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	PreCalculus Introduction Page
	This site introduces my PreCalculus publication.
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	Copyright: Trevor Roseborough (c) 2003, All Rights Reserved	top^

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This first release article is copyright material and cannot be reproduced without the permission of the author. License fees for electronic or printed copies may be paid through the web site. This applies to the main text and diagrams. However, the mathematical equations that are all shown in italic print may be used for any purpose, including educational and commercial, by permission of the author, Trevor Roseborough.

 
 
 
  PreCalculus(c)2003TRoseborough
PreCalculus

Introduction

This publication contains my views and has three main purposes: 1. Simplify the introduction and learning of the calculus. 2. Introduce the M(x) and R(x) functions. 3. And provide a geometric explanation of the relationship between the integral and derivative.

Proportion and Change

Mathematics is the study of proportion and how they change.

Anything we can measure is a dimension. The measurements we take are simply relative differences in proportion. Proportion is the quality that forms a continuum from zero to infinity with one being the base to compare the proportion. The numbers are not magic; they are only letters in an alphabet that we use to spell exact or approximate proportions. For example, three is a simple and exact proportion, as are pi and four. Each number we write is exact and precise but our use of it may be approximate.

When we count our money the number is usually exact. When a carpenter cuts a board to length it is accurate enough for building with but always somewhat approximate due to practical limits. If a board is cut too short by one tenth of the desired length it won’t work. If it is too short by one percent it may or may not work well. If it is off by one-thousandth the length or less it will almost certainly work because it is within acceptable tolerance of the job. The numbers we write are perfect proportions, however our use of them may be just approximations. A proportion does not always have an exact representation in decimal or fractional numbers, but it is no more or less important because of this.

How proportions (or numbers) change is the kind of change we study in mathematics.

Numbers and Curves

On a standard Cartesian coordinate system we plot an equation and draw a curve. The curve shows us how one number y changes in relation to a given number x. A curve that moves from left to right without ever becoming truly vertical (and therefore never reverses) is a simple conversion curve. One unique value (x) converts to another unique value (y). An example of the most basic conversion curve would be linear and convert one scale of distance to another, or one scale of temperature to another. (ex: F_t = 9C_t/5 + 32)

Notation

We show mathematical relationships as formulas or equations such as;

Fig. 1

The y-axis is renamed with our example F(x) but could also be any symbol such as G(x), or k(x), or whatever you desire.

The slope

The slope of a function (curve) at any point along the x-axis tells us how the function is changing at that point. It is the rate of change. Finding this information on change is what the calculus is all about. The slope is literally the rise over the run of a right triangle and can be calculated with any two points along a strait line by:

  rise / run = slope      or, DF(x) / D x = m_x     or,    (F(b) – F(a)) / (b-a) = m

However, with a curve, a problem occurs with D x and the concept of change at a single point. In geometry a point has no area just as a line has no thickness. At any single point on a curve there is no change so there is no slope. To find the slope we need to use two points. This causes us to make an approximation of the slope because our two points are the ends of a strait line that is not necessarily the same as the true slope. Fortunately, the smaller we make the difference (d ) between those two points, the closer the curve of the function comes to a strait line. This is like looking at the curve under a microscope. The higher the magnification or the smaller the field of view, the harder it is to tell the line is not strait. Eventually we cannot find any difference from one power to the next. We can pick any small value of d such that the slope is derived more accurately than our application requires. As a simple test of this method, we can substitute a value that is only one-tenth of our first d, and then, if the first was small enough, we should get the same slope to the same accuracy because the slope is always the same on a strait line. At this level the curve is essentially strait and equal to the slope.

m_x = (F(x+d) – F(x)) / d      ; where d is very small

Fig. 2

We see by the example in figure 2 if the run d is equal to1 then the slope would be off by quite a bit, but if d were a small enough fraction of 1 we could not find any difference from the true slope. There is no real limit as to how small d can be.

This method gives us a way to find the slope at any point of a given function, but what we really want is to find another equation or function that is derived from the first, which will give us the true slope at any point x. This derivative of the first or given function F(x) would itself be a separate function, such as f(x). The capital and lower case letters are used here respectively to show the given function and its related derivative.

Differentiation

Early on in the history of the calculus certain patterns were noticed for some simple functions that could be made into rules to produce the desired derivative, such as;

Another problem of the time was to calculate the area bounded by curves rather than strait lines. Early efforts would find the area under the curve of a function by first dividing the x dimension into small units, like the d that we used to find the slope with, and then multiplying d by the value of the function at each increment of d. The sum of all those small rectangles would closely approximate the true area under the curve. This process of disintegrating and then re-integrating to find the area was accurate when d was small enough but very time consuming.

The big discovery in calculus was that the same rules used to differentiate and find a derivative function could be reversed to find a function that will give us the sum of the area under a given curve. This reverse process, or anti-differentiation as it is known, is how we solve problems of areas under a curve, which are now called integrals. Newton was the first to make the statement of this relationship, which forms the fundamental theorem of the calculus.

Leibniz was the first to publish a complete description of the calculus and he introduced the dx notation that helped in dealing with the problems in a mathematical way. At the time he would have used ‘y’ instead of F(x). For the derivative he made a blend of ‘D ’ and ‘d’ where the equation is written in similar form to that of the slope;

    ;  where f(x) is the derivative of F(x)

And for the integral he added the ‘s’-like symbol (as in Sum) such as;
 

Where, as above, f(x) is the derivative of F(x) and C is an unknown constant that has been traditionally added. (See appendix.)

The expression dx is known as the differential and has since been the standard for working with derivatives (differentiating) and integrals (anti-differentiating). All this was done with the understanding that it was true and it worked but without proof or a geometric explanation for why.

Why

I have found that the explanation lies in the relationship of two similar triangles drawn on the graph of a simple, suitable function. Simple meaning it is not the combination of two or more curves by addition, subtraction, multiplication, or division (see appendix). A suitable function is continuous without any breaks and does not reverse on itself, and therefore has only one value F(x) for each value x over the range we are interested in.

To start we draw such a function on our graph. From any point x on the axis we can move vertically to the point F(x) on the graph to create a line that we also call ‘F(x)’. We then find the slope at that point and draw it as a line so that it cuts both the x-axis at a new point M(x), and runs one unit further along x from the line F(x). We next draw a horizontal line from the point F(x) over one unit to form the line called ‘1’ and then draw a vertical line to meet the slope line and call it ‘f(x)’. For every value of x, this forms two similar triangles accept in the case where the slope is zero or infinite.

Fig. 3

M(x) is a new undefined function that always forms the slope with F(x). The triangle with sides F(x) and M(x) is similar to the triangle with sides f(x) and 1.

We see the unusual case where, starting from the right angles, the lengths of the three functions F(x), M(x), and f(x) can all vary but are always related to each other as similar triangles. So we have a simple relationship of equality between the ratios:

          ; By similar triangles

This is not much different from the formula for the derivative, dF(x)/dx = f(x). Here we could say d/dx is equal to 1/M(x). We still have to use rules to find the derivative f(x) and hence M(x), but now we have the connection between the functions. And conversely, the basic formula for the integral is found by simply multiplying both sides by M(x) to get:

F(x) = f(x) M(x)       ; The M(x) equality

This also tells us that for any function F(x), we can find both functions f(x) and M(x), or at least that they exist.

Whenever we test to find the area under any f(x) curve (the integral), we always find it is equal to the product of f(x) times M(x).  (See appendix.)

To see why we can solve the integral without the procedure of adding together many small values we need to break down M(x) further. We factor out a single x from the function M(x) and come up with a new function R(x). The R(x) function essentially gives us the inverse rise-run ratio for the slope of the given function F(x) at x.

So now the equation for the derivative is, 

And for the indefinite or general integral, 

Plotted geometrically the integral could look like one of the following:

Fig.4

R(x) is one dimension of a three dimensional solid called F(x). In each example, two different values of x form two different solids that can be subtracted from each other to find the difference. This is what happens when you integrate. [Not to scale.]

The definite integral, like the examples in figure 4 above, is where we want to solve the problem and get an answer given two real x values, a and b. It looks like this:
 

So rather than integration being a sum of a great many small areas, f(x)dx, it is instead the product of the function f(x) and its complementary function R(x) and the given value x. F(x) is the anti-derivative or anti-differential of f(x). Although the function R(x) remains the same between F(x) and f(x) it is capitalized to show its connection to F(x). For example, the simple rules applied to F(x) to find the derivative and hence R(x), have to be modified a bit if we start with f(x) to find R(x) and the integral; such as in the power rule found above:
 

Given    F(x) = x ⁿ    ;    R(x) = 1/n      but, given     f(x) = x ⁿ    ;    R(x) =1/(n+1)

For any continuous, non-reversing curve we can find the curve M(x) [and also R(x)] by graphically plotting F(x) and finding the slope at each point, then dividing F(x) by the slope to plot a separate graph of M(x). In this way we could see the curve for M(x) before we identified its mathematical description or formula. Fortunately, most functions are simple or are compounds of simpler functions for which the derivatives and their corresponding rules for differentiating have already been found. We first apply the rules to simplify a function, and then we use the rules to find the derivative. After that, we can extract M(x) or R(x) directly from the function and its derivative.
 

       or, 

In essence, calculus is the method of using rules for simplifying and differentiating a function and then applying them to your advantage to solve problems involving changing quantities. In the appendix below are some of the differentiation rules modified for the R(x) function.

Copyright 2003 (c) Trevor Roseborough

 
 
 
 
  PreCalculus (c) 2003 T Roseborough.Appendix:

I.

Table showing a few simple functions with their derivatives, the M(x) function, the basic R(x) function, and an equivalent R(x) function that incorporates the given function.

II.

Fig. A1

Use figure A1 to see how the elements or expressions of an equation are functions themselves and can be combined both graphically and mathematically. A few basic rules are shown below using the R(x) function and can be compared to the diagram.

Constant rule

A constant value on a graph is a strait horizontal line and can be thought of as a function of x to the power of zero.

Sum rule

Simple functions like those given in appendix I and multiplied by a constant only, can be differentiated by division with the M(x) or xR(x) functions. To find the derivative when we have two functions combined by addition we simply add their derivatives.

Difference rule

The reverse process is also true.

III.

Notation

The traditional form of the indefinite integral is:

The constant C was added to show that f(x) was the derivative of an initial function F(x) that might have lost a constant value in translation. The C would always cancel out and have no effect on the definite integral. This notation suggests that the derivative f(x) does not really exist without, or that it is a dependent of, the anti-derivative F(x). My view is that the C is not necessary as it is not part of the given problem and the two functions are equally interdependent. From a different perspective, a constant is really a separate function of x (x to the power of zero) with a slope of zero. It makes no sense to assume the integral is the anti-derivative of two functions combined by addition (why not three?).

Using capital and lower case letters to signify given functions and their derivatives is limited when we want to find the derivative of a derivative and so on. The problem is worse when we want to take multiple anti-derivatives. Newton used a prime notation ( where: y = 3x ; y’ = 3 ; y’’ = 0 ) which works well for multiple derivatives but not very well for the reverse process. The notation of the lower case d and the upper case D in front of the given function allows for both procedures to be done multiple times. For the differential dF(x) , this is essentially the same as using the Leibniz method without the dx expression. Unfortunately the capital D is sometimes used without the dx to mean the derivative rather than the anti-derivative, which I would choose.

dF(x) = f(x) , Df(x) = F(x) ,   ddF(x) = d²F(x) ,   DDDF(x) =D³F(x)   and so on.

IV

Area

It is easy to calculate the rectangular area under a function that is constant [f(x)=n], or the triangular area under one that is some constant factor of x [f(x)=nx].  We can also find the exact Riemann sum for these functions.  In both cases we find these areas, from 0 to x, are equal to f(x)M(x), and therefore equal to F(x). (The M(x) equality.)

For these as well as for curved functions we can test a single small Riemann interval  f(x+h)h  and find it is very close to f(x+h)M(x+h) - f(x)M(x) [which is equal to F(x+h) - F(x)]  As we decrease the size of the interval, without limit, we decrease the difference between them.  Likewise, if we sum n small intervals to find the area from x to x+nh, then as h gets smaller, the area approaches equality to F(x+nh) - F(x). And so we conclude the area under a curve f(x) from a to b is equal to F(b) - F(a).

Given:  f(x) = x³ ,  h = .001,  x = 5 , and: M(x) = x/4 ,  F(x) = x⁴/4

The interval: f(x+h)h = .001(5.001)³ =  0.1250750

The difference: F(x+h) - F(x) = (1/4)(5.001)⁴  -  (1/4)(5)⁴  =  156.3750 - 156.250 = 0.1250375

And so for h = 0.001:  (.999700)*0.1250750 = 0.1250375
 

V
FTC

Why is the area under the graph equal to its anti-derivative?

Total Area Covered function  [x²/x] [area covered with respect to x]
If we have a function that describes how much area is covered by a process at a given point x, then we have a total area covered function.  For example, imagine the Zamboni ice-cleaning machine.  We can say it cleans six square feet of ice per foot of travel or Z(x) = 6x²/x = 6x.  So when the machine has traveled a total of ten feet we know it has cleaned or covered a total of 6(10) = 60 square feet.  The function tells us the total area covered by the process at any point x.

Rate of Area Coverage function  [x²/x²] [total area covered with respect to x]
If we find the derivative of the total area covered function we then have the instantaneous rate of area coverage function.  This function tells us how fast the area below its curve is being filled in with respect to x.  In the Zamboni example above the rate is constant at six, but the derivative of another total area covered function would have the same property without necessarily being constant.  In other words, the area under the curve f(x) covered by a small increment dx at some point x along the x-axis approaches the product of this total area covered function [f(x)] times dx.  It is interesting to note that with these functions the distance or displacement dimensions cancel the area dimensions, [x²/x² = 1].

 Now, because the dimensions do not change, if we consider a given function f(x) to be a rate of area coverage function, then its anti-derivative F(x) will be the total area covered function.  At any point a along the x-axis, F(a) is equal to the total area covered at that point.  The difference between two points [F(b)- F(a)] is equal to the area under the curve of f(x) from a to b.  This is the fundamental theorem of the calculus.
 
 

Copyright 2003 (c) Trevor Roseborough
Revision #2-14Dec2003