2-B Differentiability
Instructions
This applet involves a lot of the same elements as 2-A (The Derivative at a Point), but we will consider specific examples and investigate how the instantaneous rate of change can fail to exist at certain points.
The graph of a function is shown in the applet.
- Use the input box for c or click and drag the point on the graph to change the point where you want to investigate the instantaneous rate of change.
- Adjust the slider tool for h to move the point Q around P.
- The "Secant" checkbox will show/hide the secant line between P and Q and the slope of the secant line (i.e., average rate of change).
- The "Difference Quotient" checkbox will show/hide the graph of the difference quotient function, which has an excluded value when h = 0.
- The "Tangent" checkbox will show/hide the tangent line at P and its slope.
- Use the "h \to 0" and observe the relationship between the secant and tangent lines.
2-B Differentiability
Because the derivative of a function at a point (i.e., the instantaneous rate of change) is defined in terms of a limit and because it is possible that a limit "does not exist" (i.e., as a finite, real number), there are scenarios when a derivative may fail to exist. This simply means that no finite, numerical value can be assigned as the function's instantaneous rate at that point.
The term differentiation refers to the process of finding a derivative. When such a derivative exists (i.e., as a finite real number), we say that the function is differentiable at that point, because we are able to differentiate it.
As the examples in this applet demonstration, a function can fail to be differentiable for many reasons and it often boils down to examining the one-sided limits of the difference quotient. If you move on to the next applet (on Linearization), you will see that differentiability is related to a concept called local linearity, i.e., if you zoom in close enough to a point on the graph of a differentiable function, the graph will appear to be linear (in fact it will look nearly identical to its tangent line at that point).