### MACLAURIN SERIES COSH 2X

And we’ll get closer and closer to the real e to the x as we keep adding more and more terms. This is 1, this is 1, when we’re talking about e to the x. So f of 0 is 1 plus f prime of 0 times x. It’s always equal to e to the x. So if we could somehow reconcile the negatives in some interesting way, it looks like e to the x is somehow– or at least its polynomial representation of e to the x– is somehow related to a combination of the polynomial representations of cosine of x and sine of x. Power series of arctan 2x. But that by itself isn’t the only fascinating thing.

This is equal to f, the second derivative of x. So if you wanted to approximate e, you’d say e is approximate to– well, e is e to the first power. And in the next video, I’ll show you how we can actually reconcile these three fascinating functions. Plus x to the third over 3 factorial. So let me do my best to copy and paste the whole thing. This is 1, this is 1, when we’re talking about e to the x. So if we could somehow reconcile the negatives in some interesting way, it looks like e to the x is somehow– or at least its polynomial representation of e to the x– is somehow related to a combination of the polynomial representations of cosine of x and sine of x.

If we look back at our Maclaurin representations xosh these other functions, cosine of x– let me copy and paste cosine of x. Graphs of Taylor Polynomials Abby Brown. Practice online or make a printable study sheet. It’s just, you could keep taking its derivative.

So let me use a non-pink, non-green. The n-th derivative evaluated at 0.

### Maclaurin Series — from Wolfram MathWorld

Hints help you try the next step on your own. This also leads to other interesting results. Let me copy and paste that. Is that e to x, this is just really cool– that e to the x can be approximated by 1 plus x plus x squared over 2 factorial plus x to the third over 3 factorial. Collection of teaching and learning tools built by Wolfram education experts: So I used pink and I used green.

Power series of arctan 2x. And that’s why it makes applying the Maclaurin series formula fairly straightforward. And these things that come out of the unit circle in oscillatory motion and all of those things, there starts to seem some type of pure connectedness there.

And we’ll get closer and closer to the real e to the x as we keep adding more and more terms. So if we could somehow reconcile the negatives in some interesting way, it looks like e to the x is somehow– or at least its polynomial representation of e to the x– is somehow related to a combination of the polynomial representations of cosine of x and sine of x.

So we have to find f of 0, f prime of 0, the second derivative at 0. Maclaurin series of cos x.

If I wanted to approximate e to the x using a Maclaurin series– so e to the x– and I’ll put a little approximately over here. And in the next video, I’ll show you how we can actually reconcile these three fascinating functions. Anyway, with that said, let’s take maclarin Maclaurin representation.

So that is cosine of x. So copy and paste.

And I think you see the pattern here. So do we see any relationship between these approximations? It’s always equal to e to the x. All of these things are going to be 1.

So f of 0 is 1 plus f prime of 0 times x. And what you see here is that cosine of x looks a lot like this term plus this term, although we would want to put a negative out front here.

## Taylor/Maclaurin Series Calculator

Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. So plus x squared over 2 factorial.

If you’re seeing this message, it means we’re having trouble loading external resources on our website. But I’ll leave you there in that video. And something pretty neat is starting to emerge. Function as a geometric series.

And let’s do the same thing for maclaurij sine of x that we did last video. And that’s going to be approximately equal to this polynomial evaluated at 1. And this will, to some degree, be one of the easiest functions to find the Maclaurin series representation of.

But that by itself isn’t the only fascinating thing.