222f02 - 5. (a) Find and classify the critical points of f...

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Final Examination MATH 222 December 10,2002 1. (a) Find the interval of convergence of the power series X n =1 ( x - 1) n 2 n log( n + 1) . (b) Find a power series representation about the point x = 0 for g ( x ) = 4 (1 - x ) 2 . 2. (a) Using a power series expansion for the sine function, compute Z 1 0 sin( x 2 ) dx to 3 decimal places. (b) Compute lim x 0 ( e 2 x - 1) 2 ln(1 + x ) - x 3. (a) Find the equation of the tangent plane to the surface x 2 y + y 2 z + z 2 x = 3 at the point (1 , 1 , 1). (b) Find the directional derivative of the function F ( x,y,z ) = x 2 y + y 2 z + z 2 x at the point (1 , 1 , 1) in the direction ( - 1 , 2 , 4). 4. (a) Reparametrize the curve r ( t ) = (2 t, cos t, sin t ) in terms of arc length measured from the point where t = 0. (b) For the curve in (a), find the unit tangent, unit principal normal and binormal vectors T,N,B of the Frenet-Serret formulas as well as the curvature at any point on the curve. 1
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Final Examination MATH 222 December 10,2002
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Unformatted text preview: 5. (a) Find and classify the critical points of f ( x,y ) = x 2 y-x 2-y 2-2 y as local maxima, local minima or saddle points using the test involving the second partial deriva-tives of f ( x,y ). (b) Use the Lagrange multiplier method to nd the shortest distance from the origin to the curve xy 2 = 1. 6. For each of the following double integrals (a) Z 1 Z 1 x 1 / 3 p 1-y 4 dy dx, (b) ZZ x 2 + y 2 1 ln( x 2 + y 2 ) dxdy, sketch the domain of integration and evaluate the integral. 7. Find the volume of the region bounded by the cylinder x 2 + y 2 = 2 y , the paraboloid x 2 + y 2 = z and the plane z = 0. 8. Compute ZZZ R xz dV , where R is the solid tetrahedron with vertices (0 , , 0) , (1 , , 0) , (1 , 1 , 0) , (0 , 1 , 1) . 2...
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222f02 - 5. (a) Find and classify the critical points of f...

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