1) find the maxima and minima of f(x,y,z) = 2x + y -3z subject to the constraint 2x^2+y^2+2z^2=1
2)compute the work done by the force field F(x,y,z) = x^2I + y j +y k in moving a particle along the curve r(t) = (t,sin(t),cos(t)) where t is greater than or equal -pi and less than or equal pi.
3)use green's theorem to computer the integral F . dr where F = ( y^2 + x, y^2 + y) and c is bounded below the curve y= - cos(x),, above by y = sin(x) to the left by x=0 and to the right by x = pi/2
4) find the are length of r(t) = ( 1/2t^2, 1/3t^3, 1/3t^3) where t is between 1 and 3 (greater than or equal less than or equal)
5)sketch the level curves of f(x,y) = x^2-2y^2 for k = -1,0,1
6)let R be the triangle with vertices (0,0), (pi, pi) and (pi, -pi). using the change of variables formula u = x-y and v = x+y , compute the double integral (cos(x-y)sin(x+y) dA as an integral in du and dv.
7) compute the center of mass of the solid of unit density 1 bounded (in spherical coordinates) by p=1 and by φ is greater than or equal 0 and less than or equal pi/4