A function is said to be nondifferentiable at any value where its first derivative is undefined. There are three graphical behaviors that lead to non-differentiability.

• $f$ is nondifferentiable at $a$ if $f$ is discontinuous at $a$.

• $f$ is nondifferentiable at $a$ if the slope of $f$ is different from the left and right at $a$.

• $f$ is nondifferentiable at $a$ if $f$ has a vertical tangent line at $a$.

Subsection4.3.1Exercises

Consider the function $k$ shown in Figure 4.3.1. Please note that $k$ has a vertical tangent line at $-4$.

1

There are four values where $k$ is nondifferentiable; what are these values?

Consider the function $g$ shown in Figure 4.3.3.

3

$\fd{g}$ has been drawn onto Figure 4.3.4 over the interval $\ointerval{-5}{-2.5}$. Use the piecewise symmetry and periodic behavior of $g$ to help you draw the remainder of $\fd{g}$ over $\ointerval{-7}{7}$.

4

What six-syllable word applies to $g$ at $-5$, $0$, and $5$?

5

What five-syllable and six-syllable words apply to $\fd{g}$ at $-5$, $0$, and $5$?