Taylor Polynomial Approximations.

[Easytoadd]

Hello,

I'm new here, nice to meet you guys i was in class today and just didn't understand the taylor polynomial approximation, the professor started out approximating a function by polynomials of degree N, he first showed us how a linear polynomial was a crude approximation of the function at a given point but thing i don't get is when he moved on to the next approximation by using the first derivative, than the second...i just didnt get why the final formula added up all the approximations together?...

I just didnt get why like we had to sum up all the approximations rather than just making a formula for the last approximation of the function?...if I am not making any sense I am sorry but if you guys can explain the taylor polynomial a bit better i would greatly appreciate it.



Thanks,

Moe.

 

[CompuChip]

I wrote a little bit about this some time ago... does this help you get started? https://www.physicsforums.com/blog.php?b=1758

 

[Erland]

Not sure what you mean but if you take the Taylor polynomials for f(x)=e^x about x=0 (also called the Maclaurin polynomials for e^x) they are:

P_0(x) = 1
P_1(x) = 1 + x
P_2(x) = 1 + x + (1/2)x^2
P_3(x) = 1 + x + (1/2)x^2 + (1/6)x^3
etc.

So the first k+1 terms of a Taylor polynomial of degree n is the Taylor polynomial for the same function about the same point of degree k.

If you mean that we should just take the highest degree terms of the Taylor polynomials, in this example 1, x, (1/2)x^2, (1/6)x^3 etc. these terms alone would NOT be good approximations of the original function.