Photosintese

This is calculus in a single variable and
I'm Robert Greist, Professor of
Mathematics and Electrical and Systems
Engineering at the University of
Pennsylvania. We're about to begin lecture two, bonus material. In our last lecture, we began with an expression for e to the
x. From which we derived expressions for sine of x and cosine of x. These all were in the form of an infinite series. Now, it's best to think of such objects as something like, long polynomials.
Polynomials of unbounded degree. It's easy to manipulate such objects. But of course, they're not really polynomials. On the other hand, if we turn them in to true polynomials, if we truncated these expressions, after finding a number of terms, what would happen? Let's look at the example of E to the X. The first term one is somewhat boring. 1X + x is getting a little bit closer. One+X+X squared over two, closer still, etcetera. As we keep going. Longer and longer polynomial truncations approximate increasingly well.
And if you look very close to the origin, if you look near x=0, this approximation gets very good, very quickly. It's hard to tell the difference between e to the x and a sufficiently long polynomial truncation.
This is going to be an extremely important principle for us in this class. Let's take a look at what happens with cosign. The first term in the series for cosine one.
Well it at least gets the value for zero correct but once we start going out to two or three terms of the series we see that we're getting closer and closer to what the graph of cosine looks like near x0. = zero. The same thing holds for sign as well. The fact that we have X it matches a little bit what's happening right at the origin.
But when we take third order and fifth order polynomial expressions we get better and better approximations. Now, of course, you can keep going with this, by taking more and more terms. You can approximate more closely, what is happening