Sec. 3.2: Working with the Derivative Differentiability and Continuity This is used only in special cases (i.e. functions that are continuous but not differentiable).

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Presentation transcript:

Sec. 3.2: Working with the Derivative Differentiability and Continuity This is used only in special cases (i.e. functions that are continuous but not differentiable).

Sec. 3.2: Working with the Derivative Differentiability and Continuity The existence of the limit in this alternate form requires that the one-sided limits exist and are equal.

Sec. 3.1: Introducing the Derivative Differentiability and Continuity Ex: Discuss the differentiability of f(x) = |x – 2| at x = 2.  f is not differentiable at x = 2.

Sec. 3.2: Working with the Derivative Differentiability and Continuity Ex: Discuss the differentiability of f(x) = x 1/3 at x = 0.  f f has a vertical tangent at x = 0.

Sec. 3.1: Introducing the Derivative Differentiability and Continuity

Sec. 3.2: Working with the Derivative Differentiability and Continuity Summary of the relationship between differentiability and continuity: 1.Differentiability implies continuity. (A function that is differentiable must also be continuous.) 2.Continuity does not imply differentiability. (A function that is continuous is not necessarily differentiable.) Things that destroy differentiability. a)Discontinuity b)Sharp turns (cusps) c)Vertical tangents