For questions 1-4, your construction of the surface A(x,y)=xy out of index cards should look like the model below:

4) The graph of a multivariable function is a family of graphs of a single variable function in which you "slice" the multivariable graph by letting one variable vary, while all of the other variables stay fixed. Since this is the scheme we followed in our construction, our one variable graphs are elements of a family of one-variable graphs that all lie on the multivariable graph.

We can better approximate the surface by taking more slices, or by taking slices along a different direction.

5) The slope of the graph of A along the line y=2 at the point (3,2) is just the slope of the cross-section (one-dimensional graph) at the point x=3. Taking the derivative of A(x,2) and evaluating the derivative at x=3 gives a slope of 2.

6) Write a paragraph showing how the same ideas generalize.

Last modified: Wed Oct 23 16:58:48 1996