e306fk

# e306fk - MAC 2311 Exam III and Key Prof S Hudson 1(15 pts...

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MAC 2311 November 22, 2006 Exam III and Key Prof. S. Hudson 1) (15 pts) Use the info provided to analyze and graph the function f ( x ) = x 2 - 2 x +4 x - 2 . [Draw the f and f lines and discuss intervals. Find all the HA’s, VA’s, critical points and inflection points - if they exist]. Given: f ( x ) = x ( x - 4) ( x - 2) 2 and f ( x ) = 8 ( x - 2) 3 . Also, notice that x 2 - 2 x + 4 > 0 for all x . 2) (15 pts) Water is pouring into a conical tank at the rate of 8 cubic feet per minute. If the height of the tank is 12 feet and the radius of the circular opening (at the top) is 6 feet, how fast is the water level rising when the water is 4 feet deep? Hint: V = πr 2 h/ 3. 3) (15pts) Compute the limits: a) lim x π sin( x ) π - x b) lim x 1 + ( x x - 1 - 1 ln x ) c) lim x 0 (cos x ) 1 /x 4) (10pts) State and prove Rolle’s Theorem (prove Cases 1 and 3 carefully, if not Case 2). 5) (15 pts) Mark each sentence True or False; tan( x ) has a maximum value on the interval (0 , π ). f ( x ) = | x 2 - 4 | + 1 satisfies the hypotheses of the Mean Value Thm on [ - 4 , 1].

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