  ### Home > APCALC > Chapter 5 > Lesson 5.2.5 > Problem5-103

5-103.

FUNKY FUNCTIONS REVISITED 5-103 HW eTool (Desmos). Homework Help ✎

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

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 appear differentiable at x = –0.5? Why or why not?

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, examine f ′. Use the piecewise-defined function from part (a) to write an equation for f ′(for x ≠ −0.5).

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

$\lim_{x\rightarrow -0.5^{-}}f'(x)= \ \ \ \ \ \text{ and }\lim_{x\rightarrow -0.5^{+}}f'(x)=$

4. Does $\lim\limits _ { h \rightarrow 0 ^ { - } } \frac { f ( - 0.5 + h ) - f ( - 0.5 ) } { h } = \lim\limits _ { h \rightarrow 0 ^ { + } } \frac { f ( - 0.5 + h ) - f ( - 0.5 ) } { h }$? Justify your answer.

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

Use the eTool below to help solve the problem.
Click the link at right for the full version of the eTool: Calc 5-103 HW eTool