Mathematics GRE

Q1: Given the function f: (A U B) to C, which of the following is always true?
The choice is f(A U B)=f(A) U f(B).
Why? Is there any theorem about it?

Q2: The moment with respect to yz plane of the volume in the first octant bounded by f(x,y,z) and g(x,y,z), and balabalabala....
My question is why the answer is triple integral on volume v [(x) dxdydz], why is there an x in the integral?

Q3: Given x^2 *z-2y*z^2+xy=0, find dx/dz at (1,1,1).
I know first I need to caculate df/dx and df/dz, and then I use dx/dz=(df/dz)/(df/dx), but the answer shows that dx/dz= -(df/dz)/(df/dx), why there is a negative one there? Feeling puzzle....

Thanks for your replies.

Regarding 1: http://math.stackexchange.com/questions ... the-images

Regarding 2: What is the definition of "moment"? Seems obscure to me.

Regarding 3: Notice that the equation can be written as F(x,y,z) = F(x,y,f(x,y)) = 0. By the chain rule, (dF/dx) + (dF/dz)(dz/dx) = 0. Rearranging gives the answer with the negative sign.

BreakingBad wrote:Regarding 1: http://math.stackexchange.com/questions ... the-images

Regarding 2: What is the definition of "moment"? Seems obscure to me.

Regarding 3: Notice that the equation can be written as F(x,y,z) = F(x,y,f(x,y)) = 0. By the chain rule, (dF/dx) + (dF/dz)(dz/dx) = 0. Rearranging gives the answer with the negative sign.

'Moment' seems an concept in Physics....

Regarding Q2, I would have to see the actual question to give a proper response, but moment is related to distance and the shortest distance between a point in the first octant and the yz-plane is the x-coordinate of the point.