Differentiability of a function

This applet illustrates the concept of differentiability.

The upper section shows the graph of

, and a chord from a point (

(

)) to (

(

)). Drag the points

and

to change them. What do you get as

? Recall that the derivative of

at the point

is defined as

that is, as a limit of the chord gradient function

. Click 'Show chord slope curve' to show in the lower pane a plot of the chord gradient function

, as a function of

. Click 'Show derivative' to add a plot of

. What happens to

and

? Some interesting functions to try:

The absolute value function |x|: enter this as f(x) = abs(x) Is this function differentiable at x = 0? At x ≠ 0?
A function with a single point discontinuity: f(x) = If[ 0.98 < x < 1.02, 2, x ] Is the function differentiable at x = 1?
A function with a step discontinuity: f(x) = If[x < 1, x, x+1] Is the function differentiable at x = 1?
The Dirichlet function . This function is built in as d(x). Is this function differentiable?
A variation on the Dirichlet function: . Enter this as f(x) = d(x)*x^2 + 1 Where is this function differentiable?

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