Postclass 10.3-4-solutions

Postclass 10.3-4-solutions - im(gi768 Postclass 10.3/4...

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im (gi768) – Postclass 10.3/4 – sadun – (53088) 1 This print-out should have 12 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Which, if any, of A. (4 , π/ 3) , B. (4 , 13 π/ 6) , C. ( - 4 , 7 π/ 6) , are polar coordinates for the point given in Cartesian coordinates by P (2 , 2 3)? 1. C only 2. A and B only 3. B only 4. all of them 5. B and C only 6. A only correct 7. A and C only 8. none of them 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: differs from π/ 6 by 2 π . C. FALSE: θ incorrect. 002 10.0points Which one of the following shaded regions consists only of points whose polar coordi- nates satisfy the condition π 8 < θ 5 π 4 ? 1. 2. 3. correct
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im (gi768) – Postclass 10.3/4 – sadun – (53088) 2 4. 5. 6. Explanation: Since θ > 0 corresponds to rotating counter-clockwise around the origin, while θ < 0 corresponds to rotating clockwise, we see that the region of the plane specified by π 8 < θ 5 π 4 is the shaded region shown in keywords: 003 10.0points Find a Cartesian equation for the curve given by the polar equation r + 2 sin θ = 0 . 1. ( x - 1) 2 + y 2 + 1 = 0 2. x 2 + ( y - 1) 2 = 1 3. x 2 + ( y + 1) 2 = 1 correct 4. ( x + 1) 2 + y 2 = 1 5. ( x - 1) 2 + y 2 = 1 6. ( x + 1) 2 + y 2 + 1 = 0 7. x 2 + ( y - 1) 2 + 1 = 0 8. x 2 + ( y + 1) 2 + 1 = 0 Explanation: We have to replace r and θ in the polar equation r + 2 sin θ = 0 using the relations x = r cos θ , y = r sin θ . As a first simplification, notice that r 2 + 2 r sin θ = 0 . But then x 2 + y 2 + 2 y = r 2 + 2 r sin θ = 0 .
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im(gi768)–Postclass10.3/4–sadun–(53088) 3 Consequently, by completing the square we get the Cartesian equation x 2 + ( y + 1) 2 = 1 . 004 10.0points Find a polar representation for the curve whose Cartesian equation is x 2 + ( y + 3) 2 = 9 . 1. r = 3 cos θ 2. r + 3 sin θ = 0 3. r + 3 cos θ = 0 4. r + 6 sin θ = 0 correct 5. r = 6 cos θ 6. r = 6 sin θ 7. r + 6 cos θ = 0 8. r = 3 sin θ Explanation: We have to substitute for x, y in x 2 + ( y + 3) 2 = 9 using the relations x = r cos θ , y = r sin θ . But after expansion the Cartesian equation becomes x 2 + y 2 + 6 y + 9 = 9 . Now x 2 + y 2 = r 2 , so r 2 + 6 r sin θ = 0 , which after cancellation gives the polar repre- sentation r + 6 sin θ = 0 . 005 10.0points Which one of the following polar functions has graph 1. r = 2 sec θ correct 2. θ = 2 3. r = 2 cos θ 4. r = 2 sin θ 5. r = 2 6. r = 2 csc θ Explanation: When the graph of a polar function cannot be determined directly, it is sometimes more convenient to use the relations x = r cos θ , y = r sin θ , to convert the polar form to Cartesian form
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