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quiz2sols - 2 such that cos c-2 c = 0(5 points Let f x =...

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R McDougal Name: Math 151N 5 July 2005 Quiz II Answer each question as completely as you can; remember you must show all work for full credit. You may not consult books, notes, or each other for this quiz. Good luck! 1. Find lim x →∞ x 2 + 5 - x 2 - 2 . (5 points). lim x →∞ x 2 + 5 - x 2 - 2 = lim x →∞ x 2 + 5 - x 2 - 2 x 2 + 5 + x 2 - 2 x 2 + 5 + x 2 - 2 = lim x →∞ x 2 + 5 - ( x 2 - 2) x 2 + 5 + x 2 - 2 = lim x →∞ 7 x 2 + 5 + x 2 - 2 = 0 2. Prove there exists a c [0
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Unformatted text preview: 2 ] such that cos c-2 c = 0. (5 points). Let f ( x ) = cos x-2 x . Note that f is continuous since it is the difference of two continuous functions. Further, note that f (0) = 1-0 = 1 > 0 and f ( π 2 ) = 0-π =-π < 0. Thus, by the intermediate value theorem, there exists a c ∈ [0 , π 2 ] such that f ( c ) = 0, which is to say that cos c-2 c = 0....
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