Multivariable Calculus (Math 214) Assignment

due September 29, 1999

1. Find the equations of the lines that are (a) normal and (b) tangent to the plane curve

   sin (x y) = 0

   at the point (pi, 2).

2. Let S be the surface in space that is the set of all solutions (x, y, z) of the equation

   x^{2} y + y^{2} z + z^{2} x - 3 = 0 ,

   and let P be the point (1, 1, 1), which is easily verified to be a point of S.

   1. Find a vector that is normal to S at P.

   2. Find the equation of the plane that is tangent at P to S.

3. Find the center and the radius of the circle in space that passes through the unit points on each of the three positive coordinate axes.

4. (Continuation of exercise 2 on the second previous assignment) Let L be the line passing through the point P = (1, -3, 2) that is perpendicular to the sphere

   x^{2} + y^{2} + z^{2} = 14

   at the point P.

   1. Find two distinct planes in space that contain the line L.

   2. Parametric representation of the tangent plane. Find points A, B, and C so that the plane that is tangent to the sphere at the point P may be characterized in the arithmetic of points as the set of all points of the form

      C + uA + vB

      as u and v vary over all possible pairs of real numbers.