Exercise 1802Calculus › Multivariable calculus › Partial derivativesL1Added 3 Nov 2026

Taylor expansion of order 2

#Taylor expansion#multivariable#approximation

Statement

Give the Taylor expansion of $f(x,y)=e^{x+y}$ at $(0,0)$ up to order $2$ .

Hint

$f(0)=1$ , $\nabla f (0,0)=(1,1)$ , $H(0,0)=\begin{pmatrix}1&1\\1&1\end{pmatrix}$ .

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Expression.
$f(h,k)=f(0,0)+ \nabla f(0,0) \cdot (h,k)+\frac{1}{2}(h,k) \cdot H(0,0) \cdot (h,k) +o((h^2+k^2)).$

Simplified. $f(h,k)=1+h+k+\frac{h^2}{2}+hk+\frac{k^2}{2}+o(h^2+k^2)$ .

Conclusion. The expansion matches $e^h\cdot e^k=(1+h+h^2/2+\cdots)(1+k+k^2/2+\cdots)$ to order $2$ .