Very very basic taylor series problem

[PTTB]

Homework Statement

Consider f(x) = 1 + x + 2x^2+3x^3.
Using Taylor series approxomation, approximate f(x) arround x=x0 and x=0 by a linear function

Homework Equations

The Attempt at a Solution

This is the first time that I have seen Taylor series and I am totally lost on how to do it, I have looked all around the internet for some help and I just don't have a clue on where to start. Any Help would be appreciated

 

[Gib Z]

Well start from the definition of a taylor series:
The taylor series of a function centered around a is given by
$$\sum_{n=0}^{\infty} f^n (0) \frac{(x-a)^n}{n!}$$ where f^n is the nth derivative of f, not the nth power. The 0th derivative is just the original function.

So it wants you to find 2 series, one with center 0, or a=0, and the other with centre a=x_0. Use the formula to get the taylor series centered around those points, and then since it only wants a linear function, only use the first 2 terms, ie the constant and the x terms.

 

[PTTB]

thanks a ton, i think I am starting to get this stuff, you were a huge help

i got 1+x and 1+x-2x_o^2-6x_0^3+(4x_0)x+(9x^2_0)x I am pretty sure that's rite, thanks a ton

 

[HallsofIvy]

thanks a ton, i think I am starting to get this stuff, you were a huge help

i got 1+x and 1+x-2x_o^2-6x_0^3+(4x_0)x+(9x^2_0)x I am pretty sure that's rite, thanks a ton

Actually, it makes little sense to say "Taylor series approximation". The Taylor series is exact. It is the "Taylor polynomial" that is approximate and it is my guess that this is what is intended. Of course, the Taylor Polynomial that gives a linear approximation is just the 1^st order approximation: f(x₀)+ f '(x₀)(x- x₀)- and that's just the tangent line approximation.
Since $f(x_0)= 1+ x_0+ 2x_0^2+ 3x_0^3$ and $f '(x_0)= 1+ 4x_0+ 9x_0^2$, the tangent line approximation at $x_0$ is $1+ x_0+ 2x_0^2+ 3x_0^3+ (1+ 4x_0+ 9x_0^3)(x- x_0)$. After multiplying out the last term, that gives exactly what you have.

 

[genneth]

The Taylor series is exact.

*Grumble*
*Mutterings about analytic functions and radius of convergence*
*Grumble*