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prelim2

# prelim2 - 4(14 points Find the points on the sphere x 2 y 2...

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Math 192, Prelim 2 October 25, 2007. 7:30-9:00 You are NOT allowed calculators, the text, or any other book or notes. SHOW ALL WORK! Write your name and Lecture/Section number on each booklet you use 1) Let the surfaces S 1 , S 2 be deﬁned by the equations z = x 2 + 2 y 2 + 2 and z = 5 x - y - 2, respectively. a) (2 points) Is the point (1 , 0 , 3) on these surfaces? b) (7 points) Find an equation of the plane tangent to the surface S 1 at the point (1 , 0 , 3). c) (7 points) Find a parametric equation of the line tangent to the curve of intersection of S 1 and S 2 at the point (1 , 0 , 3). 2) Consider the function f ( x, y ) = x ln y + ye x . a) (2 points) What is the domain of deﬁnition of f ? b) (4 points) Find the unit vector u describing the direction of steepest ascent of f at the point (0 , e ). c) (4 points) With u as in b), ﬁnd the directional derivative D u f (0 , e ). d) (4 points) Find the unit vectors v describing the directions of zero change of f at (0 , e ), that is, the unit vectors v such that D v f (0 , e ) = 0. 3) (14 points) Find the local minima and maxima of f ( x, y ) = x 4 + y 4 - 4 xy + 1.
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Unformatted text preview: 4) (14 points) Find the points on the sphere x 2 + y 2 + z 2 = 4 that are closest and farthest to (2 , 1 , 3). For each point you ﬁnd, decide if it is closest or farthest to (2 , 1 , 3). 5) In the plane, consider the region R intersection of the annulus { ( x, y ) : 1 ≤ x 2 + y 2 ≤ 4 } and the sector { ( x, y ) : 0 ≤ y ≤ x } . a) (4 points) Sketch the region R . b) (10 points) Compute the double integral R R R (1 + x + xy ) dxdy . 6) Let R be the region which is the intersection of the ﬁrst quadrant { ( x, y ) : x ≥ , y ≥ } and the inside of the parabola described by { ( x, y ) : x ≤ -4 y 2 + 3 } . a) (4 points) Sketch the region R . b) (10 points) If f ( x, y ) = xy 3 , compute R R R f ( x, y ) dxdy . 7) Consider the double integral R 2 R 4 y 2 y 3 x e x 2 dxdy . a) (4 points) Sketch the region of integration. b) (10 points) Compute R 2 R 4 y 2 y 3 x e x 2 dxdy ....
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