Chapter 4

Show analytically that if marginal cost is less than average cost, then the derivative of average cost with respect to quantity satisfies \(a^{\prime}(q)<0\).

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. $$ f(x)=x \ln x, \quad x>0 $$

It has been estimated that the elasticity of demand for slaves in the American South before the civil war was equal to \(0.86\) (fairly high) in the cities and equal to \(0.05\) (very low) in the countryside. \({ }^{10}\) (a) Why might this be? (b) Where do you think the staunchest defenders of slavery were from, the cities or the countryside?

If \(R\) is percent of maximum response and \(x\) is dose in \(\mathrm{mg}\), the dose-response curve for a drug is given by $$ R=\frac{100}{1+100 e^{-0.1 x}} $$ (a) Graph this function. (b) What dose corresponds to a response of \(50 \%\) of the maximum? This is the inflection point, at which the response is increasing the fastest. (c) For this drug, the minimum desired response is \(20 \%\) and the maximum safe response is \(70 \%\). What range of doses is both safe and effective for this drug?

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) \(\quad-1 \leq x \leq 4\) (b) \(\quad-3 \leq x \leq 2\)

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. \(f(x)=x^{4}-2 x^{2}\)

Show analytically that if marginal cost is greater than average cost, then the derivative of average cost with respect to quantity satisfies \(a^{\prime}(q)>0\)

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. $$ f(x)=x+2 \sin x $$

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

When production is 2000, marginal revenue is \(\$ 4\) per unit and marginal cost is \(\$ 3.25\) per unit. Do you expect maximum profit to occur at a production level above or below 2000 ? Explain.

A reasonably realistic model of a firm's costs is given by the short-run Cobb- Douglas cost curve $$ C(q)=K q^{1 / a}+F $$ where \(a\) is a positive constant, \(F\) is the fixed cost, and \(K\) measures the technology available to the firm. (a) Show that \(C\) is concave down if \(a>1\). (b) Assuming that \(a<1\), find what value of \(q\) minimizes the average cost.

For some positive constant \(C\), a patient's temperature change, \(T\), due to a dose, \(D\), of a drug is given by $$T=\left(\frac{C}{2}-\frac{D}{3}\right) D^{2}$$ (a) What dosage maximizes the temperature change? (b) The sensitivity of the body to the drug is defined as \(d T / d D\). What dosage maximizes sensitivity?

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?

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. \(f(x)=x^{4}-8 x^{2}+5\)

For the functions in problems, do the following: (a) Find \(f^{\prime}\) and \(f^{\prime \prime}\). (b) Find the critical points of \(f\). (c) Find any inflection points of \(f\). (d) Evaluate \(f\) at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of \(f\) in the interval. (e) Graph \(f\). $$ f(x)=x^{3}-3 x^{2} \quad(-1 \leq x \leq 3) $$

The demand equation for a product is \(p=45-0.01 q\). Write the revenue as a function of \(q\) and find the quantity that maximizes revenue. What price corresponds to this quantity? What is the total revenue at this price?

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. \(f(x)=x^{4}-4 x^{3}+10\)

The derivative of \(f(t)\) is given by \(f^{\prime}(t)=t^{3}-6 t^{2}+8 t\) for \(0 \leq t \leq 5\). Graph \(f^{\prime}(t)\), and describe how the function \(f(t)\) changes over the interval \(t=0\) to \(t=5 .\) When is \(f(t)\) increasing and when is it decreasing? Where does \(f(t)\) have a local maximum and where does it have a local minimum?

For the functions in problems, do the following: (a) Find \(f^{\prime}\) and \(f^{\prime \prime}\). (b) Find the critical points of \(f\). (c) Find any inflection points of \(f\). (d) Evaluate \(f\) at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of \(f\) in the interval. (e) Graph \(f\). $$ f(x)=2 x^{3}-9 x^{2}+12 x+1(-0.5 \leq x \leq 3) $$

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. \(f(x)=x^{5}-5 x^{4}+35\)

If \(U\) and \(V\) are positive constants, find all critical points of $$ F(t)=U e^{t}+V e^{-t} $$

Show that a demand equation \(q=k / p^{r}\), where \(r\) is a positive constant, gives constant elasticity \(E=r\).

For the functions in problems, do the following: (a) Find \(f^{\prime}\) and \(f^{\prime \prime}\). (b) Find the critical points of \(f\). (c) Find any inflection points of \(f\). (d) Evaluate \(f\) at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of \(f\) in the interval. (e) Graph \(f\). $$ f(x)=2 x^{3}-9 x^{2}+12 x+1(-0.5 \leq x \leq 3) $$

The following table gives the cost and revenue, in dollars, for different production levels, \(q .\) (a) At approximately what production level is profit maximized? (b) What price is charged per unit for this product? (c) What are the fixed costs of production? \begin{array}{c|c|c|c|c|c|c} \hline q & 0 & 100 & 200 & 300 & 400 & 500 \\ \hline R(q) & 0 & 500 & 1000 & 1500 & 2000 & 2500 \\ \hline C(q) & 700 & 900 & 1000 & 1100 & 1300 & 1900 \\ \hline \end{array}

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

Consumer demand for a certain product is changing over time, and the rate of change of this demand, \(f^{\prime}(t)\), in units/week, is given, in week \(t\), in the following table. $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c} \hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline f^{\prime}(t) & 12 & 10 & 4 & -2 & -3 & -1 & 3 & 7 & 11 & 15 & 10 \\ \hline \end{array} $$ (a) When is the demand for this product increasing? When is it decreasing? (b) Approximately when is demand at a local maximum? A local minimum?

A population, \(P\), growing logistically is given by $$ P=\frac{L}{1+C e^{-k t}} $$ (a) Show that $$ \frac{L-P}{P}=C e^{-k t} $$ (b) Explain why part (a) shows that the ratio of the additional population the environment can support to the existing population decays exponentially.

For the functions in problems, do the following: (a) Find \(f^{\prime}\) and \(f^{\prime \prime}\). (b) Find the critical points of \(f\). (c) Find any inflection points of \(f\). (d) Evaluate \(f\) at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of \(f\) in the interval. (e) Graph \(f\). $$ f(x)=x+\sin x \quad(0 \leq x \leq 2 \pi) $$

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

Suppose \(f\) has a continuous derivative whose values are given in the following table. (a) Estimate the \(x\) -coordinates of critical points of \(f\) for \(0 \leq x \leq 10 .\) (b) For each critical point, indicate if it is a local maximum of \(f\), local minimum, or neither. $$ \begin{array}{l|l|l|l|r|r|r|r|l|l|l|c} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline f^{\prime}(x) & 5 & 2 & 1 & -2 & -5 & -3 & -1 & 2 & 3 & 1 & -1 \\ \hline \end{array} $$

If \(p\) is price and \(E\) is the elasticity of demand for a good, show analytically that $$ \text { Marginal revenue }=p(1-1 / E) $$

For the functions in problems, do the following: (a) Find \(f^{\prime}\) and \(f^{\prime \prime}\). (b) Find the critical points of \(f\). (c) Find any inflection points of \(f\). (d) Evaluate \(f\) at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of \(f\) in the interval. (e) Graph \(f\). $$ f(x)=e^{-x} \sin x \quad(0 \leq x \leq 2 \pi) $$

An ice cream company finds that at a price of \(\$ 4.00\), demand is 4000 units. For every \(\$ 0.25\) decrease in price, demand increases by 200 units. Find the price and quantity sold that maximize revenue.

(a) Find all critical points and all inflection points of the function \(f(x)=x^{4}-2 a x^{2}+b .\) Assume \(a\) and \(b\) are positive constants. (b) Find values of the parameters \(a\) and \(b\) if \(f\) has a critical point at the point \((2,5)\). (c) If there is a critical point at \((2,5)\), where are the inflection points?

The function \(f(x)=x^{4}-4 x^{3}+8 x\) has a critical point at \(x=1\). Use the second derivative test to identify it as a local maximum or local minimum.

Find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. $$ g(x)=4 x-x^{2}-5 $$

At a price of \(\$ 8\) per ticket, a musical theater group can fill every seat in the theater, which has a capacity of \(1500 .\) For every additional dollar charged, the number of people buying tickets decreases by \(75 .\) What ticket price maximizes revenue?

Find and classify the critical points of \(f(x)=x^{3}(1-x)^{4}\) as local maxima and minima.

Elasticity of cost with respect to quantity is defined as \(E_{C, q}=q / C \cdot d C / d q\) (a) What does this elasticity tell you about sensitivity of cost to quantity produced? (b) Show that \(E_{C, q}=\) Marginal cost \(/\) Average cost.

Find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. $$ f(x)=x+1 / x \text { for } x>0 $$

The demand equation for a quantity \(q\) of a product at price \(p\), in dollars, is \(p=-5 q+4000\). Companies producing the product report the cost, \(C\), in dollars, to produce a quantity \(q\) is \(C=6 q+5\) dollars. (a) Express a company's profit, in dollars, as a function of \(q\). (b) What production level earns the company the largest profit? (c) What is the largest profit possible?

Investigate the one-parameter family of functions. Assume that \(a\) is positive. (a) Graph \(f(x)\) using three different values for \(a\). (b) Using your graph in part (a), describe the critical points of \(f\) and how they appear to move as \(a\) increases. (c) Find a formula for the \(x\) -coordinates of the critical point(s) of \(f\) in terms of \(a\). $$ f(x)=(x-a)^{2} $$

If \(q\) is the quantity of chicken demanded as a function of the price \(p\) of beef, the cross-price elasticity of demand for chicken with respect to the price of beef is defined as \(E_{\text {cross }}=|p / q \cdot d q / d p|\). What does \(E_{\text {cross }}\) tell you about the sensitivity of the quantity of chicken bought to changes in the price of beef?

Find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. $$ g(t)=t e^{-t} \text { for } t>0 $$

(a) Production of an item has fixed costs of \(\$ 10,000\) and variable costs of \(\$ 2\) per item. Express the cost, \(C\), of producing \(q\) items. (b) The relationship between price, \(p\), and quantity, \(q\), demanded is linear. Market research shows that 10,100 items are sold when the price is \(\$ 5\) and 12,872 items are sold when the price is \(\$ 4.50 .\) Express \(q\) as a function of price \(p\). (c) Express the profit earned as a function of \(q\). (d) How many items should the company produce to maximize profit? (Give your answer to the nearest integer.) What is the profit at that production level?

Investigate the one-parameter family of functions. Assume that \(a\) is positive. (a) Graph \(f(x)\) using three different values for \(a\). (b) Using your graph in part (a), describe the critical points of \(f\) and how they appear to move as \(a\) increases. (c) Find a formula for the \(x\) -coordinates of the critical point(s) of \(f\) in terms of \(a\). $$ f(x)=x^{3}-a x $$

The income elasticity of demand for a product is defined as \(E_{\text {income }}=|I / q \cdot d q / d I|\) where \(q\) is the quantity demanded as a function of the income \(I\) of the consumer. What does \(E_{\text {income }}\) tell you about the sensitivity of the quantity of the product purchased to changes in the income of the consumer?

Find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. $$ f(x)=x-\ln x \text { for } x>0 $$

A landscape architect plans to enclose a 3000 square-foot rectangular region in a botanical garden. She will use shrubs costing \(\$ 45\) per foot along three sides and fencing costing \(\$ 20\) per foot along the fourth side. Find the minimum total cost.

Investigate the one-parameter family of functions. Assume that \(a\) is positive. $$ f(x)=x^{2} e^{-a x} $$