Pre-Calc Homework Solutions 472

Pre-Calc Homework Solutions 472 - 472 Appendix A6 1 cosh 2x...

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64. (a) y 5 } 1 2 } cosh 2 x y 95 sinh 2 x L 5 E ln ˇ 5 w 0 ˇ 1 w 1 w ( w si w n w h w 2 w x ) w 2 w dx 5 E ln ˇ 5 w 0 cosh 2 x dx 5 3 } 1 2 } sinh 2 x 4 5 3 } 1 2 } 1 } e 2 x 2 2 e 2 2 x } 24 5 } 1 4 } 1 5 2 } 1 5 } 2 5 } 6 5 } (b) y 5 } 1 a } cosh ax 1 1 ( y 9 ) 2 5 1 1 sinh 2 ax 5 cosh 2 ax L 5 E b 0 ˇ co w sh w 2 w a w x w dx 5 E b 0 cosh ax dx 5 3 } sinh a ax } 4 5 } sinh a ab } 65. (a) Let E ( x ) 5 } f ( x ) 1 2 f ( 2 x ) } and O ( x ) 5 } f ( x ) 2 2 f ( 2 x ) } . Then E ( x ) 1 O ( x ) 5 } f ( x ) 1 2 f ( 2 x ) }1} f ( x ) 2 2 f ( 2 x ) }5} 2 f 2 ( x ) } 5 f ( x ). Also, E ( 2 x ) 5 } f ( 2 x ) 1 2 f ( 2 ( 2 x )) }5 } f ( x ) 1 2 f ( 2 x ) E ( x ) E ( x ) is even, and O ( 2 x ) 5 } f ( 2 x ) 2 2 f ( 2 ( 2 x )) } 52 } f ( x ) 2 2 f ( 2 x ) }52 O ( x ) O ( x ) is odd. Consequently, f ( x ) can be written as a sum of an even and an odd function. (b) Even part: } e x 1 2 e 2 x } 5 cosh x odd part: } e x 2 2 e 2 x } 5 sinh x 66. (a) If f is even, then } f ( x ) 1 2 f ( 2 x ) f ( x ) 2 2 f ( 2 x ) } 5 } 2 f 2 ( x ) } 1 } f ( x ) 2 2 f ( x ) f ( x ) 1 0 (b) If f is odd, then } f ( x ) 1 2 f ( 2 x ) f ( x ) 2 2 f ( 2 x ) } 5 } f ( x ) 2 2 f ( x ) f ( x ) 1 2 f ( x ) 0 1 f ( x ) 67. Note that } d d v t } 5 !
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