Exam 2: Limits and Derivatives

Find an equation of the tangent line to the curve $120 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2312 \left( x ^ { 2 } - y ^ { 2 } \right)$ at the point $( 4,1 )$ .

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$ . $f ( x ) = [ g ( x ) ] ^ { 4 }$

In an adiabatic process (one in which no heat transfer takes place), the pressure $P$ and volume $V$ of an ideal gas such as oxygen satisfy the equation $P ^ { 5 } V ^ { 7 } = C$ where $C$ is a constant. Suppose that at a certain instant of time, the volume of the gas is $2 \mathrm {~L}$ , the pressure is $100 \mathrm { kPa }$ , and the pressure is decreasing at the rate of $5 \mathrm { kPa } / \mathrm { sec }$ . Find the rate at which the volume is changing.

A spherical balloon is being inflated. Find the rate of increase of the surface area $S = 4 \pi r ^ { 2 }$ with respect to the radius $r$ when $r = 1 \mathrm { ft }$ .

The position function of a particle is given by $s = t ^ { 3 } - 10.5 t ^ { 2 } - 2 t , \mathrm { t } \geq 0$ When does the particle reach a velocity of $22 \mathrm {~m} / \mathrm { s }$ ?

Differentiate the function. $f ( t ) = \frac { 1 } { 3 } t ^ { 3 } - 2 t ^ { 7 } + t$

Find the equation of the tangent to the curve at the given point. $y = \sqrt { 16 + 4 \sin x } , \quad ( 0,4 )$

A television camera is positioned $4,600 \mathrm { ft }$ from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let's assume the rocket rises vertically and its speed is $680 \mathrm { ft } / \mathrm { s }$ when it has risen $2,600 \mathrm { ft }$ . If the television camera is always kept aimed at the rocket, how fast is the camera's angle of elevation changing at this moment? Round the result to the nearest thousandth.

Find the given derivative by finding the first few derivatives and observing the pattern that occurs. $\frac { d ^ { 89 } } { d x ^ { 89 } } ( \sin x )$ Select the correct answer.

Find the average rate of change of the area of a circle with respect to its radius $r$ as $r$ changes from 3 to $8 .$

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $y \sin 3 x = x \cos 3 y , \left( \frac { \pi } { 3 } , \frac { \pi } { 6 } \right)$

In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the polluted area is circular, determine how fast the area is increasing when the radius of the circle is $20 \mathrm { ft }$ and is increasing at the rate of $\frac { 1 } { 6 } \mathrm { ft } / \mathrm { sec }$ . Round to the nearest tenth if necessary.

The volume of a right circular cone of radius $r$ and height $h$ is $V = \frac { \pi } { 3 } r ^ { 2 } h$ . Suppose that the radius and height of the cone are changing with respect to time $t$ . a. Find a relationship between $\frac { d V } { d t } , \frac { d r } { d t }$ , and $\frac { d h } { d t }$ . b. At a certain instant of time, the radius and height of the cone are 12 in. and 13 in. and are increasing at the rate of $0.2 \mathrm { in } . / \mathrm { sec }$ and $0.5 \mathrm { in } . / \mathrm { sec }$ , respectively. How fast is the volume of the cone increasing?

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of intersection of the curves. Show that the curves of the given equations are orthogonal. $y - \frac { 7 } { 4 } x = \frac { \pi } { 2 } , \quad x = \frac { 7 } { 4 } \cos y$

If a cylindrical tank holds 10000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume of water remaining in the tank after $t$ minutes as $V ( t ) = 10000 \left( 1 - \frac { 1 } { 60 } t \right) ^ { 2 } , 0 \leq t \leq 60$ Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of $V$ with respect to $t$ ) as a function of $t$ .

$\text { Compute } \Delta y \text { and } d y \text { for the given values of } x \text { and } d x = \Delta x \text {. }$ $y = x ^ { 2 } , x = 1 , \triangle x = 0.5$

Find the limit. $\lim _ { \theta \rightarrow 0 } 4 \frac { \sin ( \sin 4 \theta ) } { \sec 4 \theta }$

Plot the graph of the function $f$ in an appropriate viewing window. $f ( x ) = \frac { x ^ { 4 } } { x ^ { 4 } + 1 }$

The circumference of a sphere was measured to be $86 \mathrm {~cm}$ with a possible error of $0.8 \mathrm {~cm}$ . Use differentials to estimate the maximum error in the calculated volume.