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101f05mt2 - B U Department of Mathematics Math 101 Calculus...

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Unformatted text preview: B U Department of Mathematics Math 101 Calculus I Fall 2005 Second Midterm 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) If f is continuous on [2 , 5] and 1 ≤ f ( x ) ≤ 4 , ∀ x ∈ (2 , 5) then show that 3 ≤ f (5)- f (2) ≤ 12 Solution: By mean value theorem there is some c ∈ (2 , 5)such that f ( c ) = f (5)- f (2) 5- 2 but 1 ≤ f ( c ) ≤ 4 so 1 ≤ f (5)- f (2) 5- 2 ≤ 4 hence we have 3 ≤ f (5)- f (2) ≤ 12 (b) Evaluate the integral Z x sin x 2 cos x 2 dx Solution: Let u = sin( x 2 ) then du = 2 x cos( x 2 ) I = R x sin( x 2 ) cos( x 2 ) = R u du 2 = u 2 4 + c = sin 2 ( x 2 ) 4 + c 2. Consider the function2....
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101f05mt2 - B U Department of Mathematics Math 101 Calculus...

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