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2004finalexam_deferred

# 2004finalexam_deferred - CARLETON UNIVERSITY DEFERRED FINAL...

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CARLETON UNIVERSITY DEFERRED FINAL EXAMINATION June 2006 DURATION: 3 HOURS Department Name: School of Mathematics and Statistics Course Number: MATH 2004 D Course Instructor: Dr. C. Koestler AUTHORIZED MEMORANDA Non-programmable, non-graphing calculators are allowed Students must count the number of pages in this examination question paper before begin- ning to write, and report any discrepancy to a proctor. This question paper has 11 pages. This question paper may not be taken from the examination room. This examination may be released to the library. Attempt all problems and show ALL work on these pages . You may use the backside of the page if you need more room for your answer. Page 1 of 11 FAMILY NAME: FIRST NAME: STUDENT NUMBER: THIS PART IS FOR EXAMINER’S USE ONLY Question Marks Grade 1 10 2 11 3 11 4 10 5 9 6 9 7 10 8 10 9 10 10 10 Total 100

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MATH 2004 D Deferred Final Exam – June 2006 page 2 of 11 Problem 1 ( Fourier Series ) . The periodic even function f is defined on its half period 0 x 2 by f ( x ) = x if 0 x 1 0 if 1 x 2 . [2 marks] (a ) Sketch and state the even extension of the function f to the full period - 2 x 2. x f ( x ) 1 1 f ( x ) = x if 0 x < 1 1 if 1 x 2 - x if - 1 x < 0 1 if - 2 x < 1 . 1 mark for sketch 1 mark for stating even extension Give still 1 mark, if sketch and stated extension are in accordance [3 marks] (b ) State the formula of the Fourier cosine series of the function f . f ( x ) = a 0 2 + n =1 a n cos( n π 2 x ) , 1 mark where a 0 = 1 2 2 - 2 f ( x ) dx = 2 0 f ( x ) dx 1 mark a n = 1 2 2 - 2 f ( x ) cos( n π 2 x ) dx = 2 0 f ( x ) cos( n π 2 x ) dx. 1 mark (the second equalities are valid, since f and cos( n π 2 x ) are even functions. [5 marks] (c ) Compute all coefficients of the Fourier Cosine Series and write down the resulting series.
MATH 2004 D Deferred Final Exam – June 2006 page 3 of 11 Problem 2 ( Polar Curves ) . Let the curve C be given in polar coordinates ( r, θ ) by r = 2 sin( θ ) + 2 cos( θ ) .

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