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Quiz 3 MATH 251, Section 505
Due, October, 8th, 2015
last name : . . . . . . . . . . . . . . . . . .
first name : . . . . . . . . . . . . . . . . . .
”An Aggie does not lie, cheat or steal, or tolerate those who do”
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Write up your result, detail your calculations if necessary and BOX your final answer.
Your final work have to be neat, so use pencil first if you want.
1. [8pts] Find the equation of the plane passing through A(1, 2, 3), B(−2, 4, 2) and C(1, 2, 1).
2. [8pts] Determine whether the planes P1 : 2x + y = 1 − 2z and P2 : 2x − 3z = −4y + 5 are parallel,
orthogonal, or neither. Find cosine of the angle between the planes.
3. [10pts] Find the domain of z = f (x, y) =
z ≥ 0?
p
9 − 3x2 − 3y 2 and sketch it. What is the range of f when
4. [8pts]Let C be the curve with equations r(t) = ht2 , 2t − 2, ln(t + 3)i. Find an equation of the tangent
line to C at (1, 0, 2 ln 2).
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5. [10pts]Classify the surface (reduce the equation to one of the standard forms if necessary) given by
x2 + 10x + y + z 2 − 4z + 29 = 0 and sketch it (precise axes and some points).
6. [10pts] Identify the level curves of the surface given by z = e−(2x
7. [15pts] Find the second partial derivatives of f (x, y) =
√
2 +2y 2 )
.
2
y + 5yex − sin(x2 ).
8. Given the surface xy 2 z 3 = 12 and the point P (3, 2, 1).
(a) [5pts] Find an equation of the tangent plane to the given surface at P .
(b) [8pts] Find the equation of the normal line at P to the given surface.
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y
+ ez .
x
(a) [5pts] Find the direction in which f increases most rapidly at the point (2, 1, 0).
9. Let f (x, y, z) = y 6 +
(b) [5pts] Find the directional derivative of f at the point (2, 1, 0) in the direction < 4, 2, −1 >.
10. Let f (x, y) = 2x2 y + y 3 − 2x2 − 2y 2 + 10.
(a) [10pts] Find all critical points of f .
(b) [10pts] Classify the critical points (local maximum, local minimum or saddle points).
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