Math 251 Questions for Ch.12 from past exams
1. G ( x, y , z )  3x  y  z
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J. Lewis
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a) Find an equation of the tangent plane to G ( x, y, z )  22 at the point P ( 2, 1,  3) .

b) Find the rate of change of G in the direction of w  2,  6, 4 from the
point, P( 2, 1,  3) .
c) Find the maximum rate of change of G from the point P ( 2, 1,  3) .
2. Use Clairaut's theorem to show there is no function, f(x, y) with continuous partial
derivatives and for which f x  ye x
2y
and f y  xe x
2y
.
3. Find and classify, by the 2nd derivative test, the four critical points of
f ( x, y )  x 3  xy 2  12 xy  27 x .
4. Give parametric equations for the tangent line to the curve of intersection of the
surfaces 4 x  9 y  36 z  0 and 4 x  2 y  z  48 at the point P(3, 2, 2).
You do not need to find the curve.
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2
2
2
2
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5. f ( x , y )  xe xy
a) Show that the graph of f has no local max or min.
b) Verify Clairaut's theorem in this case, that is show f xy  f yx .
6. f ( x, y )  x 3 y 2


a) Find the directional derivative of f(x,y) in the direction of the vector i  4 j from the
point P ( 1, 2,  4) .
b) What is the maximum rate of change of f(x,y) from the point P ( 1, 2,  4) ?
c) Find the equation of the tangent plane to the surface at the point P ( 1, 2,  4) .
7. f ( x, y )   x 3  xy 2  6 xy .
Find all critical points. For each critical point, give the conclusion of the 2nd derivative
test or show the test has no conclusion.
8. Use differentials to approximate ( 26  3 9 ) .
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9. z  x e
3 y
x  x( s, t )
10. z  f ( x, y )
xs t
3 2
y  y ( s, t )
y  s t
4
Find
4
2z
s 2
.
2z
Find
.
t s