Calculus Questions - HELP!!!!! 1. Consider the cube determined by the planes x=-1, x=3, y=5, y=9, z=0, and z=4. a) Give the coordinates of the eight vertices and center of the cube. b) Determine an equation of the largest sphere contained in the cube. c) Determine an equation of the largest sphere that would fit between the sphere found in (b) and the cube. 2. Determine an equation of a plane that intersects the plane x+y+z=3 at an angle of 60 degrees. 3. a) Determine an equation of the tangent plane to the surface given by x^2*y+y^2*z+z^2*x=1 at the point (1,0,1). b) Determine an equation of the line that is normal to the surface given by x^2*y+y^2*z+z^2*x=1 at the point (1,0,1). 4. Prove that u+v+w=0, then u*v=v*w, and u*w=w*v. What is the geometric interpretation of these relationships? 5. Suppose f(x,y)=A*X^3+B*X*Y+C*Y^2, where A, B, and C are constants. For what values of A, B, and C does f have a critical value at (-2,1)? Determine what type of critical point it is. 6. Determine the maximum value of f(x,y,z)=(x*y*z)^2 subject to the constraint x^2+y^2+z^2=c^2, where c is not equal to zero.