continuity and differentiability

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• DNE

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CCSS.HSF.IF.A.2

CCSS.HSF.IF.B.4

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CCSS.HSA.REI.D.10

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CCSS.HSF.IF.A.1

CCSS.HSF.IF.A.2

CCSS.HSF.IF.B.4

A polynomial function is continuous to all real numbers.

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The funciton f(x) = x²-2x+1, continuous to the interval [-7,7]

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Is the function in 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?

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CCSS.HSA.REI.D.10

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.

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.