Slice Curves
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) Text We can generalize slice curves by considering the slice above any line (y  y_{0}) = m(x  x_{0}).
Demos
Slice Curve along a Line
 
This second demonstration is a generalization of the first in that it allows you to choose an arbitrary line (y  y_{0}) = m(x  x_{0}) in the domain window. The two orange hotspots control the location of the point (x_{0},y_{0}) and the direction of the line. The graphs of the surface (x,y,f(x,y)) as well as the image of the line under f are shown in a separate window. A pink slicing plane has also been added.

Exercises 1. Analyze the slices of the general quadratic function f(x,y) = Ax^{2} + 2Bxy + Cy^{2} for various values of A, B, and C. For which A, B, and C will the range consist of all real numbers? All nonnegative numbers? All nonpositive numbers?
2. Analyze the function f(x,y) = x^{2}*y/(x^{4} + y^{2}) for (x,y) ≠ (0,0) and f(0,0) = 0. What can be said about the restriction of this funciton to a line y = mx through the origin? What is the range of this function, and at what points will the maximum value be taken on?
