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098001A - 8001A SEMESTER 1 2009 THE UNIVERSITY OF SYDNEY...

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Unformatted text preview: 8001A SEMESTER 1 2009 THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS MATH 1001 DIFFERENTIAL CALCULUS June 2009 LECTURERS: J Henderson, C Macaskill, T Morrison, M Myerscough, J Parkinson TIME ALLOWED: One and a half hours This examination has two sections: Multiple Choice and Extended Answer. The Multiple Choice Section is worth 50% of the total examination; there are 30 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 50% of the total examination; there are 3 questions; the questions are of equal value; all questions may be attempted; working must be shown. Calculators wi]l be supplied; no other calculators are permitted. THE QUESTION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM. PAGE 1 OF 16 8001A SEMESTER 1 2009 PAGE 8 OF 16 Extended Answer Section There are three questions in this section, each with a number of parts. Write your answers in the space provided below each part. If you need more space there are extra pages at the end of the examination paper. 1. (a) Find all complex solutions of 23 = ~1. (b) Find all complex solutions of iz + E = O and sketch them in the complex plane. MARKS 8001A SEMESTER 1 2009 PAGE 9 OF 16 MARKS (0) Consider the function f of two real variables given by f (as, y) = 51:33; — 2:52y2. (2') Find the partial derivatives f$(l, —2) and fy(l, -2). 2 (22') Find the directional derivative Du f (1, —2) Where u 2 3i — 4j. 2 (121') What is the direction of the maximum directional derivative at ( 1, —2)? 1 (iv) What is the slope of the normal to the level curve f(:c, y) = ~10 at (1, —2)? 1 Question 2 begins on page 10. 8001A SEMESTER 1 2009 PAGE 10 OF‘ 16 2. (a) Consider the function g of two real variables given by 9(96, 1!) = my — $- (z') Find the coordinates of all critical points of 9. (ii) Classify any critical points you find as local maximum, local minimum or saddle point. MARKS 8001A SEMESTER 1 2009 PAGE 11 OF 16 MARKS (b) (2') Find the Taylor polynomial of order 3 for f (3:) = sinr, about the point x = 0. 3 (21') Hence determine the corresponding Taylor polynomial of order 6 for sin 3:2 1 and use this to show that the integral / sin :02 dry is approximately equal to 0 13 / 42. 3 Question 3 begins on page 12. 8001A SEMESTER 1 2009 PAGE 12 OF 16 MARKS 3. (a) Calculate the following limits or ShOW they do not exist. Carefully show each step of your working. 3:1:2 + a: — 4 ' li ——————— 2 (z) z—Ieri 2933—2m2+a:—-1 _ 2 an hm. 9%—£% 2 (z,y)—+(0.0) II? + y 4 _ 2 2 (in) lim w (use polar coordinates) 2 mmewm $2+y2 8001A SEMESTER 1 2009 PAGE 13 OF 16 MARKS (b) Let f : R2 —> R be function, and suppose that f satisfies f(ta, tb) = tf(a, b) for all (a, b) E R2 and all t E R. You may assume that the chain rule applies. (2') Use the chain rule to show that fz(a, b)a + fy(a, b)b : f(a, b) for all (a, b) E R2. 2 (ii) Show that the origin (0,0,0) 6 R3 lies on every tangent plane to the surface z = f (2:,y) 2 (7321') Given that f(2, 1) = 4, find the value of the directional derivative Du f (6, 3) in the direction u 2 2i + j. 2 More space is available on the next page. ...
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