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Pre-Calc Homework Solutions 427

# Pre-Calc Homework Solutions 427 - Section 10.6 5 7 427 3.8...

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5. [ 2 3.8, 3.8] by [ 2 2.5, 2.5] The graph passes through the pole when r 5 3 cos u 5 0, which occurs when u 5 } p 2 } and when u 5 } 3 2 p } . Since the u -interval 0 # p # 1 produces the entire graph, we need only consider u 5 } p 2 } . At this point, there appears to be a vertical tangent line with equation u 5 } p 2 } (or x 5 0). Confirm analytically: x 5 (3 cos u ) cos u 5 3 cos 2 u y 5 (3 cos u ) sin u } d d u y } 5 ( 2 3 sin u )sin u 1 (3 cos u )cos u 5 3(cos 2 u 2 sin 2 u ) and } d d u x } 5 6 cos u ( 2 sin u ). At 1 0, } p 2 } 2 , } d d u x } ) u 5 p /2 5 0, and } d d u y } ) u 5 p /2 5 3(0 2 2 1 2 ) 5 2 3. So at 1 0, } p 2 } 2 , } d d u x } 5 0 and } d d u y } 0, so } d d y x } is undefined and the tangent line is vertical. 6. [ 2 3, 3] by [ 2 2, 2] A trace of the graph suggests three tangent lines, one with positive slope for u 5 } p 6 } , a vertical one for u 5 } p 2 } , and one with negative slope for u 5 } 5 6 p } . Confirm analytically: } d d u y } 5 2 6 sin 3 u sin u 1 2 cos 3 u cos u and } d d u x } 5 2 6 sin 3 u cos u 2 2 cos 3 u sin u .
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