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Unformatted text preview: L 0 / 0 4 / 2 0 0 L T H U 1 6 : 3 6 F A X 6 4 3 4 3 3 0 George M. Bergman Bechtel Auditorium }IOFFITT LIBRARY Spring 2001, Math 53M Final Examination 1. (60 points, 6 points apiece) Find the following. If an expression is undefined, say so, (a) The area of the region of the plane described in polar coordinates by the conditions 0 S O < 1 , 0 ( r ( l + e 0 . (b) A unit vector perpendicular to both (1,2,3 ) and (q,5,6). (c) The length of the curve given by x = t2, != 1t3/31- t, whcrc -l < t < l. (d) it f(s!z), gG3)), where / is a differentiable tunction of two variables and g is a differentiable function of one variable. The answer should be expressed in terms of /, B, and their derivatives and/or partial derivatives. , l , l + r (c) Jo J,_, xydydx. (t) An expression tbr lllu f(x,y,z)dv, as an iterated integral, where E= l(x,y,z\ | ,2 * y2 * ,2 s Ioo ), and ] is a continuous function. (Do not change coordinates.) (e) Jrf'dr, where F isthevectorfield (1,2,y) and C isthecurvegivenby r(r)- ( t 2 , t 3 , t 5 )...
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- Trigraph, nd, nd l et, MOFFITT L IBRARY, r-y-plane,a nd, nd u pward