Příklad 3.1

Určete Taylorův polynom druhého stupně se středem v bodě \([0,1]\) pro funkci \(f(x,y)=e^{xy+x}+\text{arccotg}(2x) + \frac{sin(xy)}{y}\)

$$\large\begin{aligned} T_2 &= f(x,y) + \frac{df(x,y)(x-x_0, y - y_0)}{1!} + \frac{d^2f(x,y)(x-x_0, y - y_0)}{2!} = \

&= f(0,1) + \left(\frac{\partial{f}}{\partial{x}}(0, 1)\cdot{x}+\frac{\partial{f}}{\partial{y}}(0, 1)\cdot{(y-1)}\right) \ &+\left(\frac{\partial^2{f}}{\partial{x^2}}(0, 1)\cdot{x^2}+2\frac{\partial^2{f}}{\partial{xy}}(0, 1)\cdot{x \cdot (y-1)}+\frac{\partial^2{f}}{\partial{y^2}}(0, 1)\cdot{y^2}\right) \

\end{aligned}$$

$$\Large\begin{aligned} \frac{\partial{f}}{\partial{x}} &= e^{xy+x}\cdot(y+1) - \frac{1}{1+x^2}\cdot2+cos(xy)\ \frac{\partial{f}}{\partial{y}} &= e^{xy+x}\cdot{x} \

\end{aligned} $$