Multi-Variable Calculus Assignment

due Friday, March 9, 2001

Exercises

1. Let f be a function of two variables. The graph G_{f} of f is the surface in space defined by the equation

f(x, y) − z = 0 .

For fixed values of a and b let P be the point (a, b, f(a, b)).

(a) Find a vector that is normal to the surface G_{f} at the point P.

(b) Find an equation for the tangent plane to G_{f} at P.

(c) Find vectors U and V for which the expression

P + uU + vV

is a parametric representation of the tangent plane with parameters u and v.

2. Find the equation of the plane that is tangent to the graph of the function

f(x, y) = 2x^{2} − xy + y^{2}

at the point of the graph corresponding to the point (2, −1) in the domain of f, i.e., the point where

x = 2, y = −1, z = f(2, −1) .

[Processed from GELLMU to L^{A}TEX on Tue 6 Mar 2001 at 16:51:38 EST]