Below is an interactive illustration of a function of x and y. The z axis is the value of the function.[br][br]The point, [math]A[/math], can be moved with the sliders which change the [math]x[/math] and [math]y[/math] values of the point. Dashed lines on the surface show the functions values along constant [math]x[/math] or [math]y[/math] values. There are several check boxes that allow you to show various parameters of the function at the point [math]A[/math].[br][br][list][*]Normal Line - shows a line perpendicular to the surface[/*][*]Tangent Plane - shows a plane that is tangent to the surface. Any line in this plane through [math]A[/math] will also be tangent to the surface.[/*][*]Tangent Line (x=x[sub]A[/sub]) - shows a line in a constant [math]x[/math] plane that is tangent to the surface. The plane is also shown lightly.[/*][*]Tangent Line (y=y[sub]A[/sub]) - shows a line in a constant [math]y[/math] plane that is tangent to the surface[/*][*]Gradient Vector - shows the gradient vector which is in an [math]xy[/math] plane and points in the direction of increasing function value[/*][*]Contour Line - shows a contour curve where the function value is constant[/*][/list][br]Moving the view around is a good way to see the three dimensional features. Try viewing the the gradient and contour line from directly above the surface. Notice the angle between the vector and the curve.[br][br]Equations for all of the items are shown when the box is checked.[br]