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Unformatted text preview: Name Part 1 (50 points) Calculus III, Final Exam f- " , ,I \ I. (24pts.) Given the points (1,-2,3),(4, 1,-2),and(-2,-3,O). Find ! . ( (a) a vector perpendicular to the plane containing these points; (b) the equation of the plane containing these points; (c) the area of the triangle whose vertices are these three points. 2. (26 points) A particle is moving along the path r(t) =(sin : 1 ,cos : 1 ,1),1;,:O. (a) (4 pts.) What is the initial point 1'(0) of this path? (b) (10 pts.) At what poinll > 0 does the path r(t) =(sin Jr 1 ,cos Jr 1 ,1) intersect the sphere . 6 6 x'+ y' +z' =5. (c) (12 pts.)}Vhat is the unit tangent vectorto the path r(t) at the point found in part (b)? .J "\(part 2 (50 pomts) ( " I. \ (30 pt~.) (a) Sketch the level curve of j(x, y) = x 2 + 4 y 2 that passes through the point /1 (x,y)- (2, I). (b) Calculate the gradient vector at (2, I) for j(x, y).lf the gradient vector is drawn with its initial point at (2, I) on the level curve of part (a), what property must it have?initial point at (2, I) on the level curve of part (a), what property must it have?...
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- Spring '08