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Rates of Change

One thing that is easy to "read off a graph" of y(x) is the slope of the curve at any given point x. Now, if y(x) is quite "curved" at the point of interest, it may seem contradictory to speak of its "slope," a property of a straight line. However, it is easy to see that as long as the curve is smooth it will always look like a straight line under sufficiently high magnification. This is illustrated in Fig. 4.4 for a typical y(x) by a process of successive magnifications.

  

Figure: A series of "zooms" on a segment of the curve y(x) showing how the curved line begins to look more and more like a straight line under higher and higher magnification.

  

Figure: A graph of the function y(x) showing how the average slope $\Delta y/\Delta x$ is obtained on a finite interval of the curve. By taking smaller and smaller intervals, one can eventually obtain the slope at a point, dy/dx.

We can also prescribe an algebraic method for calculating the slope, as illustrated in Fig. 4.5: the definition of the "slope" is the ratio of the increase in y to the increase in x on a vanishingly small interval. That is, when x goes from its initial value x₀ to a slightly larger value $x_0 + \Delta x$, the curve carries y from its initial value y₀ = y(x₀) to a new value $y_0 + \Delta y = y(x_0 + \Delta x)$, and the slope of the curve at x = x₀ is given by $\Delta y/\Delta x$ for a vanishingly small $\Delta x$. When a small change like $\Delta x$ gets really small (i.e. small enough that the curve looks like a straight line on that interval, or "small enough to satisfy whatever criterion you want," then we write it differently, as dx, a "differential" (vanishingly small) change in x. Then the exact definition of the SLOPE of y with respect to x at some particular value of x, written in conventional Mathematical language, is

$${dy \over dx} \equiv \lim_{\Delta x \to 0} {\Delta y \over ... ...im_{\Delta x \to 0} {y(x + \Delta x) - y(x) \over \Delta x}$$ (4.16)

This is best understood by an example: consider the simple function y(x) = x². Then

$$y(x + \Delta x) = (x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2$$

$$\hbox{\rm and} \quad y(x + \Delta x) - y(x) = 2x\Delta x + (\Delta x)^2.$$

Divide this by $\Delta x$ and we have

$${\Delta y \over \Delta x} = 2x + \Delta x.$$

Now let $\Delta x$ shrink to zero, and all that remains is

$${\Delta y \over \Delta x} \goestoas{\Delta x \to 0} {dy \over dx} = 2x.$$

Thus the slope [or derivative, as mathematicians are wont to call it] of y(x) = x² is dy/dx = 2x. That is, the slope increases linearly with x. The slope of the slope - which we call the curvature, for obvious reasons - is then trivially $d(dy/dx)/dx \equiv d^2y/dx^2 = 2$, a constant. Make sure you can work this part out for yourself. We have defined all these algebraic solutions to the geometrical problem of finding the slope of a curve on a graph in completely abstract terms - "x" and "y" indeed! What are x and y? Well, the whole idea is that they can be anything you want! The most common examples in Physics are when x is the elapsed time, usually written t, and y is the distance travelled, usually (alas) written x. Thus in an elementary Physics context the function you are apt to see used most often is x(t), the position of some object as a function of time. This particular function has some very well-known derivatives, namely dx/dt = v, the speed or (as long as the motion is in a straight line!) velocity of the object; and $dv/dt \equiv d^2x/dt^2 = a$, the acceleration of the object. Note that both v and a are themselves (in general) functions of time: v(t) and a(t). This example so beautifully illustrates the "meaning" of the slope and curvature of a curve as first and second derivatives that many introductory Calculus courses and virtually all introductory Physics courses use it as the example to explain these Mathematical conventions. I just had to be different and start with something a little more formal, because I think you will find that the idea of one thing being a function of another thing, and the associated ideas of graphs and slopes and curvatures, are handy notions worth putting to work far from their traditional realm of classical kinematics.