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102f04mt2

# 102f04mt2 - B U Department of Mathematics Math 102 Calculus...

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Unformatted text preview: B U Department of Mathematics Math 102 Calculus II Fall 2004 Second Midterm 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.) Find the point(s) on the hyperbolic paraboloid z = ( y- 2) 2- ( x + 1) 2 + 1 nearest to the point (-1,2,3). Solution: We want to minimize the function F ( x, y ) = ( x + 1) 2 + ( y- 2) 2 + ( z- 3) 2 with the constraint g ( x, y ) = z- ( y- 2) 2 + ( x + 1) 2- 1 = 0. Using Lagrange’s method ~ ∇ F = λ ~ ∇ g we get 2( x + 1) = 2 λ ( x + 1) 2( y- 2) =- 2 λ ( y- 2) 2( z- 3) = λ Investigating these equations we see that there are two sets of solutions: Case I: λ = 1. This implies y = 2 and z = 7 / 2. However for these y and z values the constraint equation is not solvable for any...
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102f04mt2 - B U Department of Mathematics Math 102 Calculus...

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