Problem C: The surface S with parametrization

r(u,v) = (cos(v) cos(u), cos(v) sin(u),v)
is shown to the left.

a) (4 points) Find an implicit equation g(x,y,z)=0 for this surface, the linearization L(x,y,z) of g at the point P=(1/2,1/2,Pi/4) and the equation L(x,y,z) = C with constant C=g(1/2,1/2,Pi/4). This level surface is the tangent plane at P.

b) (2 points) Why are the vectors ru(u,v) and rv(u,v) tangent to S at P? Compute them

c) (4 points) Use b) to find the equation of the tangent plane at the point P again. You should get the same plane as in a)