Chapter 4

If time, \(t,\) is in hours and concentration, \(C,\) is in \(\mathrm{ng} / \mathrm{ml}\), the drug concentration curve for a drug is given by $$C=12.4 t e^{-0.2 t}$$ (a) Graph this curve. (b) How many hours does it take for the drug to reach its peak concentration? What is the concentration at that time? (c) If the minimum effective concentration is \(10 \mathrm{ng} / \mathrm{ml}\), during what time period is the drug effective? (d) Complications can arise whenever the level of the drug is above 4 ng/ml. How long must a patient wait before being safe from complications?

The elasticity of a good is \(E=0.5 .\) What is the effect on the quantity demanded of: .(a) A \(3 \%\) price increase? (b) \(\mathrm{A} 3 \%\) price decrease?

If \(t\) is in years since \(1990,\) one model for the population of the world, \(P,\) in billions, is $$P=\frac{40}{1+11 e^{-0.08 t}}$$ (a) What does this model predict for the maximum sustainable population of the world? (b) Graph \(P\) against \(t\). (c) According to this model, when will the earth's population reach 20 billion? 39.9 billion?

Let \(b=1,\) and graph \(C=a t e^{-b t}\) using different values for \(a .\) Explain the effect of the parameter \(a\).

The elasticity of a good is \(E=2 .\) What is the effect on the quantity demanded of: (a) A \(3 \%\) price increase? (b) A \(3 \%\) price decrease?

Investigate the effect of the parameter \(C\) on the logistic curve $$P=\frac{10}{1+C e^{-t}}$$ Substitute several values for \(C\) and explain, with a graph and with words, the effect of \(C\) on the graph.

The revenue from selling \(q\) items is \(R(q)=500 q-q^{2}\) and the total cost is \(C(q)=150+10 q .\) Write a function that gives the total profit earned, and find the quantity which maximizes the profit.

If \(t\) is in hours, the drug concentration curve for a drug is given by \(C=17.2 t e^{-0.4 t} \mathrm{ng} / \mathrm{ml} .\) The minimum effective concentration is \(10 \mathrm{ng} / \mathrm{ml}\). (a) If the second dose of the drug is to be administered when the first dose becomes incffective, when should the second dose be given? (b) If you want the onset of effectiveness of the second dose to coincide with termination of effectiveness of the first dose, when should the second dose be given?

What are the units of elasticity if: (a) Price \(p\) is in dollars and quantity \(q\) is in tons? (b) Price \(p\) is in yen and quantity \(q\) is in liters? (c) What can you conclude in general?

The following table shows the total sales, in thousands, since a new game was brought to market. (a) Plot this data and mark on your plot the point of diminishing returns. (b) Predict total possible sales of this game, using the point of diminishing returns. $$\begin{array}{c|c|c|c|c|c|c|c|c}\hline \text { Month } & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\\\\hline \text { Sales } & 0 & 2.3 & 5.5 & 9.6 & 18.2 & 31.8 & 42.0 & 50.8 \\\\\hline\end{array}$$

Revenue is given by \(R(q)=450 q\) and cost is given by \(C(q)=10,000+3 q^{2} .\) At what quantity is profit maximized? What is the total profit at this production level?

There are many brands of laundry detergent. Would you expect the elasticity of demand for any particular brand to be high or low? Explain.

The cost of producing \(q\) items is \(C(q)=2500+12 q\) dollars. (a) What is the marginal cost of producing the \(100^{\text {th }}\) item? the \(1000^{\text {th }}\) item? (b) What is the average cost of producing 100 items? 1000 items?

Write a paragraph explaining why sales of a new product often follow a logistic curve. Explain the benefit to the company of watching for the point of diminishing returns.

Would you expect the demand for high-definition television sets to be elastic or inelastic? Explain.

The cost function is \(C(q)=1000+20 q .\) Find the marginal cost to produce the \(200^{\text {th }}\) unit and the average cost of producing 200 units.

(a) Draw a logistic curve. Label the carrying capacity \(L\) and the point of diminishing returns \(t_{0}\) (b) Draw the derivative of the logistic curve. Mark the point \(t_{0}\) on the horizontal axis. (c) A company keeps track of the rate of sales (for example, sales per week) rather than total sales. Explain how the company can tell on a graph of rate of sales when the point of diminishing returns is reached.

(a) Graph a function with two local minima and one local maximum. (b) Graph a function with two critical points. One of these critical points should be a local minimum, and the other should be neither a local maximum nor a local minimum.

There is only one company offering local telephone service in a town. Would you expect the elasticity of demand for telephone service to be high or low? Explain.

The following table gives the percentage, \(P\), of households with cable television between 1977 and \(2003 .^{15}\) $$\begin{array}{c|r|r|r|r|r|r|l}\hline \text { Year } & 1977 & 1978 & 1979 & 1980 & 1981 & 1982 & 1983 \\\\\hline P & 16.6 & 17.9 & 19.4 & 22.6 & 28.3 & 35.0 & 40.5 \\\\\hline \text { Year } & 1984 & 1985 & 1986 & 1987 & 1988 & 1989 & 1990 \\\\\hline P & 43.7 & 46.2 & 48.1 & 50.5 & 53.8 & 57.1 & 59.0 \\ \hline \text { Year } & 1991 & 1992 & 1993 & 1994 & 1995 & 1996 & 1997 \\\\\hline P & 60.6 & 61.5 & 62.5 & 63.4 & 65.7 & 66.7 & 67.3 \\\\\hline \text { Year } & 1998 & 1999 & 2000 & 2001 & 2002 & 2003 & \\\\\hline P & 67.4 & 68.0 & 67.8 & 69.2 & 68.9 & 68.0 & \\\\\hline\end{array}$$ (a) Explain why a logistic model is reasonable for this data. (b) Estimate the point of diminishing returns. What limiting value \(L\) does this point predict? Does this limiting value appear to be accurate, given the percentages for 2002 and \(2003 ?\) (c) If \(t\) is in years since \(1977,\) the best fitting logistic function for this data turns out to be $$P=\frac{68.8}{1+3.486 e^{-0.237 t}}$$ What limiting value does this function predict? (d) Explain in terms of percentages of households what the limiting value is telling you. Do you think your answer to part (c) is an accurate prediction?

(a) Graph a polynomial with two local maxima and two local minima. (b) What is the least number of inflection points this function must have? Label the inflection points.

Let \(C(q)\) represent the cost, \(R(q)\) the revenue, and \(\pi(q)\) the total profit, in dollars, of producing \(q\) items. (a) If \(C^{\prime}(50)=75\) and \(R^{\prime}(50)=84,\) approximately how much profit is earned by the \(51^{\text {st }}\) item? (b) If \(C^{\prime}(90)=71\) and \(R^{\prime}(90)=68,\) approximately how much profit is earned by the \(91^{\text {st }}\) item? (c) If \(\pi(q)\) is a maximum when \(q=78,\) how do you think \(C^{\prime}(78)\) and \(R^{\prime}(78)\) compare? Explain.

The total cost of production, in thousands of dollars, is \(C(q)=q^{3}-12 q^{2}+60 q,\) where \(q\) is in thousands and \(0 \leq q \leq 8\) (a) Graph \(C(q) .\) Estimate visually the quantity at which average cost is minimized. (b) Determine analytically the exact value of \(q\) at which aver cost is minimized.

The Tojolobal Mayan Indian community in Southern Mexico has available a fixed amount of land." The proportion, \(P,\) of land in use for farming \(t\) years after 1935 is modeled with the logistic function $$P=\frac{1}{1+3 e^{-0.0275 t}}$$ (a) What proportion of the land was in use for farming in \(1935 ?\) (b) What is the long-run prediction of this model? (c) When was half the land in use for farming? (d) When is the proportion of land used for farming increasing most rapidly?

Graph a function which has a critical point and an inflection point at the same place.

If \(t\) is in minutes since the drug was administered, the concentration, \(C(t)\) in \(\mathrm{ng} / \mathrm{ml}\), of a drug in a patient's bloodstream is given by $$C(t)=20 t e^{-0.03 t}$$ (a) How long does it take for the drug to reach peak concentration? What is the peak concentration? (b) What is the concentration of the drug in the body after 15 minutes? After an hour? (c) If the minimum effective concentration is \(10 \mathrm{ng} / \mathrm{ml}\), when should the next dose be administered?

In the spring of \(2003,\) SARS (Severe Acute Respiratory Syndrome) spread rapidly in several Asian countries and Canada. Table 4.9 gives the total number, \(P\), of SARS cases reported in Hong Kong \(^{17}\) by day \(t,\) where \(t=0\) is March 17,2003. (a) Find the average rate of change of \(P\) for each interval in Table 4.9 (b) In early April \(2003,\) there was fear that the disease would spread at an ever-increasing rate for a long time. What is the earliest date by which epidemiologists had evidence to indicate that the rate of new cases had begun to slow? (c) Explain why an exponential model for \(P\) is not appropriate. (d) It turns out that a logistic model fits the data well. Estimate the value of \(t\) at the inflection point. What limiting value of \(P\) does this point predict? (e) The best-fitting logistic function for this data turns out to be $$P=\frac{1760}{1+17.53 e^{-0.1408 t}}$$ What limiting value of \(P\) does this function predict? Total number of SARS cases in Hong Kong by day \(t\) (where \(t=0\) is March 17,2003) $$\begin{array}{c|c|c|c|c|c|c|c}t & P & t & P & t & P & t & P \\\\\hline 0 & 95 & 26 & 1108 & 54 & 1674 & 75 & 1739 \\\5 & 222 & 33 & 1358 & 61 & 1710 & 81 & 1750 \\\12 & 470 & 40 & 1527 & 68 & 1724 & 87 & 1755 \\\19 & 800 & 47 & 1621 & & & & \\\\\hline\end{array}$$

You are the manager of a firm that produces slippers that sell for \(\$ 20\) a pair. You are producing 1200 pairs of slippers each month, at an average cost of \(\$ 2\) each. The marginal cost at a production level of 1200 is \(\$ 3\) per pair. (a) Are you making or losing money? (b) Will increasing production increase or decrease your average cost? Your profit? (c) Would you recommend that production be increased or decreased?

True or false? Give an explanation for your answer. The global maximum of \(f(x)=x^{2}\) on every closed interval is at one of the endpoints of the interval.

During a flood, the water level in a river first rose faster and faster, then rose more and more slowly until it reached its highest point, then went back down to its preflood level. Consider water depth as a function of time. (a) Is the time of highest water level a critical point or an inflection point of this function? (b) Is the time when the water first began to rise more slowly a critical point or an inflection point?

Using a calculator or computer, graph the functions in Problems \(8-13 .\) Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph. $$f(x)=x^{3}+6 x+1$$

Table 4.3 shows marginal cost, \(M C,\) and marginal revenue, \(M R\) (a) Use the marginal cost and marginal revenue at a production of \(q=5000\) to determine whether production should be increased or decreased from \(5000 .\) (b) Estimate the production level that maximizes profit. $$\begin{array}{c|r|r|r|r|r|r} \hline q & 5000 & 6000 & 7000 & 8000 & 9000 & 10000 \\ \hline M R & 60 & 58 & 56 & 55 & 54 & 53 \\ \hline M C & 48 & 52 & 54 & 55 & 58 & 63 \\ \hline \end{array}$$

The demand for a product is given by \(q=200-2 p^{2}\) Find the elasticity of demand when the price is \(\$ 5 .\) Is the demand inelastic or elastic, or neither?

The average cost per item to produce \(q\) items is given by $$ a(q)=0.01 q^{2}-0.6 q+13, \text { for } q>0 $$ (a) What is the total cost, \(C(q),\) of producing \(q\) goods? (b) What is the minimum marginal cost? What is the practical interpretation of this result? (c) At what production level is the average cost a minimum? What is the lowest average cost? (d) Compute the marginal cost at \(q=30 .\) How does this relate to your answer to part (c)? Explain this relationship both analytically and in words.

Plot the graph of \(f(x)=x^{3}-e^{x}\) using a graphing calculator or computer to find all local and global maxima and minima for: (a) \(-1 \leq x \leq 4\) (b) \(-3 \leq x \leq 2\)

Using a calculator or computer, graph the functions in Problems \(8-13 .\) Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph. $$f(x)=x^{3}-6 x+1$$

This problem shows how a surge can be modeled with a difference of exponential decay functions. (a) Using graphs of \(e^{-t}\) and \(e^{-2 t},\) explain why the graph of \(f(t)=e^{-t}-e^{-2 t}\) has the shape of a surge. (b) Find the critical point and inflection point of \(f\)

The demand for a product is given by \(p=90-10 q .\) Find the elasticity of demand when \(p=50 .\) If this price rises by \(2 \%,\) calculate the corresponding percentage change in demand.

The marginal cost at a production level of 2000 units of an item is 810 per unit and the average cost of producing 2000 units is \(\$ 15\) per unit. If the production level were increased slightly above 2000 , would the following quantities increase or decrease, or is it impossible to tell? (a) Average cost (b) Profit

For \(f(x)=x^{3}-18 x^{2}-10 x+6,\) find the inflection point algebraically. Graph the function with a calculator or computer and confirm your answer.

Using a calculator or computer, graph the functions in Problems \(8-13 .\) Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph. $$f(x)=3 x^{5}-5 x^{3}$$

The marginal cost and marginal revenue of a company are \(M C(q)=0.03 q^{2}-1.4 q+34\) and \(M R(q)=30\) where \(q\) is the number of items manufactured. To increase profits, should the company increase or decrease production from each of the following levels? (a) 25 items (b) 50 items (c) 80 items

A curve representing the total number of people, \(P\), infected with a virus often has the shape of a logistic curve of the form $$P=\frac{L}{1+C e^{-k t}}$$ with time \(t\) in weeks. Suppose that 10 people originally have the virus and that in the early stages the number of people infected is increasing approximately exponentially, with a continuous growth rate of \(1.78 .\) It is estimated that, in the long run, approximately 5000 people will become infected. (a) What should we use for the parameters \(k\) and \(L ?\) (b) Use the fact that when \(t=0,\) we have \(P=10,\) to find \(C\) (c) Now that you have estimated \(L, k,\) and \(C,\) what is the logistic function you are using to model the data? Graph this function. (d) Estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the value of \(P\) at this point?

School organizations raise money by selling candy door to door. The table shows \(p,\) the price of the candy, and \(q\) the quantity sold at that price. $$ \begin{array}{c|c|c|c|c|c|c|c} \hline p & \$ 1.00 & \$ 1.25 & \$ 1.50 & \$ 1.75 & \$ 2.00 & \$ 2.25 & \$ 2.50 \\\ \hline q & 2765 & 2440 & 1980 & 1660 & 1175 & 800 & 430 \\ \hline \end{array} $$ (a) Estimate the elasticity of demand at a price of \(\$ 1.00\). At this price, is the demand elastic or inelastic? (b) Estimate the elasticity at each of the prices shown. What do you notice? Give an explanation for why this might be so. (c) At approximately what price is elasticity equal to \(1 ?\) (d) Find the total revenue at each of the prices shown. Confirm that the total revenue appears to be maximized at approximately the price where \(E=1\)

Find the inflection points of \(f(x)=x^{4}+x^{3}-3 x^{2}+2\).

Using a calculator or computer, graph the functions in Problems \(8-13 .\) Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph. $$f(x)=e^{x}-10 x$$

A manufacturing process has marginal costs given in the table; the item sells for \(\$ 30\) per unit. At how many quantities, \(q,\) does the profit appear to be a maximum? In what intervals do these quantities appear to lie? $$\begin{array}{r|r|r|r|r|r|r|r} \hline q & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline \text { MC (S/unit) } & 34 & 23 & 18 & 19 & 26 & 39 & 58 \\ \hline \end{array}$$

Find the point where the following curve is steepest: $$y=\frac{50}{1+6 e^{-2 t}} \quad \text { for } t \geq 0$$

Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither. $$f(x)=x^{2}-5 x+3$$

Using a calculator or computer, graph the functions in Problems \(8-13 .\) Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph. $$f(x)=x \ln x, \quad x>0$$