Section 7: Exponential Growth and Decay

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 8.7 Exponential Growth and Decay 847 Version: Fall 2007 8.7 Exponential Growth and Decay Exponential Growth Models Recalling the investigations in Section 8.3, we started by developing a formula for discrete compound interest. This led to another formula for continuous compound interest, P ( t ) = P 0 e rt , (1) where P 0 is the initial amount (principal) and r is the annual interest rate in decimal form. If money in a bank account grows at an annual rate r (via payment of interest), and if the growth is continually added in to the account (i.e., interest is continuously compounded), then the balance in the account at time t years is P ( t ) , as given by formula ( 1 ). But we can use the exact same analysis for quantities other than money. If P ( t ) represents the amount of some quantity at time t years, and if P ( t ) grows at an annual rate r with the growth continually added in, then we can conclude in the same manner that P ( t ) must have the form P ( t ) = P 0 e rt , (2) where P 0 is the initial amount at time t = 0 , namely P (0) . A classic example is uninhibited population growth . If a population P ( t ) of a certain species is placed in a good environment, with plenty of nutrients and space to grow, then it will grow according to formula ( 2 ). For example, the size of a bacterial culture in a petri dish will follow this formula very closely if it is provided with optimal living conditions. Many other species of animals and plants will also exhibit this behavior if placed in an environment in which they have no predators. For example, when the British imported rabbits into Australia in the late 18th century for hunting, the rabbit population exploded because conditions were good for living and reproducing, and there were no natural predators of the rabbits. Exponential Growth If a function P ( t ) grows continually at a rate r > 0 , then P ( t ) has the form P ( t ) = P 0 e rt , (3) where P 0 is the initial amount P (0) . In this case, the quantity P ( t ) is said to exhibit exponential growth , and r is the growth rate . Copyrighted material. See: 1
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848 Chapter 8 Exponential and Logarithmic Functions Version: Fall 2007 Remarks 4. 1. If a physical quantity (such as population) grows according to formula ( 3 ), we say that the quantity is modeled by the exponential growth function P ( t ) . 2. Some may argue that population growth of rabbits, or even bacteria, is not really continuous. After all, rabbits are born one at a time, so the population actually grows in discrete chunks. This is certainly true, but if the population is large, then the growth will appear to be continuous. For example, consider the world population of humans. There are so many people in the world that there are many new births and deaths each second. Thus, the time difference between each 1 unit change in the population is just a tiny fraction of a second, and consequently the discrete growth will act virtually the same as continuous growth. (This is analogous to the almost identical results for continuous compounding and discrete daily compounding that we found in Section 8.3; compounding each second or millisecond would be even
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