Question

Describe the level curves of the function. Sketch the level curves for the given \(c\) -values. $$ f(x, y)=x^{2}+2 y^{2}, \quad c=0,2,4,6,8 $$

Short answer

The level curves of the function \(f(x, y)=x^{2}+2 y^{2}\) for values \(c = 0, 2, 4, 6, 8\) correspond to a point at the origin and a series of increasingly larger ellipses with the same center, growing as the value of \(c\) grows. Therefore, the level curves represent circular paths around the origin.

Step-by-step solution

Formulate the General Equation for the Level Curves

Retain the general structure of the function \(f(x, y)=x^{2}+2 y^{2}\), setting it equal to the constant \(c\), which represents the 'height' of our three-dimensional surface. So, we have \(x^{2}+2 y^{2} = c\). This is our general equation, which we can solve for different values of \(c\).

Construct Level Curves for Different Values of \(c\)

Substitute each of the given \(c\)-values, \(0,2,4,6,8\), into the level curve equation from the first step. For \(c = 0\), we get the equation \(x^2+2y^2 = 0\). For \(c = 2\), our equation is \(x^2+2y^2 = 2\), and so on.

Transform Each Equation into a Recognizable Form

Rework each equation into a form that can be easily recognized and graphed. In general, the equation \(x^2 + ay^2 = c\) describes an ellipse for \(a > 0\) and \(c > 0\). For \(c = 0\), the equation describes a single point at the origin. Our equations thus represent ellipses with the same center but different radii. The equation for \(c = 2\) for example can be simplified to \(x^2 + y^2 = 1\) and \(2y^2 = 1\), or \(x^2 + (\sqrt{2}y)^2 = 1\), an ellipse with semi-major axis \(1\) and semi-minor axis \(\frac{1}{\sqrt{2}}\), and so on for other \(c\)-values.

Sketch the Level Curves

Using the results from Step 3, sketch each level curve on a separate graph. Notice that as the value of \(c\) increases, the corresponding level curves become larger. With \(c = 0\), the curve is a single point at the origin. With \(c = 2\), the curve is an ellipse with semi-major axis \(1\) and semi-minor axis \(\frac{1}{\sqrt{2}}\). The size of the ellipse grows for following values of \(c\) until the largest ellipse corresponding to the value \(c = 8\).