At time $t = 0$, a jogger is running at a velocity of 300 meters per minute. The jogger is slowing down with a negative acceleration that is directly proportional to time $t$. This brings the jogger to a stop in 10 minutes. (a) Write an expression for the velocity of the jogger at time $t$. V (b) What is the total distance traveled by the jogger in that 10 -minute interval?

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 $t = 2$.

Two runners, $A$ and $B$, run on a straight racetrack for $0 \leq t \leq 10$ seconds. The graph above, which consists of two line segments, shows the velocity, in meters per second, of Runner $A$. The velocity, in meters per second, of Runner $B$ is given by the function $v$ defined by $v ( t ) = \frac { 24 t } { 2 t + 3 }$.
(a) Find the velocity of Runner $A$ and the velocity of Runner $B$ at time $t = 2$ seconds. Indicate units of measure.
(b) Find the acceleration of Runner $A$ and the acceleration of Runner $B$ at time $t = 2$ seconds. Indicate units of measure.
(c) Find the total distance run by Runner $A$ and the total distance run by Runner $B$ over the time interval $0 \leq t \leq 10$ seconds. Indicate units of measure.

A particle moves along the $x$-axis so that its velocity $v$ at time $t$, for $0 \leq t \leq 5$, is given by $v ( t ) = \ln \left( t ^ { 2 } - 3 t + 3 \right)$. 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 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 coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let $h$ be the depth of the coffee in the pot, measured in inches, where $h$ is a function of time $t$, measured in seconds. The volume $V$ of coffee in the pot is changing at the rate of $-5\pi\sqrt{h}$ cubic inches per second. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r^2 h$.)
(a) Show that $\dfrac{dh}{dt} = -\dfrac{\sqrt{h}}{5}$.
(b) Given that $h = 17$ at time $t = 0$, solve the differential equation $\dfrac{dh}{dt} = -\dfrac{\sqrt{h}}{5}$ for $h$ as a function of $t$.
(c) At what time $t$ is the coffeepot empty?

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 so that its velocity at any time $t \geqq 0$ is given by $v ( t ) = 1 - \sin ( 2 \pi t )$. (a) Find the acceleration $a ( t )$ of the particle at any time $t$. (b) Find all values of $t , 0 \leqq t \leqq 2$, for which the particle is at rest. (c) Find the position $x ( t )$ of the particle at any time $t$ if $x ( 0 ) = 0$.

A particle moves along the $x$-axis so that its velocity $v$ at time $t \geq 0$ is given by $v ( t ) = \sin \left( t ^ { 2 } \right)$. The graph of $v$ is shown above for $0 \leq t \leq \sqrt { 5 \pi }$. The position of the particle at time $t$ is $x ( t )$ and its position at time $t = 0$ is $x ( 0 ) = 5$. (a) Find the acceleration of the particle at time $t = 3$. (b) Find the total distance traveled by the particle from time $t = 0$ to $t = 3$. (c) Find the position of the particle at time $t = 3$. (d) For $0 \leq t \leq \sqrt { 5 \pi }$, find the time $t$ at which the particle is farthest to the right. Explain 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.

Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiable function $B$ models Ben's position on the track, measured in meters from the western end of the track, at time $t$, measured in seconds from the start of the ride. The table below gives values for $B(t)$ and Ben's velocity, $v(t)$, measured in meters per second, at selected times $t$.

\begin{tabular}{ c } $t$

(seconds)

& 0 & 10 & 40 & 60 \hline

$B(t)$

(meters)

& 100 & 136 & 9 & 49 \hline

$v(t)$

(meters per second)

& 2.0 & 2.3 & 2.5 & 4.6 \hline \end{tabular}
(a) Use the data in the table to approximate Ben's acceleration at time $t = 5$ seconds. Indicate units of measure.
(b) Using correct units, interpret the meaning of $\int_{0}^{60} |v(t)|\, dt$ in the context of this problem. Approximate $\int_{0}^{60} |v(t)|\, dt$ using a left Riemann sum with the subintervals indicated by the data in the table.
(c) For $40 \leq t \leq 60$, must there be a time $t$ when Ben's velocity is 2 meters per second? Justify your answer.
(d) A light is directly above the western end of the track. Ben rides so that at time $t$, the distance $L(t)$ between Ben and the light satisfies $(L(t))^2 = 12^2 + (B(t))^2$. At what rate is the distance between Ben and the light changing at time $t = 40$?

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

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 a line so that its acceleration for $t \geq 0$ is given by $a ( t ) = \frac { t + 3 } { \sqrt { t ^ { 3 } + 1 } }$. If the particle's velocity at $t = 0$ is 5, what is the velocity of the particle at $t = 3$ ?
(A) 0.713
(B) 1.134
(C) 6.134
(D) 6.710
(E) 11.710

For $t \geq 0$, a particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v ( t ) = 1 + 2 \sin \left( \frac { t ^ { 2 } } { 2 } \right)$. The particle is at position $x = 2$ at time $t = 4$.
(a) At time $t = 4$, is the particle speeding up or slowing down?
(b) Find all times $t$ in the interval $0 < t < 3$ when the particle changes direction. Justify your answer.
(c) Find the position of the particle at time $t = 0$.
(d) Find the total distance the particle travels from time $t = 0$ to time $t = 3$.

Two particles move along the $x$-axis. For $0 \leq t \leq 8$, the position of particle $P$ at time $t$ is given by $x_P(t) = \ln\left(t^2 - 2t + 10\right)$, while the velocity of particle $Q$ at time $t$ is given by $v_Q(t) = t^2 - 8t + 15$. Particle $Q$ is at position $x = 5$ at time $t = 0$.
(a) For $0 \leq t \leq 8$, when is particle $P$ moving to the left?
(b) For $0 \leq t \leq 8$, find all times $t$ during which the two particles travel in the same direction.
(c) Find the acceleration of particle $Q$ at time $t = 2$. Is the speed of particle $Q$ increasing, decreasing, or neither at time $t = 2$? Explain your reasoning.
(d) Find the position of particle $Q$ the first time it changes direction.

A particle moves along the $x$-axis with velocity given by $v ( t ) = \frac { 10 \sin \left( 0.4 t ^ { 2 } \right) } { t ^ { 2 } - t + 3 }$ for time $0 \leq t \leq 3.5$.
The particle is at position $x = - 5$ at time $t = 0$.
(a) Find the acceleration of the particle at time $t = 3$.
(b) Find the position of the particle at time $t = 3$.
(c) Evaluate $\int _ { 0 } ^ { 3.5 } v ( t ) \, dt$, and evaluate $\int _ { 0 } ^ { 3.5 } | v ( t ) | \, dt$. Interpret the meaning of each integral in the context of the problem.
(d) A second particle moves along the $x$-axis with position given by $x _ { 2 } ( t ) = t ^ { 2 } - t$ for $0 \leq t \leq 3.5$. At what time $t$ are the two particles moving with the same velocity?

The velocity of a particle, $P$, moving along the $x$-axis is given by the differentiable function $v_P$, where $v_P(t)$ is measured in meters per hour and $t$ is measured in hours. Selected values of $v_P(t)$ are shown in the table below. Particle $P$ is at the origin at time $t = 0$.

\begin{tabular}{ c } $t$

(hours)

& 0 & 0.3 & 1.7 & 2.8 & 4 \hline

$v_P(t)$

(meters per hour)

& 0 & 55 & -29 & 55 & 48 \hline \end{tabular}
(a) Justify why there must be at least one time $t$, for $0.3 \leq t \leq 2.8$, at which $v_P'(t)$, the acceleration of particle $P$, equals 0 meters per hour per hour.
(b) Use a trapezoidal sum with the three subintervals $[0, 0.3]$, $[0.3, 1.7]$, and $[1.7, 2.8]$ to approximate the value of $\int_0^{2.8} v_P(t)\, dt$.
(c) A second particle, $Q$, also moves along the $x$-axis so that its velocity for $0 \leq t \leq 4$ is given by $v_Q(t) = 45\sqrt{t}\cos\left(0.063t^2\right)$ meters per hour. Find the time interval during which the velocity of particle $Q$ is at least 60 meters per hour. Find the distance traveled by particle $Q$ during the interval when the velocity of particle $Q$ is at least 60 meters per hour.
(d) At time $t = 0$, particle $Q$ is at position $x = -90$. Using the result from part (b) and the function $v_Q$ from part (c), approximate the distance between particles $P$ and $Q$ at time $t = 2.8$.

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 $0 \leq t \leq \pi$.

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.

Let $f$ be the function whose graph goes through the point $(3, 6)$ and whose derivative is given by $f'(x) = \frac{1 + e^x}{x^2}$.
(a) Write an equation of the line tangent to the graph of $f$ at $x = 3$ and use it to approximate $f(3.1)$.
(b) Use Euler's method, starting at $x = 3$ with a step size of 0.05, to approximate $f(3.1)$. Use $f''$ to explain why this approximation is less than $f(3.1)$.
(c) Use $\int_{3}^{3.1} f'(x)\, dx$ to evaluate $f(3.1)$.

2. Two runners, $A$ and $B$, run on a straight racetrack for $0 \leq t \leq 10$ seconds. The graph above, which consists of two line segments, shows the velocity, in meters per second, of Runner $A$. The velocity, in meters per second, of Runner $B$ is given by the function $v$ defined by $v ( t ) = \frac { 24 t } { 2 t + 3 }$.
(a) Find the velocity of Runner $A$ and the velocity of Runner $B$ at time $t = 2$ seconds. Indicate units of measure.
(b) Find the acceleration of Runner $A$ and the acceleration of Runner $B$ at time $t = 2$ seconds. Indicate units of measure.
(c) Find the total distance run by Runner $A$ and the total distance run by Runner $B$ over the time interval $0 \leq t \leq 10$ seconds. Indicate units of measure.