Another service from Omega

Sketch a graph


*****

Problem:

Sketch the graph of the function,
f(x,y) = 2 x^2 + y^2

Solution:


Notice that the level curves are ellipses.

> with(plots):
> contourplot(2*x^2+y^2,x=-4..4,y=-4..4,contours=[1,2,3],scaling=CONSTRAINED);


picture a picture here


Notice that the level curves can be written as,

> 2*x^2+y^2 = c^2;

                                   2    2    2
                                2 x  + y  = c

from where we can write,

> (x/(c/sqrt(2)))^2 + (y/c)^2 = 1;

                                   2      2
                                  x      y
                               2 ---- + ---- = 1
                                   2      2
                                  c      c

and this is the equation of an ellipse with semi axis,

> a = c/sqrt(2); b = c;

                                           1/2
                                a = 1/2 c 2

                                     b = c

These ellipses grow with increasing z so the surface is,

> plot3d(2*x^2+y^2,x=-4..4,y=-4..4);


picture a picture here


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Wed Mar 21 14:06:26 EST 2001