Online Math Organized by Subject Into Topics

Level Curves

Definition (Trace) When the plane intersects the surface the result is the curve with the equation and such an intersection is called the trace of the graph of in the plane

Definition (Level Curves) The set of points in the -plane that satisfies is called the level curve of at and an entire family of level curves is generated as varies over the range of . Level curves are obtained by projecting a trace onto the -plane. Because level curves are used to show the shape of a surface, they are sometimes called contour curves.

Example (Level Curves) Sketch some level curves for the function with for the given function

    Solution. Graphing the lines we have the family of level curves corresponding to The level curves show that the function is a plane in

level curves _gr_21.gif]

Example (Level Curves) Sketch some level curves for the function with for the given function

    Solution. Graphing the ellipses in we have the family of level curves corresponding to The level curves show that the function is a elliptic paraboloid in

level curves _gr_31.gif]

Example (Level Curves) Sketch some level curves for the function Find the domain and range for the function and sketch some level curves for with and sketch the domain.

Solution. The domain of is . To sketch some level curves let's square both sides and so and some level curves are

level curves _gr_41.gif]

Here's a graph of from two different viewpoints.

level curves _gr_43.gif]