Tangent Planes and Linear Approximation in 2
One of the key concepts of calculus is that of linear approximation and
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
a region centered at (x0, y0, f(x0, y0)), going out a distance of del
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
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 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