Section 3'4: Differentiability and Applications '

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Section 3.4 Differentiability and Applications .

• HWK 31-37 45,47, pp. 192-4

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Questions?

• Problems 20 and 22, page 193.

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Definition of the Derivative

• Given a function f(x), the derivative of f(x) is
  the function f(x) that measures, at every value
  of xa, the slope of the tangent line to y f(x)
  at the point (a,f(a)).

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Definition of the Derivative

• Since the derivative is a limit, the issue of
  whether it exists at a given point or not is a
  legitimate question.
• We say that a function y f(x) is differentiable
  at x a if f(a) exists.

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Places where a function is not differentiable

• Places where the function is not continuous.

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Places where a function is not differentiable

• Places where the function is not continuous.
• Places where the tangent line is vertical.

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Places where a function is not differentiable

• Places where the function is not continuous.
• Places where the tangent line is vertical.
• Cusps

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Places where a function is not differentiable

• Places where the function is not continuous.
• Places where the tangent line is vertical.
• Cusps

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Applications

• Problems 46 and 48 (page 194)