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\def\R{{\bf R}}
\def\x{{\bf x}}
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\def\S{\Sigma_{i=1}^n}
\def\Sk{\Sigma_{i, j=1}^n}
\def\Sjk{\Sigma_{i_1,..., i_k=1}^n}




HW (due 4/4) 4.1: 2, 9, 11, 14, 24, 27, 28, 31; 4.2, 1. 13. 28-34

$f(x, y) = ln(xy)$.

$\nabla f(x, y) = Df(x, y) = ({1 \over x}, {1 \over y})$ 

$Hf (x, y) = \left(\matrix{-x^{-2} & 0 \cr 0 & -y^{-2}}\right)$

Let $\a = (1, 1)$ 

Then the tangent plane to $f(x, y) = ln(x, y)$ at $\a = (1, 1)$ is 

$p_1(x, y) = f(1, 1) + Df(1, 1) 
\left(\matrix{ x - 1 \cr y - 1}\right)$
$= 0 + (1, 1) \left(\matrix{ x - 1 \cr y - 1}\right)$

$= x - 1 + y - 1 = x + y - 2$.

$p_2(x, y) = f(\a) + Df(\a)(\x - \a) + {1 \over 2}(\x - \a)^T Hf(\a) (\x - \a) = $


 







\end