TITLE: Sketch a graph
# !b2 Problem:
Sketch the graph of the function,
!c0 f(x,y) = 2 x^2 + y^2

!b3 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);
# 
!im ellipses.gif
# 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);
# !im elipara.gif

>