940 14 A Look Back In this text we have studied a variety of functions: polynomial functions (including linear and quadratic functions), rational functions, exponential and logarithmic functions, trigonometric functions, and the inverse trigonometric functions. For each of these, we found their domain and range, intercepts, symmetry, if any, and asymptotes, if any, and we graphed them. We also discussed whether these functions were even, odd, or neither and determined on what intervals they were increasing and decreasing. We also discussed the idea of their average rate of change. A Look Ahead In calculus, other properties are discussed, such as finding limits of functions, determining where functions are continuous, finding the derivative of functions, and finding the integral of functions. In this chapter, we give an introduction to these properties. After you have completed this chapter, you will be well prepared for a first course in calculus. Thomas Malthus on Population Growth In the late 1700s, the British economist Thomas Malthus presented a report that criticized those who thought that life was going to continue to improve for humans. Malthus put his report together quickly and titled it An Essay on the Principle of Population as it Affects the Future Improvement of Society, with Remarks on the Speculations of Mr. Godwin, M. Condorcet, and Other Writers . Malthus argued that because the human population tends to increase geometrically (1, 2, 4, 16, and so on) and that food supplies will likely only increase arithmetically (1, 2, 3, 4, and so on), populations will naturally be held in check due to food shortages. Malthus also suggested that there are other checks on population growth (and he considered these natural and a good thing). Nonetheless, he was concerned that poverty is inevitable and will continue. Malthus used historical data to suggest that population growth has been doubling every twenty-five years in the United States (still in the early stages of development back in the late 18th Century). Malthus surmised that the youth of the country along with the vast amount of areas conducive to farming would lead to a birth rate that exceeded most countries in the world. Malthus believed there are two “checks” that control the population growth. One type are called preventative checks—these are checks that decrease the birth rate. The second type are called positive checks—these are checks that increase the death rate. Positive checks include war, famine, and natural disasters. Malthus believed that fear of famine was a major reason the birth rate may decrease. After all, who would want to have a child knowing the child may suffer from hunger, or worse, starvation? —See the Internet-based Chapter Project— Outline 14. 1 Investigating Limits Using Tables and Graphs 14. 2 Algebraic Techniques for Finding Limits 14. 3 One-sided Limits; Continuity 14. 4 The Tangent Problem; The Derivative 14. 5 The Area Problem; The Integral Chapter Review Chapter Test Chapter Project A Preview of Calculus: The Limit, Derivative, and Integral of a Function Credit: mykhailo pavlenko/Shutterstock
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