prelim2_spring2005

prelim2_spring2005 - ~ j b ± 3 points If f x y z is a...

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Fall 05: Ignore Questions 6-7-8 which are on sections NOT COVERED by our prelim 2. Math 192, Prelim 2 April 14, 2005 1) a) (7 points) Give an example of a function f ( x, y ) that has a local maximum at (0 , 0). Show that your example is valid. b) (7 points) Give an example of a function f ( x, y ) that has a saddle point at (0 , 0). Show that your example is valid. 2) (11 points) Determine all the maxima and minima of the function f ( x, y ) = x 2 + xy on the region of points ( x, y ) satisfying x 2 + xy + y 2 1. 3) (11 points) Find the average distance from a point in the interior (or on the boundary) of a circle of radius R to the center of the circle. 4) (11 points) Let C be a circle of radius 2 + 2 2 centered at the origin. Find the area of the region outside C but inside the region bound by r = 2 + cos θ . 5) The questions below are true/false. You need not give reasons. You will be penalized for wrong answers, just like the SAT’s. So do not guess!! a) ( ± 3 points) If f ( x, y ) is given and ~u is a unit vector in two dimensions then the directional derivative of f in the direction of ~u at ( x 0 , y 0 ) is f x ( x 0 , y 0 ) ~ i + f y ( x 0 , y
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Unformatted text preview: ) ~ j . b) ( ± 3 points) If f ( x, y, z ) is a scalar function then the expression grad(div( f )) is meaningful. c) ( ± 3 points) If ~ F ( x, y, z ) is a vector field then the expression curl(div( ~ F )) is meaningful. d) ( ± 3 points) The integrals Z 10 Z x/ 10 e x cos y dy dx and Z 1 Z 10 10 y e x cos y dx dy are equal. 6) (11 points) Consider the intersection C of the cylinder x 2 + y 2 = 4 and the plane x + y + z = 5. Find R C ~ F · d~ r where ~ F ( x, y, z ) = y ~ i + z ~ j + x ~ k and C is oriented counterclockwise when looking down upon it. 7) a) (7 points) Compute Z 2 Z √ 2 x-x 2 Z 1 z p x 2 + y 2 dz dy dx . b) (7 points) Z 1-1 Z √ 1-x 2 Z √ 1-x 2-y 2-√ 1-x 2-y 2 p x 2 + y 2 + z 2 dz dy dx . 8) (11 points) Let ~ f ( x, y, z ) = (2 x + y 2 + 3 x 2 y ) ~ i + (2 xy + x 3 + 3 y 2 ) ~ j . a) Show ~ F is conservative. b) Let C be the curve ~ r ( t ) = t ~ i + t sin t ~ j with 0 ≤ t ≤ π . Find Z C ~ F · d~ r ....
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This note was uploaded on 06/01/2008 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell.

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