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### midterm 2005

Course: MATH 102, Spring 2011
School: HKUST
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102 Midterm HKUST MATH Examination Problem 1 (a) Assume a, b and c are three dimensional vectors and if Multivariable and Vector Calculus (a b) c = a + b + c. 21 Dec 2005 Use sux notation to nd , and in terms of the vectors a, b and c. Can you say something about the direction of the vector (a b) c. Answer ALL 8 questions (b) Let a be a constant vector and r = (x, y, z ), use sux notation to...

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102 Midterm HKUST MATH Examination Problem 1 (a) Assume a, b and c are three dimensional vectors and if Multivariable and Vector Calculus (a b) c = a + b + c. 21 Dec 2005 Use sux notation to nd , and in terms of the vectors a, b and c. Can you say something about the direction of the vector (a b) c. Answer ALL 8 questions (b) Let a be a constant vector and r = (x, y, z ), use sux notation to evaluate Time allowed 180 minutes (i) r, (ii) (a r), (iii) (a r). Problem 2 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 (a) Sketch and describe the parametric curve C are illegible for the grader cannot be given credit. r = t cos t i + t sin t j + (2 t) k, Note that you can work on both sides of the paper and do not detach pages from this exam packet 0 t 2. Show the direction of increasing t. Find the project curve C onto the yz -plane. or unstaple the packet. (b) Find a change of parameter t = g ( ) for the semicircle r(t) = cos t i + sin t j, 0 t such that (i) the semicircle is traced counterclockwise as varies over the interval [0, 1], Student Name: (ii) the semicircle is traced clockwise as varies over the interval [0, 0.5]. Student Number: Problem 3 Tutorial Session: If a wheel with radius a rolls along a at surface without slipping, a point P on the rim of the wheel traces a curve C , nd the parametric equation of the point P . Suppose that the point P on the wheel is initially at the origin. Find also the arc length of the curve C if the wheel makes one complete turn (no need to carry out the integration). Question No. Marks 1 /20 2 /20 3 /20 4 /20 5 /20 6 /20 7 /20 8 /20 Total /160 Problem Verify 4 (a) the formula for the arc length element is cylindrical coordinates, dr dt ds = 2 + (r (t))2 d dt 2 + dz dt 2 dt. (b) Find a similar formula as in (a) for the arc length element in spherical coordinates. (c) Use part (b) or otherwise, nd the arc length of the curve in spherical coordinates: = 2t, = ln t, = /6; 1 t 5. 1 Problem 5 2 2 2xy (x y ) Let f (x, y ) = x2 + y 2 0 if (x, y ) = (0, 0), if (x, y ) = (0, 0). (a) Is the function continuous at (0, 0)? (b) Calculate fx (x, y ), fy (x, y ), fxy (x, y ) and fyx (x, y ) at point (x, y ) = (0, 0). Also calculate these derivatives at (0, 0). (c) Is fyx (x, y ) continuous at (0, 0)? (d) Explain why fyx (0, 0) = fxy (0, 0). Problem 6 Find the distance from the origin to the plane x + 2y + 2z = 3, (a) using a geometric argument (no calculus), (b) by reducing the problem to an unconstrained problem in two variables, and (c) using the method of Lagrange multipliers. Problem 7 (a) What condition must the constants a, b, and c satisfy to guarantee that lim (x,y )(0,0) xy ax2 + bxy + cy 2 exists. Prove your answer. (b) Find 2 f (y 2 , xy, x2 ) in terms of partial derivatives of the function f . yx Problem 8 (a) Find the equation of the tangent plane at the point (1, 1, 0) to the surface x2 2y 2 + z 3 = ez . (b) The temperature at a point (x, y ) on a metal plate in xy -plane is T (x, y ) = x 2 + y 3 degrees Celsius. (i) Find the rate of change of temperature at (1, 1) in the direction of a = 2 i + j. (ii) An ant at (1, 1) wants to walk in the direction in which the temperature decreases most rapidly. Find a unit vector in that direction. (c) Let C be the curve x2/3 + y 2/3 = a2/3 on the xy -plane, nd the parametric equation of the curve C . Hence nd the tangent line to the curve C at (a, 0). 2
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