Exam 9: Applications of the Derivative

The radius r of a sphere is increasing at a rate of 3 inches per minute. Find the rate of change of volume when r = 17 inches. Round your answer to one decimal place.

Boat docking. Suppose that a boat is being pulled toward a dock by a winch that is 24 ft above the level of the boat deck. If the winch is pulling the cable at a rate of 23 ft/min, at what rate is the boat approaching the dock when it is 32 ft from the dock? Use the figure below.

Find the second derivative for the function $f ( x ) = \frac { 7 x } { 7 x + 3 }$ and solve the equation $f ^ { \prime \prime } ( x ) = 0$ .

A point is moving along the graph of the function $y = \frac { 1 } { 8 x ^ { 2 } + 3 }$ such that $\frac { d x } { d t } = 5$ centimeters per second. Find dy/dt when $x = 1$ .

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If $y = f ( x ) g ( x ) \text {, then } y ^ { \prime } = f ^ { \prime } ( x ) g ^ { \prime } ( x )$

The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.

A fast-food restaurant determines the cost model, $C = 0.6 x + 5500,0 \leq x \leq 40000$ and revenue model, $R = \frac { 1 } { 30000 } \left( 55000 x - x ^ { 2 } \right)$ for $0 \leq x \leq 40000$ where x is the number of hamburgers sold. Determine the intervals on which the profit function is increasing and on which it is decreasing.

Determine the open intervals on which the graph of $y = - 3 x ^ { 3 } + 8 x ^ { 2 } + 8 x - 5$ is concave downward or concave upward.

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

The number of people who donated to a certain organization between 1975 and 1992 can be modeled by the equation $D ( t ) = - 10.82 t ^ { 3 } + 209.508 t ^ { 2 } - 167.402 t + 9774.134$ donors, where t is the number of years after 1975. Find the inflection point(s) from $t = 0$ through $t = 17$ , if any exist.

For the function $f ( x ) = 2 x ^ { 3 } - 12 x ^ { 2 } + 2$ : (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Then use a graphing utility to confirm your results.

Find the second derivative of the function. $f ( x ) = 3 x ^ { \frac { 4 } { 7 } }$

Suppose the resident population P(in millions) of the United States can be modeled by $P = 0.00000583 t ^ { 3 } + 0.005003 t ^ { 2 } + 0.13776 t + 4.658 , - 6 \leq t \leq 193$ , where $t = 0$ corresponds to 1800. Analytically find the minimum and maximum populations in the U.S. for $- 6 \leq t \leq 193$ .

Find the rate of change of x with respect to p. $p = \frac { 2 } { 0.00001 x ^ { 3 } + 0.1 x } x \geq 0$

Identify the open intervals where the function $f ( x ) = x \sqrt { 24 - x ^ { 2 } }$ is increasing or decreasing.

An airplane flying at an altitude of 5 miles passes directly over a radar antenna. When the airplane is 50 miles away (s = 50), the radar detects that the distance s is changing at a rate of 280 miles per hour. What is the speed of the airplane? Round your answer to the nearest integer.

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

Find all relative extrema of the function $f ( x ) = x ^ { 8 / 9 } - 8$ . Use the Second Derivative Test where applicable.

Find the value $g ^ { \prime \prime } ( 9 )$ for the function $g ( t ) = 5 t ^ { 6 } + 5 t ^ { 4 } + 6$ .

Find the third derivative of the function $f ( x ) = x ^ { 5 } - 3 x ^ { 4 }$ .