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Derivatives (2012)

Page history last edited by Joe Redish 11 years, 6 months ago

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.   Many of the key physics concepts covered in this class such as force, pressure, or motion are connected to one another through differences. To be concrete lets look at two examples:  (1) A difference in pressure between two different positions 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;  (2) when forces are applied to an object that results in a change of velocity of the object with time.  For both examples, 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.  In calculus classes  you focus on the mechanics of taking derivatives rather than on thinking about how to make sense of them. 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.  In other situations, x will stand for a dependent variable -- the position of something particular, which may vary with time and so we might have x = f(t).  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! [This argument is very similar to saying that instead of an y vs x graph we also study x vs t or y vs t graphs!]

 The point about a derivative is that it is the ratio of two changes.  The function f  depends on a variable x.  The derivative helps you answer the following question:  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 Δfx.  If Δx is smaller than any change in the variable x 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 what mathematicians mean when they change from Δx 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.

 

(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 can still smooth the curve in some way and take the derivative of the smoothed curve and get information about the rate of change of the force in the important time interval highlighted in redl

 

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 graph instead of in any derivatives.

 

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 very concept of position becomes fuzzy.

 

So if your math course fussed about limits, it's fine to learn about it for the sake of the math --  limits allow the math of derivatives to be proven and introduced very nicely,   However, 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.

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