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The function $f(x) = Ae^{kx}$, where $A$ and $k$ are positive constants is said to have "constant, relative" rate of change.

How does it differ from the statement that it has a constant rate of change?

Thanks,

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2 Answers

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It means that the rate of change, relative to the actual value, is constant. It always increases (or decreases) by a fixed percentage in a given time interval.

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The function $y=mx$ has a constant rate of change $y'=\frac{\Delta y}{\Delta x}=m$ (for $\Delta x \to 0$).This means that the rate of change $\frac{\Delta y}{\Delta x}$ is the same for any value of $x$. It is a linear function.

Your function is such that the rate of change at a point divided by the value of the function at this point is constant: $\frac{y'}{y}= k$. The function satisfies the differential equation $y'=ky$. So you can see that the rate of change at a point is proportional to the value of the function at this point. It is an exponential function.

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