Multi-Variable Calculus Assignment

due Tuesday, March 6, 2001

 Exercises

1. Let f be the function defined in R^{2} by

   f(x, y) = x^{2} - x y + y^{2} - x + 3 ,

   and let (u, v) be a given unit vector in R^{2}. Find the following:

   1. A formula for the directional derivative of f at the point (1, -2) in the direction of (u, v).

   2. The maximum value of the various directional derivatives of f at the point (1, -2).

2. Find a parametric representation for the line tangent at the point (2, 1, -2) to the piece of curve that is the intersection of the surface

   x y - z^{2} + 2 = 0

   with the sphere

   x^{2} + y^{2} + z^{2} = 9 .

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