Pre-Calc Homework Solutions 429

# Pre-Calc Homework Solutions 429 - Section 10.6 11 y dy d...

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11. y 5 2 sin 2 u } d d u y } 5 4 sin u cos u 5 2 sin 2 u x 5 2 sin u cos u 5 sin 2 u } d d u y } 5 2 cos 2 u } d d u y } 5 0 when u 5 0, } p 2 } , p , and } d d u x } 5 0 when u 5 } p 4 } , } 3 4 p } . They are never both zero. For u 5 0, } p 2 } , p the curve has horizontal asymptotes at (0, 0) [ y 5 0 sin 0 5 0], 1 2, } p 2 } 23 y 5 2 sin } p 2 } 5 2 4 , and (0, p ) [ y 5 0 sin p 5 0]. For u 5 } p 4 } , } 3 4 p } the curve has vertical asymptotes at 1 ˇ 2 w , } p 4 } 2 [ x 5 ˇ 2 w cos } p 4 } 5 1] and 1 ˇ 2 w , } 3 4 p } 23 x 5 ˇ 2 w cos } 3 4 p } 5 2 1 4 . This information can be summarized as follows. Horizontal at: (0, 0) [ y 5 0], 1 2, } p 2 } 2 [ y 5 2], (0, p ) [ y 5 0] Vertical at: 1 ˇ 2 w , } p 4 } 2 [ x 5 1], 1 ˇ 2 w , } 3 4 p } 2 [ x 5 2 1] 12. } d d u y } 5 4 sin 2 u 1 (3 2 4 cos u )cos u 5 4(sin 2 u 2 cos 2 u ) 1 3 cos u 5 2 8 cos 2 u 1 3 cos u 1 4 } d d u x } 5 4 sin u cos u 2 (3 2 4 cos u )sin u 5 sin u (8 cos u 2 3) 5 4 sin 2 u 2 3 sin u } d d u y } 5 0 when cos u 5 } 2 3 6 2 1 ˇ 6 1 w 3 w 7 w } , i.e., when u < 0.405, 2.146, 4.137, or 5.878 (values solved for with a graphing calculator). } d d u x } 5 0 when u 5 0, p or 2 p (then sin u 5 0) or when u 5 cos 2 1 1 } 3 8 } 2 < 1.186 or 2 p 2 cos 2 1 1 } 3 8 } 2 < 5.097 (then 8 cos u 2 3 5 0). So there is a horizontal tangent line for u < 0.405, r < 2 0.676 [the line y < 2 0.676 sin 0.405 < 2 0.267], for u < 2.146, r < 5.176 [the line y < 5.176 sin 2.146 < 4.343], for u < 4.137, r < 5.176 [the line y < 5.176 sin 4.137 < 2 4.343], and for
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