Taylor Polynomial. Understanding.

[beaf123]

Homework Statement

Last exam in my school this exircise was given:

From norweagen:
" Decide the Taylor polynomial of second degree of x=0 of the function:

f(x) = 3x^3 + 2x^2 + x + 1

I found the Taylor polynomial of second degree to be: 2X^2+X+1, which is correct.

If I get an exircise like this on the exam I thought maybe it would give me some bonus points if I could show it a the graph. what exactly does the Taylor polynomial tell me?

How f(x) behave near x=0?

And found out that the polynomial of third degree or (infinity) gives me back the original function. So what's the point in doing this on a function like this? To get an easier functiopn to work with?

And for a function like f(x) = e^x, which has infinitly many derivatives. The sum of all derivivatives estimate the function compleatly?

And what is that good for?

Homework Equations

The Attempt at a Solution

 

[Kreizhn]

You are correct in that the full Taylor expansion of an polynomial always returns to you the original polynomial. Notice that your degree two Taylor polynomial is also your original polynomial, but without the terms with degree higher than 2.

The idea of Taylor polynomials is that any smooth function can be approximated arbitrarily well with just polynomials. In my opinion, this is a truly beautiful fact. Why should we expect this? Are there other functions we can use instead of polynomials? Maybe sines and cosines?

As for their practical uses:
As you said, they describe the behaviour of the function around a point. They show up in applied mathematics and physics a fair bit when people want to make approximations to functions. They show up in numerical mathematics, again for approximation purposes. Power series are used to solve differential equations, and the Taylor series is a special kind of power series which converges to a known function. Combinatorial enumeration uses power series, and sometimes Taylor series show up there (though only formally). The Taylor series of real functions is used to extend the idea of evaluating functions on operators; for example, we use the Taylor series expansion of the exponential to define the matrix exponential, matrix logarithm, etc.

There are classes of functions which are infinitely differentiable, but are not analytic (that is, their Taylor series do not always converge in a neighbourhood of their domain). Hence some people study analytic functions themselves: this lends itself to the difference between, say, smooth and analytic manifolds. You will see in complex analysis though that every differentiable function is in fact holomorphic! (another word that means analytic).

Taylor's theorem is actually a special case of a far more powerful theorem called the Stone-Weierstrass Theorem. If X is a compact Hausdorff space and $B \subseteq C(X,\mathbb R)$ (continuous functions from X to $\mathbb R$) is a unital subalgebra which separates points, the B is dense in $C(X,\mathbb R)$.

 

[beaf123]

Beautiful answer! Thank you very much:-)