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