22.10. EXERCISES 485

(a) sin(x2 + y

)+ z2 at (0,π/2,1) in direction of (1,1,2).

(b) x(x+y)+ sin(zx) at (1,0,0) in the direction of (2,−1,0).

(c) zsin(x)+ y at (0,1,1) in the direction of (1,1,3).

4. Find the tangent plane to the indicated level surface at the indicated point.

(a) x2y+ z3 = 2 at (1,1,1)

(b) zsin(x2y)+2x+y = 2sin1+4 at (1,1,2)

(c) cos(x)+ zsin(x+ y) = 1 at(−π, 3π

2 ,2)

5. The point(

1,1,√

2)

is a point on the level surface x2 + y2 + z2 = 4. Find the lineperpendicular to the surface at this point.

6. The level surfaces x2 +y2 + z2 = 4 and z+x2 +y2 = 4 have the point(√

22 ,

√2

2 ,1)

inthe curve formed by the intersection of these surfaces. Find a direction vector for thiscurve at this point. Hint: Recall the gradients of the two surfaces are perpendicularto the corresponding surfaces at this point. A direction vector for the desired curveshould be perpendicular to both of these gradients.

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. ∗ Suppose f1 (x,y,z) = 0 and f2 (x,y,z) = 0 are two level surfaces which intersectin a curve which has parametrization, (x(t) ,y(t) ,z(t)). Find a system of differen-tial equations for (x(t),y(t),z(t)) where the point determined by (x(t),y(t),z(t)) as tvaries, moves over the curve.