HW_3_Key - Miller, Kierste Homework 3 Due: Sep 15 2007,...

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Miller, Kierste – Homework 3 – Due: Sep 15 2007, 3:00 am – Inst: JEGilbert 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. The due time is Central time. 001 (part 1 oF 1) 10 points ±ind all values oF t in (0 , π ) For which the tangent line to the graph oF x ( t ) = t + cos 2 t, y ( t ) = t - cos 2 t, is horizontal. 1. t = π 4 , 3 π 4 2. t = π 12 , 5 π 12 3. t = π 3 , 2 π 3 4. t = 5 π 12 , 7 π 12 5. t = π 6 , 5 π 6 6. t = 7 π 12 , 11 π 12 correct Explanation: AFter di²erentiating, we see that x 0 ( t ) = 1 - 2 sin 2 t, y 0 ( t ) = 1 + 2 sin 2 t. Thus dy dx = y 0 ( t ) x 0 ( t ) = 1 + 2 sin 2 t 1 - 2 sin 2 t . Now the tangent line will be horizontal when y 0 ( t ) = 1 + 2 sin 2 t = 0 , hence when sin 2 t = - 1 / 2 . ±or t in (0 , π ), thereFore, the tangent line will be horizontal when t = 7 π 12 , 11 π 12 . keywords: derivative, tangent line, vertical, horizontal, trig Function, parametric curve 002 (part 1 oF 1) 10 points Locate the points given in polar coordinates by P 4 , 1 3 π · , Q 4 , 5 6 π · , R 1 , 1 2 π · among 2 4 - 2 - 4 2 4 - 2 - 4 1. P : Q : R : 2. P : Q : R : 3. P : Q : R : 4. P : Q : R : 5. P : Q : R : correct 6. P : Q : R : Explanation: To convert From polar coordinates to Carte- sian coordinates we use x = r cos θ , y = r sin θ . ±or then the points P 4 , 1 3 π · , Q 4 , 5 6 π · , R 1 , 1 2 π ·
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Miller, Kierste – Homework 3 – Due: Sep 15 2007, 3:00 am – Inst: JEGilbert 2 correspond to P : Q : R : in Cartesian coordinates. keywords: polar coordinates, Cartesian coor- dinates, change of coordinates, 003 (part 1 of 1) 10 points Which, if any, of A. (4 , π/ 3) , B. ( - 4 , 7 π/ 6) , C. (4 , 13 π/ 6) , are polar coordinates for the point given in Cartesian coordinates by P (2 , 2 3)? 1. B and C only 2. A and B only 3. none of them 4. C only 5. all of them 6. B only 7. A and C only 8. A only correct Explanation: To convert from Cartesian coordinates to polar coordinates we use the relations: x = r cos θ , y = r sin θ , so that r 2 = x 2 + y 2 , tan θ = y x . For the point P (2 , 2 3) in Cartesian co- ordinates, therefore, one choice of r and θ is r = 4 and θ = π/ 3, but there are equivalent solutions for r < 0 as well as values of θ dif- fering by integer multiples of π . For the given choices we thus see that A. TRUE: solution noted already. B. FALSE: θ incorrect. C. FALSE: di±ers from π/ 6 by 2 π . keywords: T/F, polar coordinaates, Cartesian coordinates 004 (part 1 of 1) 10 points A point P is given in Cartesian coordinates by P ( - 1 , 1). Find polar coordinates ( r,θ ) of this point with r < 0 and 0 θ < 2 π . 1.
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This note was uploaded on 05/08/2008 for the course M 408 M taught by Professor Gilbert during the Fall '07 term at University of Texas at Austin.

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HW_3_Key - Miller, Kierste Homework 3 Due: Sep 15 2007,...

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