Expert: Scotto - 7/12/2010

Question
"Hi,  I hope you can help me, Iam taking calculus III this summer and I am having a little trouble understanding some of the problems, they are not configure as the ones in class

The first problem. Given the surface: Z^2=X^2+XY^2Z Find dz/dx and dz/dy. What do I have to do with this surface, the Z^2 at the beginning troubles me.Do I have to take square roots first?


2.Consider the plane through the two point(4,2)(0,1) that is perpendicular to the xy-plane. Find the slpe of the tangent line at the point(4,2,4) to the curve of the intersection of this plane and the graph of the function f(x,y)=(x-y)^2
This problem does not have the direction vector, what do I have to do?

Can you recomend a book that talks about calculus3 topics, one that explains these concepts in a clear way, good examples.

Thanks in advance

Answer
To find dZ/dX and dZ/dY, they are found by taking the derivative as is.
The derivative of Z² is 2ZZ' and XY²Z is found by the product rule.

For X (Y is a constant), we have 2Z(dZ/dX) = 2X  +  XY²(dZ/dX) + Y²Z and
for Y (X is a constant), we have 2Z(dZ/dY) = 0   +  XY²(dZ/dY) + 2XYZ.

In the 1st equation, subtract off the XY²(dZ/dX) from both sides of the equation.
Once this has been done, factor it out, then divide both sides by the multiplier out front of dZ/dX and that will be the solution for dZ/dX.  Do the same to the 2nd equation to find dZ/dY.


2. graphing: gives picture of where the answer is, but doesn't give exact answer most of the time
substitution: way that is most understood
elimination: solving one equation for a variable and putting it into the other equations

for graphing, it gives a picture of where the answer is
for substituion, it is used to solve problems with two or three variables in algebra
for elimination, it is usually used in higher math