midterm 2006(2) sol

midterm 2006(2) sol - HKUST MATH 102 Second Midterm...

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MATH 102 Second Midterm Examination Multivariable and Vector Calculus 15 Dec 2006 Problem 1 (a) x + 2 y - z = 10 (b) x 2 + ( y - 1) 2 + z 2 = 1 which is the equation of a sphere with center (0 , 1 , 0) and radius 1. (c) The distance between two planes is d = ± ± ± ± ( a - b ) · ( u × v ) k u × v k ± ± ± ± . If they meet, then d = 0, i.e. ( a - b ) · ( u × v ) = 0 . Problem 2 (a) r 1 describes the equation of a parabola in the plane x = - 2. [vertex at ( - 2 , 0 , - 1), opening upward]. The required tangent line is r ( t ) = - 2 i + j + t ( j + 2 k ) . (b) From y 2 4 2 + z 2 2 2 = 1 , we know that this is an elliptic cylinder with its axis equal to the x -axis, therefore we let y = 4 cos θ z = 2 sin θ and from x + y = 4 , we have x = 4 - 4 cos θ where 0 6 θ 6 2 π . The curve of intersection is an ellipse on the x + y = 4 plane!! 4 4 2 y z x x + y = 4 y 2 4 2 + z 2 2 2 = 1 quarter of the ellipse The required projection curve C onto the xz -plane is ( x - 4) 2 16 + z 2 4 = 1 which is an ellipse centered at (4

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This note was uploaded on 03/13/2012 for the course MATH 102 taught by Professor Jimmyfung during the Spring '11 term at HKUST.

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midterm 2006(2) sol - HKUST MATH 102 Second Midterm...

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