ISM_T11_C10_G - 660 Chapter 10 Conic Sections and Polar...

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660 Chapter 10 Conic Sections and Polar Coordinates 48. (a) (b) (c) (d) (e) 49. (a) r 4 cos cos ; r 1 cos r 1 0 r 4r 4 (r 2) 0 # ## œ± Ê œ ± Ê œ ± ± Ê œ ± ² Ê ± œ )) ) rr 44 Š‹ r 2; therefore cos 1 (2 ) is a point of intersection Ê œ œ± œ± Ê œ Ê ß 1 1 2 4 # (b) r 0 0 4 cos cos 0 , or is on the graph; r 0 0 1 cos œÊ œ Ê œÊœ Ê! ß ! ß œÊœ± # # # ) ) 11 1 1 33 ˆ‰ ˆ cos 1 0 (0 0) is on the graph. Since ( 0) for polar coordinates, the graphs Ê œ Ê œ Ê ß !ß œ !ß 1 # intersect at the origin. 50. (a) Let r f( ) be symmetric about the x-axis and the y-axis. Then (r ) on the graph (r ) is on the œß Ê ß ± ) graph because of symmetry about the x-axis. Then ( r ( )) ( r ) is on the graph because of ±ß±± œ ±ß symmetry about the y-axis. Therefore r f( ) is symmetric about the origin. œ ) (b) Let r f( ) be symmetric about the x-axis and the origin. Then (r ) on the graph (r ) is on the Ê ß ± ) graph because of symmetry about the x-axis. Then ( r ) is on the graph because of symmetry about ±ß± ) the origin. Therefore r f( ) is symmetric about the y-axis. œ ) (c) Let r f( ) be symmetric about the y-axis and the origin. Then (r ) on the graph ( r ) is on the Ê ± ß ± ) graph because of symmetry about the y-axis. Then ( ( r) ) (r ) is on the graph because of ±± ß± œ ß± symmetry about the origin. Therefore r f( ) is symmetric about the x-axis. œ ) 51. The maximum width of the petal of the rose which lies along the x-axis is twice the largest y value of the curve on the interval 0 . So we wish to maximize 2y 2r sin 2 cos 2 sin on 0 . Let ŸŸ œ œ ) ) ) 1 1 4 4 f( ) 2 1 2 sin (sin ) 2 sin 4 sin f ( ) 2 cos 12 sin cos . Then ) ) ) ) ) ) ) ) ) œœ ± œ ± Ê œ ± ab #$ w # f ( ) 0 2 cos 12 sin cos 0 (cos ) 1 6 sin 0 cos 0 or 1 6 sin 0 or w# # # # ) ) ) ) ) ) ) œÊ ± ± œ ± 1 sin . Since we want 0 , we choose sin f( ) 2 sin 4 sin ) ) ) ) œŸ Ÿ œ Ê œ ± „" ±" $ 1 66 4 ÈÈ 1 2 4 . We can see from the graph of r cos 2 that a maximum does occur in the œ œ "" È 6 26 9 ) interval 0 . Therefore the maximum width occurs at sin , and the maximum width œ 1 4 6 ±" " È is . 9 È 52. We wish to maximize y r sin 2(1 cos )(sin ) 2 sin 2 sin cos . Then ² œ² ) ) ) ) 2 cos 2(sin )( sin ) 2 cos cos 2 cos 2 sin 2 cos 2 cos 4 cos 2; thus dy d ) ± ² œ±² ± ) ) ) ))) # 0 4 cos 2 cos 2 = 0 2 cos cos 1 0 (2 cos 1)(cos 1) 0 cos dy d ) ² ± Ê ² ±œÊ ± ²œÊ œ " # ) ) ) ) ) or cos 1 , , . From the graph, we can see that the maximum occurs in the first quadrant so 1 œ± Ê œ 5 we choose . Then y 2 sin 2 sin cos . The x-coordinate of this point is x r cos ) ² œ œ 1 1 1 3 3 3 È # 2 1 cos cos . Thus the maximum height is h occurring at x . œ œ œ ˆ 3 3 # È
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Section 10.7 Area and Lengths in Polar Coordinates 661 10.7 AREA AND LENGTHS IN POLAR COORDINATES 1. A (4 2 cos ) d 16 16 cos 4 cos d 8 8 cos 2 d œ± œ ±± œ ± ± '' ' 00 0 22 2 11 1 "" ± ## # )) ) ) ) ) ) ab ± ˆ‰ 1c o s 2 ) (9 8 cos cos 2 ) d 9 8 sin sin 2 18 ± œ œ ' 0 2 1 ) ) 1 ± " # ! 2 1 2. A [a(1 cos )] d a 1 2 cos cos d a 1 2 cos d œ ± ±œ ± ± ' 0 2 1 " ± # # ) ) ) ) ) o s 2 ) 2 cos cos 2 d a 2 sin sin 2 a ± œ ± " " # # # # # # !
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This note was uploaded on 10/13/2011 for the course MATHEMATIC 103 taught by Professor Thommas during the Spring '11 term at LCC Intl University.

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ISM_T11_C10_G - 660 Chapter 10 Conic Sections and Polar...

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