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Section 4.6
Functions Having a Value At Which There Is No
Rate of Change at a Moment

You cannot presume that a function has a rate of change at every moment. A simple example is $f(x) = |x ‑ 2| + 1$. It’s graph has a cusp at $x = 2$. The graph of f over any interval that contains $x = 2$ in its interior (that is, 2 is not an endpoint) will contain that cusp. The cusp will always keep the function from having a constant rate of change over such intervals. The animation in Figure 4.7 illustrates this.




Figure 4.6.1. A function cannot have a rate of change at any moment where its graph has a cusp.

Figure 4.6.1 illustrates why the function $ y = \left\lvert x-2\right\rvert + 1$ cannot have a rate of change at the moment that $x = 2$. The function’s rate of change changes abruptly at $x = 2$ no matter how small an interval we make. Therefore it will never have a rate of change that is essentially constant over an interval containing $x = 2$.

The rate of change will be -1 to the left of $x = 2$ and +1 to the right of $x = 2$. Any interval containing $x = 2$ will contain an interval over which the function has a rate of change of -1 and an interval over which the function has a rate of change of +1. The function will never have a constant rate change over an entire interval that contains $x = 2$.