Question

In Exercises 57-60, use a graphing utility to graph six level curves of the function. $$ f(x, y)=x^{2}-y^{2}+2 $$

Short answer

The level curves for the function \(f(x, y)=x^{2}-y^{2}+2\) can be found by setting the function equal to different constants and solving for x and y. The level curves can then be represented graphically.

Step-by-step solution

Understanding the Function

The first step is understanding our function \(f(x, y)=x^{2}-y^{2}+2\). This is a quadratic function with two variables, x and y. The function values change with changes in x and y values.

Setting up level curves

To find the level curves of this function, we will set the function equal to different constants which we can select as per our needs. This usually includes both positive and negative values and possibly zero. For example, let's start with the level curve where function value is zero, \(0=x^{2}-y^{2}+2\), which simplifies to \(y^{2} = x^{2} + 2\)

Solving for Additional Level Curves

Let's do this for two more constants, +3 and -3, as examples. We will get two more equations, firstly \(3=x^{2}-y^{2}+2\), which simplifies to \(y^{2} = x^{2} - 1\) and secondly \(-3=x^{2}-y^{2}+2\), which simplifies to \(y^{2} = x^{2} + 5\). You can repeat this process for other constants as well.

Graphing the Level Curves

Finally, to graph these level curves, plot the equations we have derived. Each equation represents a set of points (x,y) where the function value is constant, i.e., the chosen constants. Then, all the points on the graph of each equation make up a level curve of the function.