A two-variable quadratic approximation

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Problem

For $f(x,y)=\log x+xy+y^2$ , form the second-order Taylor approximation at $(1,0)$ in increments $(h,k)$ , then use it to approximate $f(1.1,0.2)$ .

Reveal answer or reference solution

At $(1,0)$ , $f=0$ , $\nabla f=(1,1)$ , and

H=\begin{bmatrix}-1&1\\1&2\end{bmatrix}

Thus

T_2(1+h,k)=h+k+\frac12(-h^2+2hk+2k^2)

At $h=0.1$ and $k=0.2$ this is

0.3+\frac12(-0.01+0.04+0.08)=0.355=\frac{71}{200}

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A two-variable quadratic approximation

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