Taylor series | Chapter 11, Essence of calculus

When I first learned about Taylor series, I definitely

didn't appreciate just how important they are.

But time and time again they come up in math, physics,

and many fields of engineering because they're one of the most

powerful tools that math has to offer for approximating functions.

I think one of the first times this clicked for me as a

student was not in a calculus class but a physics class.

We were studying a certain problem that had to do with the potential energy of a

pendulum, and for that you need an expression for how high the weight of the

pendulum is above its lowest point, and when you work that out it comes out to be

proportional to 1 minus the cosine of the angle between the pendulum and the vertical.

The specifics of the problem we were trying to solve are beyond the point here,

but what I'll say is that this cosine function made the problem awkward and unwieldy,

and made it less clear how pendulums relate to other oscillating phenomena.

But if you approximate cosine of theta as 1 minus theta squared over 2,

everything just fell into place much more easily.

If you've never seen anything like this before,

an approximation like that might seem completely out of left field.

If you graph cosine of theta along with this function, 1 minus theta squared over 2,

they do seem rather close to each other, at least for small angles near 0,

but how would you even think to make this approximation,

and how would you find that particular quadratic?

The study of Taylor series is largely about taking non-polynomial

functions and finding polynomials that approximate them near some input.

The motive here is that polynomials tend to be much easier to deal

with than other functions, they're easier to compute,

easier to take derivatives, easier to integrate, just all around more friendly.

So let's take a look at that function, cosine of x,

and really take a moment to think about how you might construct a quadratic

approximation near x equals 0.

That is, among all of the possible polynomials that look like c0 plus c1

times x plus c2 times x squared, for some choice of these constants, c0,

c1, and c2, find the one that most resembles cosine of x near x equals 0,

whose graph kind of spoons with the graph of cosine x at that point.

Well, first of all, at the input 0, the value of cosine of x is 1,

so if our approximation is going to be any good at all,

it should also equal 1 at the input x equals 0.

Plugging in 0 just results in whatever c0 is, so we can set that equal to 1.

This leaves us free to choose constants c1 and c2 to make this

approximation as good as we can, but nothing we do with them is

going to change the fact that the polynomial equals 1 at x equals 0.

It would also be good if our approximation had the same

tangent slope as cosine x at this point of interest.

Otherwise the approximation drifts away from the

cosine graph much faster than it needs to.

The derivative of cosine is negative sine, and at x equals 0,

that equals 0, meaning the tangent line is perfectly flat.

On the other hand, when you work out the derivative of our quadratic,

you get c1 plus 2 times c2 times x.

At x equals 0, this just equals whatever we choose for c1.

So this constant c1 has complete control over the

derivative of our approximation around x equals 0.

Setting it equal to 0 ensures that our approximation

also has a flat tangent line at this point.

This leaves us free to change c2, but the value and the slope of our

polynomial at x equals 0 are locked in place to match that of cosine.

The final thing to take advantage of is the fact that the cosine graph

curves downward above x equals 0, it has a negative second derivative.