On-line Math 21

1.1 Tangent and Velocity Problems

We certainly know what we mean by a line tangent to a curve at a point. The line goes through that point, and the slope is the same as that of the curve. But, how do we measure that slope? Let's say, to make the idea seem more precise - and to see how we will compute this measurement - let's say that the curve is the graph of a function y = f(x) , that is, the curve is the set of points (x,f(x)) . We want to find the slope at, say, x = 2 . How do we do it? We could just ``carefully measure'' it, but that would probably have some error, and what do we mean by how we are measuring it? We need a procedure that will lead to the slope of the curve. The way to do it is to find the slope of something simpler, a line, and relate that line's slope to that of the curve. The relationship is that the line we look at is one near the tangent line, which we construct (here is the appeal to the special cases we already know about) by taking the line through (2,f(2)) and a nearby point, (x,f(x)) , for x near 2. This is called a secant line.

As we take x closer to 2, the secant line gets closer to the tangent, and the slope of the secant gets closer to that of the tangent. Again, in the limit as xŽ 2 , the slope of this approximating line (whose slope we know directly) approaches that of the curve at 2. The secant line's slope is an approximation of the slope of the curve, the limit is exactly what we mean by the slope:

slope

: =

lim
xŽ 2

{slope of secant}

lim
xŽ 2

f(x)-f(2)

x-2

Now, what that word limit means is something we have to make more precise later on.

This animation shows you the idea of moving the second point (x,f(x)) along the curve towards (2,f(2)) , (which we do by simply moving the x nearer to 2 ). Notice how the secant line gets closer to being tangent to the curve at (2,f(2)) as x approaches 2 .

Example 1 Let's find the slope of the function f(x): = x²-3x at x = 2 . Taking a number of points nearer and nearer 2 gives us a clearer and clearer idea of the slope at 2 exactly:

x = 3;   f(3)-f(2)
3-2

=

2
x = 2.5;   f(2.5)-f(2)
2.5-2

=

1.5
x = 2.2;   f(2.2)-f(2)
2.2-2

=

1.2
x = 2.1;   f(2.1)-f(2)
2.1-2

=

1.1
x = 2.002;   f(2.002)-f(2)
2.002-2

=

1.002

I think I get the pattern now. But why don't I just plug in x = 2 ? Because the formula doesn't make sense. It's 0/0. To get a secant line you need 2 points, and that would be the same point at both ends. Certainly, however, the slope of the curve has to be 1 at x = 2 .

Velocity:

A similar idea is behind our notion of velocity. We can see immediately how fast we are going, we can find the instantaneous velocity of a car we are driving, just by looking at the speedometer. But what is that really measuring? The average velocity makes sense, it's the distance traveled over a specific amount of time. But the speedometer doesn't measure that, it has a reading at each instant, and it doesn't calculate an average velocity.

But the idea of instantaneous velocity of a moving car, at a specific time, has to be interpreted as another limiting process, taking average velocities over shorter and shorter time intervals:

velocity v: =
lim
DtŽ 0
Ddistance
Dt

where Dt means ``the change in time'', D = change in .

If you plot the distance on the vertical axis, and time on the horizontal, then you have the same picture as before, and the velocity is the slope of that graph.

File translated from T_EX by T_TH, version 2.61.
On 25 Sep 2000, 00:02.