326 CHAPTER 17. THE DERIVATIVE OF A FUNCTION OF MANY VARIABLES
17.11 Exercises1. Find the gradient of f =
(a) x2y+ z3 at (1,1,2)
(b) zsin(x2y)+2x+y at (1,1,0)
(c) u ln(x+ y+ z2 +w
)at (x,y,z,w,u) = (1,1,1,1,2)
(d) sin(xy)+ z3 at (1,π,1)
(e) ln(x+ y2
)z
(f) z ln(4+ sin(xy)) at the point (0,π,1)
2. Find the directional derivatives of f at the indicated point in the direction(
12 ,
12 ,
1√2
).
(a) x2y+ z3 at (1,1,1)
(b) zsin(x2y)+2x+y at (1,1,0)
(c) xy+ z2 +1 at (1,2,3)
(d) sin(xy)+ z at (0,1,1)
(e) xy + z at (1,1,1).
(f) sin(sin(x+ y))+ z at the point (1,0,1).
3. Find the directional derivatives of the given function at the indicated point in theindicated direction.
(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 ,√
22 ,1
)in
the 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.