Question 8

Suppose there is a surface S whose equation we don't know, but we want to find an equation for the tangent plane to S at the point (1, 2, 1). We do know that the curves with vector equations

r1(t) =< cos(t) 2, 3sin(t) + 2, 1 t >

and

r2(s) =< 1 + sin(s), 3cos(s) 1, 2s 2 + 1 >

lie on the surface S.

(a). Find equations for the tangent lines to r1(t) and to r2(s) at the point (1, 2, 1).

(b). Find an equation for the tangent plane to the surface S at the point (1, 2, 1).