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(b) lim x } g f ( ( x x ) ) } 5 lim x 5 lim x } ˇ x ˇ 2 w 2 w x 2 w a w 2 w } 5 lim x ! 1 § 2 § } a x § 2 2 } § 5 1 (c) lim x } g f ( ( x x ) ) } 5 lim x 5 lim x } ˇ x ˇ 2 w 2 w x 2 w a w 2 w } 5 lim x ! 1 § 2 § } a x § 2 2 } § 5 1 Section 3.8 Derivatives of Inverse Trigonometric Functions (pp. 157–163) Exploration 1 Finding a Derivative on an Inverse Graph Geometrically 1. The graph is shown at the right. It appears to be a one-to- one function [ 2 4.7, 4.7] by [ 2 3.1, 3.1] 2. f 9 ( x ) 5 5 x 4 1 2. The fact that this function is always positive enables us to conclude that f is everywhere increasing, and hence one-to-one. 3. The graph of f 2 1 is shown to the right, along with the graph of f . The graph of f 2 1 is obtained from the graph of f by reflecting it in the line y 5 x . [ 2 4.7, 4.7] by [ 2 3.1, 3.1] 4. The line L is tangent to the graph of f 2 1 at the point (2, 1). [ 2 4.7, 4.7] by [ 2 3.1, 3.1] 5. The reflection of line L is tangent to the graph of f at the point (1, 2). [ 2 4.7, 4.7] by [ 2 3.1, 3.1] 6. The reflection of line L is the tangent line to the graph of y 5 x 5 1 2 x 2 1 at the point (1, 2). The slope is } d d y x } at x 5 1, which is 7. 7. The slope of L is the reciprocal of the slope of its reflection 1 since } D D y x } gets reflected to become } D D x y } 2 . It is } 1 7 } . 8. } 1 7 } Quick Review 3.8 1. Domain: [ 2 1, 1] Range: 3 2 } p 2 } , } p 2 } 4 At 1: } p 2 } 2. Domain: [ 2 1, 1] Range: [0, p ] At 1: 0 3. Domain: all reals Range: 1 2 } p 2 } , } p 2 } 2 At 1: } p 4 } 4. Domain: ( 2‘ , 2 1] < [1, ) Range: 3 0, } p 2 } 2 < 1 } p 2 } , p 4 At 1: 0 5. Domain: all reals Range: all reals At 1: 1 6. f ( x ) 5 y 5 3 x 2 8 y 1 8 5 3 x x 5 } y 1 3 8 } Interchange x and y : y 5 } x 1 3 8 } f 2 1 ( x ) 5 } x 1 3 8 } 7. f ( x ) 5 y 5 ˇ 3 x w 1 w 5 w y 3 5 x 1 5 x 5 y 3 2 5 Interchange x and y : y 5 x 3 2 5 f 2 1 ( x ) 5 x 3 2 5 2 } b a } ˇ x 2 w 2 w a w 2 w }} 2 } b a } ) x ) } b a } ˇ x 2 w 2 w a w 2 w }} } b a } ) x ) Section 3.8 111

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8. f ( x ) 5 y 5 } 8 x } x 5 } 8 y } Interchange x and y : y 5 } 8 x } f 2 1 ( x ) 5 } 8 x } 9. f ( x ) 5 y 5 } 3 x x 2 2 } xy 5 3 x 2 2 ( y 2 3) x 5 2 2 x 5 } y 2 2 2 3 } 5 } 3 2 2 y } Interchange x and y : y 5 } 3 2 2 x } f 2 1 ( x ) 5 } 3 2 2 x } 10. f ( x ) 5 y 5 arctan } 3 x } tan y 5 } 3 x } , 2 } p 2 } , y , } p 2 } x 5 3 tan y , 2 } p 2 } , y , } p 2 } Interchange x and y : y 5 3 tan x , 2 } p 2 } , x , } p 2 } f 2 1 ( x ) 5 3 tan x , 2 } p 2 } , x , } p 2 } Section 3.8 Exercises 1. } d d y x } 5 } d d x } cos 2 1 ( x 2 ) 5 2 } ˇ 1 w 2 w 1 ( x w 2 ) w 2 w } } d d x } ( x 2 ) 5 2 } ˇ 1 w 1 2 w x w 4 w } (2 x ) 5 2 } ˇ 1 w 2 2 w x x w 4 w } 2. } d d y x } 5 } d d x } cos 2 1 1 } 1 x } 2 5 2 } d d x } 1 } 1 x } 2 5 2 1 2 } x 1 2 } 2 5 } ) x ) ˇ x 1 2 w 2 w 1 w } 3. } d d y t } 5 } d d t } sin 2 1 ˇ 2 w t 5 } d d t } ( ˇ 2 w t ) 5 4.
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