I have a question asking me to describe the level curves for the following functions f(x,y) = x^2 + 2y^2 - 3x= 0
and
f(x,y) = -6
How do I go about doing this, for the second case, would the level curve be just a line or an empty set?
I have a question asking me to describe the level curves for the following functions f(x,y) = x^2 + 2y^2 - 3x= 0
and
f(x,y) = -6
How do I go about doing this, for the second case, would the level curve be just a line or an empty set?
The level curves are the sets where the functions values are constant.
So for the first one set
$\displaystyle f(x,y)=c \iff x^2+2y^2-3x=c $
Now complete the square on the x terms to get
$\displaystyle \left(x-\frac{3}{2} \right)^2+2y=c+\frac{9}{2}$
$\displaystyle \left(x-\frac{3}{2} \right)^2+2y=\frac{2c+9}{2}$
$\displaystyle \frac{\left(x-\frac{3}{2} \right)^2}{\frac{2}{2c+9}}+\frac{y}{2c+9}=1$
This is a familiy of ellipses.
For the 2nd one think of the definition. Where is the function constant