Comparing Rate-of-Change to Closed Form Equations (again)
Recall that a closed form equation is something like
Y = 3x2 – 9
OrY = 4 sin (x-2) + 6
In your math classes, you have learned the canonical forms of lots of different kinds of closed-form equations. You know that a parabola can always be written as something like y = a x2 + bx + c, and a line can always be written as y = mx + b, and something that looks like y = sin (x) will have sinusoidal undulations.
Figuring out what a rate-of-change equation (or system of equations) will look like is a little different. Of course you can always just mechanically iterate the equations. But especially without a computer, this can take a lot of time (!!!), and even with a computer, it won’t allow you to make interesting predictions such as the threshold effect above. So, to analyze a rate-of-change equation (or system of equations), the important steps are:
- Figure out under what conditions the rates are positive or negative
- Determine where change is fast and where it is slow.
- Determine if there are any equilibrium points – this can happen if the rate of change depends on only the size of ITS OWN compartment, and is positive below some size and negative above that size. If there is an equilibrium, then once the compartment reaches that size it will stay there. There are also more complicated equilibria possible that we’re not dealing with here.
- Determine whether a given compartment peaks and then declines. This can happen if the rate of change switches from positive to negative at some threshold that does not depend only on the size of the that compartment.
As you can see, interpreting rate-of-change equations is as much an art as a science, and we’re barely scratching the surface here.
Copyright University of Maryland, 2007
You may link to this site for educational purposes.
Please do not copy without permission
requests/questions/feedback email: mathbench@umd.edu