Working with Graphs of Functions of Two Variables Worksheet by Russell Blyth - blythrd@slu.edu QyZJKHJlc3RhcnRHJSpwcm90ZWN0ZWRHIiIiLUkld2l0aEc2IjYjSSZwbG90c0c2JEYkSShfc3lzbGliR0YoISIi

Some graphs can be understood in terms of simpler graphs - perhaps as translated or flipped versions. Consider the following function. f := x^2+y^2; plot3d(f, x = -3 .. 3, y = -3 .. 3, axes = normal); Predict the shape and location of the following functions before graphing them. g := x^2+y^2+13; h := (x-2)^2+(y+1)^2; k := 2-((x+1)^2); Write your predictions here, using geometric descriptions: Then plot. You may find it helpful to adjust the x and y ranges and use view= lowz..highz to get good graphs. plot3d(g, x = -3 .. 3, y = -3 .. 3, axes = normal); plot3d(h, x = -3 .. 3, y = -3 .. 3, axes = normal); plot3d(k, x = -3 .. 3, y = -3 .. 3, axes = normal); Click on the triangle to open the next section.

We can understand a function of two variables better if we let one of the variables vary while the other takes a fixed value. f1 := (x, y) -> x*y; First we sketch the graph alone. plot3d(f1(x, y), x = -3 .. 3, y = -3 .. 3, axes = normal); Next we fix (say) the value of y to be 2, and graph the curve traced out on the surface. spacecurve([t, 2, f1(t, 2)], thickness = 2, color = black, t = -3 .. 3, axes = normal); Next we plot the surface and the curve together, and also show the cross-section projected in the xz-plane. Rotate the 3-D plot to match it with the projected plot if the cross-section. (It may be helpful to click on the "1:1" button.) g1 := plot3d(f1(x, y), x = -3 .. 3, y = -3 .. 3, axes = normal, style = patchnogrid): g2 := spacecurve([t, 2, f1(t, 2)], thickness = 3, color = black, t = -3 .. 3, axes = normal): display({g1, g2}); plot(f1(x, 2), x = -3 .. 3, color = black); Now plot also the plane y= 2. g3 := plot3d([u, 2, v], u = -3 .. 3, v = -9 .. 9, color = yellow, style = patchnogrid); display({g1, g2, g3}); Note that the cross-section of the surface in the plane y = 2 gives a straight line on this surface. Why? (Answer this question on the next line: ) Now try the cross-section of the surface in the plane x = -1. Rotate the 3-D plot to match it with the projected plot if the cross-section. (It may be helpful to click on the "1:1" button.) gg1 := plot3d(f1(x, y), x = -3 .. 3, y = -3 .. 3, axes = normal, style = patchnogrid): gg2 := spacecurve([-1, t, f1(-1, t)], thickness = 3, color = black, t = -3 .. 3, axes = normal): display({gg2, gg1}); plot(f1(-1, y), y = -3 .. 3, color = black); Next we add in the plane x = -1. gg3 := plot3d([-1, u, v], u = -3 .. 3, v = -9 .. 9, color = yellow, style = patchnogrid): display({gg2, gg3, gg1});

1. Find cross-sections of the function z = g(x, y) = sin(x-y) in the planes x = 1/2 and y = -2. Start by defining the function g. Show the cross-section planes as well as the curves of intersection. (Avoid reusing variable names from earlier in the worksheet.)