midterm2009-solution

midterm2009-solution - Please write out the honor pledge...

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Unformatted text preview: Please write out the honor pledge and sign it: NAME (print): Instructor / class section: MAT 203 – Midterm October 28, 2009 7:30-9:30PM Information Please read and sign the exam conditions first before turning the page: • No books / notes / calculators / collaborations are allowed. • The midterm has to be completed in a single time stretch of 120 minutes . No interruptions ! • There are 5 problems. • Each problem is worth 20 points. • The total score is 100 points. I have read these conditions and will follow them (initials): Score: Problem 1: points Problem 2: points Problem 3: points Problem 4: points Problem 5: points 1 Problem 1 (20 points) 1. ( 10 pts ) Find the equation of the tangent plane to the level surface of the function f given by f ( x, y, z ) = x 2 + 5 yz + cos(2 πz ) sin(2 πx ) e y at the point ( x, y, z ) = (0 , 1 , 1) 2. ( 5 pts ) Find the equation of the line passing through P = (3 , 4 , 5) and perpendicular to this plane. 3. ( 5 pts ) What is the distance of P from the tangent plane? Solution: 1. The gradient of f is given by: ∇ f = (2 x +2 π cos(2 πz ) cos(2 πx ) e y , 5 z +cos(2 πz ) sin(2 πx ) e y , 5 y − 2 π sin(2 πz ) sin(2 πx ) e y ) ∇ f (0 , 1 , 1) = (2 eπ, 5 , 5) So the equation of the tangent plane is: (2 eπ, 5 , 5) · ( x, y − 1 , z − 1) = 0 2. The line is parameterized by: (3 , 4 , 5) + t (2 eπ, 5 , 5) 3....
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This note was uploaded on 11/05/2011 for the course MAT 203 at Princeton.

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midterm2009-solution - Please write out the honor pledge...

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