Exam 2: Limits and Derivatives

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. $y - \frac { 7 } { 4 } x = \frac { \pi } { 2 } , \quad x = \frac { 7 } { 4 } \cos y$

The sides of a square baseball diamond are $90 \mathrm { ft }$ long. When a player who is between the second and third base is $30 \mathrm { ft }$ from second base and heading toward third base at a speed of $24 \mathrm { ft } / \mathrm { sec }$ , how fast is the distance between the player and home plate changing? Round to two decimal places.

Find the differential of the function at the indicated number. $f ( x ) = e ^ { 7 x } + \ln ( x + 8 ) ; x = 0$

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 $t$ .

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?

Identify the "inside function" $u = f ( x )$ and the "outside function" $y = g ( u )$ . Then find $d y / d x$ using the Chain Rule. $y = \sqrt { x ^ { 2 } - 2 }$

If $f ( t ) = \sqrt { 9 t + 1 }$ , find $f ^ { \prime \prime } ( 5 )$ .

The quantity $Q$ of charge in coulombs $C$ that has passed through a point in a wire up to time $t$ (measured in seconds) is given by $Q ( t ) = t ^ { 3 } - 3 t ^ { 2 } + 4 t + 3$ Find the current when $t = 1 \mathrm {~s}$ .

Find an equation of the tangent line to the curve $x e ^ { y } + x + 2 y = 2 \text { at } ( 1,0 )$

Find the derivative of the function. $f ( x ) = \frac { 2 \sqrt { x } } { x ^ { 2 } + 9 }$

Find $\frac { d y } { d x }$ by implicit differentiation. $8 \sqrt { x } + \sqrt { y } = 8$

$s ( t )$ is the position of a body moving along a coordinate line, where $t \geq 0$ , and $s ( t )$ is measured in feet and $t$ in seconds. $s ( t ) = - 3 + 2 t - t ^ { 2 }$ a. Determine the time(s) and the position(s) when the body is stationary. b. When is the body moving in the positive direction? In the negative direction? c. Sketch a schematic showing the position of the body at any time $t$ .

$s ( t )$ is the position of a body moving along a coordinate line; $s ( t )$ is measured in feet and $t$ in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s ( t ) = \frac { 3 t } { t ^ { 2 } + 1 } ; \quad t = 2$

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 $r = 1 \mathrm { ft }$ .

If $g ( x ) = \sqrt { 2 - 3 x }$ , use the definition of derivative to find $g ^ { \prime } ( x )$

The mass of part of a wire is $x ( 1 + \sqrt { x } )$ kilograms, where $x$ is measured in meters from one end of the wire. Find the linear density of the wire when $x = 36 \mathrm {~m}$ .

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.

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

Find the derivative of the function. $f ( x ) = - x ^ { 2 } + x + 2$

Differentiate. $y = \frac { \sin x } { 3 + \cos x }$