91Ó°ÊÓ

For each demand equation, use implicit differentiation to find \(d p / d x\). $$ 8 p^{2}+2 p+100=x $$

Sketch the graph of each function "by hand" after making a sign diagram for the derivative and finding all open intervals of increase and decrease. \(f(x)=x^{3}(x-5)^{2}\)

If a government wants to increase revenue, why not just increase the tax rate? Explain.

Graph each function using a graphing calculator by first making a sign diagram for just the first derivative. Make a sketch from the screen, showing the coordinates of all relative extreme points and inflection points. Graphs may vary depending on the window chosen. $$ f(x)=x^{3 / 2} $$

BUSINESS: Cigarette Production Per capita cigarette production in the United States during recent decades is approximately given by \(f(x)=-0.6 x^{2}+12 x+945\), where \(x\) is the number of years after \(1980,0 \leq x \leq 20 .\) Find the year when per capita cigarette production was at its greatest.

Graph each function using a graphing calculator by first making a sign diagram for just the first derivative. Make a sketch from the screen, showing the coordinates of all relative extreme points and inflection points. Graphs may vary depending on the window chosen. $$ f(x)=x^{-1 / 2} $$

In a revenue, cost, and profit problem, is maximizing the revenue the same as maximizing the profit? Explain.

Sketch the graph of each rational function after making a sign diagram for the derivative and finding all relative extreme points and asymptotes. $$ f(x)=\frac{6}{x+3} $$

BUSINESS: Timber Value The value of a timber forest after \(t\) years is \(V(t)=480 \sqrt{t}-40 t \quad\) (for \(0 \leq t \leq 50\) ). Find when its value is maximized.

For each demand equation, use implicit differentiation to find \(d p / d x\). $$ x p^{2}=96 $$