Question

Use a graphing utility to graph six level curves of the function. $$ g(x, y)=\frac{8}{1+x^{2}+y^{2}} $$

Short answer

The level curves of the function are represented by six equations: \( \frac{8}{1 + x^{2} + y^{2}} = 1 \), \( \frac{8}{1 + x^{2} + y^{2}} = 2 \), \( \frac{8}{1 + x^{2} + y^{2}} = 3 \), \( \frac{8}{1 + x^{2} + y^{2}} = 4 \), \( \frac{8}{1 + x^{2} + y^{2}} = 5 \), \( \frac{8}{1 + x^{2} + y^{2}} = 6 \). These give us six level curves which can be graphed using a graphing utility.

Step-by-step solution

Understanding Level Curves

A level curve, also called a contour curve, of a function f of two variables is a curve where the function is constant. In terms of a graph of a function, these are the lines along which the surface doesn't rise or fall, which means the function has the same value.

Determining Constants for Level Curves

Each level curve will represent a different constant value of the function. For this problem, six level curves need to be graphed, therefore six different constant values need to be considered. Since the function \( g(x, y) = \frac{8}{1 + x^{2} + y^{2}} \) varies between 0 (for high values of x and y) and 8 (for x = y = 0), good values for the level curves could be 1, 2, 3, 4, 5 and 6.

Setting up Equations for Level Curves

Now we need to express our constants as equations in terms of our function. The general form of these equations would be \( \frac{8}{1 + x^{2} + y^{2}} = constant \). Substituting our chosen constants, we get six equations: \( \frac{8}{1 + x^{2} + y^{2}} = 1\), \( \frac{8}{1 + x^{2} + y^{2}} = 2\), \( \frac{8}{1 + x^{2} + y^{2}} = 3\), \( \frac{8}{1 + x^{2} + y^{2}} = 4\), \( \frac{8}{1 + x^{2} + y^{2}} = 5\), \( \frac{8}{1 + x^{2} + y^{2}} = 6\). Each of these equations represents a level curve. The next step would be to plot these level curves.

Graphing the Level Curves

The final step is to graph each of these curves on the same graph, which will represent different constant 'heights' along our function \( g(x, y) \). This procedure might be best served by using a graphing utility, plug in each curve and generate the plot.