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101f02fin

# 101f02fin - B U Department of Mathematics Math 101 Calculus...

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B U Department of Mathematics Math 101 Calculus I Fall 2002 Final Exam Calculus 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. Show that f ( x ) = x 2 + 1 x - 1 + x 4 x - 2 has at least one root in the open interval (1,2) (Hint: Check continuity and the behaviour at the end points). Solution: lim x + f ( x ) = + . lim x →-∞ f ( x ) = -∞ . Therefore there exists an element c in (1 , 2) such that f ( c ) = 0. 2. Compute f 0 (0) by using the deﬁnition of derivative if f ( x ) = ± 1 x 2 R x 0 sin( t 2 ) dt if x 6 = 0 0 if x = 0 Solution: f 0 (0) = lim x 0 1 x 3

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101f02fin - B U Department of Mathematics Math 101 Calculus...

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