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Unformatted text preview: Math 20C  Final (Lecture D, Winter 2008) Duration: 3 hours Please close your books and turn off your phones. You can use one page of handwritten notes. To get full credit, you should support your answers. 1. Consider the points A (1 , 3 , 2) and B (3 , 5 , 4). a) (2 points) Calculate the length of the segment AB and the area of the triangle OAB , where O is the origin. Solution.  AB  = radicalbig (3 1) 2 + (5 3) 2 + (4 2) 2 = 4 + 4 + 4 = 12 = 2 3 . vector OA vector OB = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vector i vector j vector k 1 3 2 3 5 4 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = 2 vector i + 2 vector j 4 vector k. Area ( OAB ) =  vector OA vector OB  2 = 2 2 + 2 2 + 4 2 2 = 24 2 = 6 . b) (3 points) Find an equation of the set of all points equidistant from the points A and B . Describe the set. Solution. Let P ( x, y, z ) be a point equidistant to A and B . It means that  PA  =  PB  which implies radicalbig ( x 1) 2 + ( y 3) 2 + ( z 2) 2 = radicalbig ( x 3) 2 + ( y 5) 2 + ( z 4) 2 . Squaring and expanding the brackets, we get that x 2 2 x + 1 + y 2 6 y + 9 + z 2 4 z + 4 = x 2 6 x + 9 + y 2 10 y + 25 + z 2 8 z + 16 Canceling the x 2 , y 2 , z 2 we get that 2 x 6 y 4 z + 14 = 6 x 10 y 8 z + 50 which implies 4 x + 4 y + 4 z = 36 Thus, x + y + z = 9. This is a plane. 2. Let f ( x, y ) = xe x + y . a) (2 points) Calculate the first order partial derivatives f x and f y . Solution. f x = 1 e x + y + xe x + y = (1 + x ) e x + y . f y = xe x + y . b) (3 points) Calculate the second order partial derivatives f xx , f xy , f yx and f yy . Solution. f xx = 1 e x + y + (1 + x ) e x + y = (2 + x ) e x + y . f xy = f yx = (1 + x ) e x + y . f yy = xe x + y . 3.a) (2 points) Find parametric equations for the line through (5 , 1 , 0) that is perpendicular to the plane 2 x y + z = 1....
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 Winter '08
 Helton
 Math

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