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Mathematics 23 Calculus III Spring 2014Modified Old Final ExaminationWarning:This is just one old examination.The upcomingfinal examination may be very different.The final examination will take place Monday, May 5, 8:00–11:00 a.m.The examination will be cumulative.The roomassignments are:110–113 Packard 101210–213 Packard 101310–313 Packard 101410–413 Packard 466Accommodations: To be announced1.Tis the triangle with vertices (1,1,2), (0,-1,3), and (2,-2,0).a. Find the area ofT.b. Find the equation of the plane containingT.2. EvaluateRC13√ydsfor the space curveCgiven by the vectorfunctionr(t) =i+t3j+t4k, 1≤t≤2.3. Letf(x, y) be a differentiable function for which the directionalderivatives satisfyDuf(1,2) = 9 andDvf(1,2) =-2, whereu=35i+45jandv=-i. Find the gradient∇f(1,2).4. Use the linear approximation (or differential) to find the approx-imate value off(2.04,2.95,3.02) whenf(x, y, z) =x2ycos(πz).5. Find the absolute maximum and minimum values off(x, y) =3x2+ 3y2-2xyon the diskx2+y2≤1.6. Find the volume of the region in space bounded below by thexy-plane, above by the surfacez= 1-y2, and which lies withinthe cylinderx2+y2= 1.7. Find the centroid of the solid hemispherex2+y2+z2≤9,z≥0. You may use symmetry and that the volume of a sphereof radiusRis43πR3.8. Find the mass of the region bounded above by the spherex2+y2+z2= 1 and below by the conez2=x2+y2,z≥0, if thedensity at each point equals its distance from the origin.9. Evaluate the line integralRCF·drifF=hy2,2xy-eyiandCis