# Assignment 6

Problem 1.  Read sections 10.3 and 10.4

Problem 2.  Find the equation of the plane tangent to the graph of

$\displaystyle z = \ln \frac{\cos x}{\cos y}$

at the point $(\pi/4, \pi/4, 0)$.

Problem 3.  For this problem we will use the function $z = f(x,y)$ from Problem 2.

(a) Compute the derivative matrix $Df$.

(b) Compute the linear approximation $L(x,y)$ for $f(x,y)$ at the point $(\pi/4, \pi/4, 0)$.

Problem 4.  Given the function $F(x,y,z) = (x^2y - z, \cos(xyz)\,)$, compute the derivative matrix $DF$.

Problem 5.  (a) Given a function $f:\mathbb{R}^2 \to \mathbb{R}$, explain why its derivative matrix $Df$ is a $1 \times 2$ “row matrix.”

(b) If instead of arranging the partials of $f(x,y)$ in a matrix we arrange them in a vector, then we do not call the resulting vector expression a “matrix.”  Instead we call it the gradient of $f(x,y)$ and we notate it as

$\displaystyle \text{the gradient of } f = \nabla f$.

Compute the derivative matrix and compute the gradient, $\nabla f$, of the function $f(x,y) = x^2+y^2$.

(c ) Find al points where $\nabla f = \vec{0}$.

(d) What do the points in part (c) have to do with the graph of $z = f(x,y)$?