# Assignment 9

Group 1 Question

The set of all points $\displaystyle S = \left\{ (x,y,z) : (x-1)^2 + y^2 + \frac{z^2}{4} = 1\right\}$ forms a sphere-like object called an ellipsoid centered at the point $(1, 0, 0)$ (a picture of this surface is shown below).

Find the points on this ellipsoid that are closest to and farthest away from the point the origin.

Group 2 Question

Find the equation for the tangent plane to the surface determined by

$\displaystyle y^z + x^z + x^y = 3$

at the point $(1, 1, 1)$.

Group 3 Question

Find the maximum and minimum values of $f(x,y,z) = xyz$ over all points in the region of space $W = \left\{(x,y,z) : 2x^2+2y^2+z^2 \leq 2 \right\}$.

Group 4 Question

Find the linear approximation of the function

$\displaystyle F(u, v) = \left(\,e^u, \arctan\left(\frac{v^2}{1+u^2}\right), e^v\right)$

at the point \$latex (0, 1)4.

Group 5 Question

Let $f(x,y,z)$ be a real-valued function that satisfies

$\displaystyle \nabla f\,(1, -1, \sqrt{2}) = \langle 1, 2, -2 \rangle$

(and note that there are many functions that satisfy this).  Given this information, compute the partial derivative $\partial f/\partial \theta$ at the point with spherical coordinates $(\rho, \theta, \varphi) = (2, -\pi/4, \pi/4)$.

Group 6 Question

For this question let $A$ be a symmetric, $2\times 2$ matrix of real numbers

$\displaystyle A = \left[ \begin{array}{cc} a& b \\ b & c \end{array}\right]$

and define a function $f:\mathbb{R}^2\to\mathbb{R}$ by

$\displaystyle f(x,y) = f(\vec{x}) = \vec{x}\cdot\,A\vec{x}$.

(a) Show that the gradient of $f$ satisfies $\nabla f = 2A\vec{x}$.

(b) Compute the Hessian matrix of $f$.

(c) Show that $f$ has a critical point $\vec{x}_0 = (x_0, y_0)$ \textit{on the unit circle} precisely when

$\displaystyle A\vec{x}_0 = \lambda\,\vec{x}_0$

for some $\lambda \in \mathbb{R}$.

(d) (Bonus) What is a vector $\vec{x}_0$ that satisfies an equation like that in part (c) called?