(937, #9) Match the equations with the graphs. Give reasons for your answer.
1. r(u,v) = < u cos(v), u sin(v), v >
2. r(u,v) = < u cos(v), u sin(v), u >
3. x = (u-sin(u)) cos(v), y = (1-cos(u)) sin(v), z = u
4. x = (1-u)(3+cos(v))cos(4*Pi*u),
y = (1-u)(3+cos(v))sin(4*Pi*u), z = 3 u + (1-u) sin(v)
I

II

III

IV

Solution: The relations of equation 2, x =u cos(v), y = u sin(v), z = u imply that which is the equation of a cone. So, equation 2 corresponds to III.
Equation 1 has a circular domain: . Thus, by elimination, equation 1 corresponds to I.

In equation 3 the values of x, y, and z are all small when u and v are small. In equation 4 the value of x is about 3 when u and v are small. So, by comparison, equation 3 corresponds to II; equation 4 corresponds to IV.