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Quiz 3 MATH 251, Section 506
Due, February, 19th, 2015
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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. [10pts] Find an equation of the plane passes through the point (1, 0, 2) and parallel to the plane
2x + y + 5z = 4.
2. [10pts] Determine whether the planes P1 : x + y = 1 + z and P2 : 2x + 4z = 3y + 5 are parallel,
orthogonal, or neither. Find cosine of the angle between the planes.
3. [20pts]Classify the surface (reduce the equation to one of the standard forms if necessary) and sketch
it (precise axes and some points).
(a) (x − 1)2 + 4z 2 = y 2 .
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(b) x2 + 4x + 2y + z 2 − 10z + 29 = 0.
4. Let C be the curve with equations r(t) = h2 − t2 , 2t − 1, ln ti.
(a) [4pts] What is the point where C intersects the xz-plane ?
(b) [8pts] Find an equation of the tangent line to C at (1, 1, 0).
5. [10pts] Identify the level curves of the surface given by z = e−(x
2 +y 2 )
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6. [15pts]Find the second partial derivatives of f (x, y) = y 3 + 5y 2 e4x − cos(x2 ).
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7. Let f (x, y, z) = zexy .
(a) [4pts] Find the direction in which f increases most avidly at the point (0, 1, 2).
(b) [8pts] Find the directional derivative of f at the point (0, 1, 2) in the direction u = h2, 1, −2i.
8. Let f (x, y) = 3x2 y + y 3 − 3x2 − 3y 2 .
(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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