Multi-Variable Calculus Assignment

due Tuesday, March 20, 2001

Exercises

1. Find the maximum value of the function

f(x, y) = 2x^{2} − xy + y^{2}

on the disk x^{2} + y^{2} = 1.

2. Find the point on the surface

x^{2} − y + z^{2} = 0

that is closest to the point (0, 0, 5).

Optional Exercise

due Wednesday, March 21

Show that the curve

x^{2} − y + z^{2} = 0
x − y + z = 0

is an ellipse, and find the coordinates of its foci. (Recall that A_{1} and A_{2} are the foci of an ellipse E if the sum of the distance from P to A_{1} and the distance from P to A_{2} is is a positive constant as P varies through the points of E.