calassignment 2-solutions

calassignment 2-solutions - miller (zdm77) – assignment 2...

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Unformatted text preview: miller (zdm77) – assignment 2 – luecke – (58600) 1 This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Locate the points given in polar coordinates by P parenleftBig 2 , 1 6 π parenrightBig , Q parenleftBig 1 , 1 2 π parenrightBig R parenleftBig 4 , 5 6 π parenrightBig , among 2 4- 2- 4 2 4- 2- 4 b u b c r u t r s 1. b u t b c P : Q : R : 2. b u t b c P : Q : R : 3. b u t b c P : Q : R : 4. b u t b c P : Q : R : correct 5. b u t b c P : Q : R : 6. b u t b c P : Q : R : Explanation: To convert from polar coordinates to Carte- sian coordinates we use x = r cos θ , y = r sin θ . For then the points P parenleftBig 2 , 1 6 π parenrightBig , Q parenleftBig 1 , 1 2 π parenrightBig R parenleftBig 4 , 5 6 π parenrightBig , correspond to b u t b c P : Q : R : in Cartesian coordinates. keywords: polar coordinates, Cartesian coor- dinates, change of coordinates, 002 10.0 points Which, if any, of A. (4 , π/ 6) , B. (4 , 7 π/ 3) , C. (- 4 , 4 π/ 3) , are polar coordinates for the point given in Cartesian coordinates by P (2 , 2 √ 3)? 1. all of them 2. A and C only 3. none of them 4. C only 5. B and C only correct 6. B only 7. A and B only 8. A only 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 miller (zdm77) – assignment 2 – luecke – (58600) 2 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. FALSE: θ incorrect. B. TRUE: differs from π/ 3 by 2 π . C. TRUE:- 4 cos(4 π/ 3) = 2 ,- 4 sin(4 π/ 3) = 2 √ 3 . 003 10.0 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. parenleftBig- √ 2 , 5 π 4 parenrightBig 2. parenleftBig- √ 2 , 7 π 4 parenrightBig 3. parenleftBig- √ 3 , 3 π 4 parenrightBig 4. parenleftBig- √ 3 , 7 π 4 parenrightBig 5. parenleftBig- √ 3 , 5 π 4 parenrightBig 6. parenleftBig- √ 2 , 3 π 4 parenrightBig correct Explanation: Since the relationship between Cartesian coordinates and polar coordinates is x = r cos θ , y = r sin θ , the point P (1 ,- 1) in Cartesian coordinates can be given in polar coordinates as P parenleftBig- √ 2 , 3 π 4 parenrightBig , 004 10.0 points Find a Cartesian equation for the curve given by the polar equation r + 4 sin θ = 0 . 1. x 2 + ( y- 2) 2 + 4 = 0 2. x 2 + ( y- 2) 2 = 4 3. ( x + 2) 2 + y 2 + 4 = 0 4. ( x + 2) 2 + y 2 = 4 5. ( x- 2) 2 + y 2 + 4 = 0 6. ( x- 2) 2 + y 2 = 4 7. x 2 + ( y + 2) 2 + 4 = 0 8. x 2 + ( y + 2) 2 = 4 correct Explanation: We have to replace r and θ in the polar equation r + 4 sin θ = 0 using the relations x = r...
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This note was uploaded on 09/30/2008 for the course M 408M taught by Professor Gilbert during the Fall '07 term at University of Texas at Austin.

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calassignment 2-solutions - miller (zdm77) – assignment 2...

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