Math 3298 - Hampton - Lab 5 - Due 10/16

Critical points and level curves

Exercises

1.  The family of surfaces z = 2+x^2+ A y^2 + 4 xy + x y^2, which depend upon the parameter A, all have a critical point at (x,y) = (0,0).  Describe this critical point for three different values of A, one of which is negative and the other two positive (not A=1).  In particular, decide whether the critical point is a local maximum, minimum, or saddle for those values.  Use functions such as Plot3D, ContourPlot, Manipulate, and Plot to illustrate your analysis.  Some examples for A = 1 are shown below.

f[x_,y_,A_]:=2+x^2+A y^2 +4 x y+x y^2

Plot3D[f[x,y,1],{x,-2,2},{y,-2,2}]

Plot3D[f[x,y,1],{x,-2,2},{y,-2,2},ViewPoint->{1,2,.1},PlotPoints->50,Mesh->False]

ContourPlot[f[x,y,1],{x,-2,2},{y,-2,2},Contours->20]

Here are cross-sections through the critical point in the x direction and y direction:

Plot[f[x,0,1],{x,-2,2}]
Plot[f[0,y,1],{y,-2,2}]

And here is one along the line y = - x:

Plot[f[x,-x,1],{x,-2,2}]

Finally, here are examples showing the level curves a little above and below the critical value of 2.

ContourPlot[f[x,y,1]==2-1/100,{x,-1,1},{y,-1,1},PlotPoints->200]

ContourPlot[f[x,y,1]==2+1/100,{x,-1,1},{y,-1,1},PlotPoints->200]