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samplefinal - Sample Final Examination 1 For each of the...

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Unformatted text preview: Sample Final Examination 1. For each of the following series find (i) the radius of convergence and (ii) what happens at the endpoints of the interval of convergence. ( a ) ∞ X n =1 (- 1) n x 2 n n 4 n , ( b ) ∞ X n =0 x 3 n 64 n · √ n + 1 . 2. (a) Find the Maclaurin series of F ( x ) = Z x e- t 2 / 2 dt. and evaluate F (0 . 1) correct to 5 decimal places. (b) Without using l’Hˇopital’s Rule, compute lim x → ( e 2 x- 1) 2 ln(1 + x )- x 3. Let g ( x,y ) = x 2 y x 2 + y 2 , h ( x,y ) = xy x 2 + y 2 if ( x,y ) 6 = (0 , 0) and g (0 , 0) = h (0 , 0) = 0. Show that g is continuous at (0 , 0) while h is not. 4. (a) Re-parametrize 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 as well as the curvature at any point on the curve. 5. (a) Find the equation of the tangent plane and normal line to the surface z = 3 xe y- x 3- e 3 y at the...
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