Activity 1.7.4.
In this activity, we explore two different functions and classify the points at which each is not differentiable.
For (a)-(c), let \(g\) be the function given by the rule \(g(x) = |x|\text{.}\) Note that for part (d) we use the function \(f\) given by the graph in (d).
(a)
Reasoning graphically (that is, by discussing the graph of \(g(x)=|x|\)), explain why \(g\) is differentiable at every point \(x\) such that \(x \ne 0\text{.}\)
(b)
Use the limit definition of the derivative to show that \(g'(0) = \lim_{h \to 0} \frac{|h|}{h}\text{.}\)
(c)
(d)
Let \(f\) be the function that we have previously explored in Preview Activity 1.7.1, whose graph is given again in the following figure.
State all values of \(a\) for which \(f\) is not differentiable at \(x = a\text{.}\) For each, provide a reason for your conclusion.
(e)
True or false: if a function \(p\) is differentiable at \(x = b\text{,}\) then \(\lim_{x \to b} p(x)\) must exist. Why?
