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) $2 \cos \left( t ^ { 2 } \right) - 4 t ^ { 2 } \sin \left( t ^ { 2 } \right)$

25. Consider the curve in the $x y$-plane represented by $x = e ^ { t }$ and $y = t e ^ { - t }$ for $t \geq 0$. The slope of the line tangent to the curve at the point where $x = 3$ is
(A) 20.086
(B) 0.342
(C) - 0.005
(D) - 0.011
(E) - 0.033

1993 AP Calculus BC: Section I

1. If $y = \arctan \left( e ^ { 2 x } \right)$, then $\frac { d y } { d x } =$
   (A) $\frac { 2 e ^ { 2 x } } { \sqrt { 1 - e ^ { 4 x } } }$
   (B) $\frac { 2 e ^ { 2 x } } { 1 + e ^ { 4 x } }$
   (C) $\frac { e ^ { 2 x } } { 1 + e ^ { 4 x } }$
   (D) $\frac { 1 } { \sqrt { 1 - e ^ { 4 x } } }$
   (E) $\frac { 1 } { 1 + e ^ { 4 x } }$
2. The interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { 3 ^ { n } }$ is
   (A) $- 3 < x \leq 3$
   (B) $- 3 \leq x \leq 3$
   (C) $- 2 < x < 4$
   (D) $- 2 \leq x < 4$
   (E) $0 \leq x \leq 2$
3. If a particle moves in the $x y$-plane so that at time $t > 0$ its position vector is $\left( \ln \left( t ^ { 2 } + 2 t \right) , 2 t ^ { 2 } \right)$, then at time $t = 2$, its velocity vector is
   (A) $\left( \frac { 3 } { 4 } , 8 \right)$
   (B) $\left( \frac { 3 } { 4 } , 4 \right)$
   (C) $\left( \frac { 1 } { 8 } , 8 \right)$
   (D) $\left( \frac { 1 } { 8 } , 4 \right)$
   (E) $\left( - \frac { 5 } { 16 } , 4 \right)$
4. $\int x \sec ^ { 2 } x d x =$
   (A) $\quad x \tan x + C$
   (B) $\frac { x ^ { 2 } } { 2 } \tan x + C$
   (C) $\sec ^ { 2 } x + 2 \sec ^ { 2 } x \tan x + C$
   (D) $\quad x \tan x - \ln | \cos x | + C$
   (E) $\quad x \tan x + \ln | \cos x | + C$
5. What is the volume of the solid generated by rotating about the $x$-axis the region enclosed by the curve $y = \sec x$ and the lines $x = 0 , y = 0$, and $x = \frac { \pi } { 3 }$ ?
   (A) $\frac { \pi } { \sqrt { 3 } }$
   (B) $\pi$
   (C) $\pi \sqrt { 3 }$
   (D) $\frac { 8 \pi } { 3 }$
   (E) $\quad \pi \ln \left( \frac { 1 } { 2 } + \sqrt { 3 } \right)$
6. If $s _ { n } = \left( \frac { ( 5 + n ) ^ { 100 } } { 5 ^ { n + 1 } } \right) \left( \frac { 5 ^ { n } } { ( 4 + n ) ^ { 100 } } \right)$, to what number does the sequence $\left\{ s _ { n } \right\}$ converge?
   (A) $\frac { 1 } { 5 }$
   (B) 1
   (C) $\frac { 5 } { 4 }$
   (D) $\left( \frac { 5 } { 4 } \right) ^ { 100 }$
   (E) The sequence does not converge.
7. If $\int _ { a } ^ { b } f ( x ) d x = 5$ and $\int _ { a } ^ { b } g ( x ) d x = - 1$, which of the following must be true? I. $f ( x ) > g ( x )$ for $a \leq x \leq b$ II. $\quad \int _ { a } ^ { b } ( f ( x ) + g ( x ) ) d x = 4$ III. $\quad \int _ { a } ^ { b } ( f ( x ) g ( x ) ) d x = - 5$
   (A) I only
   (B) II only
   (C) III only
   (D) II and III only
   (E) I, II, and III
8. Which of the following is equal to $\int _ { 0 } ^ { \pi } \sin x d x$ ?
   (A) $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \cos x d x$
   (B) $\quad \int _ { 0 } ^ { \pi } \cos x d x$
   (C) $\quad \int _ { - \pi } ^ { 0 } \sin x d x$
   (D) $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \sin x d x$
   (E) $\int _ { \pi } ^ { 2 \pi } \sin x d x$

1993 AP Calculus BC: Section I

[Figure]

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) $( 3,4 )$

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 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

The coordinates $( x , y )$ of a moving point P are given by the following functions in time $t$:
$$\begin{aligned} & x = 4 t - \sin 4 t \\ & y = 4 - \cos 4 t \end{aligned}$$
(1) The derivatives of $x$ and $y$ with respect to $t$ are
$$\begin{aligned} \frac { d x } { d t } & = \mathbf { A } ( \mathbf { A } - \mathbf { B } \cos 4 t ) \\ \frac { d y } { d t } & = \mathbf { C } \sin 4 t . \end{aligned}$$
Hence we have
$$\left( \frac { d x } { d t } \right) ^ { 2 } + \left( \frac { d y } { d t } \right) ^ { 2 } = \mathbf { D E } \sin ^ { 2 } \mathbf { F } t$$
(2) As the point P moves from the time $t = 0$ to the time $t = 2 \pi$, its speed $v$ is maximized a total of $\mathbf { G }$ times. Let us denote by $t _ { 0 }$ the moment of the first time the speed is maximized and the moment of the last time it is maximized by $t _ { 1 }$. Then
$$t _ { 0 } = \frac { \mathbf { H } } { \mathbf { I } } \pi , \quad t _ { 1 } = \frac { \mathbf { J } } { \mathbf { I } } \pi$$
and the maximum speed is $v = \mathbf { L }$.
(3) For $t _ { 0 }$ and $t _ { 1 }$ in (2), the distance that point P moves during the period from $t = t _ { 0 }$ to $t = t _ { 1 }$ is $\mathbf{MN}$.

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 $\kappa$.