Tangent Planes and Linear Approximation in 2 Variables

One of the key concepts of calculus is that of linear approximation and local linearity.  Roughly, this says that at any function nice enough to be used in a standard calculus class, if we zoom in far enough, the tangent plane gives a good approximation of the function.  With any concept in multivariable calculus, it is worthwhile to review the same concept, linear approximation, in single variable calculus.
• The applet has two graphics windows.  The full window graphs in the window specified in the control window.  The Zoom Window plots a region centered at (x0, y0, f(x0, y0)), going out a distance of del in each direction.
• In the Zoom Window, the point P1can be moved by dragging.
• The differences in the x, y, and z coordinates of P0 and P1 are delx, dely, and delz respectively.
• The error is the difference between the value of f(x1) and the z value generated by the tangent plane.

Things to notice:
• The value of del can either be typed in to the box on the control panel.  The "vcr buttons" can also be used to scroll through values of del.  As del gets smaller, we zoom in on the graph.  Eventually the graph of the function and of its tangent line become indistinguishable.
• The equation of the tangent plane will be  LinApprox(x,y)=f(x0, y0)+f_x(x0,y0)*(x-x0)+f_y(x0, y0)*(y-y0).
• One measure of the goodness of an approximation is the size of the region where the error  is small enough.  With the default function, find the region where the tangent plane is within .01 of the graph of the function.
• A second measure of the goodness of an approximation is the size of the region where the ratio of the error and dely is small.  Find the region where the error between the tangent plane and the graph is less than 10% of delz, the distance z has changed from the center of the approximation.
• The size of the region where the tangent line is a good approximation can change as we change the base point.  Describe how the size of the good region changes as we change the value of x0 and y0.

A worksheet on using the tangent plane to understand local linearity is available.

This applet was designed as by modifying an applet from a demo by Tom Banchoff at Brown University.  It is used with permission.   Go to the Banchoff Applet Help page.