what is the growth rate and continuous growth rate?

The problem: $\;f(t) = 4 \cdot 2^{\,t/5}$

I know continuous is e^k, but this problem doesn't seem to work for that. Is the initial value, 4, able to be used to find the growth rates?

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1 Answer

The relative continuous growth rate of $f(t)$ is defined as $$\frac{f'(t)}{f(t)}.$$ Your function is $f(t)=4 \cdot 2^{t/5}$, with $f'(t)=4\cdot (1/5)\ln(2)2^{t/5}.$ So its relative growth rate is $(1/5)\ln(2)$. Note how the initial value 4 "cancelled out" in finding the relative continuous growth rate.

However you must be sure if instead you just want the continuous growth rate, i.e. the derivative. Some texts when they write "growth rate" implicitly mean relative growth rate, and this is a common usage for exponential functions, to mean relative growth rate and just call it the growth rate.

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the growth rates need to be a percent though. – Tyler Zika Dec 12 '12 at 3:58
The growth rate is as I stated. It may be expressed as a percent, but that is irrelevant, since for example 30% means exactly the same as $30 \cdot \frac{1}{100}$ which is $.3$. – coffeemath Dec 12 '12 at 9:03