Taylor/Polynomial Approximation Question

[Lebombo]

If I have a polynomial function f(x) and I want to find an approximate polynomial g(x). I can apply the Nth order Taylor Polynomial of f centered at some value a.

So suppose I have f(x) = 5 - 6x + 20x^3 + 10x^5

and I want to find an approximate function, g(x), centered at 5.

From what I gather, this means that after I create the Taylor Polynomial of 5 degrees, when I plug in 5 to both f(x) and g(x), they will be equal values. That is, f(5) = g(5).

However, does this also mean that f(x) = g(x) for all values of x (not just the centered value) if one polynomial is approximated (Taylor polynomial) from another polynomial? (as opposed to a polynomial being approximated from a non-polynomial)?

Note: I'm not considering infinite series in this question, only nth term Taylor polynomials.

 

[Mark44]

If I have a polynomial function f(x) and I want to find an approximate polynomial g(x). I can apply the Nth order Taylor Polynomial of f centered at some value a.

So suppose I have f(x) = 5 - 6x + 20x^3 + 10x^5

and I want to find an approximate function, g(x), centered at 5.

From what I gather, this means that after I create the Taylor Polynomial of 5 degrees, when I plug in 5 to both f(x) and g(x), they will be equal values. That is, f(5) = g(5).

However, does this also mean that f(x) = g(x) for all values of x (not just the centered value) if one polynomial is approximated (Taylor polynomial) from another polynomial? (as opposed to a polynomial being approximated from a non-polynomial)?

Note: I'm not considering infinite series in this question, only nth term Taylor polynomials.

Your "approximation" polynomial g(x) will agree exactly with f(x) for all values. All you're really doing is changing from a polynomial that is a sum of powers of x (f) to one that is a sum of powers of x - 5.

 

[Dick]

If I have a polynomial function f(x) and I want to find an approximate polynomial g(x). I can apply the Nth order Taylor Polynomial of f centered at some value a.

So suppose I have f(x) = 5 - 6x + 20x^3 + 10x^5

and I want to find an approximate function, g(x), centered at 5.

From what I gather, this means that after I create the Taylor Polynomial of 5 degrees, when I plug in 5 to both f(x) and g(x), they will be equal values. That is, f(5) = g(5).

However, does this also mean that f(x) = g(x) for all values of x (not just the centered value) if one polynomial is approximated (Taylor polynomial) from another polynomial? (as opposed to a polynomial being approximated from a non-polynomial)?

Note: I'm not considering infinite series in this question, only nth term Taylor polynomials.

If your expansion around x=5 is degree 5, like the original polynomial, the 'approximate' polynomial will be exact everywhere. It will be the same as the original polynomial if you expand it out, because the terms in the Taylor expansion vanish after degree 5. If it's less than degree 5 then it will only be a good approximation near x=5.