What is the definition of exponential growth?


Exponential growth/decay is the process in which a quantity increases/decays by an increasing fraction/excess of itself over time.

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The model of this is ##x##(sub n) = ##x(1+r)^n##, where ##r## is the rate of growth or decay (negative if decay, positive if growth), ##x## is the initial value, ##n## is what entry you are on, and ##x##(sub n) is the entry that you are on’s value. As you can see, if ##r < 0##, the rate is becoming increasingly fractional, and if ##r## is positive, the value is increasing.

This is best taught with an example. Lets say that you are being pushed on a swing. Every swing after you stop being pushed you are decelerated by the air resistance. Hypothetically, after a second, you are be 90% as fast, after 2 seconds you are 90 percent of your speed after the 1st push, et cedera. Here, the ratio ##r## is ##-10%##, and hence we can model the speed of you at any time with ##x##(sub n)##=x(.9)^n##, where we can just place n in for how many swings have been completed, and find your speed.

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