In each case, evaluate the directional derivative off for the points and directions specified: (a) f(x,y,z)=3x−5y+2z at (2,2,1) in the direction of the outward normal to the sphere x
2
+y
2
+z
2
=9 (b) f(x,y,z)=x
2
−y
2
at a general point of the surface x
2
+y
2
+z
2
=4 in the direction of the outward normal at that point. (c) f(x,y,z)=x
2
+y
2
−z
2
at (3,4,5) along the curve of intersection of the two surfaces 2x
2
+2y
2
−z
2
=25 and x
2
+y
2
=z
2
. 4. (a) Find a vector V(x,y,z) normal to the surface z=
x
2
+y
2
+(x
2
+y
2
)
3/2
at a general point (x,y,z) of the surface, (x,y,z)
=(0,0,0). (b) Find the cosine of the angle 0 between V(x,y,z) and the z-axis and determine the limit of cosθ as (x,y,z)→(0,0,0). 5. The two equations e
u
cosv=x and e
u
sinv=y define u and v as functions of x and y, say u=U(x,y) and v=V(x,y). Find explicit formulas for U(x,y) and V(x,y), valid for x>0, and show that the gradient vectors ∇U(x,y) and VV(x,y) are perpendicular at each point (x,y). 6. Let f(x,y)=
∣xy∣
. (a) Verify that ∂f/∂x and ∂f∣∂y are both zero at the origin. (b) Does the surface z=f(x,y) have a tangent plane at the origin? [Hint: Consider the section of the surface made by the plane x=y.]