FINAL-v1-solutions

FINAL-v1-solutions - Math233 Final Exam - Solutions Fall...

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Unformatted text preview: Math233 Final Exam - Solutions Fall 2006 Problem 1. Determine if the line that passes through the point (0 ,- 3 , 8) and is parallel to the line given by x = 10 + 3 t,y = 12 t and z =- 3- t passes through the xz-plane. If it does give the coordinates of that point. The equation of the line is x = 3 t,y =- 3 + 12 t and z = 8- t . It passes through the xz-plane if there is a t such that y = 0 , i.e. when- 3 + 12 t = 0 , which gives t = 1 / 4 . We plug this value of the parameter into the equation of the line to find the coordinate of the intersecting point, we get (3 / 4 , , 31 / 4) . Problem 2. Find the equation of the plane that contains the following intersecting lines: L 1 : x- 1 3 = y + 3 5 = z- 4- 1 ; L 2 : x = 2 t + 1 y = t- 3 z = 4 . First, notice that the point (1 ,- 3 , 4) belongs to both lines, showing that the lines intersect. Next, we can find a normal vector ~n for the plane by taking the cross product of the two direction vectors of the lines. L 1 has direction vector ~u = (3 , 5 ,- 1) and L 2 has direction vector ~v = (2 , 1 , 0) . Therefore ~n = ~u ~v = (1 ,- 2 ,- 7) . The equation of the plane is then ( x- 1)- 2( y + 3)- 7( z- 4) = 0 , or x- 2 y- 7 z + 21 = 0 . Problem 3. Consider the function f ( x,y ) = 2 x 2- y 2 + 6 y defined on the disk x 2 + y 2 16 . (a) Find all critical points and the value of f at those points....
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FINAL-v1-solutions - Math233 Final Exam - Solutions Fall...

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