ^ { 3 }$.
A. Find the velocity of particle $H$ at time $t = 1$. Show the work that leads to your answer.
B. During what open intervals of time $t$, for $0 < t < 5$, are particles $H$ and $J$ moving in opposite directions? Give a reason for your answer.
C. It can be shown that $v _ { J } ^ { \prime } ( 2 ) > 0$. Is the speed of particle $J$ increasing, decreasing, or neither at time $t = 2$ ? Give a reason for your answer.
D. Particle $J$ is at position $x = 7$ at time $t = 0$. Find the position of particle $J$ at time $t = 2$. Show the work that leads to your answer.

A car is traveling on a straight road with velocity $55\,\mathrm{ft/sec}$ at time $t = 0$. For $0 \leq t \leq 18$ seconds, the car's acceleration $a(t)$, in $\mathrm{ft/sec}^2$, is the piecewise linear function defined by the graph above.
(a) Is the velocity of the car increasing at $t = 2$ seconds? Why or why not?
(b) At what time in the interval $0 \leq t \leq 18$, other than $t = 0$, is the velocity of the car $55\,\mathrm{ft/sec}$? Why?
(c) On the time interval $0 \leq t \leq 18$, what is the car's absolute maximum velocity, in $\mathrm{ft/sec}$, and at what time does it occur? Justify your answer.
(d) At what times in the interval $0 \leq t \leq 18$, if any, is the car's velocity equal to zero? Justify your answer.

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?

A particle moves along the $x$-axis so that its velocity at time $t$, for $0 \leq t \leq 6$, is given by a differentiable function $v$ whose graph is shown above. The velocity is 0 at $t = 0 , t = 3$, and $t = 5$, and the graph has horizontal tangents at $t = 1$ and $t = 4$. The areas of the regions bounded by the $t$-axis and the graph of $v$ on the intervals $[ 0,3 ] , [ 3,5 ]$, and $[ 5,6 ]$ are 8, 3, and 2, respectively. At time $t = 0$, the particle is at $x = - 2$.
(a) For $0 \leq t \leq 6$, find both the time and the position of the particle when the particle is farthest to the left. Justify your answer.
(b) For how many values of $t$, where $0 \leq t \leq 6$, is the particle at $x = - 8$ ? Explain your reasoning.
(c) On the interval $2 < t < 3$, is the speed of the particle increasing or decreasing? Give a reason for your answer.
(d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer.

Caren rides her bicycle along a straight road from home to school, starting at home at time $t = 0$ minutes and arriving at school at time $t = 12$ minutes. During the time interval $0 \leq t \leq 12$ minutes, her velocity $v(t)$, in miles per minute, is modeled by the piecewise-linear function whose graph is shown above.
(a) Find the acceleration of Caren's bicycle at time $t = 7.5$ minutes. Indicate units of measure.
(b) Using correct units, explain the meaning of $\int_{0}^{12} |v(t)| \, dt$ in terms of Caren's trip. Find the value of $\int_{0}^{12} |v(t)| \, dt$.
(c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer.
(d) Larry also rides his bicycle along a straight road from home to school in 12 minutes. His velocity is modeled by the function $w$ given by $w(t) = \frac{\pi}{15} \sin\left(\frac{\pi}{12} t\right)$, where $w(t)$ is in miles per minute for $0 \leq t \leq 12$ minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer.