高等数学 多元泰勒 通式 f(x,y)=limn→∞∑i=0ndnf(x0,y0)n! f(x,y)=\lim_{ n \to \infty } \sum_{i=0}^{n} \frac{ \mathrm{d}^{n}f(x_{0},y_{0})}{n!} f(x,y)=n→∞limi=0∑nn!dnf(x0,y0) 皮亚诺余项 f(x,y)=∑i=0ndnf(x0,y0)n!+o(ρn) f(x,y)=\sum_{i=0}^{n} \frac{ \mathrm{d}^{n}f(x_{0},y_{0})}{n!}+o(\rho^{n}) f(x,y)=i=0∑nn!dnf(x0,y0)+o(ρn) 拉格朗日余项 f(x,y)=∑i=0ndnf(x0,y0)n!+dn+1f(x0+θΔx,y0+θΔy)(n+1)! f(x,y)=\sum_{i=0}^{n} \frac{ \mathrm{d}^{n}f(x_{0},y_{0})}{n!}+\frac{ \mathrm{d}^{n+1}f(x_{0}+\theta\Delta x,y_{0}+\theta\Delta y)}{(n+1)!} f(x,y)=i=0∑nn!dnf(x0,y0)+(n+1)!dn+1f(x0+θΔx,y0+θΔy) 特例：拉格朗日中值定理 f(x,y)−f(x0,y0)=∂f∂x(x0+θΔx,y0+θΔy)Δx+∂f∂y(x0+θΔx,y0+θΔy)Δyf(x,y)-f(x_{0},y_{0})=\frac{ \partial f }{ \partial x } (x_{0}+\theta\Delta x,y_{0}+\theta\Delta y)\Delta x+\frac{ \partial f }{ \partial y }(x_{0}+\theta\Delta x,y_{0}+\theta\Delta y)\Delta yf(x,y)−f(x0,y0)=∂x∂f(x0+θΔx,y0+θΔy)Δx+∂y∂f(x0+θΔx,y0+θΔy)Δy