Single-Variable Form

Let f(x)f(x) be a C3C^3 function on some open set II and x0∈Ix_0\in I, then for any x∈Ix\in I, we have

f(x)=f(x0)+f′(x0)(x−x0)+f′′(x0)2!(x−x0)2+12!∫x0x(x−τ)2f′′′(τ)dτf(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\frac{1}{2!}\int_{x_0}^x (x-\tau)^2 f'''(\tau) d\tau

Multi-Variable Form

f(x,y)=f(x0,y0)+[fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)]+12![fxx(x−x0)2+2fxy(x−x0)(y−y0)+fyy(y−y0)2]+ Remainder\begin{aligned} f(x,y)=& f(x_0,y_0)\\ +& \Big[ f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0) \Big]\\ +& \frac{1}{2!}\Big[ f_{xx}(x-x_0)^2 + 2f_{xy}(x-x_0)(y-y_0) + f_{yy}(y-y_0)^2 \Big]\\ +& \text{ Remainder} \end{aligned}