This applet follows the same approach as
the multivariable continuity applet,
trying to build intuition the definition of
differentiability for a function of two variables.
The definition of continuity can be understood in terms of boxes
and approximations of constant functions. If a function is
continuous, for any given error height, we can find a width where the
function is within a box. (To understand the definition we draw a
rectangle in the one variable case and a hat box in the two variable
case. In either case this is a constant function plus or
minus the error.)
The definition of differentiability can be understood in terms of
cones and linear functions (planes). For functions of one variable,
differentiability says that for any given error in slopes of the
tangent line, we
can find a width where the function is within a cone of two
lines. (The cone
is the candidate tangent line with the slopes wiggled up and down by
the error tolerance.)
For functions of two variables, wiggling the tangent plane
produces a cone. If a function is
differentiable, for any given error in slopes of the tangent plane, we
can find a width where the function is within the cone.
As with continuity, to prove
differentiability at a point we would need to prove a rule that gives a
delta for any positive epsilon, with the function trapped by the
approximating cone. As with continuity, we will probably be
convinced if we can find the deltas that go along with the epsilon
values of .1, .01, and .001.
Click on the button below to launch the
The applet has a control window and 3 graphics windows.
The first graphical window is the domain window. It
contains shows a grid of points, a big red dot (the hot spot with
coordinates (C_1,C_2)), a green ring or radius ε centered at the
hotspot, and a yellow dot at the edge of the disk. These two dots
are hot spots and can moved with a mouse click and drag.
The values of the coordinates are on the control panel. The
values of ε and δ can either be typed in or shifted with the
The graph window shows a graph of the function, with a red dot at
the graph of the hot spot and a green patch for the graph of the green
circle of radius &epsilon. It also has two halves of the
approximating cone above and below the graph of the hot
spot. With checkboxes on the control panel you can remove
the cone or add in a small section of the candidate tangent plane.
The close up window zooms in on the disks. For a given
ε, you want to find a δ where the green surface is trapped
between the two halves of the pink cone.
Things to notice:
Sometimes it is hard to understand differentiability since almost
functions we can think of are differentiable almost everywhere.
Consider the function f(x,y) = 2*x*y/sqrt(x^2+y^2). (This
function in polar coordinates translates to r*sin(2*theta).) It
is differentiable everywhere except at the origin. Compare what
at the origin with what happens away from the origin.