17.11. EXERCISES 327

7. For v a unit vector, recall that Dv f (x) = ∇ f (x) ·v. It was shown above that thelargest directional derivative is in the direction of the gradient and the smallest in thedirection of −∇ f . Establish the same result using the geometric description of thedot product, the one which says the dot product is the product of the lengths of thevectors times the cosine of the included angle.

8. The point(

1,1,√

2)

is on the level surface x2 + y2 + z2 = 4 and the level surface

y2 + 2z2 = 5. Find an equation for the line tangent to the curve of intersection ofthese two surfaces at this point.

9. ∗In a slightly more general setting, suppose f1 (x,y,z) = 0 and f2 (x,y,z) = 0 are twolevel surfaces which intersect in a curve which has parametrization, (x(t) ,y(t) ,z(t)).Find a system of differential equations for (x(t),y(t),z(t)) where as t varies, the pointdetermined by (x(t),y(t),z(t)) moves over the curve.