Multivariable Calculus (Math 214) Assignment

due Monday, October 25, 1999




  1. The curve in space that is given parametrically by the function

    g(t) = (t, t^{2}, t^{3})

    is called a twisted cubic.

    1. Find a parametric representation of the line that is tangent to it when t=0.

    2. Find a parametric representation of the line that is tangent to it when t=1.

    3. Find a parametric representation of the line that is tangent to it when t=-1.

    4. Do these tangent lines all line in the same plane?

  2. Recall that the graph of a function f of two variables is the surface in space that is the set G_{f} of solutions in space of the equation f(x, y) - z = 0. This question asks you to express various things in terms of a given function f and its derivatives. First, persuade yourself that G_{f} is precisely the surface in space that is parameterized by the function

    R(u, v) = (u, v, f(u, v)) .

    1. Let P be the point on this surface for which u = 3 and v = 5. Express the point P in terms of f.

    2. Find the vector A that is tangent at the point P to the curve that is parameterized by the function alpha(t) = R(t, 5).

    3. Find the vector B that is tangent at the point P to the curve that is parameterized by the function beta(t) = R(3, t).

    4. Remembering that f is given and that the objects P, A, and B are constant, expand the parametric formula

      P + u A + v B

      for the plane tangent at P to G_{f} in terms of the previously computed values of A, B, and P.

    5. Find, in terms of f, a vector that is normal to the plane parameterized by the formula of the preceding part.

    6. How does the normal that you found in the preceding part relate to the gradient of the function

      F(x, y, z) = f(x, y) - z ?

  3. Find the maximum value of the function

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

    on the unit sphere x^{2} + y^{2} + z^{2} = 1.

  4. Given a subset S of space that is either a curve or a surface and a point P in space that is not in S, why must a line segment of minimum length drawn from P to a point of S always be normal to S ? Can the same statement be made about a line segment of maximum length, if there is one, drawn from P to a point of S ?

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