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 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.

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?

4. A squirrel starts at building $A$ at time $t = 0$ and travels along a straight, horizontal wire connected to building $B$. For $0 \leq t \leq 18$, the squirrel's velocity is modeled by the piecewise-linear function defined by the graph above.
(a) At what times in the interval $0 < t < 18$, if any, does the squirrel change direction? Give a reason for your answer.
(b) At what time in the interval $0 \leq t \leq 18$ is the squirrel farthest from building $A$ ? How far from building $A$ is the squirrel at that time?
(c) Find the total distance the squirrel travels during the time interval $0 \leq t \leq 18$.
(d) Write expressions for the squirrel's acceleration $a ( t )$, velocity $v ( t )$, and distance $x ( t )$ from building $A$ that are valid for the time interval $7 < t < 10$.

WRITE ALL WORK IN THE EXAM BOOKLET.

1. Consider the differential equation $\frac { d y } { d x } = \frac { x + 1 } { y }$.
   (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and for $- 1 < x < 1$, sketch the solution curve that passes through the point $( 0 , - 1 )$. (Note: Use the axes provided in the exam booklet.) [Figure]
   (b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the $x y$-plane for which $y \neq 0$. Describe all points in the $x y$-plane, $y \neq 0$, for which $\frac { d y } { d x } = - 1$.
   (c) Find the particular solution $y = f ( x )$ to the given differential equation with the initial condition $f ( 0 ) = - 2$.
2. Two particles move along the $x$-axis. For $0 \leq t \leq 6$, the position of particle $P$ at time $t$ is given by $p ( t ) = 2 \cos \left( \frac { \pi } { 4 } t \right)$, while the position of particle $R$ at time $t$ is given by $r ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 9 t + 3$.
   (a) For $0 \leq t \leq 6$, find all times $t$ during which particle $R$ is moving to the right.
   (b) For $0 \leq t \leq 6$, find all times $t$ during which the two particles travel in opposite directions.
   (c) Find the acceleration of particle $P$ at time $t = 3$. Is particle $P$ speeding up, slowing down, or doing neither at time $t = 3$ ? Explain your reasoning.
   (d) Write, but do not evaluate, an expression for the average distance between the two particles on the interval $1 \leq t \leq 3$.

WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM

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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.