Multivariable Calculus (Math 214) Assignment

due Monday, October 18, 1999




  1. Let f and g be the functions defined in space by

    f(x, y, z) = x^{2} + y and g(x, y, z) = x^{3} - x y^{2} - x + z .

    Let C be the set of points in space satisfying the simultaneous equations

    f(x, y, z) = 0 and g(x, y, z) = 0 .

    1. Find the point P in the set C with first coordinate 1.

    2. Find a vector that is normal at P to the surface f=0.

    3. Find a vector that is normal at P to the surface g=0.

    4. Under the assumption that C is a curve and that any tangent to C at the point P must lie in the intersection of the tangent plane at P to the surface f = 0 with the tangent plane at P to the surface g=0, find a vector that is tangent to the curve C.

    5. Find a parameterization of the curve C.

    6. Find a vector tangent to C at the point of C corresponding to an arbitrary given value of the parameter in your answer to the previous part.

  2. If a and b are given positive constants what type of plane curve is given parametrically by

    (x,y) = (a cos t, b sin t) ?

    Find an equation of the line that is tangent to this curve at the point on the curve corresponding to parameter value t = pi/4 when a = 1 and b = 2.

  3. Recall that the graph of a function f of two variables is the surface in space that is the set of solutions in space of the equation f(x, y) - z = 0. 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) .

  4. Find the maximum value of the function

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

    on the unit circle (x, y) = ( cos t, sin t).

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