Differentiability

Is the function is the following graph differentiable at x = 0? Why?

• Yes. The limit from the left and right of 0 both approach 0.

• No, $\lim_{x\rightarrow0}\left|x\right|\$ does not exist.

• No, $\lim_{h\rightarrow0}\frac{\left(\left|x+h\right|-\left|x\right|\right)}{h}$ does not exist.

• Yes, the function is differentiable because there is a constant rate of change of -1 from $\left(-\infty,0\right)$ and 1 from $\left(0,\infty\right)$ .

Which of the following would be a valid reason the above function is non-differentiable at x = 0?

• The graph contains a corner.

• The graph contains a cusp.

• The graph contains a discontinuity.

• The graph contains a vertical tangent.

Tags

CCSS.8.F.A.3

If  $f\left(x\right)$  has a derivative at x = a, then f is continuous at x = a.

• True

• False

For what value(s) of x is the function continuous but not differentiable?

• x = -1

• x = 0

• x = 2

• x = 3; x = -2

For what values of x in the domain of the above function does the function to appear differentiable?

• (-2, 3)

• All real numbers.

• [-2, 3]

• $x\ge0$

Where is the graph not differentiable?

• x = -2, -1, 0, 1
• x = -2, -1, 1
• x = -2, -1
• x = -1, 

Where is the graph continuous yet NOT differentiable?

• x = a, b, c, d
• x = b, c, d
• x = a, b, 
• x = b, d

True or False: Continuity implies differentiability 

• True
• False

True or False: Differentiability implies continuity 

• True
• False

Where is the graph continuous yet NOT differentiable?

• x = a, b, c, d
• x = b, c, d
• x = a, b, 
• x = b, d