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e306k - MAC 2311 Exam III and Key April 3 2006 Prof S...

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MAC 2311 April 3, 2006 Exam III and Key Prof. S. Hudson 1) (10pt) Analyze the function f ( x ) = 3 x 4 - 4 x 3 . Find all the special points, and the intervals where it is increasing, concave up (etc). 2) (10pt) Sketch a graph of y = (2 x 2 - 8) / ( x 2 - 16) including all asymptotes. Some hints: The function is even. It has x -intercepts at -2 and +2. The derivative is y = - 48 x/ ( x 2 - 16) 2 . Also, y = 48(16 + 3 x 2 ) / ( x 2 - 16) 3 is never zero. f (0) = 1 / 2 and f (5) = 14 / 3. 3) (5pt each) Compute the derivative of each: a) y = ln( x 1+ x 2 ) b) y = ( x - 1 x +1 ) 1 / 5 (use log-diff) c) y = cot - 1 ( x ). d) y = 2 x 4) (10pts) A 13-ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2 ft per second, how fast will the foot be moving away from the wall when the top is 5 ft above ground ? 5) (15pts) Answer True or False: If f > 0 on [3,4] then f is concave up there. If f (6) = 0 then x = 6 is an inflection point of f . If f is defined on the closed interval [3,4] then it has a maximum value there. If f ( x ) = ln( x ) on [0.5,2.5] then f has an inverse function.
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