Taylor Polynomial

Taylor Polynomial

The Taylor polynomial of degree $n$ approximates $f(x)$ near point $a$:

P_n(x) = sum_{k=0}^n rac{f^{(k)}(a)}{k!} cdot (x-a)^k

Where $f^{(k)}(a)$ is the $k$-th derivative of $f$ at $a$.

Building Intuition

• $P_0(x) = f(a)$ — constant (matches value at $a$)
• $P_1(x) = f(a) + f'(a)(x-a)$ — linear (also matches slope)
• $P_2(x)$ — also matches curvature
• Each higher term matches one more derivative at $a$

Exact for Polynomials

A degree-$n$ Taylor polynomial reproduces any degree-$n$ polynomial exactly:

$f(x) = x^2$ at $a=0$:

P_2(x) = f(0) + f'(0)x + rac{f''(0)}{2!} x^2 = 0 + 0 + rac{2}{2} x^2 = x^2 checkmark

Numerical nth Derivative

Use recursive central differences:

f^(0)(x) = f(x)
f^(n)(x) ≈ (f^(n-1)(x+h) - f^(n-1)(x-h)) / (2h)

This is accurate to $O(h^2)$ at each level. Keep $n leq 4$ and $h approx 10^{-3}$ for best results.

Your Task

Implement double taylor_poly(double (*f)(double), double a, double x, int n, double h).

Use recursive nth_deriv and a loop-based factorial. Helper stubs are provided.