SalasSV_07_08_ex

SalasSV_07_08_ex - 7.8 THE HYPERBOLIC SINE AND COSINE 439...

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7.8 THE HYPERBOLIC SINE AND COSINE 439 EXERCISES 7.8 Differentiate the function. 1. y = sinh x 2 . 2. y = cosh ( x + a ). 3. y = cosh ax . 4. y = (sinh ax )( cosh ax ). 5. y = sinh x cosh x 1 . 6. y = sinh x x . 7. y = a sinh bx b cosh ax . 8. y = e x ( cosh x + sinh x ). 9. y = ln | sinh ax | . 10. y = ln | 1 cosh ax | . 11. y = sinh ( e 2 x ). 12. y = cosh (ln x 3 ). 13. y = e x cosh 2 x . 14. y = tan 1 (sinh x ). 15. y = ln ( cosh x ). 16. y = ln (sinh x ). 17. y = ( sinh x ) x . 18. y = x cosh x . Verify the identity. 19. cosh 2 t sinh 2 t = 1. 20. sinh( t + s ) = sinh t cosh s + cosh t sinh s . 21. cosh( t + s ) = cosh t cosh s + sinh t sinh s . 22. sinh 2 t = 2 sinh t cosh t . 23. cosh 2 t = cosh 2 t + sinh 2 t = 2 cosh 2 t 1 = 2 sinh 2 t + 1. 24. cosh( t ) = cosh t ; the hyperbolic cosine function is even. 25. sinh( t ) = − sinh t ; the hyperbolic sine function is odd. Find the absolute extreme values. 26. y = 5 cosh x + 4 sinh x . 27. y = − 5 cosh x + 4 sinh x . 28. y = 4 cosh x + 5 sinh x . 29. Show that for each positive integer n ( cosh x + sinh x ) n = cosh nx + sinh nx . 30. Verify that y = A cosh cx + B sinh cx satis fi es the equation y c 2 y = 0.
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