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Mathematics 23 Final Exam, December 14, 2002Answer all questions andshow all work. Simplify your answers.1. Find a vector perpendicular to the plane containing the points (1,2,1),(2,4,2),and (1,0,1).2. For the functionf(x, y) =xe-2y+ 6y, finda. the maximum rate of change offat (1,0).b. the direction in which the maximum rate of change at (1,0) occurs.3. LetEbe the solid abovez=√x2+y2and belowx2+y2+z2=z. Set up anintegral (in spherical coordinates) which gives the volume ofE.4. Find the volume of the largest rectangular box in the first octant with threefaces in the coordinate planes and one vertex in the planex+ 3y+ 2z= 6.5.a.Find the tangent plane to the ellipsoidx2+ 2y2+ 3z2= 1 at the point(1,0,0).b. Find the points on the ellipsoidx2+ 2y2+ 3z2= 1 where the tangent plane isparallel to the plane 3x-y+ 3z= 1.6. a. Find the length of the curveCgiven byr(t) =hcost,sint, ti,0≤t≤1.b. What is the curvature ofCatr(to), wheretois an arbitrary point 0≤to≤1?