(Polynomial approximation) Why does the nth derivative of both the function and the polynomial has to be equal at center?

Question

I've watched many videos about Taylor/Maclaurin polynomial but no tutor ever explained why it has to be f⁽ⁿ⁾(c) = p⁽ⁿ⁾(c) at center x = c.

I've seen the behavior of the graph of p(x) and f(x) when they are approximating and how each higher degree polynomial approximate the function more accurate around the center. My confusion is at the part "around the center", since the higher derivatives of both functions only match at the center:

1/ How does that make the vicinity of both functions match, since the derivatives of the f(x)'s and the p(x)'s around the center doesn't equal (only the derivative of the center is equal)?

2/ How does the approximation get better and better once the "n" of f⁽ⁿ⁾(c) = p⁽ⁿ⁾(c) start increasing? Like, how does f⁽²⁾(c) = p⁽²⁾(c) provide better approximation than f⁽¹⁾(c) = p⁽¹⁾(c), and how does f⁽³⁾(c) = p⁽³⁾(c) does an even better job? And it keeps going... The Youtube tutors I've watched only tell: "Higher derivatives of both functions match will give better approximation" but never explain how that is the case.

Thank you all. Any help is greatly appreciated!

Answer 1

The rough idea is that if two functions have the same $n$ first derivatives at $a$, then their difference is of the order $(x-a)^{n+1}$. Do you see why, for $x$ close to $a$, a $(x-a)^2$ is much smaller than a $(x-a)$ ?

The rigorous foundation for that is Taylor's theorem.

But you can see already that for polynomials. If two polynomials $f$ and $g$ have the same $n$ first derivatives at $a$, it means that $a$ is a root of order $n+1$ of the difference polynomial $f-g$. Hence $f-g = (x-a)^{n+1}$ times something.

Try it with $-x^3 + 6 x^2 - 8 x + 3$ and $x^3 - 2 x + 1$, for $a=1$. First show that they have the same derivatives up to the second, and then show that their difference can be factored out by $(x-1)^3$.

So essentially what Taylor's theorem tells you is that nice functions (which are $n$ times differentiable) obey the same principles as polynomials.

I agree that it's initially wierd to see the equality condition being only at the center $a$. But the derivative itself is something that sums up what happens in smalls neighborhoods of $a$ if the first derivative agrees at $a$, it means that the function has to nearly agree near $a$.