For $0 \leq t \leq 5$, a particle is moving along a curve so that its position at time $t$ is $( x ( t ) , y ( t ) )$. At time $t = 1$, the particle is at position $( 2 , - 7 )$. It is known that $\frac { d x } { d t } = \sin \left( \frac { t } { t + 3 } \right)$ and $\frac { d y } { d t } = e ^ { \cos t }$.
(a) Write an equation for the line tangent to the curve at the point $( 2 , - 7 )$.
(b) Find the $y$-coordinate of the position of the particle at time $t = 4$.
(c) Find the total distance traveled by the particle from time $t = 1$ to time $t = 4$.
(d) Find the time at which the speed of the particle is 2.5. Find the acceleration vector of the particle at this time.

Train $A$ runs back and forth on an east-west section of railroad track. Train A's velocity, measured in meters per minute, is given by a differentiable function $v _ { A } ( t )$, where time $t$ is measured in minutes. Selected values for $v _ { A } ( t )$ are given in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 2 & 5 & 8 & 12 \hline

$v _ { A } ( t )$

(meters/minute)

& 0 & 100 & 40 & - 120 & - 150 \hline \end{tabular}
(a) Find the average acceleration of train $A$ over the interval $2 \leq t \leq 8$.
(b) Do the data in the table support the conclusion that train $A$'s velocity is $-100$ meters per minute at some time $t$ with $5 < t < 8$? Give a reason for your answer.
(c) At time $t = 2$, train $A$'s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train $A$, in meters from the Origin Station, at time $t = 12$. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time $t = 12$.
(d) A second train, train $B$, travels north from the Origin Station. At time $t$ the velocity of train $B$ is given by $v _ { B } ( t ) = - 5 t ^ { 2 } + 60 t + 25$, and at time $t = 2$ the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train $A$ and train $B$ is changing at time $t = 2$.

Johanna jogs along a straight path. For $0 \leq t \leq 40$, Johanna's velocity is given by a differentiable function $v$. Selected values of $v(t)$, where $t$ is measured in minutes and $v(t)$ is measured in meters per minute, are given in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 12 & 20 & 24 & 40 \hline $v(t)$ & 0 & 200 & 240 & -220 & 150 \hline

(meters per minute)

& & & & & \hline \end{tabular}
(a) Use the data in the table to estimate the value of $v'(16)$.
(b) Using correct units, explain the meaning of the definite integral $\int_0^{40} |v(t)|\, dt$ in the context of the problem. Approximate the value of $\int_0^{40} |v(t)|\, dt$ using a right Riemann sum with the four subintervals indicated in the table.
(c) Bob is riding his bicycle along the same path. For $0 \leq t \leq 10$, Bob's velocity is modeled by $B(t) = t^3 - 6t^2 + 300$, where $t$ is measured in minutes and $B(t)$ is measured in meters per minute. Find Bob's acceleration at time $t = 5$.
(d) Based on the model $B$ from part (c), find Bob's average velocity during the interval $0 \leq t \leq 10$.

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?

A particle moving along a curve in the plane has position $( x ( t ) , y ( t ) )$ at time $t$, where $$\frac { d x } { d t } = \sqrt { t ^ { 4 } + 9 } \text { and } \frac { d y } { d t } = 2 e ^ { t } + 5 e ^ { - t }$$ for all real values of $t$. At time $t = 0$, the particle is at the point $( 4,1 )$.
(a) Find the speed of the particle and its acceleration vector at time $t = 0$.
(b) Find an equation of the line tangent to the path of the particle at time $t = 0$.
(c) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 3$.
(d) Find the $x$-coordinate of the position of the particle at time $t = 3$.

An object moving along a curve in the $xy$-plane is at position $(x(t), y(t))$ at time $t$, where $$\frac{dx}{dt} = \sin^{-1}\left(1 - 2e^{-t}\right) \text{ and } \frac{dy}{dt} = \frac{4t}{1 + t^{3}}$$ for $t \geq 0$. At time $t = 2$, the object is at the point $(6, -3)$. (Note: $\sin^{-1} x = \arcsin x$)
(a) Find the acceleration vector and the speed of the object at time $t = 2$.
(b) The curve has a vertical tangent line at one point. At what time $t$ is the object at this point?
(c) Let $m(t)$ denote the slope of the line tangent to the curve at the point $(x(t), y(t))$. Write an expression for $m(t)$ in terms of $t$ and use it to evaluate $\lim_{t \rightarrow \infty} m(t)$.
(d) The graph of the curve has a horizontal asymptote $y = c$. Write, but do not evaluate, an expression involving an improper integral that represents this value $c$.

A diver leaps from the edge of a diving platform into a pool below. The figure above shows the initial position of the diver and her position at a later time. At time $t$ seconds after she leaps, the horizontal distance from the front edge of the platform to the diver's shoulders is given by $x(t)$, and the vertical distance from the water surface to her shoulders is given by $y(t)$, where $x(t)$ and $y(t)$ are measured in meters. Suppose that the diver's shoulders are 11.4 meters above the water when she makes her leap and that $$\frac{dx}{dt} = 0.8 \quad \text{and} \quad \frac{dy}{dt} = 3.6 - 9.8t,$$ for $0 \leq t \leq A$, where $A$ is the time that the diver's shoulders enter the water.
(a) Find the maximum vertical distance from the water surface to the diver's shoulders.
(b) Find $A$, the time that the diver's shoulders enter the water.
(c) Find the total distance traveled by the diver's shoulders from the time she leaps from the platform until the time her shoulders enter the water.
(d) Find the angle $\theta$, $0 < \theta < \frac{\pi}{2}$, between the path of the diver and the water at the instant the diver's shoulders enter the water.

A particle is moving along a curve so that its position at time $t$ is $(x(t), y(t))$, where $x(t) = t^2 - 4t + 8$ and $y(t)$ is not explicitly given. Both $x$ and $y$ are measured in meters, and $t$ is measured in seconds. It is known that $\frac{dy}{dt} = te^{t-3} - 1$.
(a) Find the speed of the particle at time $t = 3$ seconds.
(b) Find the total distance traveled by the particle for $0 \leq t \leq 4$ seconds.
(c) Find the time $t$, $0 \leq t \leq 4$, when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time? Give a reason for your answer.
(d) There is a point with $x$-coordinate 5 through which the particle passes twice. Find each of the following.
(i) The two values of $t$ when that occurs
(ii) The slopes of the lines tangent to the particle's path at that point
(iii) The $y$-coordinate of that point, given $y(2) = 3 + \frac{1}{e}$

At time $t$, a particle moving in the $xy$-plane is at position $(x(t), y(t))$, where $x(t)$ and $y(t)$ are not explicitly given. For $t \geq 0$, $\frac{dx}{dt} = 4t + 1$ and $\frac{dy}{dt} = \sin\left(t^2\right)$. At time $t = 0$, $x(0) = 0$ and $y(0) = -4$.
(a) Find the speed of the particle at time $t = 3$, and find the acceleration vector of the particle at time $t = 3$.
(b) Find the slope of the line tangent to the path of the particle at time $t = 3$.
(c) Find the position of the particle at time $t = 3$.
(d) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 3$.

For $t \geq 0$, a particle is moving along a curve so that its position at time $t$ is $(x(t), y(t))$. At time $t = 2$, the particle is at position $(1, 5)$. It is known that $\frac{dx}{dt} = \frac{\sqrt{t+2}}{e^{t}}$ and $\frac{dy}{dt} = \sin^{2} t$.
(a) Is the horizontal movement of the particle to the left or to the right at time $t = 2$? Explain your answer. Find the slope of the path of the particle at time $t = 2$.
(b) Find the $x$-coordinate of the particle's position at time $t = 4$.
(c) Find the speed of the particle at time $t = 4$. Find the acceleration vector of the particle at time $t = 4$.
(d) Find the distance traveled by the particle from time $t = 2$ to $t = 4$.

For $t \geq 0$, a particle is moving along a curve so that its position at time $t$ is $( x ( t ) , y ( t ) )$. At time $t = 2$, the particle is at position $( 1,5 )$. It is known that $\frac { d x } { d t } = \frac { \sqrt { t + 2 } } { e ^ { t } }$ and $\frac { d y } { d t } = \sin ^ { 2 } t$.
(a) Is the horizontal movement of the particle to the left or to the right at time $t = 2$ ? Explain your answer. Find the slope of the path of the particle at time $t = 2$.
(b) Find the $x$-coordinate of the particle's position at time $t = 4$.
(c) Find the speed of the particle at time $t = 4$. Find the acceleration vector of the particle at time $t = 4$.
(d) Find the distance traveled by the particle from time $t = 2$ to $t = 4$.

Train $A$ runs back and forth on an east-west section of railroad track. Train A's velocity, measured in meters per minute, is given by a differentiable function $v _ { A } ( t )$, where time $t$ is measured in minutes. Selected values for $v _ { A } ( t )$ are given in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 2 & 5 & 8 & 12 \hline

$v _ { A } ( t )$

(meters/minute)

& 0 & 100 & 40 & - 120 & - 150 \hline \end{tabular}
(a) Find the average acceleration of train $A$ over the interval $2 \leq t \leq 8$.
(b) Do the data in the table support the conclusion that train $A$'s velocity is $-100$ meters per minute at some time $t$ with $5 < t < 8$? Give a reason for your answer.
(c) At time $t = 2$, train $A$'s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train $A$, in meters from the Origin Station, at time $t = 12$. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time $t = 12$.
(d) A second train, train $B$, travels north from the Origin Station. At time $t$ the velocity of train $B$ is given by $v _ { B } ( t ) = - 5 t ^ { 2 } + 60 t + 25$, and at time $t = 2$ the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train $A$ and train $B$ is changing at time $t = 2$.

At time $t \geq 0$, a particle moving along a curve in the $xy$-plane has position $( x ( t ) , y ( t ) )$ with velocity vector $v ( t ) = \left( \cos \left( t ^ { 2 } \right) , e ^ { 0.5 t } \right)$. At $t = 1$, the particle is at the point $( 3 , 5 )$.
(a) Find the $x$-coordinate of the position of the particle at time $t = 2$.
(b) For $0 < t < 1$, there is a point on the curve at which the line tangent to the curve has a slope of 2. At what time is the object at that point?
(c) Find the time at which the speed of the particle is 3.
(d) Find the total distance traveled by the particle from time $t = 0$ to time $t = 1$.

Johanna jogs along a straight path. For $0 \leq t \leq 40$, Johanna's velocity is given by a differentiable function $v$. Selected values of $v ( t )$, where $t$ is measured in minutes and $v ( t )$ is measured in meters per minute, are given in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 12 & 20 & 24 & 40 \hline

$v ( t )$

(meters per minute)

& 0 & 200 & 240 & - 220 & 150 \hline \end{tabular}
(a) Use the data in the table to estimate the value of $v ^ { \prime } ( 16 )$.
(b) Using correct units, explain the meaning of the definite integral $\int _ { 0 } ^ { 40 } | v ( t ) | \, dt$ in the context of the problem. Approximate the value of $\int _ { 0 } ^ { 40 } | v ( t ) | \, dt$ using a right Riemann sum with the four subintervals indicated in the table.
(c) Bob is riding his bicycle along the same path. For $0 \leq t \leq 10$, Bob's velocity is modeled by $B ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 300$, where $t$ is measured in minutes and $B ( t )$ is measured in meters per minute. Find Bob's acceleration at time $t = 5$.
(d) Based on the model $B$ from part (c), find Bob's average velocity during the interval $0 \leq t \leq 10$.

At time $t$, the position of a particle moving in the $xy$-plane is given by the parametric functions $( x ( t ) , y ( t ) )$, where $\frac { d x } { d t } = t ^ { 2 } + \sin \left( 3 t ^ { 2 } \right)$. The graph of $y$, consisting of three line segments, is shown in the figure above. At $t = 0$, the particle is at position $( 5,1 )$.
(a) Find the position of the particle at $t = 3$.
(b) Find the slope of the line tangent to the path of the particle at $t = 3$.
(c) Find the speed of the particle at $t = 3$.
(d) Find the total distance traveled by the particle from $t = 0$ to $t = 2$.

For time $t \geq 0$, a particle moves in the $x y$-plane with position $( x ( t ) , y ( t ) )$ and velocity vector $\left\langle ( t - 1 ) e ^ { t ^ { 2 } } , \sin \left( t ^ { 1.25 } \right) \right\rangle$. At time $t = 0$, the position of the particle is $( - 2,5 )$.
(a) Find the speed of the particle at time $t = 1.2$. Find the acceleration vector of the particle at time $t = 1.2$.
(b) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 1.2$.
(c) Find the coordinates of the point at which the particle is farthest to the left for $t \geq 0$. Explain why there is no point at which the particle is farthest to the right for $t \geq 0$.

A particle moving along a curve in the $x y$-plane is at position $( x ( t ) , y ( t ) )$ at time $t > 0$. The particle moves in such a way that $\frac { d x } { d t } = \sqrt { 1 + t ^ { 2 } }$ and $\frac { d y } { d t } = \ln \left( 2 + t ^ { 2 } \right)$. At time $t = 4$, the particle is at the point $( 1,5 )$.
(a) Find the slope of the line tangent to the path of the particle at time $t = 4$.
(b) Find the speed of the particle at time $t = 4$, and find the acceleration vector of the particle at time $t = 4$.
(c) Find the $y$-coordinate of the particle's position at time $t = 6$.
(d) Find the total distance the particle travels along the curve from time $t = 4$ to time $t = 6$.

For $0 \leq t \leq \pi$, a particle is moving along the curve shown so that its position at time $t$ is $(x(t), y(t))$, where $x(t)$ is not explicitly given and $y(t) = 2\sin t$. It is known that $\frac{dx}{dt} = e^{\cos t}$. At time $t = 0$, the particle is at position $(1, 0)$.
(a) Find the acceleration vector of the particle at time $t = 1$. Show the setup for your calculations.
(b) For $0 \leq t \leq \pi$, find the first time $t$ at which the speed of the particle is 1.5. Show the work that leads to your answer.
(c) Find the slope of the line tangent to the path of the particle at time $t = 1$. Find the $x$-coordinate of the position of the particle at time $t = 1$. Show the work that leads to your answers.
(d) Find the total distance traveled by the particle over the time interval $0 \leq t \leq \pi$. Show the setup for your calculations.

A particle moving along a curve in the $xy$-plane has position $(x(t), y(t))$ at time $t$ seconds, where $x(t)$ and $y(t)$ are measured in centimeters. It is known that $x'(t) = 8t - t^2$ and $y'(t) = -t + \sqrt{t^{1.2} + 20}$. At time $t = 2$ seconds, the particle is at the point $(3, 6)$.
(a) Find the speed of the particle at time $t = 2$ seconds. Show the setup for your calculations.
(b) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 2$. Show the setup for your calculations.
(c) Find the $y$-coordinate of the position of the particle at the time $t = 0$. Show the setup for your calculations.
(d) For $2 \leq t \leq 8$, the particle remains in the first quadrant. Find all times $t$ in the interval $2 \leq t \leq 8$ when the particle is moving toward the $x$-axis. Give a reason for your answer.

As shown in the figure, there is a rectangular piece of paper ABCD with $\overline { \mathrm { AB } } = 9$ and $\overline { \mathrm { AD } } = 3$. Using the line connecting point E on segment AB and point F on segment DC as the fold line, the paper is folded so that the orthogonal projection of point B onto the plane AEFD is point D. When $\overline { \mathrm { AE } } = 3$, the angle between the two planes AEFD and EFCB is $\theta$. Find the value of $60 \cos \theta$. (Given that $0 < \theta < \frac { \pi } { 2 }$ and the thickness of the paper is negligible.) [4 points]

The position $\mathrm { P } ( x , y )$ of a point P moving on the coordinate plane at time $t ( 0 < t < \pi )$ is given by $$x = \sqrt { 3 } \sin t , \quad y = 2 \cos t - 5$$ At time $t = \alpha ( 0 < \alpha < \pi )$, the velocity $\vec { v }$ of point P and $\overrightarrow { \mathrm { OP } }$ are parallel. What is the value of $\cos \alpha$? (Here, O is the origin.) [4 points]
(1) $\frac { 1 } { 10 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 3 } { 10 }$
(4) $\frac { 2 } { 5 }$
(5) $\frac { 1 } { 2 }$

The position $( x , y )$ of a point P moving on the coordinate plane at time $t ( t \geq 0 )$ is $$x = 1 - \cos 4 t , y = \frac { 1 } { 4 } \sin 4 t.$$ When the speed of point P is maximum, find the magnitude of the acceleration of point P. [3 points]

A particle in three-dimensional space moves along the path: $\vec { r } ( t ) = \left\langle t ^ { 2 } , \sin ( t ) , e ^ { t } \right\rangle$. Find the time when the particle changes direction.