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)).

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

   2. Find an equation for the tangent plane to G_{f} at P.

   3. Find vectors U and V for which the expression

      P + u U + v V

      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) = 2 x^{2} - x y + 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) .

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