Problem:Sketch the graph of the function, |
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);
 |
| 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);
 |