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Unformatted text preview: HKUST MATH 102 Midterm One Examination Multivariable and Vector Calculus 30 Oct 2007 Answer ALL 5 questions Time allowed – 120 minutes Directions – This is a closed book examination. No talking or whispering are allowed. Work must be shown to receive points. An answer alone is not enough. Please write neatly. Answers which are illegible for the grader cannot be given credit. Note that you can work on both sides of the paper and do not detach pages from this exam packet or unstaple the packet. Student Name: Student Number: Tutorial Session: Question No. Marks 1 /20 2 /20 3 /20 4 /20 5 /20 Bonus /5 Total /100 Problem 1 (20 points) Your Score: 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 ). Solution: (a) Let b u = ( u 1 , u 2 , u 3 ), then u 1 x + u 2 y + u 3 z = 0, this is a plane with its normal in the direction of u and passing through the origin. (b) ( r- a ) · ( r- b ) = k r · r- ( a + b ) · r + a · b = k k r- a + b 2 k 2 = k- a · b + k a + b k 2 4 = k + k a- b k 2 4 ∴ It is a sphere center at ( a + b ) / 2 with radius k + k a- b k 2 / 4 1 / 2 . (c) Note that p = ( r · b u ) b u is the vector compo- nent of r onto the vector u and r- ( r · b u ) b u is the vector component of r orthogonal to u and the norm of this vector is a constant k . Therefore, we can conclude that it is a circular cylinder of radius k with its axis parallel to u . u r p – 1 – Problem 2 (20 points) Your Score: (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)....
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