A Lorenz curve is given by y = L(x), where

A Lorenz curve is given by y = L(x), where 0 ≤ x ≤ 1 represents the lowest fraction of the population of a society in terms of wealth and 0 ≤ y ≤ 1 represents the fraction of the total wealth that is owned by that fraction of the society. For example, the Lorenz curve in the figure shows that L(0.5) = 0.2, which means that the lowest 0.5 (50%) of the society owns 0.2 (20%) of the wealth. (See the Guided Project of Wealth for more on Lorenz curves.)

a. A Lorenz curve y = L(x) is accompanied by the line y = x, called the line of perfect equality. Explain why this line is given this name.

b. Explain why a Lorenz curve satisfies the conditions L(0) = 0, L(1) = 1, L(x) ≤ x, and L'(x) ≥ 0 on [0, 1].

c. Graph the Lorenz curves L(x) = xp corresponding to p = 1.1, 1.5, 2, 3, 4. Which value of p corresponds to the most equitable of wealth (closest to the line of perfect equality)? Which value of p corresponds to the least equitable of wealth? Explain. 

d. The information in the Lorenz curve is often summarized in a single measure called the Gini index, which is defined as follows. Let A be the area of the region between y = x and y = L(x) (see figure) and let B be the area of the region between y = L(x) and the x-axis. Then the Gini index is

e. Compute the Gini index for the cases L(x) = xp and p = 1.1, 1.5, 2, 3, 4.

f. What is the smallest interval [a, b] on which values of the Gini index lie for L(x) = xp with p ≥ 1? Which endpoints of [a, b] correspond to the least and most equitable of wealth?

g. Consider the Lorenz curve described by L(x) = 5x2/6 + x/6. Show that it satisfies the conditions L(0) = 0, L(1) = 1, and L'(x) ≥ 0 on [0, 1]. Find the Gini index for this function.

Consider the parabola y = x2. Let P, Q, and

Consider the parabola y = x2. Let P, Q, and R be points on the parabola with R between P and Q on the curve. Let ℓP, ℓQ, and ℓR be the lines tangent to the parabola at P, Q, and R, respectively (see figure). Let P’ be the intersection point of ℓQ and ℓR, let Q’ be the intersection point of ℓP and ℓR, and let R’ be the intersection point of ℓP and ℓQ. Prove that Area ΔPQR = 2 • Area ΔP’Q’R’ in the following cases.

a. P(-a, a2), Q(a, a2), and R(0, 0), where a is a positive real number

b. P(-a, a2), Q(b, b2), and R(0, 0), where a and b are positive real numbers

c. P(-a, a2), Q(b, b2), and R is any point between P and Q on the curve

Suppose a dartboard occupies the square {(x, y): 0 ≤

Suppose a dartboard occupies the square {(x, y): 0 ≤ |x| ≤ 1, 0 ≤ |y| ≤ 1}. A dart is thrown randomly at the board many times (meaning it is equally likely to land at any point in the square). What fraction of the dart throws land closer to the edge of the board than the center? Equivalently, what is the probability that the dart lands closer to the edge of the board than the center? Proceed as follows.

a. Argue that by symmetry, it is necessary to consider only one quarter of the board, say the region R: {(x, y): |x| ≤ y ≤ 1}.

b. Find the curve C in this region that is equidistant from the center of the board and the top edge of the board (see figure).

c. The probability that the dart lands closer to the edge of the board than the center is the ratio of the area of the region R1 above C to the area of the entire region R. Compute this probability.

Let f(x) = xp and g(x) = x1/q, where p

Let f(x) = xp and g(x) = x1/q, where p > 1 and q > 1 are positive integers. Let R1 be the region in the first quadrant between y = f(x) and y = x and let R2 be the region in the first quadrant between y = g(x) and y = x. 

a. Find the area of R1 and R2 when p = q, and determine which region has the greater area.

b. Find the area of R1 and R2 when p > q, and determine which region has the greater area.

c. Find the area of R1 and R2 when p < q, and determine which region has the greater area.

Consider the functions f(x) = a sin 2x and g(x)

Consider the functions f(x) = a sin 2x and g(x) = (sin x)/a, where a > 0 is a real number.

a. Graph the two functions on the interval [0, π/2], for a = 1/2, 1, and 2.

b. Show that the curves have an intersection point x* (other than x = 0) on [0, π/2] that satisfies cos x* = 1/(2a2), provided a > 1/√2.

c. Find the area of the region between the two curves on [0, x*] when a = 1.

d. Show that as a →1/√2+. The area of the region between the two curves on [0, x*] approaches zero.

Consider the cubic polynomial f(x) = x(x – a)(x –

Consider the cubic polynomial f(x) = x(x – a)(x – b), where 0 ≤ a ≤ b.

a. For a fixed value of b, find the functionFor what value of a (which depends on b) is F(a) = 0?

b. For a fixed value of b, find the function A(a) that gives the area of the region bounded by the graph of f and the x-axis between x = 0 and x = b. Graph this function and show that it has a minimum at a = b/2. What is the maximum value of A(a), and where does it occur (in terms of b)?

A strong west wind blows across a circular running track.

A strong west wind blows across a circular running track. Abe and Bess start running at the south end of the track, and at the same time, Abe starts running clockwise and Bess starts running counterclockwise. Abe runs with a speed (in units of mi/hr) given by u(φ) = 3 – 2 cos φ and Bess runs with a speed given by v(θ) = 3 + 2 cos θ, where φ and θ are the central angles of the runners.

a. Graph the speed functions u and v, and explain why they describe the runners’ speeds (in light of the wind).

b. Compute the average value of u and v with respect to the central angle.

c. Challenge: If the track has a radius of 1/10 mi, how long does it take each runner to complete one lap and who wins the race?

Kelly started at noon (t = 0) riding a bike

Kelly started at noon (t = 0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15/(t + 1)2 (decreasing because of fatigue). Sandy started at noon (t = 0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20/(t + 1)2 (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours. 

a. Make a graph of Kelly’s distance from Niwot as a function of time. 

b. Make a graph of Sandy’s distance from Berthoud as a function of time.

c. When do they meet? How far has each person traveled when they meet?

d. More generally, if the riders’ speeds are v(t) = A/(t + 1)2 and u(t) = B/(t + 1)2 and the distance between the towns is D, what conditions on A, B, and D must be met to ensure that the riders will pass each other?

e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).

The daily discharge of the Spokane River as it flows

The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t2 + 37.46t + 722.47 (April) and
r2(t) = 0.90t2 – 69.06t + 2053.12 (June),

where the discharge is measured in millions of cubic feet per day and t = 1 corresponds to the first day of the month (see figure).

a. Determine the total amount of water that flows through Spokane in April (30 days).

b. Determine the total amount of water that flows through Spokane in June (30 days).

c. The Spokane River flows out of Lake Coeur d’Alene, which contains approximately 0.67 mi3 of water. Determine the percentage of Lake Coeur d’Alene’s volume that flows through Spokane in April and June.

The owners of an oil reserve begin extracting oil at

The owners of an oil reserve begin extracting oil at time t = 0. Based on estimates of the reserves, suppose the projected extraction rate is given by Q'(t) = 3t2 (40 – t)2, where 0 ≤ t ≤ 40, Q is measured in millions of barrels, and t is measured in years. 

a. When does the peak extraction rate occur?

b. How much oil is extracted in the first 10, 20, and 30 years?

c. What is the total amount of oil extracted in 40 years?

d. Is one-fourth of the total oil extracted in the first one-fourth of the extraction period? Explain.