If we want to examine how good an approximation the degree Taylor polynomial is on the interval, we could just look at the graph of the error on that interval: If you want hours of harmless amusement, double-click on one of these graphs and it will animate the whole set.) If you would like to show all of these together on a single graph, you could use the Show command (remember, "fred" is the name we gave to this set of graphs):Īnother graph that might be useful to look at is, where is the degree Taylor series (you take the absolute value with Abs).: (I use the PlotRange command so that all the graphs show the same range on the Y axis, to make them easier to compare.
The way this works is the term that sets k equal to 3 through 13, counting by 2 (so ), where is the degree of the Taylor series. If we want to generate a bunch of Taylor polynomials and graph them along with the original function, we can use the Table function: Let's graph and the Taylor series together on the same graph:
(Notice the Evaluate command we also had to add in you must do this whenever you graph a Taylor series.) To make your series usable, you need to make it a "normal" polynomial: This is because of this extra "error term". However, notice that it still didn't give you an answer. Notice that the way to "plug in" is to use the notation. This is cool, but not very useful when you need to plug in a number: However, notice the last term: This means the "error" term is "on the order of". Mathematica gives you the Taylor series for expanded around c ( here) of degree 5.
There is a catch, however: If you just type: Now, if we want to work with finite Taylor polynomials, Mathematica has a built in function for that called Series. To work with Taylor series in Mathematica, we first have to define the function we want to approximate and the "center" of the interval where we want to approximate it:
Note: In class I said this lab would consist of two parts (this was to be the second part), but I realized that the other part I was planning for you to use really wasn't workable for you, so this is it.
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