Describe this very hard integral as a taylor-polynomial

[Nikitin]

Homework Statement

Find the polynomal that approximates the integral of (t^2)e^-t^2 from 0 to x, [0,1], with an error less than 10^-3

Homework Equations

You need to use taylor theorem & remainder estimation theorem

The Attempt at a Solution

I set up the tailor series with the first 4 terms (1/e, 0, (2/e)(x-1)^2, (1/6)*(x-1)^4), but where am I supposed to go now? I integrated the series, but I still don't know if I got enough terms.

Hell, what does it mean to integrate something from 0 to x? x isn't even a number! Does the book mean from 0 to 1, since t is an element between 0 and 1? Until I figure what the hell the book is on about, I can't use the remainder estimation theorem.

I am extremely confused atm and I think the book I have is pretty bad at explaining.

 

[lurflurf]

x is a number in [0,1]
estimate the function
F(x)=\int_0^x (t^2)e^{-t^2} dt
by another function G(x) such that
|G(x)-F(x)|<10^-3
whenever
0<=x<=1

 

[Nikitin]

uhm, I'm having trouble finding the error.. I now understand the problem, but I still am unable to use the remainder estimation theorem because:

1) I cannot find M, the biggest value f derived n times, without digital tools. This is because I get increasingly complicated functions as I keep deriving them, and there is no pattern in F(x)'s taylor series.
2) I am a bit unsure on how it works, because I've never used it before.

 

[lurflurf]

This problem is tedious if done certain ways.

The quick way is to first find the Taylor polynomials of e^x call it p(x) then
(t^2)p(-t^2) is the Taylor polynomial of (t^2)e^-t^2
The error is easily estimated by observing the series is alternating, so the first omitted term is a good estimate of the error.

 

[Nikitin]

uhm, so it's allowed to just substitute stuff in and out, and multiply the polynomials with t^2? I had no idea... I just wish our professor would be so kind to go through this stuff before the assignments are due.. sigh

Anyway, THANKS allot!