a Find a vector perpendicular to the plane through...

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PRACTICE PROBLEMS FOR THE FINAL EXAM (1) (a) Find a vector perpendicular to the plane through (0 , 2 , 0), (2 , 3 , 1) and (1 , 2 , 4). (b) Give an equation for the plane in part (a). (2) Find a parametric equation of the line of intersection of the planes P 1 defined by 2 x + 3 y z = 0 and P 2 defined by x + 3 y + z = 1. (3) Find the arclength of the curve −→ r ( t ) = e t cos t −→ i + e t sin t −→ j with t [0 , 2 π ]. (4) The position function for the motion of a particle is given by −→ r ( t ) = ( t, 2 t, t 2 ) . (a) Find the velocity and acceleration vectors. (b) Find the tangential component a T of acceleration. (c) Find the curvature when t = 0. (5) Let f ( x, y, z ) = x 2 y + xyz 3 . Find each of the following. (a) ∂f ∂x (b) 2 f ∂x 2 (c) 2 f ∂z∂x (1 , 2 , 3) (6) Find the tangent plane to the surface defined by x 2 + y 2 z 2 4 = 1 at the point (1 , 1 , 2). (7) Let f be the function f ( x, y ) = x 2 3 y 2 . (a) Find the directional derivative of f at the point (2 , 3) in the direction of the vector −→ a = (− 4 , 3 ) . (b) Find the maximal rate of change of f at the point (2 , 3) and the direction in which it occurs. (8) (a) Find the absolute maximum and the absolute minimum values of the function f (

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