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- A point moves along the curve of intersection of the paraboloid z=x^2+5y^2 and the plane x=3. At what rate is z changing with y when the point is at (3,-1,14)?If V=R\power{3}, and W\index{1} is the xy plane and let W\index{2} is yz-plane: W\index{1}={(x,y,0):x,y∈R} W\index{2}={(0,y,z):y,z∈R} then V is not the direct sum of W\index{1} and W\index{2}.Find the distance between the point (0, 0, 0) and the plane 2x + 3y + z = 12.
- Find the points on the hyperboloid 9x^2 - 45y^2 + 5z^2 = 45 where the tangent plane is parallel to the plane x + 5y - 2z = 7The upper half of the unit sphere x2+y2+z2=1 is z=Square root of 1-x2-y2. Find its centroid?C is the hexagon in the xy-plane with vertices (5, −5), (5, 5), (0, 10), (−5, 5), (−5, −5), and (0, −10), directed counter-clockwise. Evaluate integralC (3y/(x^2+y^2) dx − 3x/(x^ 2 + y^2) dy)
- Find the extreme values of ƒ(x, y, z) = x2yz + 1 on the intersection of the plane z = 1 with the sphere x2 + y2 + z2 = 1Find the trace of the plane 3x − 2y + 5z = 4 in the xy-plane.Find the orthogonal trajectories of the conicoid (x + y)z = 1 of the conics in which it is cut by the system of the plane x − y + z = K where K is a parameters