review3f10 - MAC 2311 Test Three Review, Fall 2010 Exam...

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MAC 2311 Test Three Review, Fall 2010 Exam covers Lectures 21 { 27, Sections 3.10 - 4.5 1. Find all critical numbers and local extrema of f ( x ) = (3 x ¡ x 3 ) 1 3 and g ( x ) = x 2 + 8 x ¡ 1 . 2. (a) State Rolle’s Theorem and the Mean Value Theorem (MVT). (b) Find each value of c that satisfles Rolle’s Theorem on the following intervals. If not possible, state why. 1) f ( x ) = sin x + cos 2 x on [ …; 2 ] 2) f ( x ) = 2 ¡ x 2 = 3 on [ ¡ 1 ; 1] 3) f ( x ) = tan x on [0 ;… ]. (c) Use Rolle’s Theorem to show that f ( x ) = 3 x ¡ cos x ¡ 1 has exactly one real root. (d) Find each value of c implied by the MVT for f ( x ) = x x ¡ 5 on [0 ; 4]. (e) Find the value of c implied by the MVT for f ( x ) = x + ln x on [1 ;e ]. (f) Use Theorem 5, Sec. 4.2, to verify that sin 2 x + cos 2 x = 1. Hint: let f ( x ) = sin 2 x + cos 2 x . 3. If f (0) = 4 and f 0 ( x ) • ¡ 2 for 0 x 6, flnd the largest possible value of f (6). 4. Find the maximum and minimum values of f ( x ) = x 2 ¡ 8ln x on [1 ;e ]. 5. Find the following limits, using L’Hospital’s rule if it applies: a) lim x ! 0
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review3f10 - MAC 2311 Test Three Review, Fall 2010 Exam...

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