Differentiability Applet - 1 Variable

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 applet.
• 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 arrow buttons.
• 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 all 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 happens at the origin with what happens away from the origin.

The method used in this applet can be generalized to look at the meaning of differentiability for functions of 2 variables.

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.