A particle moves along the $X$-axis so that at time $t$ its position is given by $x ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 9 t + 11$. (a) What is the velocity of the particle at $t = 0$ ? (b) During what time intervals is the particle moving to the left? (c) What is the total distance traveled by the particle from $t = 0$ to $t = 2$ ?

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 $t = 2$.

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 $0 \leq t \leq 2$.

$V ( t ) = t \cos t , t \geq 0 , y ( 0 ) = 3$
a) $t \cos t > 0,0 \leq t \leq 5$
$t \cos t = 0$
$t = 0$
$$\begin{gathered} \cos t = 0 \quad t \quad \cdots \quad t \\ t = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \quad a _ { - } \frac { \pi } { 2 } \quad \frac { \pi } { 2 } \quad E \end{gathered}$$
$( 0 , \pi / 2 ) , ( 3 \pi / 2,5 )$
b) $$\begin{aligned} \Lambda ( t ) & = V ( t ) \\ \Lambda ( t ) & = \cos t + t - \sin t ) \\ & = \cos t - t \sin t \end{aligned}$$
() $Y ( t ) = \int Y ( t ) d t$
$$\begin{aligned} & y ( t ) = \int t \cos t d t \text { let } \quad \frac { d } { d } = t \quad \frac { d v } { d } \\ & = \cos t \\ & = t \sin t - \int \sin t d t \quad \frac { d t } { d t } \quad v \\ & \end{aligned}$$
$Y ( t ) = + \sin t - ( - \cos t ) + c$
$= t \sin t + \cos t + c$
$y | 0 | = 3$
$j = \cos 0 + c$
$c = 2$
$Y ( t ) = t \sin t + \cos t + 2$
d) $V ( t ) = t \cos t$
$t \cos t = 0$
$t \neq 0 \quad \cos t = 0$
$$t = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \ldots$$
$Y \left( \frac { \pi } { 2 } \right) = \frac { \pi } { 2 } \sin \frac { \pi } { 2 } + \cos \frac { \pi } { 2 } + 2 = \frac { \pi } { 2 } + 0 + 2 = \frac { \pi } { 2 } + 2$

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 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 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 $v$ 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.

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 $t = 4$.

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 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.

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 $t = 3$.

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?

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.

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 $t = 4$.