FUNKY FUNCTIONS REVISITED 

1. Graph $f ( x ) = | x ^ { 3 } + 0.125 |$ and rewrite $f(x)$ as a piecewise function.

   Hint (a):

   There will be two pieces. The graph indicates that $x = −0.5$ is the boundary point.

2. Zoom in at $x = −0.5$ on your graphing calculator and carefully examine the curve. Does $f(x)$ appear differentiable at $x = −0.5$? Why or why not?

   Hint (b):

   Does the slope from the left of $x = −0.5$ appear to agree with the slope from the right?

3. To confirm whether or not $f ( x ) = | x ^ { 3 } + 0.125 |$ is differentiable at $x = −0.5$, we need to examine $f^\prime(x)$. Use the piecewise function from part (a) to find $f^\prime(x)$ for $x ≠ −0.5$.

   Hint (c):

   Find the derivative of each piece of your piecewise function.
   Then evaluate these limits. Do they agree?

4. Does $\lim\limits_ { h \rightarrow 0 ^ { - } } \frac { f ( - 0.5 + h ) - f ( - 0.5 ) } { h } = \lim\limits_ { h \rightarrow 0 ^ { + } } \frac { f ( - 0.5 + h ) - f ( - 0.5 ) } { h }$? State a conclusion.

   Hint (d):

   Notice Hana's definition of the derivative! This question is asking $f^\prime(-0.5)$ from the left agrees with $f^\prime(-0.5)$ from the right.
   Looking at the graphs, do the slopes appear to agree from both sides of $−0.5$?