Graph f(x) =
and rewrite f as a piecewise-defined function.
There will be two pieces. The graph indicates that x = −0.5 is the boundary point.
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?
To confirm whether or not f(x) =
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?
? 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