101s06fin - B U Department of Mathematics Math 101 Calculus...

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B U Department of Mathematics Math 101 Calculus I Spring 2006 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. (a) State the Mean Value Theorem. (b) Consider the function g : R R defined by g ( t ) = ± 2 if t ≤ - 2 t if t > - 2 Is the Mean Value Theorem satisfied in [ - 2 , 2]? Explain. Solution: (a) For a function f , which is continuous on [ a, b ] and differentiable on ( a, b ), there exits c ( a, b ) such that f 0 ( c ) = f ( b ) - f ( a ) b - a . (b) Clearly g 0 ( t ) = ± 0 if t < - 2 1 if t > - 2 . So, for all t ( - 2 , 2), g 0 ( t ) = 1. On the other hand, g (2) - g ( - 2) 2 - ( - 2) = 0, but there is no c ( - 2 , 2) for which g 0 ( c ) = 0. So the Mean Value Theorem is not satisfied for g . The reason for this fact is that g is not continuous at - 2.
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2. Let f be a function differentiable at x = 0 satisfying the relation f ( x + y ) = f ( x ) f ( y ) and f (0) = 1 . Find f 0 ( x ) and f 00 ( x ) in terms of f 0 (0) and f ( x ) . Solution:
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This note was uploaded on 11/16/2011 for the course MATH 102 taught by Professor Soysal during the Winter '09 term at Boğaziçi University.

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101s06fin - B U Department of Mathematics Math 101 Calculus...

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