Differentiability and Piecewise Functions

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Presentation transcript:

Differentiability and Piecewise Functions

What are the three things that make a function not differentiable ? Not continuous at the point Vertical tangent line at the point A cusp at the point.

Where is the derivative undefined? 1. …at a point of discontinuity. Example: f(x) is not differentiable at x = ____

Where is the derivative undefined? 2. …where there is a vertical tangent line. Example: y = x1/3 is not differentiable at x = ___

3. …if the graph has a sharp point (cusp) Where is the derivative undefined? 3. …if the graph has a sharp point (cusp) f(x) is not differentiable at x = ____ f(x) is not differentiable at x = ____

Determine the value(s) of x at which the function is not differentiable. Give the reason.

Derivatives of piecewise functions… If then, f ’(x)= f ’(5)= f ’(-2)=

Is the function continuous? Justify your answer. Is the function differentiable at x = 1? Justify your answer.

Determine whether the function is differentiable at x = 3 Determine whether the function is differentiable at x = 3. Justify your answer.

Derivatives of Absolute value functions… If then, f ’(x)= f ’(5)= f ’(-2)=

Is the function differentiable everywhere?