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Unformatted text preview: 436 Chapter 10 Review
25. 23. d 1 cos sin 2 d 1 sin sin cos cos cos 2 2 2 dx d 1 cos cos 2 d d 1 sin cos sin cos sin 2 2 2 dy Solve 0 for with a graphing calculator: the solutions d dy d [ 1.5, 1.5] by [ 1, 1] The tips have Cartesian coordinates
1 2 1 2 1 2 , 1 2 1 2 , . From the are 0, 2.243, 4.892, 7.675, 10.323, and 4 . Using r sin reveals 1.739. , 1 2 , 1 2 , 1 2 , and , the middle four solutions to y horizontal tangent lines at y Solve are 0,
dx d curve's symmetries, it is evident that the tangent lines at those points have slopes of 1, 1, 1, and 1, respectively. 0.443 and y 0 for with a graphing calculator: the solutions 3.531, 2 , 9.035, 11.497, and 4 . So the equations of the tangent lines are y
1 2 1.070, x 2, x
2 1 2 1 2 or Using the middle five solutions to find x vertical tangent lines at x Where 2, x r cos reveals y 1.104. y y y y x
1 1 x 0.067, and x 0, 4 ), close dy dx and both equal zero ( dt dt or inspection of the plot shows that the tangent lines are horizontal, with equation y using L'Hpital's rule.) 24.
dy d dx d d [2(1 d d [2(1 d 2, x x
1 2 1 2 0. (This can be confirmed or 2 2, and x 2
1 2 sin )sin ] sin )cos ] 2 sin 2 sin 2 4 sin cos 2 cos y y x or 2 cos2 4 sin2
dy Solve d 2 sin2 26. 0 for :
6 2 the solutions are , , 5 3 , and . 6 2 1 and 2 [ 3, 3] by [ 2, 2] Using the first, third, and fourth solutions to find y y Solve r sin 4.
dx d reveals horizontal tangent lines at y 0 for (by first using the quadratic formula to
2 find sin ): the solutions are two solutions to find x lines at x zero
2 3 2 3 , 7 11 , and . Using the last 6 6 As the plot shows, the curve crosses the xaxis at (x, y)coordinates ( 1, 0) and (1, 0), with slope 1 and 1, respectively. (This can be confirmed analytically.) So the equations of the tangent lines are y 0 (x 1) y x 1 and y 0 x 1 y x 1. 27. r cos r sin x y, a line 28. r r2 x2 3 cos 3r cos y2 3x
3 2 2 r cos reveals vertical tangent
dy dx and both equal dt dt 2.598. Where , inspection of the plot shows that the tangent 0. (This can be confirmed line is vertical, with equation x using L'Hpital's rule.) 3x
9 4 x2 x y2 9 4 3 2 2 3 , 0 , radius 2 y2 a circle center 3 2 ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN

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