Chapter 4

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

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?

The rate of sales of an automobile anti-theft device are given in the following table. (a) When is the point of diminishing returns reached? (b) What are the total sales at this point? (c) Assuming logistic sales growth, use your answer to part (b) to estimate total potential sales of the device. $$ \begin{array}{c|c|c|c|c|c|c} \hline \text { Months } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Sales per month } & 140 & 520 & 680 & 750 & 700 & 550 \\ \hline \end{array} $$

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?

Dwell time, \(t\), is the time in minutes that shoppers spend in a store. Sales, \(s\), is the number of dollars they spend in the store. The elasticity of sales with respect to dwell time is 1.3. Explain what this means in simple language.

A rumor spreads among a group of 400 people. The number of people, \(N(t)\), who have heard the rumor by time \(t\) in hours since the rumor started to spread can be approximated by a function of the form $$ N(t)=\frac{400}{1+399 e^{-0.4 t}} $$ (a) Find \(N(0)\) and interpret it. (b) How many people will have heard the rumor after 2 hours? After 10 hours? (c) Graph \(N(t)\). (d) Approximately how long will it take until half the people have heard the rumor? Virtually everyone? (e) Approximately when is the rumor spreading fastest?

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?

A demand function is \(p=400-2 q\), where \(q\) is the quantity of the good sold for price \(\$ p\). (a) Find an expression for the total revenue, \(R\), in terms of \(q\) (b) Differentiate \(R\) with respect to \(q\) to find the marginal revenue, \(M R\), in terms of \(q .\) Calculate the marginal revenue when \(q=10\). (c) Calculate the change in total revenue when production increases from \(q=10\) to \(q=11\) units. Confirm that a one-unit increase in \(q\) gives a reasonable approximation to the exact value of \(M R\) obtained in part (b).

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 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.

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.

(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.

(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.

During an illness a person ran a fever. His temperature rose steadily for eighteen hours, then went steadily down for twenty hours. When was there a critical point for his temperature as a function of time?

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

Graph a function with the given properties. Has no local or global maxima or minima.

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 average cost is minimized.

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

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

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

Graph a function with the given properties. 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.

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?

Marginal revenue and marginal cost are given in the following table. Estimate the production levels that could maximize profit. Explain. $$ \begin{array}{c|c|c|c|c|c|c} \hline q & 1000 & 2000 & 3000 & 4000 & 5000 & 6000 \\ \hline M R & 78 & 76 & 74 & 72 & 70 & 68 \\ \hline M C & 100 & 80 & 70 & 65 & 75 & 90 \\ \hline \end{array} $$

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 hegan to rise more slowly a critical point or an inflection point?

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.

Sketch the graph of a function on the inter val \(0 \leq x \leq 10\) with the given properties. Has local minimum at \(x=3\), local maximum at \(x=8\), but global maximum and global minimum at the endpoints of the interval.

A company estimates that the total revenue, \(R\), in dollars. received from the sale of \(q\) items is \(R=\ln \left(1+1000 q^{2}\right)\). Calculate and interpret the marginal revenue if \(q=10\).

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.

When 1 got up in the morning \(I\) put on only a light jacket because, although the temperature was dropping, it seemed that the temperature would not go much lower. But I was wrong. Around noon a northerly wind blew up and the temperature began to drop faster and faster. The worst was around \(6 \mathrm{pm}\) when, fortunately, the temperature started going back up. (a) When was there a critical point in the graph of temperature as a function of time? (b) When was there an inflection point in the graph of temperature as a function of time?

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 Tojolobal Mayan Indian community in Southern Mexico has available a fixed amount of land. \({ }^{16}\) 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?

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.

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. $$ f(x)=x^{3}-6 x+1 $$

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 \(E=1\).

Sketch the graph of a function on the inter val \(0 \leq x \leq 10\) with the given properties. Has local and global minimum at \(x=3\), local and global maximum at \(x=8\).

An agricultural worker in Uganda is planting clover to increase the number of bees making their home in the region. There are 100 bees in the region naturally, and for every acre put under clover, 20 more bees are found in the region. (a) Draw a graph of the total number, \(N(x)\), of bees as a function of \(x\), the number of acres devoted to clover. (b) Explain, both geometrically and algebraically, the shape of the graph of: (i) The marginal rate of increase of the number of bees with acres of clover, \(N^{\prime}(x)\). (ii) The average number of bees per acre of clover, \(N(x) / x\)

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. \(f(x)=x^{2}-5 x+3\)

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. $$ f(x)=x^{3}+6 x+1 $$

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.

Sketch the graph of a function on the inter val \(0 \leq x \leq 10\) with the given properties. Has global maximum at \(x=0\), global minimum at \(x=10\), and no other local maxima or minima.

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

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. \(f(x)=x^{3}-3 x+10\)

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. $$ f(x)=3 x^{5}-5 x^{3} $$

A grapefruit is tossed straight up with an initial velocity of \(50 \mathrm{ft} / \mathrm{sec}\). The grapefruit is 5 feet above the ground when it is released. Its height at time \(t\) is given by $$y=-16 t^{2}+50 t+5$$ How high does it go before returning to the ground?

Figure \(4.65\) shows the average cost, \(a(q)=b+m q\). (a) Show that \(C^{\prime}(q)=b+2 m q\). (b) Graph the marginal cost \(C^{\prime}(q)\).

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. \(f(x)=2 x^{3}+3 x^{2}-36 x+5\)

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 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?

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. \(f(x)=\frac{x^{3}}{6}+\frac{x^{2}}{4}-x+2\)