"Consider the approximate formulate: $f'(x) \approx 3/(2h^3) \int_{-h}^{h} tf(x+t) dt$. Determine its error term."
I was thinking taking the taylor series of $f(x+t)$ to help me actually integrate the integral, but when would I stop the taylor series expansion?
Consider $$ F_x(r) = \int_0^r tf(x+t) dt $$ then via the Taylor formula: $$ F_x(r) = 0 + 0f(x+0)\times r + A_xr^2 + \frac 16 (2f'(x+0) + 0f''(x+0))r^3 + B_xr^4 + O(r^5) \\=A_xr^2 + \frac 13f'(x)r^3 + B_xr^4 + O(r^5)\\ \frac2{3h^3}\int_{-h}^h tf(x+t)dt = \frac2{3h^3}(F_x(h) - F_x(-h)) = f'(x) + O(h^2)\\ $$
