978 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function 4. Use the result from 3 to predict the population in 1961. What was the actual population in 1961? 5. Approximate the instantaneous growth of population in 1970, 1980, 1990, 2000, and 2010. What is happening to the instantaneous growth rate as time passes? Is Malthus’ contention of a geometric growth rate accurate? 6. Using the numerical derivative function on your graphing utility, graph ( ) = ′ Y f t , 2 where ( ) ′ f t represents the derivative of ( ) f t with respect to time. Y2 is the growth rate of the population at any time t. 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 ( ) = Y f t . 1 7. Evaluate ( ) →∞ f t lim . t This limiting value is the carrying capacity of Earth. What is the carrying capacity of Earth? 8. 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? Chapter Project 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’ conjecture were true, geometric growth of the world’s population would imply that = + − P P r r 1, where is the growth rate t t 1 where Pt is the population in year t. 1. 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, and so on, until you have entered all the years and the corresponding populations up to the current year. 2. Graph ( ) = Y f t , 1 where ( ) f t represents the logistic growth function of best fit found in 1. 3. Approximate the instantaneous rate of growth of population in 1960 using the derivative function on a graphing utility. Internet-based Project 16. Write the integral that represents the shaded area. Do not attempt to evaluate. y x 8 4 4 8 f(x) = –x2 + 5x + 3 17. An object is moving along a straight line according to some position function ( ) = s s t . The distance s (in feet) of the object, from its starting point after t seconds is given in the table. Find the average rate of change of distance from = t 3 to = t 6 seconds. t s 0 0 1 2.5 2 14 3 31 4 49 5 89 6 137 7 173 8 240 Credit: mykhailo pavlenko/ Shutterstock
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