[This is a reprint of a 1997 posting]
Higher Viewpoints: Part Two
From Algebra to Calculus:The Emergence of the Power Rule
A Thought Experiment
DRAFT VERSION 2
by David Fleischacker
Copyright © 1997. All rights reserved
September 4, 1997 (Originally written in 1992)
(First presented as supplementary notes in a seminar on INSIGHT held September 13, 1993)
The following is an exercise in creating a dynamic image which leads to the insight underpinning the power rule in calculus. This image is a particular “play” with algebraic equations and geometric graphs and definitions. Furthermore, I have intentionally set up diagrams, or symbols, in particular ways so as to illustrate the importance of images in order to get the insight. This exercise does not explicitly distinguish between the rules of calculus and the rules of algebra, but all this is not a far step once the exercises have been performed.
The general outline of the paper begins with some definitions. An understanding of arithmetic and some other basic definitions in math are presupposed. Once some key definitions are established, then we proceed to the setting up of the dynamic image and the thought experiment which leads to the power rule.
Part I. Some Preliminary Definitions (skip to part II)
(1) The Definition of a Point:
The definition we are using for a “point” is that any “x” and “y” on a coordinate system will define a point. The coordinate system in this case is two dimensional. Here is a general diagram of it;
y-axis
5|
4|
3|………. * (5,3)
2|
1|________________
0 1 2 3 4 5 6 7 x-axis
The y-axis is the vertical line and the x-axis is the horizontal line on the coordinate system. Technically the two lines are perpendicular and intersect at a point which we have label (0,0). Every point will be given the form (x,y) where x is the number on the x-coordinate and y is the number on the y-coordinate. Thus, the point identified by the “*” on the graph is 5 units on the x-coordinate and 3 units on the y-coordinate (5,3).
The slope of a line is found by taking a segment of the line and measuring its rise and dividing by the measure of its run. The rise is the distance on the y-coordinate axis in a given segment on the line itself. The run is the distance on the x-coordinate axis in the same given segment (d) of the line. So, to get the slope of a line, simply select two points on that line [such as (x1,y1) and (x2,y2) in the diagram. Examine how you can figure out the slope from knowing two points on a line.].^{(1)}
One may wonder why such numbers are used. The reason depends upon the problem one is solving. For now, let us say that we are just putting the numbers into a type of pattern, and later the reason will become clear.
The definition (geometrical) of a tangent is a line which passes through a curve on one and only one point on that curve. Thus, to “tilt” the line one way or the other would necessarily result in the contact of a second point on the line with a second point on the curve.
(4) Tangent and the Curve: An important clue
Notice, when the tangent moves to points “higher” on the curve, the slope of the tangent increases. In other words, the ratio of rise/run increases. When the tangent is moved lower on the curve, then the slope decreases.^{(2)} This raises the question about the existence of a relationship between the slope of these tangents to the curve.
(5) Definition of a function:
The next step is to introduce the notion of function. Instead of a curve, one can actually figure out an algebraic function for the curve. Here, we cannot enter into the tricks of how that is performed. But we can go in the reverse direction, namely start with a function and then draw a curve using it.
A function equates variables to one another through the familiar operations of addition, subtraction, multiplication, division, roots, and powers. So, in the equation Y = X^{2},^{(3)} the function uses equality and the operation of “powers” in order to relate two variables, namely “x” and “y.” In this equation, “x” and “y” are fixed, such that if you know “x” or “y” you can calculate the other (I do not wish to discuss imaginary numbers or other problems which arise in this activity, for we are staying with real numbers). So, if x equals 2, then y equals the square of 2, or 4. If x equals 3, then y = 9. One can set this up in a graph (see “PLOTTING A FUNCTION.”)^{(4)}
In the diagram to the right, the function is plotted as a curve. One simply plots a point where “x” and “y” meet on the coordinate system. In addition to the six numbers plugged into the function one could include many more. This curve then approximates to the function, and the more points one calculates and the more dots one marks, the closer the approximation (If one could plot the infinity of points on the curve, one would have a continuous curve which would entirely represent the function, but since the curve is material and imaginative, it only approximates, hence the imagination struggles to keep up with intelligence).
Part II: Image and Insight underpinning the Power Rule
(1) The Slope of a Tangent
Before we move to the actual image that leads from algebra to calculus, we need to discuss how one arrives at the slope of a tangent of a curve. You may ask why, and again you will have to wait and see. It is simply another way of organizing the data or numbers for the purpose of understanding the Power Rule. The following set of diagrams will reveal one way to approach the slope of a tangent.
The exercise is to locate the point on the curve through which the tangent line passes, call it point A. Then choose any other point on that curve (5,25) and draw a line from point A to your chosen point. Since you have two points, you can figure out the slope (m2).
Then select a point closer to point A. Perhaps move to the other side (2,4). Although you cannot tell from the diagram, slope m1 is closer to the tangent slope (m3) than is slope m2. As one gets closer to point A, you will find a convergence upon some slope. From this convergence, you can actually approximate the slope of the tangent (m3).^{(5)}
An example will reveal this convergence. We shall use the function “y = x^{2}.” Let us say that we are interested in the slope of the tangent of this function at point (3,9). So, we need to approach the slope by drawing lines through points on the curve which are increasingly closer to (3,9). As the points approach (3,9) from both sides of the point, the lines drawn from (3,9) to those points will increasingly approach the slope of the tangent at (3,9).^{(6)}
Destination(3,9)^{(8)} | Selected Point(x_{1},y_{1})^{(7)} | Calculation^{(9)}of Slope (m)rise/run = (9-y_{1})/(3-x_{1}) = m |
a. (3,9)b. (3,9)
c. (3,9) d. (3,9) e. (3,9) f. (3,9) g. (3,9) h. (3,9) i. (3,9) |
(1,1)(2,4)
(4,16) (2.5,6.25) (3.5,12.25) (2.75,7.5625) (3.25,10.5625) (2.95,8.7025) (3.05,9.3025) |
(9-1)/(3-1) = 8/2 = 4/1 —thus 4 is the slope(9-4)/(3-2) = 5/1
(9-16)/(3-4) = -7/-1=7/1 (9.00-6.25)/(3.0-2.5) = 2.75/.5 = 5.5/1 (9-12.25)/(3-3.5) = -3.25/-.5 = 6.5/1 (9-7.5625)/(3-2.75) = 1.4375/.25 = 5.75/1 (9-10.5625)/(3-3.25) = -1.5625/-.25 = 6.25/1 (9-8.7025)/(3-2.95) = .2975/.05 = 5.95/1 (9-9.3025)/3-3.05) = -.3025/-.05 = 6.05/1 |
Notice: As we moved from step “a” to step “i” you can see that the point (x_{1},y_{1}) approaches the point (3,9) and the slope (m) approaches 6. So, perhaps the slope of the tangent at 3,9 on the function “y = x^{2} is 6. It at least approaches that number. If one continues to bring the points closer to (3,9), one will find that the number likewise continues to approach 6.^{(10)}
The basic question is “what is the relationship between the slope of a tangent line and the curve itself?” A clue was given earlier, when we noticed a correlation between the location of the point on the curve and the slope of the tangent through that point. Obtaining an insight into this will be gained through a series of hypotheses about this relationship that serve as the playground for our inquiry.
(1) Hypothesis Number 1
In the next pieces of data, let us say that we have performed the above activity for the points (4,16), (5,25), (6,36), (7,49) on the same function and found the various approximations to slopes.
Slope (m) at (x,y)
(x,y)(3,9)
(4,16) (5,25) (6,36) (7,49) |
rise/run (m)6/1
8/1 10/1 12/1 14/1 |
Are there any patterns? Examine the numbers in both columns. There are many relations which could be examined, but to move toward our goal, notice the relationship between the “x” in the left column and the slopes in the right (each is boldfaced below).
(x,y)(3,9)
(4,16) (5,25) (6,36) (7,49) |
rise/run (m)6/1
8/1 10/1 12/1 14/1 |
What is the relationship? The relationship appears to be 2*x or 2x^{(11)} (“*” means multiply, and in 2x, the multiplication symbol is implied).
x * 2 = m
3 * 2 = 6
4 * 2 = 8
5 * 2 = 10
6 * 2 = 12
7 * 2 = 14 ^{(12)}
Let the “2x” be named the “slope function” because it is the equation which relates the “x” to the slope of the tangent which passes through the point on a function at (x,y). Once again, we could ask whether this has significance. To ascertain this significance, return to the original equation of the function. It is “y = x^{2}.” Do you see any pattern?
Both the square^{(13)} in the function, and, on the other hand, the “slope function” have two’s in them. Perhaps the relationship between the slope of the tangent and the function involves the power which in this case is 2. To get the slope of any tangent on the function at any point (x,y), you simply multiply the power of the function by the “x”.
(2) Hypothesis Number 2
Let us turn to another function that is not complicated, such as “y = x^3”. If you perform all the suppositions and operations done on the earlier function, this is what you get
(x,y)(1,1)
(2,8) (3,27) (4,64) (5,125) |
rise/run (m)3/1
12/1 27/1 48/1 75/1 |
The pattern is not exactly the same. The relationship between “x” and the slope of any tangent is not 2x. In looking at the first point, (1,1), maybe it is 3x. But, in trying to multiply the x-coordinated in the second point (2,8) times 3, the number is six, not twelve which was the approximated slope of the tangent at this point. Let us draw up a quick list, placing 3x alongside the (x,y) and the slope (m)
(x,y)(1,1)
(2,8) (3,27) (4,64) (5, 125) |
rise/run (m)3/1
12/1 27/1 48/1 75/1 |
3x3
6 9 12 15 |
Disappointed? The relationship between the function and the slope of its tangent is not simply multiplying the power by “x.” Look at the numbers again for a pattern. Try another function. Perhaps “y = x^{4}” and add 4x alongside so that it will be consistent with the two earlier diagrams. This will keep things simple.
(x,y)(1,1)
(2,16) (3,81) (4,256) (5,625) |
rise/run (m)4/1
32/1 108/1 256/1 500/1 |
4×4
8 12 16 20 |
Set this up in the same manner as the first two sets because keeping a consistency in the setups improves the chances of recognizing patterns. Sit back again, and look at the numbers.
Look at the “y = x^{3}” data again.
(x,y)(1,1)
(2,8) (3,27) (4,64) (5, 125) |
rise/run (m)3/1
12/1 27/1 48/1 75/1 |
3x3
6 9 12 15 |
Notice that if you multiply the “3x” by the “x” again, you get the slope.
3x * x = m
3 * 1 = 3
6 * 2 = 12
9 * 3 = 27
12 * 4 = 48
15 * 5 = 75
Then turn to the “y = x^{4}” data.
(x,y)(1,1)
(2,16) (3,81) (4,256) (5,625) |
rise/run (m)4/1
32/1 108/1 256/1 500/1 |
4×4
8 12 16 20 |
Notice that the pattern does not follow when you multiply 4x times x.
4x * x does not equal m, except when “x” is 1.
4 * 1 = 4 does follow the pattern
8 * 2 = 16 does not equal 32, which is the slope
12 * 3 = 36 does not equal 108
16 * 4 = 64 does not equal 256
20 * 5 = 100 does not equal 500
Look at the numbers again. Notice that if you multiply the outcome of what you just did (4, 16, 36, 64, and 100) with “x”, you get the slope.
4 * 1 = 4
16 * 2 = 32
36 * 3 = 108
64 X 4 = 256
100 X 5 = 500
Now let’s see. To get the slope of the tangent when the function was “y = x^{2}” then the “x” was only multiplied once, by the power. When the function was “y = x^{3}” then the “x” was multiplied twice, once by the power and then by itself. When it was “y = x^{4}” then the “x” was multiplied three times, once by the power and twice by itself. If you carry out the same activities with the function “y = x^{5}“, you will find a similar pattern.This time the “x” was multiplied four times, once by the power and three times by itself.
Notice the pattern? Not only do you have to multiply the x more times when the powers of the function increase, but the times you multiply happen to be exactly one less than the power. You compile the pattern as follows;
if y = x^{2}, then the slope of a tangent on that function is 2 times x or 2x.
if y = x^{3}, then 3 times x times x or 3x^{2}.
if y = x^{4}, then 4 times x times x times x or 4x^{3}.
if y = x^{5}, then 5 times x times x times x times x or 5x^{4}.
What this pattern solves is the slope of a tangent on a function by finding what was called the “slope function.” If you think about it more, a simple rule can be devised from the original function. Let the power = n. Then if the curve is defined by the function y = x^{n}, then to get the slope of the tangent along this function simply multiply “x” by “n” and give the “x” the power of “n-1.”
x^{n} ———–> nx^{(n-1)}
Examine more functions and try out the rule. It should work in every applicable case. Basically, it gives you a new way to figure out the slope of the tangent on a curve at any point you would like to examine. Simply carry out this rule, and then plug in the “x” of the point on the curve which you would like to investigate. It makes this task much easier. Instead of performing the rather involved task in finding the slope which we did earlier, now we just follow this simple rule. Not only that, but the rule is not an approximation like the slope found on page 7 (although it is still a “serial analytic principle”–see ch. 9 of INSIGHT).^{(14)} One thing that should be noted in the applicability of this rule is that it only works for simply functions like x^{2}, x^{3}, x^{4}, x^{5}, etc.. Functions like “x^{2} + 2x + 3″ do not work with this rule. Finding tangents on those more complicated functions will require more work.^{(15)}
What has been named the “slope function” in this example is, for those who have studied calculus, the derivative. The rule developed in which x^{n} ——-> nx^{(n-1)} is the familiar power rule. The process of applying the rule to a particular problem is called derivation.
This rule is only a first step in developing the mathematical viewpoint of calculus, and it, like arithmetic and algebra, has an analogous deductive and homogeous expansion.
Reponse?
1.The rise of a slope is equal to the distance on the y-axis, which, regarding segment “d,” is y2-y1. Likewise, the run of a slope is equal to the distance on the x-axis, which, regarding segment “d,” is x2-x1. Hence the algebraic definition of slope.
2.In practice, you would probably examine many curves and tangents to see if there is a pattern, not just one curve like we are doing. Using terms like “up” and “down” are really only relevant to the curve and tangents we are using. Furthermore, we are only drawing and staying in one quadrant of the coordinate system. The larger coordinate system extends into the negative y- and x-axis. These extension are not need though, for our concerns.
- Notice the “apt” symbolism. If “x” times “x” were used, it would not have the same probabilities of leading to insight into the power rule. Pay attention to the next few sections to verify this claim.
- We are staying with a simple function. This is all we need for a basic grasp of the power rule.
- It is easier to see the convergence of slopes of the lines drawn from point A to the points 1, 2, and 3 upon the slope of the tangent to point A.
- This exercise uses certain rules. One can compare these to the higher rule which eventually emerges from this kind of “play,” namely the power rule.
- Chosen point.
- The is the point at which the tangent contacts the curve.
- This is calculated from the algebraic definition of a slope on page 1.
- There is actually an algebraic equation which can be used to solve this problem definitively, but it is rather complex, and it is not needed for our purposes.
- In standard notion, although a multiplication sign is not used here, in “2x” what this means is 2 times “x.” In other places in this paper, multiplication may be signified by the capital X.
- Since any number divided by “1” is equal to the number, the slopes listed in this chart do not have the form “m/1.” So, that does not mean that we have eliminate the “run.” Instead, you should just assume the “1” is there.
- This means the power of 2.
- Proving this requires utilizing the algebraic equation used to solve the slope of a tangent on a curve. Not only is this equation more difficult to learn than the approximations we performed above, but it has many limits to its use. There is much guesswork which has to be waded through in order to solve problems using this method, whereas with calculus, the rules are very systematic.
- In calculus, the next step is usually the chain rule. In the same way that this present “thought” experiment was set up, so one could be performed with this second rule. It would be more difficult though.
***In step “a.”, draw a line from point (3,9) to (1,1). In calculating the slope of this line (under the third column), carry out the operations within parentheses first. So, in the above equation, first carry out (9-1) and (3-1), which will result in two numbers, 8 (the rise) and 2 (the run). Then divide the first number obtained with the second, resulting in a rise/run ratio of 4/1
^ The “^” means “to the power of.”