Math 262 Winter 2010 Final Exam

Math 262 Winter 2010 Final Exam - i j ? (c) Along what...

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Math 262 Winter 2010 Final Examination 16 th April 2010:2:00–5:00 pm 1) (a) Find a series representation of 1 ( ) (3 ) f x x = + in powers of ( 2) x - . (b) Find the interval of convergence of the series found in (a). 2) Find the Maclaurin series expansion of 2 0 ( ) sin( ) x S x t dt = . Hence evaluate S(0.1) with 3 10 error - . 3) Suppose that a particle moves through 3-space so that its position vector at time t is 2 3 ( ) t t t t = + + r i j k (a) Find the tangential and normal components of acceleration at time t. (b) Find the curvature of the path at the point when t=1. 4) The temperature T(x,y) at points on the xy-plane is given by T(x,y) = x 2 – 2y 2 . (a) In what direction should an ant at position (2, 1) - move if it wishes to cool off as quickly as possible? (b) At what rate would the ant experience the decrease of temperature if it moved from (2, 1) - at speed k in the direction of vector 2 - -
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Unformatted text preview: i j ? (c) Along what curve through the point (2, -1) should the ant move in order to continue to experience max. rate of cooling? 5) If sin x t s = and cos y t s = . Assuming ( , ) f x y is differentiable: (a) Find 2 ( , ) f x y s t (b) Find 2 2 ( , ) f x y s . 6) Show that the equations 2 2 3 3 2 2 1 xy zu v x z y uv xu yu xyz + + = +-= +-= can be solved for x, y and z as functions of u and v near point P where (x,y,z,u,v) = (1,1,1,1,1) and find v y u at (u,v) = (1,1) . 7) Find the maximum and minimum values of 2 2 ( , ) 1 x f x y x y = + + . 8) Use Lagrange multipliers to find the shortest distance from the origin to the surface 2 2 xyz = . 9) Find the volume of the solid bounded by the plane z = 0 and the paraboloid 2 2 1 z x y = --....
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