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Chapter Projects
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Chapter Projects
(i) What do you think will happen if the population of
Earth exceeds the carrying capacity? Do you think
that agricultural output will continue to increase at
the same rate as population growth? What effect will
urban sprawl have on agricultural output?
2.
Finding the Profit-maximizing Level of Output
The owners of Two for the Road Bicycle Shop are concerned that they are not maximizing their profits. They
wish to determine the profit-maximizing level of sales for
their shop.The following monthly data represent the number of bicycles produced and sold, total cost of production, and total revenue.
1.
World Population Thomas Malthus believed that
“population, when unchecked, increases in a geometrical
progression of such nature as to double itself every twenty-five years.” However, the growth of population is
limited because the resources available to us are limited
in supply. If Malthus’s conjecture were true, then geometric growth of the world’s population would imply that
Pt
= r + 1, where r is the growth rate.
Pt - 1
Number of
Bicycles, x
(a) Using world population data and a graphing utility,
find the logistic growth function of best fit, treating
the year as the independent variable. Let t = 0 represent 1950, t = 1 represent 1951, etc., until you have
entered all the years and the corresponding population up to the current year.
(b) Graph Y1 = f1t2, where f1t2 represents the logistic
growth function of best fit found in part (a).
(c) Determine the instantaneous rate of growth of
population in 1960 using the numerical derivative
function on your graphing utility.
(d) Use the result from part (c) to predict the population in 1961.What was the actual population in 1961?
(e) Determine the instantaneous growth of population in
1970, 1980, and 1990. What is happening to the instantaneous growth rate as time passes? Is Malthus’s
contention of a geometric growth rate accurate?
(f) Using the numerical derivative function on your
graphing utility, graph Y2 = f¿1t2, where f¿1t2 represents the derivative of f1t2 with respect to time. Y2 is
the growth rate of the population at any time t.
(g) Using the MAXIMUM function on your graphing
utility, determine the year in which the growth rate of
the population is largest. What is happening to the
growth rate in the years following the maximum?
Find this point on the graph of Y1 = f1t2.
(h) Evaluate lim f1t2. This limiting value is the carrying
t: q
capacity of Earth. What is the carrying capacity of
Earth?
Total Cost of
Production, C
Total Revenue, R
0
24,000
0
25
27,750
28,000
60
31,500
45,000
102
35,250
53,400
150
39,000
59,160
190
42,750
62,360
223
46,500
61,485
249
50,250
59,875
(a) Calculate the profit (total revenue - total cost) for
0, 25, 60, 102, 150, 190, 223, and 249 bicycles. What is
the profit-maximizing level of output?
(b) The owners want a more specific result. Draw a scatter diagram of x versus C and x versus R on the same
viewing window, treating number of bicycles as the
independent variable. Be sure to use different plotting symbols for cost and revenue.
(c) To estimate a cost function, use a graphing utility to
find the cubic function of best fit, treating total cost
as the dependent variable and number of bicycles as
the independent variable.
(d) To estimate a revenue function, use a graphing utility to find the quadratic function of best fit, treating
total revenue as the dependent variable and number
of bicycles as the independent variable.
(e) Find the profit function P1x2 = R1x2 - C1x2,
where R1x2 is the revenue function found in part (d)
and C1x2 is the cost function found in part (c).
(f) Graph Y1 = R1x2, Y2 = C1x2, and Y3 = P1x2 on the
same viewing window. Using the MAXIMUM feature of a graphing utility, determine the profit-maximizing level of output. What is the maximum profit?
How does this compare with your answer in part (a)?
(g) Using the numerical derivative function on your
graphing utility, graph Y4 = P¿1x2, where P¿1x2 represents the derivative of profit with respect to x.
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A Preview of Calculus: The Limit Derivative, and Integral of a Function
(h) Using ROOT (or ZERO), determine the value of x,
where P¿1x2 = 0. Compare this result to the result of
part (f). What do you conclude?
(j) Using the INTERSECT feature on your graphing
utility, determine the value of x at which marginal
revenue equals marginal cost.
(i) Using the numerical derivative function on your
graphing utility, graph Y5 = R¿1x2, where R¿1x2 represents the derivative of revenue with respect to x.
Economists call R¿1x2 the marginal revenue function.
Using the numerical derivative function on your
graphing utility, graph Y6 = C¿1x2. C¿1x2 is called the
marginal cost function.
(k) Interpret marginal revenue and marginal cost in the
context of the derivative. Use this interpretation to
explain to the owners why the level of output, where
marginal revenue equals marginal cost, is the profitmaximizing level of output.