Class content >Modeling with mathematics > Math recap > Values, change, and rates of change
Prerequisites
Understanding how things change is a critical part of what science is trying to understand. Moreover, the way things change is often related to what happens in interesting ways. For example, a difference in pressure between two different points in a fluid can be responsible for the motion of that fluid -- both for how the blood flows in your veins, how sap moves in a tree, and how winds are created. For another, the result of an unbalanced force on an object is responsible for change in the object's velocity -- an acceleration. For the first example, we need to understand how pressure changes as a function of position; for the second, how velocity changes as a function of time. For both, the basic concept is the derivative.
While you have already studied derivatives in your calculus class, using them in science might require a bit of a shift in perspective. Professors in calculus classes sometimes focus on the mechanics of taking derivatives rather than on thinking about how to make sense of them. Further, in math a big point is sometimes made of building a completely consistent mathematical structure. When we apply this math as a model of a real physical system, not all of the mathematical details are always relevant -- or even correct.
We'll recap the heart of the idea of derivative and how it applies in science using standard math notation -- a function f that depends on an independent variable we will call x. We will refer to the value that f takes as y. Thus we will write
y = f(x)
| Be careful not to get too attached to this notation! We will have lots of different symbols and functions. In some situations in this class, x will stand for an independent variable -- something specifying a location in space of something. In other situations, x will stand for a dependent variable -- the position of something particular, which may vary with time. In other situations, x will become an independent variable again, specifying which particular object we are referring to. This can be very confusing if you think only about the symbol and not what it is meant to represent! |
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The point about a derivative is that it is the ratio of two changes. The function f depends on a variable x. If x changes, what happens to the value of f?
We define the average change in the value taken on by f when x changes by the expression Δf/Δx. If Δx is smaller than any change in the variable x than we care to worry about, we write the changes as dx and df and define the derivative as the ratio of the small changes:
g(x) = f '(x) = df/dx.
The notation with the prime (or dot if the independent variable is time) is due to Isaac Newton, one of the inventors of the calculus. The ratio notation is due to Gottfried Leibniz who invented calculus independently from Newton at about the same time.* Newton's notation hides the fact that the derivative is a ratio and makes it hard to see the units. We'll stick with Leibniz's notation.
In creating a complete and self-consistent mathematical structure for doing calculus, mathematicians often talk about "taking limits" as the change goes to zero (referring, for example, to dx). They would see our language "smaller than any change we care to worry about" as being sloppy math. Well, it is. But we are not doing math. We are using math as a model of a system in the physical world, and the smooth ("differentiable") functions that the mathematicians like to talk about don't often exist in the physical world. In a math book you might be shown the graph of a curve representing a function. They might then "zoom in" on a little piece of the curve showing it getting straighter and straighter as you get closer and closer in. That's because the curve is assumed to be smooth. But in a real-world example, zooming in might make a curve look smooth and straight for a bit, but getting further in often starts to show something funny: like in the picture below. The closer you look at the curve, the more it looks wiggly -- like grass.
(Source: Almquist and Melosh, Fusion of biomimetic stealth probes into lipid bilayer cores,
PNAS 2010, 107:13, 5815-5820.)
In this paper, the researchers studied the properties of a lipid membrane by attaching a brush of hydrophobic molecules to the hydrophilic tip of an atomic force microscope. They then measured the force it took to push the tip through the membrane and inferred properties of the membrane.
The point is that the curve that they found on the right is not smooth. It has spikes and jiggles, and the closer you look the worse they get. Taking a derivative of this curve at this scale would be not very meaningful. We might smooth the curve in some way and take the derivative of that and get some possibly interesting information.
This is why we say that our derivative is the ratio of two changes when "the change is smaller than anything we care to worry about." In the graph above, we might only be worried about what the average force is as we start to push the probe through the membrane and how much time it takes to get through. For that, we might want to smooth the curve. If we are interested in the actual mechanism of how the individual molecules on the brush interact with the membrane, we might be interested in the details of the individual fluctuations in the grass.
In general, in physical systems we can't take a limit to 0, since as we get too small the physics of the system changes. When we get down to the micron level, we have to worry about thermal fluctuations, and when we get down to molecular sizes we have to worry about the discreteness of matter caused by its being made up of atoms. At even smaller scales, we need quantum mechanics, and the classical variables we are working with become meaningless.
So if your math course fussed about limits, it's fine to learn about it for the sake of the math -- but it's about how the math works; it doesn't have anything to do with the physical system. When thinking about using derivatives to model something in a physical system, it's much better to think about them as ratios of small changes.
This view also clarifies something that confuses many students. Although we talk about a derivative "at a point" -- so we write "g(x) = f '(x)" -- the derivative of f at the point x -- thinking in terms of the ratio of changes shows us that a derivative is really about the values of f at two different points. See the discussion of velocity for an example.
Follow-on
*A very interesting book on the battle for credit that went on between the two is Jason Bardi's book, The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time.