but you may not know why you've learned to do it. Here's why: The difference quotient plays a major role in understanding the first derivative of a function (which is one of the most important tools in modern mathematics). Roughly stated:
We've not yet defined what we mean by "tangent line," so you may not understand this statement exactly, but you should understand that we're finding the slope of a line. With that in mind, the end of the statement may seem problematic: "...at any given point." It appears to say that we're told a single point and, knowing only that point, have to calculate the slope of some line. But if you're given only a single point, how can you calculate the slope of any line at all? We need two points to define a line and to calculate slope!
There is a relatively simple way around this problem, once we understand what is meant by a "tangent" line. Toward that end, we'll begin with an idea that is closely related to tangent lines, but easier to describe: "secant" lines.
In simple language, a secant line is a line that cuts through something. You may remember the word "secant" in the context of circles. It means a line that cuts through the circle. On the other hand, a tangent line to a circle is a line that touches the circle, but doesn't cut through it. That is, it touches in exactly one place. Imagine for a moment that we're discussing the unit circle, and are concerned with the point (0,1). The tangent line at (0,1) is the horizontal line y=1.
We want to take this idea and generalize it to other graphs, but in doing so, we encounter some complications. For example, imagine we're dealing again with the unit circle, but this time only with the upper half. We said a moment ago that a tangent line touches a graph in only one place, but there are lots of different ways to get a line to touch this graph in only one place. All we need do is make the slope of the line steep enough!
For example, any of the lines in the above figure touch the graph only once, but this isn't at all what we mean when we say "tangent" line. These lines still have the "feel" of secant lines because, in a very intuitive way, they cut through the graph. What we mean by a "tangent line" is one which, in an intuitive way, moves with the graph.
So the "touch in one place" definition of tangent line isn't quite right. It needs to be refined, in order to avoid situations like the one we've just seen. Oddly enough, we refine our definition of what it means to be a "tangent" line by using an approximation. You might think that approximations - by their very definition - make things less precise, but it all depends on how they're used. The next paragraph will (hopefully) clear this up.
Return to the idea of the circle for a moment. The point (0.1, 0.99498) also lies on the unit circle. If we connect (0,1) to (0.1, 0.99498) with a secant line, that line would have a slope of -0.0502. That's a small number, which indicates that our line is rather shallow. If we choose a point that's even closer to (0,1) on the circle, the line we make has an even smaller slope. For example,
You can see that as we choose points on the circular arc which are closer and closer to (0,1), the secant lines we produce are more and more flat. That is, we come closer and closer to making the tangent line. This is the whole idea:
If we don't get too far away from our point of interest, we can approximate the tangent line by producing secant lines. As we move closer and closer, our approximation becomes better and better and (to use a technical term) in the limit we find the tangent line.
Limits are something we'll talk about in class. For now, you need only have an intuitive understanding of our goal and the construction we're using to arrive at it. To reiterate:
I've created an example (using the mathematical software Maple) which I hope will clarify the way in which secant lines approximate tangent lines (remember - in the "limit," i.e. in the long run). I'll use the graph of the function which is shown below.
If you have looked at the last of these three animations, you may have
noticed that the secant lines did not coincide (exactly) at the end.
This where the phrase "in the limit" comes to the rescue.
Had I continued the animations, getting ever closer to my point of
interest, the secant lines would coincide - they'd be the same!
Moreover, that single line would be the tangent line.
You might ask, does this new way of getting tangent lines
(as the limit of secants) agree with the more simplistic idea
that tangent lines should touch the graph only once?
The following figure is the graph of the function
f(x)=x3+x2-2x along with its tangent
line at the point (-1,2). The tangent line was obtained by the
approximation method we've been discussing.
That fits well with what we've discussed so far, but if we zoom out,
we see the following:
The line touches the graph twice! Don't let this
confuse you. In the case of the circle, the tangent line only touches
the graph once, but the circle is a very special graph. The "only
touch once" condition is not necessary in the grand scheme of things. If it really
bothers you, zoom in and, in a smaller window, the tangent line will
touch only one spot on the graph.
I began this page by recalling the difference quotient, and telling
you that it was the first step in understanding the derivative of a
function. Now that you understand tangent lines, and how they are
found using an approximation by secant lines, you are in a good
position to approach the derivative. We need only recall that the
graph of a function is the graph of the
equation
On the graph of the function f, the point which is
associated to x=a has a height of y=f(a). That is, a point on the
graph of f looks like (a, f(a)). Suppose that h is some very, very
small number (for this example, we'll choose h>0). When x=a+h,
y=f(a+h) and we find a second point on the graph:
(a+h, f(a+h)). With these two points we can make a secant line.
That's familiar! Recall the
difference quotient:
So the slope of the secant line is told to us by the difference quotient,
evaluated when x=a. Now we put the pieces together:
What we've found is that the derivative of f, evaluated at x=a, can be
approximated by the difference quotient and, as we approximate the
tangent line better and better - as we take smaller and smaller values
of h, we get closer and closer to the slope of the tangent line. In fact,
in the limit (again, that technical term) we get the slope of
the tangent line exactly.
![[Maple Plot]](images/secant_approximation2.gif)
y = f(x)
height on the graph
=
function value
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