Not sure I get the Taylor Series

[fisico30]

not sure I get the Taylor Series...

Hello Everyone.

I understand that the taylor series approximate a function locally about a point, within the radius of convergence.
If we use the Taylor series it means that we do not know the function itself.

But to find the taylor series we need the derivatives of the function. and to have the derivatives we need the function itself...

where is the problem?

Also, we could represent a function is a certain interval of interest by using orthogonal functions (legendre, sines cosines, and any other orthogonal set). Why go with the Taylor series?

 

[Mapes]

If we use the Taylor series it means that we do not know the function itself.

Not true! The Taylor series is useful in simplifying a known, complex function in a local region.

Why go with the Taylor series?

An example: the Taylor series tells us that any function is approximately quadratic around a minimum or maximum. This has tremendous implications when analyzing energy landscapes. Since a spring's energy is also quadratic around its equilibrium point, we can apply a lot of existing mathematics (e.g., simple harmonic motion) to systems near equilibrium.

 

[atyy]

But to find the taylor series we need the derivatives of the function. and to have the derivatives we need the function itself...

It's most useful if "x" is "small", and you only need one or two terms to get a good approximation. I have heard (humourously) that all of theoretical physics consists of looking for "small parameters" so they can use Taylor series.

It's also useful if you want to represent a known/unknown function in another form which may cast the problem in a more familiar form.

Also, we could represent a function is a certain interval of interest by using orthogonal functions (legendre, sines cosines, and any other orthogonal set). Why go with the Taylor series?

All the various series expansions are potentially useful, and which one to use depends on the situation. For example, in describing sound hitting the ear, some variant of the Fourier series/transform is used most often because it's close to what the ear does.