You can use the programming capability of your graphing calculator to quickly and easily perform the iterations in Newton's Method.
First, recall Newton's Method is for finding roots (or zeros) of functions. In order to use Newton's Method, you need to (1) make a first "guess" as to what you think the root is and (2) find the derivative of the function. You then use the following, easilyderived formula (where is your first guess) to arrive at your second guess, called :
We can then continue this process iteratively, using as our next "guess":
This repetitive, calculationheavy process is exactly what a calculator program would be appropriate for. To program this example into your graphing calculator, follow these steps:
Press the key on your calculator. You should get the following screen:
Use the right arrow key to move over to "NEW" to get the following screen:
Press to create a new program. You should now see this screen:
At this point, you must name your program. You may use a combination of up to eight letters and numbers, but the first character must be a letter. Note that the Alpha Lock is on so that you will automatically get letters when you press the calculator's keys (to tun off Alpha Lock, press ).
After naming the program NEWT and pressing , we get the following screen:
We are now ready to enter the steps in our program. The colon ( : ) indicates the beginning of a line of code. The first thing we want to happen when we run or "execute" the program is for the calculator to ask us for our initial guess. We do this by putting an "INPUT" line in the program. To do this, press again and use the right arrow key to move over to I / O, at which point you should see the following screen:
Since Input is what we want, simply press , and you should see the following screen:
We now need to name the variable that we want to input when we run the program. We can use any single letter we like, but X is an obvious, easy choice, so press the key, followed by to get:
We begin by entering "X" in the next line of the program:
To access the "Y Variables," press the key, the right arrow key to move over to the "YVARS" menu:
Press to go to the standard key to get the "Function" screen:
Since we want Y_{1} , just press to select it, taking us back to the program screen:
We next enter the division sign ( / ), and then go back to the "Function" screen to select Y_{2}:
.Before leaving this line of the program, we need to store this value in another variable, call it Z. We do this by pressing and then the letter Z (don't forget to press the Alpha key first). After pressing , you should see the following screen:
We're almost finished! The calculator now knows the result of Newton's Method, so we just have to instruct the program to tell us. Once again, press , and move over to I / O, and press for Disp (short for Display). Your screen should look like this:
Since the result of our calculation is what we want to display, enter Z to get:
That's it! To quit the programming mode, press .
Now, the only thing remaining is to do before we run the program is to tell it what Y_{1} and Y_{2} are. To do this, press and enter the function in Y_{1} and its derivative in Y_{2} in the usual way:
We can continue this process iteratively as long as we want, but after just a few times, we'll discover that the answer (to 10 decimal places) stops changing (see screen below). Newton's Method is that good (and fast!).
To do this process for a different function, all we need to do is change Y_{1} (the function) and Y_{2} (its derivative, and then simply run the program again.
