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3_8_exp_growth_decay

3_8_exp_growth_decay - 3.8 Exponential Growth and Decay...

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3.8 Exponential Growth and Decay Math 1271, TA: Amy DeCelles 1. Overview This section discusses several natural phenomena (population growth, radioactive decay, Newton’s law of cooling, continuously compounded interest) from a mathematical perspective. In each of these examples, there is a quantity that is changing, and in particular, the rate at which it is changing is proportional to the quantity. Putting this into symbols, if y is a quantity changing with respect to x , then the fact that the rate at which y is changing is proportional to y means: dy dx = k · y for some proportionality constant k There is a theorem that says that any quantity satisfying this relationship is an exponential. In particular: y ( x ) = y 0 · e kx where y 0 = y (0) Growth vs Decay Population growth is an example of exponential growth. The amount of bacteria in a petrie dish will increase at a rate proportional the the amount of bacteria present. So the more and more bacteria there are, the faster the population will grow. More specifically, given a set time interval,
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