1.4 SLOPE AND VELOCITY; THE HYPERREAL LINE

In Section 1.3 the slope of the line through the points (x1, y1) and (x2, y2) is shown to be the ratio of the change in y to the change in x,

If the line has the equation

y=mx+b,

Then the constant m is the slope.

What is meant by the slope of a curve? The differential calculus is needed to answer this question, as well as to provide a method of computing the value of the slope. We shall do this in the next chapter. However, to provide motivation, we now describe intuitively the method of finding the slope.

Consider the parabola

y=x².

The slope will measure the direction of a curve just as it measures the direction of a line. The slope of this curve will be different at different points on the x-axis, because the direction of the curve changes.

If (x0, y0) and (x0 + Δx, y0+Δy) are two points on the curve, then the “average slope” of the curve between these two points is defined as the ratio of the change in y to the change in x,

average slope = ____.

This is exactly the same as the slope of the straight line through the points (x0, y0) and

(x0+Δx, y0 +Δy), as shown in Figure 1.4.1.

Figure 1.4.1

Let us compute the average slope. The two points (x0, y0) and (x0+ Δx, y0+Δy) are on the curve, so

____________,

y0+Δy=(x0+Δx)²

Subtracting, Δy=(x0+Δx)² - ____

Dividing by Δx, ______________

This can be simplified,

Thus the average slope is

______________

Notice that this computation can only be carried out when Δx ≠0, because at Δx=0 the quotient Δy/Δx is undefined.

Reasoning in a nonrigorous way, the actual slope of the curve at the point (x0, y0) can be found thus. Let x be very small (but not zero). Then the point (x0+x, y0+y) is close to (x0, y0), so the average slope between these two points is close to the slope of the curve at (x0, y0);

[slope at (x0, y0)] is close to 2x0 + Δx.

We neglect the term Δx because it is very small, and we are left with

[slope at (x0, y0)] =2x0.

For example, at the point (0,0) the slope is zero, at the point (1,1) the slope is 2, and at the point (-3,9) the slope is -6.(see figure 1.4.2.)

Figure 1.4.2

The whole process can also be visualized in another way. Let t represent time, and suppose a particle is moving along the y-axis according to the equation y=t². That is, at each time t the particle is at the point t² on the y-axis. We then ask : what is meant by the velocity of the particle at time t0? Again we have the difficulty that the velocity is different at different times, and the calculus is needed to answer the question in a satisfactory way. Let us consider what happens to the particle between a time t0and a later time t0 + Δt. The time elapsed is Δt, and the distance moved is Δy=2 t0Δt + (Δt)². If the velocity were constant during the entire interval of time, then it would just be the ratio Δy/t. However, the velocity is changing during the time interval. We shall call the radio Δy/t of the distance moved to the time elapsed the “average velocity” for the interval;

The average velocity is not the same as the velocity at time t0 which we are after. As a matter of fact, for t0>0, the particle is speeding up; the velocity at time t0 will be somewhat less than the average velocity for the interval of time between t0 and t0 +t, and the velocity at time t0+t will be somewhat greater than the average.

But for a very small increment of time Δt, the velocity will change very little, and the average velocity Δy/Δt will be close to the velocity at time t0. To get the velocity v0 at time t0, we neglect the small term Δt in the formula

V ave = 2t0 + Δt,

and we are left with the value

V0 = 2 t0.

When we plot y against t, the velocity is the same as the slope of the curve y=t², and the average velocity is the same as the average slope.

The trouble with the above intuitive argument, whether stated in terms of slope or velocity, is that it is not clear when something is to be “neglected.” nevertheless, the basic idea can be made into a useful and mathematically sound method of finding the slope of a curve or the velocity. What is needed is a sharp distinction between numbers which are small enough to be neglected and numbers which aren’t. Actually, no real number except zero is small enough to be neglected. To get around this difficulty, we take the bold step of introducing a new kind of number, which is infinitely small and yet not equal to zero.

A number ε is said to be infinitely small, or infinitesimal, if

-a < ε <a

for every positive real number a. Then the only real number that is infinitesimal is zero. We shall use a new number system called the hyperreal numbers, which contains all the real numbers and also has infinitesimals that are not zero. Just as the real numbers can be constructed from the rational numbers, the hyperreal numbers can be constructed from the real numbers. This construction is sketched in the Epilogue at the end of the book. In this chapter, we shall simply list the properties of the hyperreal numbers needed for the calculus.

First we shall give an intuitive picture of the hyperreal numbers and show how they can be used to find the slope of a curve. The set of all hyperreal numbers is denoted by R*. Every real number is a member of R*, but R* has other elements too. The infinitesimal in R* are of three kinds: positive, negative, and the real number 0. The symbols Δx, Δy,…and the Greek letters ε (epsilon) and δ (delta) will be used for infinitesimals. If a and b are hyperreal numbers whose difference a -b is infinitesimal, we say that a is infinitely close to b. For example, if Δx is infinitesimal the x0+x is infinitely close to x0. If ε is positive infinitesimal, then -ε will be a negative infinitesimal. 1/ε will be an infinite positive number, that is, it will be greater than any real number. On the other hand, -1/ε will be an infinite negative number, i.e., a number less than every real number. Hyperreal numbers which are not infinite numbers are called finite numbers. Figure 1.4.3 shows a drawing of the hyperreal line. The circles represent “infinitesimal microscopes” which are powerful enough to show an infinitely small portion of the hyperreal line. The set R of real numbers is scattered among the finite numbers. About each real number c is a portion of the hyperreal line composed of the numbers infinitely close to c (shown under an infinitesimal microscope for c=0 and c=100). The numbers infinitely close to 0 are the infinitesimals.

In Figure 1.4.3 the finite and infinite parts of the hyperreal line were separated from each other by a dotted line. Another way to represent the infinite parts of the hyperreal line is with an “infinite telescope” as in Figure 1.4.4. The field of view of an infinite telescope has the same scale as the finite portion of the hyperreal line, while the field of view of an infinitesimal microscope contains an infinitely small portion of the hyperreal line blown up.

Figure 1.4.3

Figure 1.4.4

We have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus it is helpful to imagine a line in physical space as a hyperreal line. The hyperreal line is, like the real line, a useful mathematical model for a line in physical space.

The hyperreal numbers can be algebraically manipulated just like the real numbers. Let us try to use them to find slopes of curves. We begin with the parabola y=x².

Consider a real point (x0, y0) on the curve y=x². Let Δx be either a positive or a negative infinitesimal (but not zero), and let Δy be the corresponding change in y.

Then the slope at (x0, y0)is defined in the following way:

[ slope at (x0, y0) ] = [the real number infinitely close to ___ ].

We compute __ as before : ___________________________

This is hyperreal number, not a real number. Since Δx is infinitesimal, the hyperreal number

2x0 + x is infinitely close to the real number 2x0 . We conclude that

[ slope at (x0, y0 ) ] = 2x0.

Figure 1.4.5

Figure 1.4.7

The process can be illustrated by the picture in Figure 1.4.5, with the infinitesimal changes Δx and Δy shown under a microscope.

The same method can be applied to other curves. The third degree curve y=x3 is shown in Figure 1.4.6. Let(x0, y0) be any point on the curve y=x3, and let Δx be a positive or a negative infinitesimal. Let Δy be the corresponding change in y along the curve. In figure 1.4.7, Δx and Δy are shown under a microscope. We again define the slope at (x0, y0) by

[slope at (x0, y0)] = [ the real number infinitely close to_____ ].

We now compute the hyperreal number _____.

and finally _____________________.

In the next section we shall develop some rules about infinitesimal which sill enable us to show that since Δx is infinitesimal,

3x0 Δx + (Δx)²

is infinitesimal as well. Therefore the hyperreal number

____________________

is infinitely close to the real number ______ , whence

___________________

For example, at(0,0) the slope is zero, at (1,1) the slope is 3, and at (2,8) the slope is 12.

We shall return to the study of the slope of a curve in Chapter 2 after we have learned more about hyperreal numbers. Form the last example it is evident that we need to know how to show that two numbers are infinitely close to each other. This is our next topic.