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Chapter 6 | Applications of Integration
6.8 | Exponential Growth and Decay Learning Objectives 6.8.1 Use the exponential growth model in applications, including population growth and compound interest. 6.8.2 Explain the concept of doubling time. 6.8.3 Use the exponential decay model in applications, including radioactive decay and Newton’s law of cooling. 6.8.4 Explain the concept of half-life. One of the most prevalent applications of exponential functions involves growth and decay models. Exponential growth and decay show up in a host of natural applications. From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature. In this section, we examine exponential growth and decay in the context of some of these applications. Exponential Growth Model Many systems exhibit exponential growth. These systems follow a model of the form y = y 0 e kt , where y 0 represents the initial state of the system and k is a positive constant, called the growth constant . Notice that in an exponential growth model, we have (6.27) y ′ = ky 0 e kt = ky . That is, the rate of growth is proportional to the current function value. This is a key feature of exponential growth. Equation 6.27 involves derivatives and is called a differential equation. We learn more about differential equations in Introduction to Differential Equations (http://cnx.org/content/m53696/latest/) . Rule: Exponential Growth Model Systems that exhibit exponential growth increase according to the mathematical model y = y 0 e kt , where y 0 represents the initial state of the system and k >0 is a constant, called the growth constant . Population growth is a common example of exponential growth. Consider a population of bacteria, for instance. It seems plausible that the rate of population growth would be proportional to the size of the population. After all, the more bacteria there are to reproduce, the faster the population grows. Figure 6.79 and Table 6.1 represent the growth of a population of bacteria with an initial population of 200 bacteria and a growth constant of 0.02. Notice that after only 2 hours (120 minutes), the population is 10 times its original size!
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