Section+3.4+Taylor+Expansion

Since I (Mrs. Tyson) am writing the page today. I'm going to write primarily in the first person singular and sometimes in the first person plural. I'm also going to create this page using the widget just so that I get a little more practice with that.

Involuntary History Lesson
I started out the class day today by reminding students that we've been on a little journey through polynomials this unit. As a piece of that journey we were exposed to other forms of a polynomial in 3.1 FYE #4. We learned that one of the forms was called the Taylor expansion and that we could find a Taylor expansion using a computer algebra system. We've also looked at the fact that we can trap a polynomial between two horizontal lines: when media type="custom" key="11008512"is a polynomial (and hence continuous) and media type="custom" key="11008550", then I can make media type="custom" key="11008556" as close to 5 as I want by keeping media type="custom" key="11008558" near 2, and we saw that it most often looked like a line when we zoomed in closely. This idea of looking at lines was continued yesterday when we looked at the secant line that connected 2 points on the graph of a function. We figured out that we could find the slope of the secant line by doing long division in some cases.

Learning a little more about the Wiki
As a piece of this review, I displayed the wiki entry from yesterday and talked about both the content and the formatting. I also drew attention to the new pages under the general tab that talk about different ways to typeset mathematics in the wiki and the pros and cons thereof.

Beginning the new material:
Today we looked finding the Taylor expansion of a polynomial at media type="custom" key="11008584". Today I declared that we were finding the Taylor expansion because I am a big stinky meany and I said we had to learn how to do it. Tomorrow I claim that I will try to explain better why we want to find it, but that you may still just think that I am a big stinky meany. So if I want to find the Taylor expansion of media type="custom" key="11008632" at media type="custom" key="11008630"or in powers of media type="custom" key="11008634", I have three options
 * 1) Iterated Long Division by media type="custom" key="11008644"
 * 2) Iterated Synthetic Division by media type="custom" key="11008650"
 * 3) Method of Undetermined Coefficients
 * 4) CAS does the work for me.

I tried to demo all of these methods today, but didn't get as far as I would like during either class.

Iterated Long Division
I'm stymied about how to typeset this on the web (or anywhere else for that matter). Suggestions will be gratefully accepted. Meanwhile I'll plan to scan my handwritten notes and post those tomorrow. (That took way longer than tomorrow, but here I go with uploading scanned images and melding it with the text and seeing what happens.



So this long division shows me that media type="custom" key="11129958" and now I iterate that long division by dividing the quotient media type="custom" key="11129964" by media type="custom" key="11129978"

So the next piece of long division looks like:

This long division shows me that media type="custom" key="11130038" When I plug that back into what I learned from the first instance of long division, I learn that

media type="custom" key="11130126" and distributing the media type="custom" key="11130120" gives me that media type="custom" key="11130168" Now I have just one more instance of long division.

This long division shows me that media type="custom" key="11130158" which when I plug back into the last line from before the division gives me that media type="custom" key="11130180" which is the taylor expansion of media type="custom" key="11130188" at media type="custom" key="11130198" which we can confirm by looking at the output from wolfram alpha.. and looking at the graph from Wolfram Alpha we see how the taylor expansion seems to approximate the graph of the polynomial. Iterated Long Division will always work to get the Taylor expansion of a polynomial. It is however LONG and often fraught with sign errors (or brain errors where the teacher becomes convinced that media type="custom" key="11008698". On the plus side, since the work is carefully spread out, you can work your way up to the Taylor expansion one step at a time making sure that you understand why this new expression is equal to the original function.

Iterated Synthetic Division
Still stymied on that typesetting thing...



Iterated synthetic division is much easier to complete without making mistakes, AS LONG AS you know how to do synthetic division. Remember that the point (not the factor) goes in the upper left box. Remember that you bring down the first number, multiply and record above the line in the next column, and then add the column. Remember to **BOX** the remainder of the division or when you iterate you might try to use that number again.

On the other hand it seems less intuitive to convert the results of the iterated synthetic division into the Taylor Expansion. Maybe with practice this will get easier.

Method of Undetermined Coefficients
I might never want to actually do this by hand, but I do know how to typeset it. So if I want a Taylor Expansion for media type="custom" key="11008742"at media type="custom" key="11008746", that means I want to write media type="custom" key="11008772" and solve for media type="custom" key="11008774". I'm going to start by expanding the right hand side to get media type="custom" key="11008816" Now inspection tells me that media type="custom" key="11008832" using the first two of those equations, I can solve for media type="custom" key="11008838" Now that I know media type="custom" key="11008846", I can use the final equation and find that media type="custom" key="11008862" Thus I can finally write that media type="custom" key="11008876" This method of undetermined coefficients takes along time just as iterated long division takes a long time. This method also relies on the ability to expand media type="custom" key="11008888" quickly and accurately. I'm willing to do it for a quadratic and maybe a cubic, but I wouldn't want to do anything higher than that by hand.

A different technique for undetermined coefficients was done on 11/1 and should appear in the notes on media type="custom" key="11130236"

At this point I was out of time, so I guess I'll be talking about this some more tomorrow. I won't have to talk about how to find the Taylor Expansion with the CAS, because we did do that in Section 3.1.