Problem on parametrized Surfaces

Problem A of Week 4

Problem A: Lets look at the parametrized surface

r(t,s) =  < t cos(s), t sin(s), s t >

where s is between 0 and 6Pi and t is between 0 and 1. We call it the rose surface.
a) (2 points) If t is kept constant (like for example t=1), we obtain a grid curve on the surface. Identify these space curves algebraically and by name. Similarly, if s is kept constant (like for example s=0), we obtain grid curves on the surface. Identify these space curves by formula and by name.
b) (3 points) Find a formula for the curve obtained by intersecting the surface with the plane z=1. Sketch these planar curves in the plane. Find also the intersection of the surface with the plane y=0 as well as with the plane x=0 and sketch them in the xz-plane and yz-plane.

c) (3 points) Sketch the surface by making use of part a) and b). You are welcome to use use Mathematica if you like (*) but you can do the problem without technology.
d) (1 points) If the parameter range of the s parameter is increased from [0,6Pi] to [0,10 Pi], the shape of the rose changes. Check what applies

A: The surface gains more turns (it appears as if the rose gets more leaves)
B: The rose gets taller (the z coordinate is the "up" direction)
C: The rose gets wider (the x,y coordinates increase)

No explanations are needed in d). Just note the letters in A,B,C which apply.
e) (1 points) If the range of the t parameter is increased from [0,1] to [0,3], the shape of the rose changes. Check what applies.

A: The surface gains more turns (it appears as if the rose gets more leaves)
B: The rose gets taller (the z coordinate is the "up" direction)
C: The rose gets wider (the x,y coordinates increase)

No explanations are needed in e). Just write down the letters in A,B,C which apply.

(*) Here is an example on how to use it: to plot a cylinder in Mathematica, enter

ParametricPlot3D[ { Cos[s], Sin[s],t },{s,0,2Pi},{t,0,1}]

into a new cell, click into the cell, hold the shift key and hit return. See the lab page.