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Unformatted text preview: Gauthier, Joseph Homework 3 Due: Sep 15 2007, 3:00 am Inst: JEGilbert 1 This printout should have 20 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Find all values of t in (0 , ) for which the tangent line to the graph of x ( t ) = t + sin 2 t, y ( t ) = t sin 2 t, is vertical. 1. t = 4 , 3 4 2. t = 3 , 2 3 3. t = 6 , 5 6 correct 4. t = 7 12 , 11 12 5. t = 12 , 5 12 6. t = 5 12 , 7 12 Explanation: After differentiating, we see that x ( t ) = 1 + 2 cos 2 t, y ( t ) = 1 2 cos 2 t. Thus dy dx = y ( t ) x ( t ) = 1 + 2 cos 2 t 1 2 cos 2 t . Now the tangent line will be vertical when x ( t ) = 1 2 cos 2 t = 0 , hence when cos 2 t = 1 / 2 . For t in (0 , ), therefore, the tangent line will be vertical when t = 6 , 5 6 . 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  1 , 1 2 , Q 4 , 2 3 , R 2 , 1 4 , 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 , y = r sin . For then the points P  1 , 1 2 , Q 4 , 2 3 , R 2 , 1 4 , Gauthier, Joseph 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 , 4 / 3) , B. (4 , / 6) , C. (4 , 7 / 3) , are polar coordinates for the point given in Cartesian coordinates by P (2 , 2 3)? 1. B and C only 2. C only 3. none of them 4. A and C only correct 5. all of them 6. A and B only 7. A only 8. B only Explanation: To convert from Cartesian coordinates to polar coordinates we use the relations: x = r cos , y = r sin , so that r 2 = x 2 + y 2 , tan = y x . For the point P (2 , 2 3) in Cartesian co ordinates, therefore, one choice of r and is r = 4 and = / 3, but there are equivalent solutions for r < 0 as well as values of dif fering by integer multiples of . For the given choices we thus see that A. TRUE: 4 cos(4 / 3) = 2 , 4 sin(4 / 3) = 2 3 . B. FALSE: incorrect. C. TRUE: differs from / 3 by 2 . 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, ) of this point with r < 0 and 0 < 2 ....
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This homework help was uploaded on 03/19/2008 for the course M 408M taught by Professor Gilbert during the Fall '07 term at University of Texas at Austin.
 Fall '07
 Gilbert

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