(a) A point moves on the hyperbola $3 x ^ { 2 } - y ^ { 2 } = 23$ so that its $y$-coordinate is increasing at a constant rate of 4 units per second. How fast is the $x$-coordinate changing when $x = 4$ ? (b) For what values of $k$ will the line $2 x + 9 y + k = 0$ be normal to the hyperbola $3 x ^ { 2 } - y ^ { 2 } = 23$ ?

If $( x + 2 y ) \cdot \frac { d y } { d x } = 2 x - y$, what is the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 3,0 )$ ?
(A) $- \frac { 10 } { 3 }$
(B) 0
(C) 2
(D) $\frac { 10 } { 3 }$
(E) Undefined

A particle moves along the curve defined by the equation $y = x^{3} - 3x$. The $x$-coordinate of the particle, $x(t)$, satisfies the equation $\dfrac{dx}{dt} = \dfrac{1}{\sqrt{2t+1}}$, for $t \geq 0$ with initial condition $x(0) = -4$.
(a) Find $x(t)$ in terms of $t$.
(b) Find $\dfrac{dy}{dt}$ in terms of $t$.
(c) Find the location and speed of the particle at time $t = 4$.

The fuel consumption of a car, in miles per gallon (mpg), is modeled by $F ( s ) = 6 e ^ { \left( \frac { s } { 20 } - \frac { s ^ { 2 } } { 2400 } \right) }$, where $s$ is the speed of the car, in miles per hour. If the car is traveling at 50 miles per hour and its speed is changing at the rate of 20 miles/hour$^{2}$, what is the rate at which its fuel consumption is changing?
(A) 0.215 mpg per hour
(B) 4.299 mpg per hour
(C) 19.793 mpg per hour
(D) 25.793 mpg per hour
(E) 515.855 mpg per hour

If $g ( x ) = \int _ { \sin x } ^ { \sin ( 2 x ) } \sin ^ { - 1 } ( t ) d t$, then
[A] $g ^ { \prime } \left( \frac { \pi } { 2 } \right) = - 2 \pi$
[B] $g ^ { \prime } \left( - \frac { \pi } { 2 } \right) = 2 \pi$
[C] $g ^ { \prime } \left( \frac { \pi } { 2 } \right) = 2 \pi$
[D] $g ^ { \prime } \left( - \frac { \pi } { 2 } \right) = - 2 \pi$

A particle is moving in a straight line such that its velocity is increasing at $5 \mathrm{~m}\mathrm{~s}^{-1}$ per meter. The acceleration of the particle is $\_\_\_\_$ $\mathrm{m}\mathrm{~s}^{-2}$ at a point where its velocity is $20 \mathrm{~m}\mathrm{~s}^{-1}$.