Homework 9
Math 352, Fall 2014
Due Date: Friday, November 7
~
1. Let S be the surface parameterized by X(u,
v) = (u2 + 6v, 3u + 9v, u2 + 2u + v 2 ).
~
(a) Find the critical point for the function X.
(b) Find a point on S whose tangent plane is parallel to the xy-plane.
2. Let P be the parabola x = z 2 − 1 in the xz-plane, and let S be the surface obtained
by rotating P around the z-axis.
~ : R2 → R3 for the surface S.
(a) Find a parametrization X
~ What is happening at the correspond(b) Find the critical points for the function X.
ing points on the surface?
3. Let F~ : R3 → R2 be the function F~ (x, y, z) = x, x2 + y 2 + z 2 .
(a) Find the set of critical points for F~ .
(b) Let F1 (x, y, z) = x and F2 (x, y, z) = x2 + y 2 + z 2 be the component functions of F~ .
What are the level surfaces for these functions?
(c) What is the geometric relationship between the level surfaces for F1 and F2 at
the critical points? Explain this phenomenon.
4. Let F~ : R2 → R2 be the function F~ (x, y) = 2x3 + y 3 , x2 + 2y 2 . Then the set of critical
points for F~ is the union of three lines. Find the equations of these lines.
~ : R2 → R3 be a regular parametrization of a surface S, and suppose that
5. Let X


1 1


~
~
X(3,
9) = (5, 3, 1)
and
dX(3,
9) = 2 1
2 0
(a) Find the angle between the u and v coordinate lines for S at the point (5, 3, 1).
(b) Find the equation of the tangent plane to S at the point (5, 3, 1).
~
(c) Find ~x 0 (9) if ~x is the curve ~x(t) = X(3,
t).
~ t2 ).
(d) Find ~y 0 (3) if ~y is the curve ~y (t) = X(t,
~
(e) Estimate the value of X(3.02,
9.05).
~
6. Let S be the helicoid X(u,
v) = (u cos v, u sin v, v).
√
√
√
~ v 2, π/4 , and dX
~ 2, π/4 .
2, π/4 , X
√
√
~ u 2, π/4 and X
~ v 2, π/4 .
(b) Express the vector (1, 5, 2) as a linear combination of X
~u
(a) Compute X
(c) Find the formula for any space curve ~x(t) that lies on the helicoid S for which
~x(0) = (1, 1, π/4) and ~x 0 (0) = (1, 5, 2).