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**Unformatted text preview: **Math 192, Final December 6, 2007. 2:00-4:30 You are NOT allowed calculators, the text, or any other book or notes except for a one recto/verso letter format sheet. SHOW ALL WORK! Write your name and Lecture/Section number on each booklet you use 1) Let f ( x, y ) = x 2 + y 2- 2 x and consider the region R in the upper half-plane { y ≥ } enclosed by the curves x 2 + y 2 = 1 and x 2 + ( y/ 2) 2 = 1. a) (4 points) Sketch the region R . b) (12 points) Find the absolute minimum and maximum values of f in the region R and the points at which they occur. 2) (16 points) Consider the surfaces S 1 : z = x 2- y 2 and S 2 : xyz + 30 = 0. Find a parametric equation of the tangent line to the curve of intersection of S 1 and S 2 at the point (- 3 , 2 , 5). 3) In the plane, consider the vector field F ( x, y ) = e x (1- cos y ) i + e x (sin y- y ) j and the region R = { ( x, y ) : 0 ≤ x ≤ π, ≤ y ≤ sin x } . Let C be the closed curve bounding the region R , oriented counter-clockwise....

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