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 $\int _ { 0 } ^ { 24 } v ( t ) \, dt$. Using correct units, explain the meaning of $\int _ { 0 } ^ { 24 } v ( t ) \, dt$.
(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?

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 below.

\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}
(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 $\int_{10}^{70} v(t)\, dt$ in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate $\int_{10}^{70} v(t)\, dt$.
(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.

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 below.

$t$ (seconds)	0	10	20	30	40	50	60	70	80
$v(t)$ (feet per second)	5	14	22	29	35	40	44	47	49

(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 $\int_{10}^{70} v(t)\, dt$ in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate $\int_{10}^{70} v(t)\, dt$.
(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.

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.

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50. The equation of motion of an object with mass $500\,\text{g}$ moving along the $x$-axis is, in SI units, $\vec{F} = (6 - 3t)\hat{i}$. The net average force exerted on this object during the time interval $t_1 = 1\,\text{s}$ to $t_2 = 3\,\text{s}$, in newtons, is:
(1) $3\hat{i}$ (2) $-3\hat{i}$ (3) $6\hat{i}$ (4) $-6\hat{i}$
\hrule
Space for calculations
%% Page 5

44-- The position--time graph of a particle moving along the $x$-axis with constant acceleration is shown below. What is the magnitude of the average speed of the particle in the first 7 seconds?
\begin{minipage}{0.45\textwidth} [Figure: position-time graph with x(m) axis showing value 24 at top, curve passing through origin, reaching a minimum near t=2, then rising to 24 at t=7; t(s) axis shows values 2 and 7] \end{minipage} \begin{minipage}{0.45\textwidth} (1) $\dfrac{25}{8}$
(2) $\dfrac{25}{7}$
(3) $\dfrac{23}{8}$
(4) $\dfrac{23}{7}$ \end{minipage}

46. The figure below shows the acceleration–time graph of a moving object that at moment $t = 0\,\text{s}$ has velocity $\vec{V} = +\!\left(8\,\dfrac{\text{m}}{\text{s}}\right)\hat{i}$ and has been moving. What is the average velocity of the object in these 8 seconds (in meters per second)?
[Figure: acceleration-time graph with $a\,(\frac{\text{m}}{\text{s}^2})$ on vertical axis and $t\,(\text{s})$ on horizontal axis. The graph shows $a = +2$ from $t=0$ to $t=3$, then $a = -6$ from $t=3$ to $t=8$.]

• [(1)] $12$
• [(2)] $15$
• [(3)] $\dfrac{43}{4}$
• [(4)] $\dfrac{53}{6}$

A particle starts from the origin at time $t = 0$ and moves along the positive $x$-axis. The graph of velocity with respect to time is shown in figure. What is the position of the particle at time $t = 5s$?
(1) $10 m$
(2) $9 m$
(3) $6 m$
(4) $3 m$