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sample_midtermisu03

sample_midtermisu03 - MTH 255 VECTOR CALCULUS II SAMPLE...

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MTH 255 VECTOR CALCULUS II SAMPLE PROBLEMS FOR MIDTERM I 0. Review your class notes and work through the HW problems! 1. Let c be the curve with vector equation r ( t )= e t sin t i + e t cos t j . a) Find the arc length function s = s ( t ) for c measured from t = 0, and reparametrize c with respect to s . b) Find the curvature of c at the point x = 0, y = 1. 2. Let F 1 ( x, y, z )= xz i + xyz j - y 2 k and F 2 ( x, y, z )= xy 2 i + x 2 y j + cos z k . a) Show that there is no function f = f ( x, y, z ) so that F 1 ( x, y, z )= f ( x, y, z ). b) Find a function g = g ( x, y, z ) so that F 2 ( x, y, z )= g ( x, y, z ). c) Evaluate the integral ± c F 2 · d r , where the curve c is given by r ( t )= e cos t i + t 4 sin t j + t k , 0 t 2 π . 3. a) Find the maximum rate of change of the function f ( x, y, z )= x 2 +4 y 2 +9 z 2 at the point P (2 , - 1 , 0) and the direction in which it occurs. b) Next consider the ellipsoid x 2 +4 y 2 +9 z 2 = 1. Find the equation of the tangent plane to this surface at the point P (1 / 2 , 1 / 3 , - 11 / 18).
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