So this rule will equal -6 when x is either 2 or 3. This particular situation
has many applications, but to list these applications is not the purpose
of this essay.
High school students spend much of their time learning to manipulate variables and expressions in order to find the roots of the function. But unfortunately, the functions that they will encounter in their occupations are very rarely nice. In fact, it may be the case that most of the functions they deal with will not be factorable or easily manipulated. Take for example the function f(x) = .5e^x - 5x + 2. How can a highschool student go about finding the roots of this equation. Most likely, in these cases an estimate of the root with accuracy to a certain number of decimal places will be sufficient. This esitmation could be very difficult if not impossible to do by hand, depending on how the student is at calculus and calculating numbers. A mcuh better way to handle these situations is to use technology. This essay will demonstrate four different ways to find the roots of equations. Only one will not be applicable to high schools students who ave not had calculus. Throughout this demonstration I will be finding the roots of
Hence from this graph we can see that this is indeed a function, and
that it has two roots in [0, 5]. We can estimate the roots to be about 1
and 3, but this is not very precise. To improve the precision, we can use
the technology to zoom in on one root at a time by changing the axes. If
I consider the larger root first, I can "zoom in" and obtain the
From this picture we can see that the root is just larger than 3.4. To
obtain more precision, I can zoom in again, and I can continue this process
until I reach my desired precision or the precision on my machine becomes
inadequate. The next several graphs show estimates of the larger root up
to the precision of the machine.
So, by this last graph, we see that the larger roots is approximately
3.401795803857806749. This gives us accuracy to 17decimal places. Again
most students should be able to do this with no problem. The same thing
can be done to approximate the smaller root.
So, after 24 iterations we see that the smaller root can be found in
the [.578300476, .578300595]. So we can estimate one root to be about .5783005.
One of the first things to be noticed about this process is that it takes
a lot of iterations to get a high degree of precision. It might be asked,
is there a way to get the roots in fewer iterations. Well, as a matter of
fact there is.
Now we can take the zero of this line as an approximation to the zero
of f. This zero can be represented by (c' represents the zero)
We can now take the point c' as our starting point and repaeat the process.
In this way we get a sequence of values that come closer and closer to the
root of f. Maybe the following picture will help.
Since this process can be done with iterations, we can attempt this process
using a spreadsheet. I have found approaximations of the two roots of f(x)
= .5e^x - 5x + 2 using several different starting values. In the first data
set, I started with x = 0 and then x = 1. It can be seen that in both trials
I found the root to be approximately x = .578300577, which agrees with our
In the second data set, I started with x = 3, x = 3.5 and x = 4. In all
three trials, I found that the root is at approximately x = 3.4017958.
To set up the spreadsheet to do this algorithm, let x be the starting
value and let the spreadsheet compute f(x) and f'(x). Then simply let your
next x-value be x - f(x)/f'(x). Simply drag down in all three columns and
let the computer do the work.
Unfortunately, there are some problems with Newton's method aside from the fact that the student must know what he derivative is. First, the x-values may run away in a sense (a). This might occur when the x-axis is an asymptote. Hopefully, the picture below will clarify this. Second, we might choose an x-value that when evaluated in the derivative give us 0 (b). The problem here is that we want the tangent line to intersect the x-axis so that we can approximate the root. If x has a horizontal tangent line, then we can't do this. Third, we might choose an x, that is the beginning of a cycle(c). Again it is hoped that the picture will clarify this.
Now, if your students don't know calculus there is a method similar to
Newton's method, only it does not use the derivative of a function.
So, for this method we will need two intial guesses and our iteration
revolves around the use of two numbers. The nice thing about this, is that
we do not need to know the derivative of the function, since we can set
up our iteration in the following manner.
Again, using a spreadsheet makes our computations easier and quicker.
Again, it can be seen that we achieved "good" approximations
of the two roots. (Look in the last entry in the b column for the approximation),
and it took considerably fewer entries than the bisection method.
Cheney, W. & Kincaid, D. (1994). Numerical Mathematics and Computing.
Brooks/Cole: Pacific Grove, CA.