I was recently introduced to Taylor polynomials as a part of my academic degree and I think I don't understand the idea of it. In one hand, I can see why it's useful when it comes to the exponent function or the trigonometry functions (namely, when the function does not return values "explicitly", in contrast to polynomials). However, as a part of my homework I was asked, several times, to calculate Taylor polynomial for functions which were simply polynomials(). Then, assuming I understand the motivation to use Taylor polynomials, I wondered why would someone would want to calculate that ()?
I would like if someone could shed some light. Thanks
You can get a linear or polynomial approximation to any differentiable function using Taylor's Formula.A polynomial is easier to deal with than others.
a simliar question to yours, what is the motivation to derivative?
Taylor's Formula is a common tool.Here is an example.
We know $f'(x)>=0,\forall \ x\in[a,b]$ means $f(x)$ increases in [a,b].The proof uses Lagrange's mean value theoerm which can be regarded as a one-order Taylor's Formula.We use $f'$ to get the property of $f$.
Using the simple form(polynomial),you can get many properties of derivative.
Consider some property called convex about the second derivative. https://en.wikipedia.org/wiki/Convex_function