The elasticity of a good is $E=2$. What is the effect on demand of:

(a) A $3 \%$ price increase?

(b) A $3 \%$ price decrease?

Find the exact price that maximizes revenue for sales of the product.

If $R$ is percent of maximum response and $x$ is dose in $\mathrm{mg}$, the dose-response curve for a drug is given by

$$R=\frac{100}{1+100 e^{-0.1 x}} .$$

(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?

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. ${ }^{10}$

(b) Where do you think the staunchest defenders of slavery were from, the cities or the countryside?

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:

(b) $-3 \leq x \leq 2$

If $R$ is percent of maximum response and $x$ is dose in $\mathrm{mg}$, the dose-response curve for a drug is given by

$$R=\frac{100}{1+100 e^{-0.1 x}} .$$

(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.

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) $-1 \leq x \leq 4$

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.

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?

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.

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

The demand for yams is given in Problem 12.

(d) Show that $E=1$ at the price you found in part (b).

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.

(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?

The demand for yams is given in Problem 12.

(c) What quantity is sold at the price you found in part (b), and what is the total revenue?

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.

(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.

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.

(b) Use the fact that when $t=0$, we have $P=10$, to find $C$.