A curve is defined by the parametric equations $x ( t ) = t ^ { 2 } + 3$ and $y ( t ) = \sin \left( t ^ { 2 } \right)$. Which of the following is an expression for $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $t$?
(A) $- \sin \left( t ^ { 2 } \right)$
(B) $- 2 t \sin \left( t ^ { 2 } \right)$
(C) $\cos \left( t ^ { 2 } \right) - 2 t ^ { 2 } \sin \left( t ^ { 2 } \right)$
(D) $2 \cos \left( t ^ { 2 } \right) - 4 t ^ { 2 } \sin \left( t ^ { 2 } \right)$

A particle moves in the $xy$-plane so that its position at any time $t$, $0 \leq t \leq \pi$, is given by $x(t) = \frac{t^2}{2} - \ln(1+t)$ and $y(t) = 3\sin t$.
(a) Sketch the path of the particle in the $xy$-plane below. Indicate the direction of motion along the path.
(b) At what time $t$, $0 \leq t \leq \pi$, does $x(t)$ attain its minimum value? What is the position $(x(t), y(t))$ of the particle at this time?
(c) At what time $t$, $0 < t < \pi$, is the particle on the $y$-axis? Find the speed and the acceleration vector of the particle at this time.

An object moving along a curve in the $xy$-plane has position $(x(t), y(t))$ at time $t$ with $$\frac{dx}{dt} = \cos\left(t^3\right) \text{ and } \frac{dy}{dt} = 3\sin\left(t^2\right)$$ for $0 \leq t \leq 3$. At time $t = 2$, the object is at position $(4,5)$.
(a) Write an equation for the line tangent to the curve at $(4,5)$.
(b) Find the speed of the object at time $t = 2$.
(c) Find the total distance traveled by the object over the time interval $0 \leq t \leq 1$.
(d) Find the position of the object at time $t = 3$.

A particle moves in the $xy$-plane so that its position at any time $t$, for $-\pi \leq t \leq \pi$, is given by $x(t) = \sin(3t)$ and $y(t) = 2t$.
(a) Sketch the path of the particle in the $xy$-plane provided. Indicate the direction of motion along the path.
(b) Find the range of $x(t)$ and the range of $y(t)$.
(c) Find the smallest positive value of $t$ for which the $x$-coordinate of the particle is a local maximum. What is the speed of the particle at this time?
(d) Is the distance traveled by the particle from $t = -\pi$ to $t = \pi$ greater than $5\pi$? Justify your answer.

The position of a roller coaster car at time $t$ seconds can be modeled parametrically by $$\begin{aligned} & x ( t ) = 10 t + 4 \sin t \\ & y ( t ) = ( 20 - t ) ( 1 - \cos t ) , \end{aligned}$$ where $x$ and $y$ are measured in meters, over the time interval $0 \leq t \leq 18$ seconds. The derivatives of these functions are given by $$\begin{aligned} & x ^ { \prime } ( t ) = 10 + 4 \cos t \\ & y ^ { \prime } ( t ) = ( 20 - t ) \sin t + \cos t - 1 \end{aligned}$$
(a) Find the slope of the path at time $t = 2$. Show the computations that lead to your answer.
(b) Find the acceleration vector of the car at the time when the car's horizontal position is $x = 140$.
(c) Find the time $t$ at which the car is at its maximum height, and find the speed, in $\mathrm { m } / \mathrm { sec }$, of the car at this time.
(d) For $0 < t < 18$, there are two times at which the car is at ground level ( $y = 0$ ). Find these two times and write an expression that gives the average speed, in $\mathrm { m } / \mathrm { sec }$, of the car between these two times. Do not evaluate the expression.

A particle moves in the $xy$-plane so that the position of the particle at any time $t$ is given by $$x(t) = 2e^{3t} + e^{-7t} \text{ and } y(t) = 3e^{3t} - e^{-2t}.$$
(a) Find the velocity vector for the particle in terms of $t$, and find the speed of the particle at time $t = 0$.
(b) Find $\frac{dy}{dx}$ in terms of $t$, and find $\lim_{t \rightarrow \infty} \frac{dy}{dx}$.
(c) Find each value $t$ at which the line tangent to the path of the particle is horizontal, or explain why none exists.
(d) Find each value $t$ at which the line tangent to the path of the particle is vertical, or explain why none exists.

The position of a particle moving in the $x y$-plane is given by the parametric equations $x ( t ) = t ^ { 3 } - 3 t ^ { 2 }$ and $y ( t ) = 12 t - 3 t ^ { 2 }$. At which of the following points $( x , y )$ is the particle at rest?
(A) $( - 4,12 )$
(B) $( - 3,6 )$
(C) $( - 2,9 )$
(D) $( 0,0 )$
(E) $( 3,4 )$

On the coordinate plane, as shown in the figure, for a point P on the circle $x ^ { 2 } + y ^ { 2 } = 1$, let $\theta \left( 0 < \theta < \frac { \pi } { 4 } \right)$ be the angle that the line segment OP makes with the positive direction of the $x$-axis. Let Q be the point where the line passing through P and parallel to the $x$-axis meets the curve $y = e ^ { x } - 1$, and let R be the foot of the perpendicular from Q to the $x$-axis. Let T be the intersection point of the line segment OP and the line segment QR, and let $S ( \theta )$ be the area of triangle ORT. When $\lim _ { \theta \rightarrow + 0 } \frac { S ( \theta ) } { \theta ^ { 3 } } = a$, find the value of $60 a$. [4 points]

For a real number $0 < t < 41$, the curve $y = x ^ { 3 } + 2 x ^ { 2 } - 15 x + 5$ and the line $y = t$ intersect at three points. Let the point with the largest $x$-coordinate be $( f ( t ) , t )$ and the point with the smallest $x$-coordinate be $( g ( t ) , t )$. Let $h ( t ) = t \times \{ f ( t ) - g ( t ) \}$. What is the value of $h ^ { \prime } ( 5 )$? [4 points]
(1) $\frac { 79 } { 12 }$
(2) $\frac { 85 } { 12 }$
(3) $\frac { 91 } { 12 }$
(4) $\frac { 97 } { 12 }$
(5) $\frac { 103 } { 12 }$

For the curve represented parametrically by $$x = \ln(t^3 + 1), \quad y = \sin\pi t$$ where $t > 0$, find the value of $\frac{dy}{dx}$ at $t = 1$. [3 points]
(1) $-\frac{1}{3}\pi$
(2) $-\frac{2}{3}\pi$
(3) $-\pi$
(4) $-\frac{4}{3}\pi$
(5) $-\frac{5}{3}\pi$

If $x = 3\tan t$ and $y = 3\sec t$, then the value of $\frac{d^2y}{dx^2}$ at $t = \frac{\pi}{4}$, is:
(1) $\frac{1}{6}$
(2) $\frac{1}{6\sqrt{2}}$
(3) $\frac{1}{3\sqrt{2}}$
(4) $\frac{3}{2\sqrt{2}}$

If $x = 2 \sin \theta - \sin 2 \theta$ and $y = 2 \cos \theta - \cos 2 \theta , \theta \in [ 0,2 \pi ]$, then $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at $\theta = \pi$ is:
(1) $\frac { 3 } { 4 }$
(2) $- \frac { 3 } { 8 }$
(3) $\frac { 3 } { 2 }$
(4) $- \frac { 3 } { 4 }$

Let $x ( t ) = 2 \sqrt { 2 } \cos t \sqrt { \sin 2 t }$ and $y ( t ) = 2 \sqrt { 2 } \sin t \sqrt { \sin 2 t } , t \in \left( 0 , \frac { \pi } { 2 } \right)$. Then $\frac { 1 + \left( \frac { d y } { d x } \right) ^ { 2 } } { \frac { d ^ { 2 } y } { d x ^ { 2 } } }$ at $t = \frac { \pi } { 4 }$ is equal to
(1) $\frac { - 2 \sqrt { 2 } } { 3 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { - 2 } { 3 }$

If the angle made by the tangent at the point $(x_0, y_0)$ on the curve $x = 12(t + \sin t \cos t)$, $y = 12(1 + \sin t)^2$, $0 < t < \frac{\pi}{2}$, with the positive $x$-axis is $\frac{\pi}{3}$, then $y_0$ is equal to
(1) $63 + 2\sqrt{2}$
(2) $37 + 4\sqrt{3}$
(3) 27
(4) 48

Problem 4

I. In the two-dimensional orthogonal $x y$ coordinate system, consider the curve $L$ represented by the following equations with the parameter $t ( 0 \leq t \leq 2 \pi )$. Here, $a$ is a positive real constant.
$$\begin{aligned} & x ( t ) = a ( t - \sin t ) \\ & y ( t ) = a ( 1 - \cos t ) \end{aligned}$$

1. Obtain the length of the curve $L$ when $t$ varies in the range of $0 \leq t \leq 2 \pi$.
2. Obtain the curvature at an arbitrary point of the curve $L$.

Here, $t = 0$ and $t = 2 \pi$ are excluded.
II. In the three-dimensional orthogonal $x y z$ coordinate system, consider the curved surface represented by the following equations with the parameters $u$ and $v$ ( $u$ and $v$ are real numbers).
$$\begin{aligned} & x ( u , v ) = \sinh u \cos v \\ & y ( u , v ) = 2 \sinh u \sin v \\ & z ( u , v ) = 3 \cosh u \end{aligned}$$

1. Express the curved surface by an equation without the parameters.
2. Sketch the $x y$-plane view at $z = 5$ and the $x z$-plane view at $y = 0$, respectively, of the curved surface. In the sketches, indicate the values at the intersection with each of the axes.
3. Express the unit normal vector $\boldsymbol { n }$ of the curved surface by $u$ and $v$. Here, the $z$-component of $\boldsymbol { n }$ should be positive.
4. Let $\kappa$ be the Gaussian curvature at the point $u = v = 0$. Calculate the absolute value of $\kappa$.