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Unformatted text preview: B U Department of Mathematics Math 101 Calculus I Fall 2005 Final Exam This archive is a property of Bo˘ gazi¸ ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1. Assume that f ( x ) is defined for all x such that | x | ≤ 1 and satisfies x ≤ f ( x ) ≤ x + x 2 for all x with | x | ≤ 1 . Prove that f (0) exists and has the value 1 . Solution: ≤ f (0) ≤ 0 implies f (0) = 0. f (0) = lim x → f ( x )- f (0) x = lim x → f ( x ) x x → + , x > 1 ≤ f ( x ) x ≤ 1 + x lim x → + f ( x ) x = 1 x →- , x < 1 ≥ f ( x ) x ≥ 1 + x lim x →- f ( x ) x = 1 so f (0) = 0 2. An isosceles triangle is drawn with a vertex at the origin, its base parallel to and above the x-axis and the vertices of its base on the curve 12 y = 36- x 2 . Find the largest possible area of...
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