Find the Gini index for the given Lorenz curve. $L(x)=x^{n}$ (for $n>1$)

Find the Gini index for the given Lorenz curve. $L(x)=x^{2}$

Find the Gini index for the given Lorenz curve. $L(x) = x^3$

Researchers are interested in the relationship between cigarette smoking and lung cancer. Suppose an adult male is randomly selected from a particular population. The following table shows the probabilities of some events related to this chance process.

$$\begin{array}{|l|c|} \hline \text { Event } & \text { Probability } \\ \hline \text { Smokes } & 0.25 \\ \hline \text { Smokes and gets cancer } & 0.08 \\ \hline \begin{array}{l} \text { Does not smoke and } \\ \text { does not get cancer } \end{array} & 0.71 \\ \hline \end{array}$$

Make a two-way table or a Venn diagram to display the sample space.

For the demand function $d(x)$ and supply function $s(x)$ : $d(x)=300-0.4 x, s(x)=0.2 x$

a. Find the market demand (the positive value of $x$ at which the demand function intersects the supply function).

b. Find the consumers' surplus at the market demand found in part (a).

c. Find the producers' surplus at the market demand found in part (a).

Money is transferred continuously into an account at the constant rate of $\$ 5,000$ per year. The account earns interest at the annual rate of $5 \%$ compounded continuously. How much will be in the account at the end of 3 years?

The manager of a shoe store determines that the price p (dollars) for each pair of a popular brand of sports sneakers is changing at the rate of $p^{\prime}(x)=\frac{-300 x}{\left(x^{2}+9\right)^{3 / 2}}$ when x (hundred) pairs are demanded by consumers. When the price is $75 per pair, 400 pairs (x = 4) are demanded by consumers. a. Find the demand (price) function p(x). b. At what price will 500 pairs of sneakers be demanded? At what price will no sneakers be demanded? c. How many pairs will be demanded at a price of$90 per pair?

If the average value of f(x) on an interval is a number c, what will be the average value of the function -f(x) on that interval?

The slope f'(x) at each point (x, y) on a curve y = f(x) is given, along with a particular point (a, b) on the curve. Use this information to find f(x). $f^{\prime}(x)=\frac{2 x}{1+3 x^{2}};$ (0, 5)

For the consumers ' demand functions $D(q)$ in exercise.

(a) Find the total amount of money consumers are willing to spend to get $q_0$ units of the commodity.

(b) Sketch the demand curve and interpret the consumers' willingness to spend in part (a) as an area.

$D(q)=50 e^{-0.04 q}$ dollars per unit; $q_0=15$ units

Derive each formula by using integration by parts on the left-hand side. (Assume n > 0.) $\int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x$

Derive each formula by using integration by parts on the left-hand side. (Assume n > 0.) $\int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x$

BUSINESS: Revenue If a company's marginal revenue function is $\operatorname{MR}(x)=x e^{x / 4}$, find the revenue function. [Hint: Evaluate the constant C so that revenue is 0 at x=0.]

An investment will generate income continuously at the constant rate of $1,200 per year for 5 years. If the prevailing annual interest rate remains fixed at 5% compounded continuously, what is the present value of the investment?