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Pictorial Definition of the Derivative
The following two statements are equivalent:
Explanation of the Animation In this animation we'll look at 3 functions:
The Numbers Let Xmin and Xmax be the leftmost and rightmost values of x plotted on any given graph. These two numbers appear on the bottom of the graph. Let Yup and Ydown be the upper and lower values of y on a given graph. These numbers appear on the right. Since the scale of both axes is always the same, we always have Xmax - Xmin = Yup - Ydown. Let Ymax = f(Xmax), and Ymin = f(Xmin) (don't get confused: these are in general NOT maximum and minimum values of y, just the values of y on the graph corresponding to the maximum and minimum values of x). Let Xmid be the x-coordinate of the point you clicked, and let Ymid = f(Xmid). After you click and the graph is magnified and redrawn this point gets plotted on the middle of the screen (at which point Xmid = (Xmax + Xmin)/2, and Ymid = (Yup + Ydown)/2 (got it?). Note: the points (Xmax, Ymax) and (Xmin, Ymin) may or may not be graphed in the black box. For example, if Ymax > Yup, then the graph disappears out the top of the box before it gets plotted. Anyway, the final two numbers appearing in the animation are:
One final point: all six numbers shown are only accurate to two places after the decimal point. The Derivative Because of the finite accuracy of the computer we don't have to magnify things infinitely to determine if a given function is differentiable at a given point. I don't think there's any point on any of the graphs that requires more than 7 clicks (so the magnification would be 47 = 16384) before it either does or doesn't settle down to a straight line. After 7 clicks the values of Xmin and Xmax will be the same (to within 2 places after the decimal), as will be the values of Ydown and Yup. That is, you're left with one x value and one y value, which now correspond to the x- and y-coordinates of the point you zoomed in on, which is also the point in the middle of the black box. If the graph appears to be a straight line, then the left slope and right slope will also equal each other, and in that case that common value is the derivative of the function at the corresponding value of x. The 3 Functions The line function (2nd button) is the easiest. Since its equation is y = 1.6x - 2, the slope between any two points on the graph will always be 1.6, so the left slope and right slope values will both be 1.6 at all magnifications and at all points. The polynomial is also differentiable everywhere, so at every point by the time you've magnified through 7 clicks you should see a straight line plot. However, in this case the value of the magnified final slope (derivative) changes from point to point. Finally the pointy graph has 3 points where the graph looks pointy (has a nonsmooth corner). If you magnify these points 7 clicks or 700 clicks they will still not result in what appears to be a straight line. At best in will appear to be two lines coming together at a corner. These 3 points on this graph correspond to places where this peculiar function is not differentiable. No straight line at high (theoretically infinite) magnification means no derivative. Every other point of this function that appears in its initial plot is differentiable (but there are infinitely many other pointy points on the graph that don't appear in the black box where the function is not differentiable - however, mathematicians would say this function is differentiable "almost everywhere". Extrema Ok, we should know that a positive slope means an increasing function, and a negative slope a decreasing function. Let's suppose the slope (derivative) of a given function is positive immediately to the left of a given point, and negative immediately to the right. That means it must be bigger than the points immediately to the left and right, ie., at least in its local neighborhood it's a maximum value on the curve. We call this a local maximum. If the function is differentiable at the point, then that derivative must be zero. But it may not be differentiable. In general, however, any local maximum (or minimum - defined similarly) must be a point where the derivative is either zero or not defined. Well, you can read words like this in books, but you can't interact with a graph in a book. Go to the animation and see if you can find the coordinates of the two local maxima and one local minimum of the polynomial, and two local maxima and two local minima of the pointy curve. You should be able to prove theoretically these experimental results by taking derivatives and finding intersections. |