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Read aloud Readability settings Prev Up Next \(\newcommand{\Z}{\mathbb{Z}}
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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 nondifferentiability.
\(f\) is nondifferentiable at
\(a\) if
\(f\) is discontinuous at
\(a\text{.}\)
\(f\) is nondifferentiable at
\(a\) if the slopes of
\(f\) from the left and from the right at
\(a\) are different.
\(f\) is nondifferentiable at
\(a\) if
\(f\) has a vertical tangent line at
\(a\text{.}\)
Exercises Exercises
Exercise Group.
Consider the function
\(k\) shown in the first figure. Please note that
\(k\) has a vertical tangent line at
\(-4\text{.}\)
1. There are four values where
\(k\) is nondifferentiable; what are these values?
2. Draw
\(\fd{k}\) onto the second figure.
Exercise Group.
Consider the function
\(g\) shown in the first figure.
3.
\(\fd{g}\) has been drawn onto the second figure 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
\(5\text{?}\)
