Exam 9: Applications of the Derivative

A point is moving along the graph of the function $y = 9 x ^ { 2 } + 4$ such that $\frac { d x } { d t } = 3$ centimeters per second. Find dy/dt for the given values of x. (a) $x = 4~~~~~~~~~~~~~~~~~~~~~~~~~~$ (b) $x = 7$

Profit. Suppose that the monthly revenue and cost (in dollars) for x units of a product are $R = 200 x - \frac { x ^ { 2 } } { 10 } \text { and } C = 6000 + 10 x$ At what rate per month is the profit changing if the number of units produced and sold is 100 and is increasing at a rate of 10 units per month?

Find $\frac { d y } { d x }$ for the equation $\sqrt { x y } = x - 20 y$ by implicit differentiation and evaluate the derivative at the point $( 100,4 )$ .

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

Find the indicated derivative. Find $y ^ { ( 4 ) } \text { if } y = x ^ { 7 } - 3 x ^ { 3 } \text {. }$

A brick becomes dislodged from the Empire State Building (at a height of 1175 feet) and falls to the sidewalk below. Write the position s(t), velocity v(t), and acceleration a(t) as functions of time.

Find the $f ^ { ( 6 ) } ( x )$ of $f ^ { ( 4 ) } ( x ) = \left( x ^ { 2 } + 1 \right) ^ { 2 }$ .

Approximate the critical numbers of the function shown in the graph and determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown.

Find the rate of change of x with respect to p. $p = \sqrt { \frac { 200 - x } { 2 x } } , \quad 0 < x \leq 200$

The lengths of the edges of a cube are increasing at a rate of 7 ft/min. At what rate is the surface area changing when the edges are 22 ft long?

For the given function, find the critical numbers. $y = \frac { x ^ { 4 } } { 4 } - \frac { x ^ { 3 } } { 3 } - 4$

A baseball diamond has the shape of a square with sides 90 feet long (see figure). A player running from second base to third base at a speed of 26 feet per second is 80 feet from third base. At what rate is the player's distance s from home plate changing? Round your answer to one decimal place.

Find the slope of the graph at the given point. $x ^ { 2 } + y ^ { 2 } = 4$

Find $\frac { d y } { d x }$ for the equation $\frac { x + 4 y } { 5 x - 6 y } = 6$ .

Assume that x and y are differentiable functions of t. Find dx/dt given that $x = - 2$ , $y = - 8$ , and $d y / d t = 6$ $y ^ { 2 } - x ^ { 2 } = 60$

Volume and radius. Suppose that air is being pumped into a spherical balloon at a rate of $8 \mathrm { in. } ^ { 3 } / \mathrm { min }$ At what rate is the radius of the balloon increasing when the radius is 3 in.?

Find $d y / d x$ implicitly and explicitly(the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating $d y / d x$ at the point.

Identify the open intervals where the function $f ( x ) = 4 x ^ { 2 } - 5 x - 5$ is increasing or decreasing.

Find the second derivative for the function $f ( x ) = 3 x ^ { 3 } + 27 x ^ { 2 } - 12 x - 32$ and solve the equation $f ^ { \prime \prime } ( x ) = 0$ .

Find dy/dx for the following equation: $8 x + y ^ { 2 } - 7 y + 9 = 0$