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Activity4.3Nondifferentiability

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 \(a\text{.}\)

Subsection4.3.1Exercises

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

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<<SVG image is unavailable, or your browser cannot render it>>

Figure4.3.1\(y=\fe{k}{x}\)
Figure4.3.2\(y=\fe{\fd{k}}{x}\)
1

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

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

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<<SVG image is unavailable, or your browser cannot render it>>

Figure4.3.3\(y=\fe{g}{x}\)
Figure4.3.4\(y=\fe{\fd{g}}{x}\)
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 \(5\text{?}\)