Surfaces
Reading Assignment: 7.1
Recommended Problems: 7.1: 1, 3, 5, 6, 7, 11, 12, 13, 16, 17, 18, 19, 23, 26
1) The parametric surface given by X(s, t )   s cos t , s sin t , t  is called a helicoid. (Note that it
differs from the cone’s parameterization in the z-coordinate.) It is like a spiral staircase. Sketch
a portion of this surface that is two units wide (0  s  2) and two complete twists (0  t  4).
Find its area.
2) Consider the graph of z  xy x  2 . We’ll look at the point (x, y, z) = (2, 1, 4)
a) Find an equation for its tangent plane the “old way” (with partial derivatives).
b) Parameterize the surface.
c) Find an equation for its tangent plane the “new way” (with the T vectors).
3) The graph of a function z = f(x, y) can always be parameterized as (x, y, f(x, y)). Show that in
this case the “old way” (section 2.3, formula 4) and “new way” (section 7.1, formula 5) for
finding the tangent plane at a point always agree.
4) Find the surface area of the part of the surface z = 16 – x2 – y2 which lies above the xy-plane. .
5) Let (x(t), y(t)), a  t  b, be a parametric curve in the plane with y(t)  0 for all t. Rotate this
curve around the x-axis (like in BC2!). Parameterize, and find an integral formula for the area
of, the resulting surface.