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 $\left(0,\infty\right)$ .

Is the above function differentiable for all values of x? Why?

• Yes, the slope of the tangent line can be calculated for all values of x.

• Yes, the function is continuous for all values of x so the function is differentiable.

• No, the function has a sharp turn in the first quadrant so if is not differentiable.

• No, the function is discontinuous so it is not differentiable.

In which quadrant of the graph above will there be a value of x that is non-differentiable?

• 1

• 2

• 3

• 4

Tags

CCSS.8.F.A.3

Is the above function differentiable at x = 0? Why?

• No. The $\lim_{h\rightarrow0}\frac{\left(\left(x+h\right)^{\frac{2}{3}}-\left(x\right)^{\frac{2}{3}}\right)}{h}$ does not exist.

• Yes. The $\lim_{h\rightarrow0}\frac{\left(\left(x+h\right)^{\frac{2}{3}}-\left(x\right)^{\frac{2}{3}}\right)}{h}$ exists.

• Yes. The $\lim_{x\rightarrow0}\ x^{\frac{2}{3}}=0$ from the left and the right.

• No. The $\lim_{x\rightarrow0}\ x^{\frac{2}{3}}=0$ doesn't exist since the left and right hand limits are not the same.

Is the above function differentiable at

x = 0? Why?

• Yes. The derivative would be equal to zero since there is a vertical tangent.

• No. The derivative would not exist at x = 0 since there is a vertical tangent.

• No. The derivative does not exist since $\lim_{x\rightarrow0}\ x^{\frac{1}{3}}$ does not exist.

• Yes since $\lim_{h\rightarrow0}\ \frac{\left(x+h\right)^{\frac{1}{3}}-x^{\frac{1}{3}}}{h}$ exists.

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

Find all points of the domain of  $y\ =\ \left|x-2\right|+3$  where  $\frac{dy}{dx}$  does not exist. 

•  $\frac{dy}{dx}$  exists for all real numbers.

•  $\frac{dy}{dx}$  does not exist at (0,0).

•  $\frac{dy}{dx}$  does not exist at (2,3).

•   $\frac{dy}{dx}$  does not exist at (-2,3).

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

Graph of a function is given.Where is the function continuous yet NOT differentiable?

• x = a, b, c, d

• x = b, c, d

• x = a, b,

• x = b, d