Rates of Change

The next concept we need to put in our toolbox is the notion of "rate of change". The example that most of us were first exposed to was speed being the rate of change of distance. My first science course, and probably yours as well, contained the formula,

speed=distance/time .

This is the prototype of all rate of change formulae. In general a rate of change may be the change in anything divided by the corresponding change in a related variable. The "slope" of a graph of y vs x for example is the change in y divided by the corresponding change in x. It is called the rate of change in y with respect to x.

Since x is the independent variable, we will pick two points on the x axis to be the interval over which we will calculate the rate of change. The difference between these x values is called "delta x". The upper case Greek letter delta (D) is used as a symbol of this difference. Customarily we subtract the lower x value from the higher to get Dx.

 linear slope

For each of the chosen x values there will be a corresponding value of y. We get Dy by subtracting the y corresponding with the lower x value from that corresponding to the upper x value. The ratio Dy over Dx is the rate of change, and for a straight line, the slope of that line. Run the Linear Rate of Change display to experiment with this. I have chosen a Dx of 0.5 units for this display You may select the lower value of x using the cursor. The y cursor value will be ignored. As you will see, the slope of a linear function is a constant.

nonlinear slope

Now suppose that y is a nonlinear function of x. Would that alter our idea of rate of change? In the nonlinear case compare the rate of change at various locations along the x axis. We will use the Gaussian function as a model of a non-linear function. Notice that the rate of change we get is some sort of average rate of change over the interval. Run the Non-Linear Rate of Change display to try out this idea.

We will find it useful to be able to associate a rate of change with a particular value of the independent variable. In the linear case this is easy since the rate of change is constant everywhere. In the nonlinear case we need to do some more thinking about the situation. Not only does the rate of change depend on where along the x axis we look, it also depends on how we select the interval over which to measure it. Suppose we looked at shorter and shorter intervals such that delta x was smaller than any small number you could choose, approaching zero itself. What would happen to the rate of change Dy / Dx. Run the Infinitesimal Delta X display to see.

What we saw happen in the last display was the rate of change approaching a limit as Dx got smaller and smaller. That limiting value is called the "derivative" of y with respect to x at the value of x picked for the left end of the Dx interval. If we draw a line which just touches the curve at a point, the slope of that line is the derivative of y with respect to x at that point. Such a line is called a "tangent" to the curve. The derivative of y with respect to x is written dy/dx, symbolizing that it is the limiting value of Dy over Dx as Dx approaches zero.

There are some assumptions about the nature of the function we are working with which make possible this idea of letting Dx approach zero and finding the limiting value of Dy over Dx. For this to work the function must be continuous, meaning that there is a y for every real number x in the domain of the function. And it must be smooth, meaning that as I look at smaller and smaller Dx, the absolute value of the corresponding Dy also decreases continuously. There are some badly behaved functions which do not meet these conditions but we will steer clear of them.

Now that we have defined the derivative of a function at a point, we could calculate it for every point on a curve. Or at least for enough points on a curve to get some idea what dy/dx would look like as a function of x. Let's work with the quadratic function. Run the Quadratic Derivative display to do this.

The question comes up, if
y=10*x-x^2 ,

what is the function dy/dx. We have seen the graph of dy/dx in the last display. It is a straight line cutting the y axis at 10 and the x axis at 5. We know the equation of such a line from basic algebra. It is
y=10-2*x .
If you experimented with a great many functions of form
you would discover that the derivative is always
n*a*x^(n-1) .
Verify that this formula gives the correct derivative of
y=10*x-x^2 ,

noting that the derivative of a sum is the sum of the derivatives and remembering that

x^0=1 .

Here is how that verification we talked about above goes. The function


has two terms in it, the 10*x and the -x^2. Applying the n*a*x^(n-1) rule, 10*x^1 yields 1*10*x^(1-1) or 10*x^0 or 10. For the second term -x^2 yields 2*(-1)*x^(2-1) or 2*x. Since the derivative of a sum (10*x)+(-x^2) is the sum of the derivatives,

d(10*x-x^2)/dx = 10-2*x
as required.

Just to round out this discussion of the rate of change of one variable with respect to another let's consider the derivative of two more functions. First the sinusoidal function. Run the Sinusoidal Derivative display to get an idea of the shape of the derivative curve.

Notice that the sine function and its derivative have the same cyclic shape only the derivative lags the sine by a quarter cycle. You may, depending on your recollection of trigonometry, recognize the derivative of the sine function as the cosine function. In any event that is what it is.

Next take a look at the Gaussian Derivative display.

The derivative of many different functions have been studied and the form of the derivative tabulated. Also there are rules for how the derivatives of composite functions are to be calculated from the derivatives of basic functions. An example of this was my statement that the derivative of a sum is the sum of the derivatives. Learning how to calculate derivatives, and why, and when, is the point of a course in differential calculus which this is not. I bring it up here because the derivative is the rate of change of a function and we will want to take advantage of that fact somewhere in our discussion.

What we got out of our discussion so far was that the rate of change of a function with respect to a variable, is itself a function of that variable. Therefore we may conclude that we might find a rate of change of a rate of change. And so we can. The series of illustrations below show the curves for first and second derivatives of the quadratic, sinusoidal and Gaussian functions.

quadratic second derivative

sinusoidal second derivative

Gaussian second derivative

You may run the displays from which these illustrations were taken by clicking on the links here. In these displays we present the derivatives, already plotted to save you the time of generating the curve by means of the mouse button.

Quadratic Second Derivative

Sinusoidal Second Derivative

Gaussian Second Derivative

Remember I warned you that we would have to deal with some fundamentals whose application was not immediately obvious. Well here we are introducing exotic stuff like second derivatives of purely mathematical functions, not the sort of thing you probably want to think about. So take a break and think about something else for a while. These ideas will come back again when we begin poking around in the laws of nature.

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