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). ∇f(x, y) = Df(x, y) = (\frac{1}{x}, \frac{1}{y})

Hf(x, y) = ( \begin{matrix}−x^{−2} & 0 \\ 0 & −y^{−2} \end{matrix} )

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) ( \begin{matrix}x − 1 \\ y − 1 \end{matrix} ) = 0 + (1, 1) ( \begin{matrix}x − 1 \\ y − 1 \end{matrix} )

= x − 1 + y − 1 = x + y − 2. p_{2}(x, y) = f(a) + Df(a)(x − a) + \frac{1}{2}(x − a)^{T} Hf(a)(x − a) =