Pre-Calc Homework Solutions 71

Pre-Calc Homework Solutions 71 - Section 3.1 dy dx y(x h...

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11. } d d y x } 5 lim h 0 } y ( x 1 h h ) 2 y ( x ) } 5 lim h 0 5 lim h 0 5 lim h 0 5 lim h 0 (4 x 1 2 h 2 13) 5 4 x 2 13 At x 5 3, } d d y x } 5 4(3) 2 13 52 1, so the tangent line has slope 2 1 and passes through (3, y (3)) 5 (3, 2 16). y 52 1( x 2 3) 2 16 y 52 x 2 13 12. Let f ( x ) 5 x 3 . f 9 (1) 5 lim h 0 } f (1 1 h h ) 2 f (1) } 5 lim h 0 } (1 1 h h ) 3 2 1 3 } 5 lim h 0 5 lim h 0 (3 1 3 h 1 h 2 ) 5 3 (a) The tangent line has slope 3 and passes through (1, 1). Its equation is y 5 3( x 2 1) 1 1, or y 5 3 x 2 2. (b) The normal line has slope 2} 1 3 } and passes through (1, 1). Its equation is y 52} 1 3 } ( x 2 1) 1 1, or y 52} 1 3 } x 1 } 4 3 } . 13. Since the graph of y 5 x ln x 2 x is decreasing for 0 , x , 1 and increasing for x . 1, its derivative is negative for 0 , x , 1 and positive for x . 1. The only one of the given functions with this property is y 5 ln x . Note also that
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