Chapter 12 Review
Math 251
J. Lewis
1. Verify Clairaut's theorem for f ( x , y )  xe
xy 2
.
2. Find the equation of the tangent plane to each function at the given point.
a)
f ( x , y )  x 2  y 2  2 xy 3
b)
f ( x , y )  ln(1  x 2 y )
P (1,1,4 )
P (1, e  1,1)
3. a) Find the maximum rate of change of f ( x, y )  x e from the point P(1,0,1) and
in what direction does it occur? (Give a unit vector for the direction.)
2 xy
b) Find the directional derivative of f(x,y) in the direction of <4,1>.
4. Show that z = f(x,y) has no local extreme points.
a) f ( x , y )  xe x
2  y2
b) f ( x, y )  x 2  xe y
5. Find and classify, if possible, all critical points of each function.
a)
f ( x , y )  3 xy 2  4 x 3  6 y 2
c)
f ( x , y )  9 xy  3 x 3  y 2
d)
f ( x , y )  6 xy 2  2 x 3  3 y 4
e)
f ( x , y )  y  e xy
f)
f ( x, y )  6 x 2 y 2  4 x 3  6 y 4
b)
f ( x , y )   3 x 2  2 xy  y 2  6 x  4 y
6. Find the equations of the tangent line to the intersection of the surfaces
z  3 x 2  9 y 2 and z 2  x 2  y 2  4 at the point P(2,1,3). You do not need to
find the intersection, only the tangent line.
7. Use the chain rule to find
a)
z  ex
8. Find
2
2z
ts
y
b)
for
2z
s 2
if
x  x ( s, t )
y  y ( s, t ) .
z  x 2 y  xy 2
z  f ( x, y )
9. Word problems 33-38 Section 12.4
x  s 2 cos t
y  s 2 sin t
10. Find an equation of the tangent plane to the given level surface at the given point.
a)
F ( x , y , z )  x 2 yz 3  96
b)
F ( x, y, z )  x 2 e y
2
z
 9e 2
P ( 2,3, 2 )
P (3,1, 2 )