Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
With a solid mathematical tool, calculus, in hand, we can set out to try to understand the phenomena of the world mathematically. Let's start with a simple example. Imagine an object in free-fall. At any given time during its fall, it will have some specific velocity, v. Furthermore, we intuitively know that the longer something falls, the faster it goes. This suggests that the velocity of the object should be expressed as a function of elapsed time, t.
To write the specific expression that will tell us the object's velocity at any point in time, let's first assume that the object begins from a state of rest. This gives us an "initial condition," of v(0) = 0, or "the velocity at time zero equals zero." The velocity of the object as it falls will then be due solely to the influence of gravity. If we multiply the time spent falling t by the acceleration due to gravity g, which is the experimentally observed rate at which the velocity of a freely falling object changes, we can determine the speed at which our object is falling at any point in time:
v(t) = gt
Notice here that what interests us is not a specific value for velocity or time, but rather the exact relationship between the two. In this example, we have a non-constant velocity. If we take the derivative of this, we should get an expression that tells us how fast velocity is changing. Doing this, we get:
(It is the derivative of a linear equation, like the first example in the table on page 12. Note that the equation is shorthand for "the derivative of v with respect to t.")
This is a very simple example of what is known as a differential equation. A differential equation is simply an equation that relates quantities with their rates of change. In this example, we see that the amount by which v changes, dv, in some small amount of time, dt, is equal to a constant, g.
To solve this equation, we are looking for a function whose derivative is the constant g. Notice that solving a differential equation does not give us a simple number, as we would expect were we to solve the equation 10 = 4x -2 for the variable x. Rather, our solution to a differential equation is a function v(t). This example is somewhat contrived because we already know that the answer will be v(t) = g. After all, that's what we started with. But if we didn't already know, how could we figure it out?
There are a variety of methods that one can use to solve different types of differential equations. No one method can solve every differential equation, and there are many differential equations that can't be solved at all. In the next example, we'll get a sense of the methods and thinking that go into solving differential equations.
Let's look at another example, one that gives us an equation involving both a quantity and its derivative. Imagine a single bacterium surrounded by nutrients—perhaps it's in a bottle of milk. Bacteria divide asexually by binary fusion, their population basically doubling at set intervals. The more bacteria there are, the more that are "born." This implies a rate of change, or growth, that is not steady, as was the case in the previous example of the velocity of a falling object. Furthermore, the rate of increase in the bacteria population depends on how many there are to begin with. If there are two bacteria initially, the first increase is by two, the second increase is by four, the third increase is by 8, etc.
Let's designate P(t) as the number of bacteria at any given time, t. The rate of change in this population is then , some small change in population over a small change in time. The rate, , depends on how many bacteria there are, P. Therefore:
The a is just a constant that is related to the specifics of the situation—what type of bacteria, how long it takes them to reproduce, etc. In this situation, we have a rate of change that is directly proportional to the quantity that is changing; in other words, we have an equation that relates a certain quantity to its derivative. This is a classic differential equation that describes exponential growth.
We could use a process known as integration to solve this by separating the variables, putting the parts having to do with P on one side of the equation and the parts having to do with t on the other side. Integration and differentiation are two of the most important concepts of calculus. Whereas differentiation seeks to explain rates of change, integration makes sense of the accumulation of an infinite number of tiny changes. Integration is in a very real sense the "opposite" of differentiation, but it can be very complicated for anything but the simplest of equations. A faster way, for our purposes, might be simply to try a few possible solutions and see if they work.
First let's try P(t) = at. According to our table from the previous section, would then be just a. Substituting this value into our differential equation we would get:
P(t) = at
Since this is true only for t = 1, let's try something else.
How about P(t) = sin at? would then be a cos(at) and we would have:
a cos at = a sin at
Again, this is true only sometimes, in much the same way that a stopped clock is right twice a day. We need something that is always true regardless of what value of t we consider. Let's try something else.
How about P(t) = e^{at}? would then be ae^{at}, which is just aP(t)! This gives us ae^{at} = ae^{at}, which is always true, no matter what t is. So the solution to our differential equation is P(t) = e^{at}.
In this example, we see again how the solution to a differential equation is a function, not a number. In our example here, this function describes how to find the population of bacteria at any point in time, even though the rate of increase is changing. It's a nice, simple expression that encompasses the complexity of the situation under examination.
In addition to integration and the "guess and check" method we just used, there are other ways of solving differential equations (sometimes nicknamed "diff EQs"), and they generally fall into two categories: exact and numerical methods. Exact methods yield exact solutions, as did the function in our example above. Numerical methods give approximations based on different algorithms. Often, however, we can discover interesting behavior regarding our situation without having to solve any equation. We can look at its qualitative behavior via what is called a phase portrait, a picture that shows a system's "phase space."
Phase space is handy because it provides a way to represent all the possible states of a system with one picture. It is a graph of the variables, such as position and velocity, that determine the state of a system. We will talk about phase space in more depth in Unit 13. For our purposes here, it suffices to say that examining graphical representations of systems of differential equations can yield a wealth of qualitative information about the system, such as whether or not it will display cyclical or synchronous behavior.
Now that we have an idea how to model certain real-life situations using equations that use both quantities and rates of change, we can tackle the issue of how synchronization arises in nature. We are going to look at one of the most basic and accessible types of synchronization, that of cyclical behavior.
Next: 12.5 Cycles