098025A - 8025A SEMESTER 1 2009 THE UNIVERSITY OF SYDNEY...

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Unformatted text preview: 8025A SEMESTER 1 2009 THE UNIVERSITY OF SYDNEY FACULTIES OF ARTS, ECONOMICS, EDUCATION, ENGINEERING AND SCIENCE MATH1901/1906 DIFFERENTIAL CALCULUS (ADVANCED) June 2009 LECTURER: C M Cosgrove TIME ALLOWED: One and a half hours This examination has two sections: Multiple Choice and Extended Answer. The Multiple Choice Section is worth 35% of the total examination; there are 20 questions; the questions are of equal value; all questions may be attempted. Answers to the Multiple Choice questions must be coded onto the Multiple Choice Answer Sheet. The Extended Answer Section is worth 65% of the total examination; there are 4 questions; the questions are of equal value; all questions may be attempted; working must be shown. Calculators will be supplied; no other calculators are permitted. THE QUESTION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM. PAGE 1 OF 8 8025A SEMESTER 1 2009 PAGE 7 OF 8 Extended Answer Section Answer these questions in the answer b00k(3) provided. Ask for extra books if you need them. MARKS 1. (a) In the complex plane, sketch the set {2 E (C l 1 3 [2+ 1 — S 2}. 3 (b) Find all real and complex solutions of the equation, 24—423 +922 —16z+20 = 0, given that 2 +73 is a root. 5 (c) Show that the function, g : (C —> (C, 2 1—> z4, is surjective but not injective. (Please keep your answer short.) 4 2. (a) Let f : R2 —> R, (r, y) 1—+ ln(a:2 +4y2), and let P be the point (3, 1) in the my—plane. Calculate the directional derivative Du f at P in the direction of the vector u = 3i — 2j. 2 Find the equation of the tangent plane to the graph of z = f (2:, y) at the point on the graph vertically above P. Express your answer in the form z 2 as: + by + c. 2 (b) Let f denote the function f : R —> R given by sing: f@)= x’ mi” 1, :1: = 0. Calculate the Taylor polynomial of order 6 of the function f about 1: = O and deduce the values of the even—order derivatives f”(0), f (4) (0) and f(6) (Hint. Use the standard Taylor polynomial ofsinz: to a suitable order. Do not try to calculate derivatives using the quotient rule or l’Hopital’s rule, for example.) 5 (c) Evaluate the limit, hmsinSr—Ssinr, m—rO $3 by using Taylor polynomials of suitable order. 3 8025A SEMESTER 1 2009 PAGE 8 OF 8 MARKS 3. (a) Find the following limits, showing the steps of your working clearly, or show that the limit does not exist. (You may use any valid method. Allow +00 and —00 as values that a limit can take.) lim V552 + 2cm; — V332 ~ 2152:, a and 17 real constants. 3 (I 311330 0) ln(sin(:1:2 + 3 2 (m) lim (In x) 3 z—d 1 + cos 7m:- (b) Prove that the graph of y = 23/5 has a vertical tangent at the origin. 3 4. (a) A cardioid is a closed plane curve having the parametric equations: :1; = R(2cos€—00320), y = R(2sin6—sin26), 0 g 6 _<_ 27r, R positive constant. Find the equation of the tangent line to the cardioid at the point (:13, y) = (R, 2R). 4 (b) The function f : R2 —> R is defined by the rule 1:53; (25,14) 75 (0, 0) flay) = $4+y2’ O, (x, y) = (0, O). Evaluate the partial derivatives, fz(07y)7 f2(010)7 fy($70)a fy(0a (The definition of partial derivative as a limit is the recommended method.) 4 ( Evaluate the mixed second derivatives, fzy(070)7 fyI<Oa (You will notice that they are not equal to each other.) 4 End of Extended Answer Section THIS IS THE LAST PAGE OF THE QUESTION PAPER. ...
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098025A - 8025A SEMESTER 1 2009 THE UNIVERSITY OF SYDNEY...

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