91Ó°ÊÓ

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

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.

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. \(f(x)=x^{-1 / 2}\)

Use implicit differentiation to find \(d p / d x\). \(x p^{2}=96\)

A recent study of the exercise habits of 17,000 Harvard alumni found that the death rate (deaths per 10,000 person-years) was approximately \(R(x)=5 x^{2}-35 x+104\), where \(x\) is the weekly amount of exercise in thousands of calories \((0 \leq x \leq 4) .\) Find the exercise level that minimizes the death rate.

In a revenue, cost, and profit problem, is minimizing the cost the same as maximizing the profit? 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. \(f(x)=x^{-3 / 2}\)

Sketch the graph of each rational function after making a sign diagram for the derivative and finding all relative extreme points and asymptotes. \(f(x)=\frac{4}{x-2}\)

Use implicit differentiation to find \(d p / d x\). \((p+5)(x+2)=120\)

If an epidemic spreads through a town at a rate that is proportional to the number of uninfected people and to the square of the number of infected people, then the rate is \(R(x)=c x^{2}(p-x)\), where \(x\) is the number of infected people and \(c\) and \(p\) (the population) are positive constants. Show that the rate \(R(x)\) is greatest when two-thirds of the population is infected.