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Activity 4.3 Nondifferentiability

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 \(a\text{.}\)

  • \(f\) is nondifferentiable at \(a\) if the slope of \(f\) is different from the left and right at \(a\text{.}\)

  • \(f\) is nondifferentiable at \(a\) if \(f\) has a vertical tangent line at \(a\text{.}\)

Subsection Exercises

Consider the function \(k\) shown in Figure 4.3.1. Please note that \(k\) has a vertical tangent line at \(-4\text{.}\)

a graph of a function whose curve has two segments; the first segment enters from the left at about (-7,4), moving rightward and slightly downward; it becomes more and more steep until at x=-4, it is briefly vertical; it then loses its steepness and continues moving downward and to the right reaching (-3,-1); here it sharply bends into a straight line, connecting to (-1,3); there is a solid dot at (-1,3), and this segment ends here; the second segment begins at (-1,-5) with a hollow dot; fromt here it connects to (4,5) with a straight line; here it sharply bends downward to a straight line with slope -4, and exits the graph near the lower right corner
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a blank graph with x-axis and y-axis each ranging from -7 to 7
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Figure 4.3.1. \(y=\fe{k}{x}\)
Figure 4.3.2. \(y=\fe{\fd{k}}{x}\)
1.

There are four values where \(k\) is nondifferentiable; what are these values?

Consider the function \(g\) shown in Figure 4.3.3.

the graph of a periodic function; the curve enters from the left at about (-7,2.5), and curves down to (-5,-2); here it sharply changes direction, moving back upward as it travels to the right; it gradually loses its steepness until it reaches (-2.5,3), where it smoothly transitions from an increasing curve to a decreasing one, and then falls down to (0,-2); here it sharply changes direction again, and repeats the same "camel hump" shape that was just described; it repreats this again, exiting the graph at about (7,2.5); all together there were three visible sharp turning points, at (-5,-2), (0,-2), and (5,-2); and there were two visible flat points, at (-2.5,3) and (2.5,3)
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only part of a curve is plotted; the curve has a hollow dot at (-5,3); then it moves rightward, curving more and more downward, until it reaches (-2.5,0); here the visible portion abruptly stops, but the reader is meant to continue drawing it
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Figure 4.3.3. \(y=\fe{g}{x}\)
Figure 4.3.4. \(y=\fe{\fd{g}}{x}\)
3.

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

4.

What six-syllable word applies to \(g\) at \(-5\text{,}\) \(0\text{,}\) and \(5\text{?}\)

5.

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

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