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Thread: Help recalling equation involving exponential growth over time

  1. #1
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    Help recalling equation involving exponential growth over time

    I recall having learned an equation involving exponential growth/decay that had something like y=initial amount x e^(2t) or something like that. Do you guys know the exact equation adn perhaps a sample problem to work with? I am prepping for a state math exam.
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  2. #2
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    Re: Help recalling equation involving exponential growth over time

    Exponential growth is of the form $\displaystyle y(t)= y_0e^{\alpha t}$. Yes, $\displaystyle y_0= y(0)$, the initial size. What $\displaystyle \alpha$ is depends upon the rate of growth. In one time unit (hour, day, year, depending on the units for t), y will have grown from $\displaystyle y(0)= y_0$ to $\displaystyle y(1)= y_0e^{\alpha}$ for a rate of $\displaystyle y(1)/y(0)= e^{\alpha}$. In fact, between year "n" and year "n+1", y will have grown from $\displaystyle y(n)= y_0e^{\alpha n}$ to $\displaystyle y(n+1)= y_0e^{\alpha(n+1)}= y_0e^{\alpha n+ \alpha}= y_0e^{\alpha n}e^{\alpha}$ again giving a rate of growth of $\displaystyle y(n+1)/y(n)= e^{\alpha}$.

    Note that, since $\displaystyle e^{\alpha t}= \left(e^{\alpha}\right)^t$, that can also be written as $\displaystyle y(x)= y_0\left(e^\alpha\right)^t= y_0r^t$ where $\displaystyle r= e^\alpha$ is that rate of growth.
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