Emat233finalWinter2006

# Emat233finalWinter2006 - Concordia University EMAT 233...

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Concordia University EMAT 233 - Final Exam Instructors: Dafni, Dryanov, Enolskii, Keviczky, Kisilevsky, Korotkin, Shnirelman Course Examiner: M. Bertola Date : May 2006. Time allowed : 3 hours. [10] Problem 1. Compute the curvature κ ( t ) of the curve C defined by ~ r ( t ) = t i + t 3 3 j + t 2 2 k . [10] Problem 2. Find points on the surface x 2 + 3 y 2 + 4 z 2 - 2 xy = 16 at which the tangent plane is parallel to the yz –plane. [10] Problem 3. Find the direction in which the function below increases most rapidly at the indicated point. Find also the maximum rate of increase. f ( x, y ) = e 2 x sin(2 y ) , P (0 , π/ 8) [10] Problem 4. The D’Alambert equation 2 ∂t 2 U ( t, x, y ) - 2 ∂x 2 U ( t, x, y ) - 2 ∂y 2 U ( t, x, y ) = 0 . describes the propagation of small waves on an elastic membrane. Show that the function defined as U ( t, x, y ) := cos( ct - ax - by ) , c = p a 2 + b 2 is a solution of the wave equation for any value of the constants a, b (where c is given by the formula written on the right in the equation above).

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