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e307k - MAC 2311 Exam III and Key April 5 2007 Prof S...

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MAC 2311 April 5, 2007 Exam III and Key Prof. S. Hudson. 1) (10pts) A spherical balloon is inflated so that its volume ( V = 4 πr 3 / 3) is increasing at a rate of 3 ft 3 /min. How fast is the diameter increasing when the radius is 1 ft.? 2) (10pts) Compute each limit (and show work); a) lim x + (1 - 2 /x ) x b) lim x 0 tan( x ) tan - 1 ( x ) 3) (25pts) Compute: a) Find dy/dx : y = e 1 /x b) Find dy/dx : y = tan - 1 ( x 3 ) c) Find dy/dx using logarithmic differentiation: y = x x d) Find dy/dx using implicit differentiation: x 2 + y 2 = 100 e) Find d 2 y/dx 2 using implicit differentiation: 2 x 2 - 3 y 2 = 4 4) (15 pts) Give an example of a function f ( x ) for each part, or explain why none exists. [Give three separate answers]. a) It has a stationary point at x = 0, but that point is not a relative extrema. b) It has a relative extrema at x = 0, but that point is not a stationary point. c) f (0) = 0 but x = 0 is not an inflection point. 5) (10pts) Prove that d dx sin - 1 ( x ) = 1 1 - x 2 using a theorem about inverse functions (or the chain rule).
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