第1.4节 斜率与速度；超实线

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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 constantmis 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 computingthe value of the slope. We shall do this in the next chapter.However, to provide motivation, we now describe intuitively themethod of finding the slope.

Considerthe parabola

y=x².

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

If(x0,y0)and (x0+Δx,y0+Δy)are two points on the curve, then the “averageslope”ofthe curve between these two points is defined as the ratio of thechange inytothe change inx,

averageslope = ____.

Thisis exactly the same as the slope of the straight line through thepoints (x0,y0)and

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

Figure1.4.1

Letus 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-____

Dividingby Δx,______________

This can be simplified,

Thusthe average slope is

Noticethat this computation can only be carried out when Δx≠0,because at Δx=0the quotient Δy/Δxisundefined.

Reasoningin a nonrigorous way, the actual slope of the curve at the point (x0,y0)can be found thus. Let x bevery 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 slopeof the curve at (x0,y0);

[slopeat(x0,y0)]is close to 2x0+Δx.

Weneglect the term Δxbecauseit is very small, and we are left with

[slopeat (x0,y0)]=2x0.

Forexample, at the point (0,0) the slope is zero, at the point (1,1) theslope is 2, and at the point (-3,9) the slope is -6.(see figure1.4.2.)

Figure1.4.2

Thewhole process can also be visualized in another way. Lettrepresenttime, and suppose a particle is moving along they-axisaccording to the equationy=t².That is, at each timettheparticle is at the pointt²onthey-axis.We then ask : what is meant by thevelocityofthe particle at timet0?Again we have the difficulty that the velocity is different atdifferent times, and the calculus is needed to answer the question ina satisfactory way. Let us consider what happens to the particlebetween a timet0anda later timet0+Δt.The time elapsed is Δt,and the distance moved is Δy=2t0Δt+(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 shallcall the radio Δy/t ofthe distance moved to the time elapsed the“averagevelocity”forthe interval;

Theaverage velocity is not the same as the velocity at timet0whichwe are after. As a matter of fact, fort0>0,the particle is speeding up; the velocity at timet0willbe somewhat less than the average velocity for the interval of timebetween t0andt0+t,and the velocity at time t0+twillbe somewhat greater than the average.

Butfor a very small increment of time Δt,the velocity will change very little, and the average velocity Δy/Δtwillbe close to the velocity at time t0.To get the velocityv0attimet0,we neglect the small term Δtinthe formula

Vave=2t0+Δt,

andwe are left with the value

V0=2t0.

Whenwe plotyagainstt,the velocity is the same as the slope of the curvey=t²,and the average velocity is the same as the average slope.

Thetrouble with the above intuitive argument, whether stated in terms ofslope 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 soundmethod of finding the slope of a curve or the velocity. What isneeded is a sharp distinction between numbers which are small enoughto 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 anew kind of number, which is infinitely small and yet not equal tozero.

Anumberεissaid to be infinitely small, or infinitesimal, if

-a< ε < a

for every positive real numbera.Then the only real number that is infinitesimal is zero. We shall usea new number system called thehyperreal numbers,which contains all the real numbers and also has infinitesimals thatare not zero. Just as the real numbers can be constructed from therational numbers, the hyperreal numbers can be constructed from thereal numbers. This construction is sketched in the Epilogue at theend of the book. In this chapter, we shall simply list the propertiesof the hyperreal numbers needed for the calculus.

Firstwe shall give an intuitive picture of the hyperreal numbers and showhow they can be used to find the slope of a curve. The set of allhyperreal numbers is denoted byR*.Every real number is a member ofR*,butR*has other elements too. The infinitesimal inR*are of three kinds: positive, negative, and the real number 0. Thesymbols Δx,Δy,…andthe Greek lettersε(epsilon)andδ(delta)will be used for infinitesimals. Ifaandbarehyperreal numbers whose differencea -bisinfinitesimal, we say thataisinfinitely close tob.For example, if Δxisinfinitesimal thex0+xisinfinitely close tox0.Ifεispositive infinitesimal, then -εwillbe a negative infinitesimal. 1/εwillbe aninfinite positive number,thatis, it will be greater than any real number. On the other hand, -1/εwillbe an infinite negative number,i.e., a number less than every real number. Hyperreal numbers whichare not infinite numbers are calledfinite numbers.Figure1.4.3 shows a drawing of the hyperreal line. The circles represent“infinitesimalmicroscopes”whichare powerful enough to show an infinitely small portion of thehyperreal line. The setRofreal numbers is scattered among the finite numbers. About each realnumbercisa portion of the hyperreal line composed of the numbers infinitelyclose toc(shownunder an infinitesimal microscope forc=0andc=100).The numbers infinitely close to 0 are the infinitesimals.

InFigure 1.4.3 the finite and infinite parts of the hyperreal line wereseparated from each other by a dotted line. Another way to representthe infinite parts of the hyperreal line is with an“infinitetelescope”asin Figure 1.4.4. The field of view of an infinite telescope has thesame scale as the finite portion of the hyperreal line, while thefield of view of an infinitesimal microscope contains an infinitelysmall portion of the hyperreal line blown up.

Figure1.4.3

Figure1.4.4

Wehaveno way of knowing what a line in physical space is really like. Itmight be like the hyperreal line, the real line, or neither. However,in applications of the calculus it is helpful to imagine a line inphysical space as a hyperreal line. The hyperreal line is, like thereal line, a useful mathematical model for a line in physical space.

Thehyperreal numbers can be algebraically manipulated just like the realnumbers. Let us try to use them to find slopes of curves. We beginwith the parabolay=x².

Considera real point (x0,y0)on the curve y=x².Let Δxbeeither a positive or a negative infinitesimal (but not zero), and letΔybethe corresponding change iny.

Thenthe slope at (x0,y0)isdefined in the following way:

[slopeat (x0,y0)] = [the real number infinitely close to___].

We compute__asbefore :

Thisis hyperreal number, not a real number. Since Δxisinfinitesimal, the hyperreal number

2x0+xisinfinitely close to the real number 2x0.We conclude that

[slopeat (x0,y0)] = 2x0.

Figure1.4.5

Figure1.4.6

Figure1.4.7

Theprocess can be illustrated by the picture in Figure 1.4.5, with theinfinitesimal changes ΔxandΔyshownunder a microscope.

Thesame method can be applied to other curves. The third degree curvey=x3is shown in Figure 1.4.6. Let(x0,y0)be any point on the curve y=x3,and let Δxbea positive or a negative infinitesimal. Let Δybethe corresponding change in y along the curve. In figure 1.4.7, ΔxandΔyareshown under a microscope. We again define the slope at (x0,y0)by

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

We now compute the hyperreal number_____.

and finally_.

Inthe next section we shall develop some rules about infinitesimalwhich sill enable us to show that since Δxisinfinitesimal,

3x0Δx+(Δx)²

isinfinitesimal as well. Therefore the hyperreal number

isinfinitely 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 afterwe have learned more about hyperreal numbers. Form the last exampleit is evident that we need to know how to show that two numbers areinfinitely close to each other. This is our next topic.