Chapter 10

A manufacturer of tennis rackets finds that the total cost \(C(x)\) (in dollars) of manufacturing \(x\) rackets/day is given by \(C(x)=400+4 x+0.0001 x^{2}\). Each racket can be sold at a price of \(p\) dollars, where \(p\) is related to \(x\) by the demand equation \(p=10-0.0004 x\). If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer.

Sketch the graph of the function, using the curve-sketching quide of this section. $$ g(x)=\frac{2}{x-1} $$

Find the inflection point(s), if any, of each function. $$ f(x)=x^{3}-2 $$

The weekly demand for the Pulsar 25 -in. color console television is given by the demand equation $$ p=-0.05 x+600 \quad(0

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=\frac{1}{x+1} $$

Find the inflection point(s), if any, of each function. $$ g(x)=x^{3}-6 x $$

A division of Chapman Corporation manufactures a pager. The weekly fixed cost for the division is $$\$ 20,0004$$, and the variable cost for producing \(x\) pagers/week is $$ V(x)=0.000001 x^{3}-0.01 x^{2}+50 x $$ dollars. The company realizes a revenue of $$ R(x)=-0.02 x^{2}+150 x \quad(0 \leq x \leq 7500) $$ dollars from the sale of \(x\) pagers/week. Find the level of production that will yield a maximum profit for the manufacturer. Hint: Use the quadratic formula.

Sketch the graph of the function, using the curve-sketching quide of this section. $$ h(x)=\frac{x+2}{x-2} $$

Find the inflection point(s), if any, of each function. $$ f(x)=6 x^{3}-18 x^{2}+12 x-15 $$

Sketch the graph of the function, using the curve-sketching quide of this section. $$ g(x)=\frac{x}{x-1} $$

Find the inflection point(s), if any, of each function. $$ g(x)=2 x^{3}-3 x^{2}+18 x-8 $$

The total monthly cost (in dollars) incurred by Cannon Precision Instruments for manufacturing \(x\) units of the model \(\mathrm{M} 1\) camera is given by the function $$ C(x)=0.0025 x^{2}+80 x+10,000 $$ a. Find the average cost function \(\bar{C}\). b. Find the level of production that results in the smallest average production cost. c. Find the level of production for which the average cost is equal to the marginal cost. d. Compare the result of part (c) with that of part (b).

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(t)=\frac{t^{2}}{1+t^{2}} $$

Find the inflection point(s), if any, of each function. $$ f(x)=3 x^{4}-4 x^{3}+1 $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=x^{2}-4 x $$

Sketch the graph of the function, using the curve-sketching quide of this section. $$ g(x)=\frac{x}{x^{2}-4} $$

Find the inflection point(s), if any, of each function. $$ f(x)=x^{4}-2 x^{3}+6 $$

Find the relative maxima and relative minima, if any, of each function. $$ g(x)=x^{2}+3 x+8 $$

Suppose the quantity demanded per week of a certain dress is related to the unit price \(p\) by the demand equation \(p=\sqrt{800-x}\), where \(p\) is in dollars and \(x\) is the number of dresses made. To maximize the revenue, how many dresses should be made and sold each week? Hint: \(R(x)=p x\).

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(t)=e^{t}-t $$

Find the inflection point(s), if any, of each function. $$ g(t)=\sqrt[3]{t} $$

Find the relative maxima and relative minima, if any, of each function. $$ h(t)=-t^{2}+6 t+6 $$

The quantity demanded each month of the Sicard wristwatch is related to the unit price by the equation $$ p=\frac{50}{0.01 x^{2}+1} \quad(0 \leq x \leq 20) $$ where \(p\) is measured in dollars and \(x\) is measured in units of a thousand. To yield a maximum revenue, how many watches must be sold?

Sketch the graph of the function, using the curve-sketching quide of this section. $$ h(x)=\frac{e^{x}+e^{-x}}{2} $$

Find the inflection point(s), if any, of each function. $$ f(x)=\sqrt[5]{x} $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=\frac{1}{2} x^{2}-2 x+4 $$

The present value of a piece of waterfront property purchased by an investor is given by the function $$ P(t)=80,000 e^{\sqrt{t / 2}-0.09 t} \quad(0 \leq t \leq 8) $$ where \(P(t)\) is measured in dollars and \(t\) is the time in years from the present. Determine the optimal time (based on present value) for the investor to sell the property. What is the property's optimal present value?

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=2-e^{-x} $$

Find the inflection point(s), if any, of each function. $$ f(x)=(x-1)^{3}+2 $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=x^{5 / 3} $$

It has been estimated that the total production of oil from a certain oil well is given by $$ T(t)=-1000(t+10) e^{-0.1 t}+10,000 $$ thousand barrels \(t\) yr after production has begun. Determine the year when the oil well will be producing at maximum capacity.

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=\frac{3}{1+e^{-x}} $$

Find the inflection point(s), if any, of each function. $$ f(x)=(x-2)^{4 / 3} $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=x^{2 / 3}+2 $$

When organic waste is dumped into a pond, the oxidation process that takes place reduces the pond's oxygen content. However, given time, nature will restore the oxygen content to its natural level. Suppose the oxygen content \(t\) days after organic waste has been dumped into the pond is given by $$ f(t)=100\left[\frac{t^{2}-4 t+4}{t^{2}+4}\right] \quad(0 \leq t<\infty) $$ percent of its normal level. a. When is the level of oxygen content lowest? b. When is the rate of oxygen regeneration greatest?

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=\ln (x-1) $$

Find the inflection point(s), if any, of each function. $$ f(x)=2 e^{-x^{2}} $$

Find the relative maxima and relative minima, if any, of each function. $$ g(x)=x^{3}-3 x^{2}+4 $$

The amount of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in the city of Long Beach is approximated by $$ A(t)=\frac{136}{1+0.25(t-4.5)^{2}}+28 \quad(0 \leq t \leq 11) $$ where \(A(t)\) is measured in pollutant standard index (PSI) and \(t\) is measured in hours, with \(t=0\) corresponding to 7 a.m. Determine the time of day when the pollution is at its highest level.

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=2 x-\ln x $$

Find the inflection point(s), if any, of each function. $$ f(x)=x e^{-2 x} $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=x^{3}-3 x+6 $$

The average revenue is defined as the function $$ \bar{R}(x)=\frac{R(x)}{x} \quad(x>0) $$ Prove that if a revenue function \(R(x)\) is concave downward \(\left[R^{\prime \prime}(x)<0\right]\), then the level of sales that will result in the largest average revenue occurs when \(\bar{R}(x)=R^{\prime}(x)\).

A city's main well was recently found to be contaminated with trichloroethylene (a cancer-causing chemical) as a result of an abandoned chemical dump that leached chemicals into the water. A proposal submitted to the city council indicated that the cost, measured in millions of dollars, of removing \(x \%\) of the toxic pollutants is given by $$ C(x)=\frac{0.5 x}{100-x} $$ a. Find the vertical asymptote of \(C(x)\). b. Is it possible to remove \(100 \%\) of the toxic pollutant from the water?

Find the inflection point(s), if any, of each function. $$ f(x)=x^{2} \ln x $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=\frac{1}{2} x^{4}-x^{2} $$

According to a law discovered by the 19th-century physician Jean Louis Marie Poiseuille, the velocity (in centimeters/second) of blood \(r \mathrm{~cm}\) from the central axis of an artery is given by $$ v(r)=k\left(R^{2}-r^{2}\right) $$ where \(k\) is a constant and \(R\) is the radius of the artery. Show that the velocity of blood is greatest along the central axis.

Find the inflection point(s), if any, of each function. $$ f(x)=\ln \left(x^{2}+1\right) $$

Find the relative maxima and relative minima, if any, of each function. $$ h(x)=\frac{1}{2} x^{4}-3 x^{2}+4 x-8 $$

A developing country's gross domestic product (GDP) from 2000 to 2008 is approximated by the function $$ G(t)=-0.2 t^{3}+2.4 t^{2}+60 \quad(0 \leq t \leq 8) $$ where \(G(t)\) is measured in billions of dollars and \(t=0\) corresponds to 2000 . Show that the growth rate of the country's GDP was maximal in 2004 .