describe the graph of the function by computing some level sets and sections F:
ID: 1942849 • Letter: D
Question
describe the graph of the function by computing some level sets and sections
F: R^2 to R, (x,y,z) to xy + z^2
Explanation / Answer
R3->R simply means that the independent variable is from R3 (in other words (x,y,z)) while the value of the function is in R (a single real number). In this particular case f(x,y,z)= xy. Since there are 4 variables, x, y, z, and the function value, f, in order to "graph" it you would need a 4-dimensional drawing. What you can, in general, do is draw, in 3-dimensions, f(x,y,z)= C for different values of C. You could then interpret the 4th variable as time and imagine the drawings as pages in a flip book (frames in an animation for more modern people). For example, taking C= 1, xy= C is a hyperbola (strictly speaking a hyperbolic cylinder with axis parallel to the z axis). xy= -1, xy= 2, xy= -2, etc. give different hyperbolic cylinders showing how the system "evolves over time". Since there is no z in the formula, you can actually draw them in a 2 dimensional xy- graph and imagine them extending into and out of the plane of the graph.
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