Pre-Calc Homework Solutions 99

# Pre-Calc Homework Solutions 99 - Section 3.6 d d 99 [g(x) d...

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(c) } d d x } } g ( x f ) ( x 1 ) 1 }5 5 At x 5 1, the derivative is 5 5 } 9 9 } 5 1. (d) } d d x } f ( g ( x )) 5 f 9 ( g ( x )) g 9 ( x ) At x 5 0, the derivative is f 9 ( g (0)) g 9 (0) 5 f 9 (1) g 9 (0) 5 1 2} 1 3 } 21 } 1 3 } 2 52} 1 9 } . (e) } d d x } g ( f ( x )) 5 g 9 ( f ( x )) f 9 ( x ) At x 5 0, the derivative is g 9 ( f (0)) f 9 (0) 5 g 9 (1) f 9 (0) 5 1 2} 8 3 } 2 (5) 4 3 0 } (f) } d d x } [ g ( x ) 1 f ( x )] 2 2 52 2[ g ( x ) 1 f ( x )] 2 3 } d d x } [ g ( x ) 1 f ( x )] At x 5 1, the derivative is 25 2 5 2 } 2 2 6 1 }52 6. (g) } d d x } [ f ( x 1 g ( x ))] 5 f 9 ( x 1 g ( x )) } d d x } [ x 1 g ( x )] 5 f 9 ( x 1 g ( x ))(1 1 g 9 ( x )) At x 5 0, the derivative is f 9 (0 1 g (0))(1 1 g 9 (0)) 5 f 9 (0 1 1) 1 1 1 } 1 3 } 2 5 f 9 (1) 1 } 4 3 } 2 5 1 2} 1 3 } 21 } 4 3 } 2 4 9 } . 58. For y 5 sin 2 x , y 95 (cos 2 x ) } d d x } (2 x ) 5 2 cos 2 x and the slope at the origin is 2. For y sin } 2 x } , y 1 2 cos } 2 x } 2 } d d x } 1 } 2 x } 2 1 2 } cos } 2 x } and the slope at the origin is 2} 1 2 } . Since the slopes of the two tangent lines are 2 and 2} 1 2 } , the lines are perpendicular and the curves are orthogonal. A graph of the two curves along with the tangents y 5 2 x and y 5 2} 1 2 } x is shown.
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## This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Fall '08 term at University of Florida.

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