Math 6C – Quiz 3 – Spring 2005
NOTE: For each problem, answer (E) if none of the given answers are correct!
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1. Find
g
 gx
x
for the function g(x, y) = ln(x2 + xy3).
(A) 1
(B) 1/2
Then gx(1, 1) = ?
(D) –1/2
(C) 3/2
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at (2, 1, ).
2. Find the linearization of f (x, y, z) = x2y2 + z3sin(xz)
(A) 4x + 8y – 12
(B) 2x + 8y – 6
L(x,y,z) = ?
(C) 2x + 6y – 8
(D) 4x + 6y – 12
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Find the gradient of f (x, y, z) = 2x2y + 3z3 at the point (1,2,1). The z-component of this
gradient is _____ .
3.
(A) 9
(B) 10
(C) 11
(D) 12
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4. Find the directional derivative of the f in problem #3, in the direction of
Du f (1,2,1)  ?
(A) 9
(B) 10
u  1  2,1,2 at the point (1,2,1).
3
(C) 11
(D) 12
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5. Find the equation of the tangent plane to x5 – xyz + z2 = 7
(A) 7x + 2y – 5z = 18
(B) 7x + 2y – 6z = 18
at the point (1,1,–2).
(C) 7x + 2y – 5z = 19
(D) 7x + 2y – 6z = 19
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6. Find the equation of the normal line to x5 – xyz + z2 = 7 at the point (1,1,–2).
 x   7t  1 
  

(A) y   2t  1
  

 z   5t  2 
 x   7t  1 
  

(B) y  2t  1
  

 z    5t  2
 x    7t  1
  

(C) y   3t  1
  

 z   5t  2 
 x   7t  1
  

(D) y  3t  1
  

 z  5t  2
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7. Find the maximum value of
(A)
2 3
2
f ( x, y)  x2  3 xy  y 2
(B)
2 3
2
(C)
on the unit disk,
x2  y 2  1 .
3 2
2
(D)
3 2
2
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8. If a tachyon field intensity is described by T(x, y, z) = 5exz – 3exy, then the tachyon field is increasing at
(0, 1, ) most rapidly in what direction ?
(A) (0, 2, )
(B) (0, 2, )
(C) (0, 1, )
(D) (2, 0, )
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9. How fast is the tachyon field increasing in the direction of maximum increase that you found in #8?
(A)
10
(B)
5
(C)
2
(D) 2
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10. Find any local extrema and saddle points of
(A) maximum at (8,7)
f ( x, y)  2x2  3 y2  5xy  3x  2 y .
(B) saddle point at (8,7)
(C) minimum at (7,8)
(D) saddle point at (7,8)
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11. Find the tangent plane to the surface defined by
(A)
1 y
(B)
z  G( x, y)  2x2 y  e xy
1 x  y
(C)
1 x  y
z=?
at (1,0,1).
(D)
1 x  y
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12. Find the equation of the tangent plane to the implicit surface defined by
at the point (1,1,1).
(A)
x  6 y  3z  4
x  5 y  3z  4
(B)
 x2  3xy 2  z3  1
x  5 y  3z  4
(C)
(D)
x  6 y  3z  4
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f (2,3)  4 and f '(2,3)  0 0 . Further suppose that the Hessian of f evaluated at (2,3),
13. Suppose that
 f xx

 f yx
f xy    2 3 

.
f yy   3  1
(A) relative maximum
One can conclude that at the point (2,3,4), the graph of f has a . . .
(B) relative minimum
(C) saddle point
(D) point of inflection
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14. Suppose that
 g xx

 g yx
g (2,3)  4 and g '(2,3)  0 0 . Further suppose that the Hessian of g evaluated at (2,3),
g xy   5  3
.

g yy    3 4 
(A) relative maximum
One can conclude that at the point (2,3,4), the graph of g has a . . .
(B) relative minimum
(C) saddle point
(D) point of inflection
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g ( x, y)  2  2x  2 y  x2  y 2
15. Find the absolute maximum value of
by the x and y axes and the line y = 9 – x.
(A) 2
over the triangle in Quadrant I bounded
(B) 3
(C) 4
(D) 5
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16. Find the equation of the tangent plane to
(A)
4 y
(B)
g ( x, y)  2  2x  2 y  x2  y 2
4 x y
(C)
at the point (1,1).
4 x
(D)
z=?
4
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17. Find the greatest value that
2
y2 .
f ( x, y)  xy attains over the closed elliptic disk x 
1
8
2
(B) –1
(A) 2
(C) –2
(D) 4
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18. Find the greatest value that
(A) 3
f ( x, y)  3x  4 y attains on the closed unit disk x2  y 2  1 .
(B) 5
(C) 7
(D) 9
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19. Find the minimum distance from the surface at
(A)
10
(B)
xyz  1 to the origin.
5
(C)
3
(D)
2
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20. What’s the largest possible product n positive numbers can have, given that their sum is 1?
(A)
1
n
(B)
1
nn
(C)
n 1
n2
(D)
n 1
nn
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