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, 2s² + 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).