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Pre-Calc Homework Solutions 92

# Pre-Calc Homework Solutions 92 - 92 Section 3.6 d sin 2x dx...

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28. Continuous: Note that g (0) 5 lim x 0 1 g ( x ) 5 lim x 0 1 cos x 5 cos (0) 5 1, and lim x 0 2 g ( x ) 5 lim x 0 2 ( x 1 b ) 5 b . We require lim x 0 2 g ( x ) 5 g (0), so b 5 1. The function is continuous if b 5 1. Differentiable: For b 5 1, the left-hand derivative is 1 and the right-hand derivative is 2 sin (0) 5 0, so the function is not differentiable. For other values of b , the function is discontinuous at x 5 0 and there is no left-hand derivative. So, there is no value of b that will make the function differentiable at x 5 0. 29. Observe the pattern: } d d x } cos x 5 2 sin x } d d x 5 5 } cos x 5 2 sin x } d d x 2 2 } cos x 5 2 cos x } d d x 6 6 } cos x 5 2 cos x } d d x 3 3 } cos x 5 sin x } d d x 7 7 } cos x 5 sin x } d d x 4 4 } cos x 5 cos x } d d x 8 8 } cos x 5 cos x Continuing the pattern, we see that } d d x n n } cos x 5 sin x when n 5 4 k 1 3 for any whole number k . Since 999 5 4(249) 1 3, } d d x 9 9 9 9 9 9 } cos x 5 sin x . 30. Observe the pattern: } d d x } sin x 5 cos x } d d x 5 5 } sin x 5 cos x } d d x 2 2 } sin x 5 2 sin x } d d x 6 6 } sin x 5 2 sin x } d
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