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midterm1 2007

# midterm1 2007 - HKUST MATH 102 Midterm One Examination...

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HKUST MATH 102 Midterm One Examination Multivariable and Vector Calculus 30 Oct 2007 Answer ALL 5 questions Time allowed – 120 minutes Problem 1 Identify the following surfaces (a) r · b u = 0. (b) ( r - a ) · ( r - b ) = k . (c) k r - ( r · b u ) b u k = k . [Hint: What are the vectors ( r · b u ) b u and r - ( r · b u ) b u ?] Here k is fixed scalar, a , b are fixed 3D vectors and b u is a fixed 3D unit vector and r = ( x, y, z ). Problem 2 (a) Find the velocity, speed and acceleration at time t of the particle whose position is r ( t ). Describe the path of the particle. r = at cos ωt i + at sin ωt j + b ln t k (b) Find the required parametrization of the first quadrant part of the circular arc x 2 + y 2 = a 2 in terms of arc length measured from (0 , a ), oriented clockwise. (c) Let C be the curve x 2 / 3 + y 2 / 3 = a 2 / 3 on the xy -plane, find the parametric equation of the curve C . Hence find the tangent line to the curve C at ( a, 0). Problem 3 (a) Assume a , b , c , x and y are three dimensional vectors and if ( a × b ) · ( b × c ) × ( c × a ) = [ x · y ] 2 .
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