This applet follows the same approach as
the 1 variable 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. (The box is a constant function plus or
minus the error.)
The definition of differentiability can be understood in terms of
cones and lines. 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. (The cone
is the tangent line with the slopes wiggled.)
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 2 graphics windows.
The f(x) window shows a graph of the function, with a red dot at
the graph of the hot spot X and a magenta patch on the graph
corresponding to the graph of a neighborhood of radius delta around
the point (X). the hotspot and magenta patch also show up on the
x-axis. With checkboxes on the control panel you can remove
the cone or add in a small section of the candidate tangent line.
The values of the coordinates of the hot spot are on the
control panel. The
values of ε and δ can either be typed in or shifted with the
The close up window zooms in on the disks. For a given
ε, you want to find a δ where the magenta curve is trapped
between the two blue lines.
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) = abs(x). It
is differentiable everywhere except at x = 0. Compare what
at the origin with what happens away from the origin.