Chapter 3 Derivatives Section 3.2 Differentiability.

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Presentation transcript:

Chapter 3 Derivatives Section 3.2 Differentiability

Quick Review

Quick Review Solutions

Quick Review

Quick Review Solutions

What you’ll learn about Why f ′(a) might fail to exist at x = a Differentiability implies local linearity Numerical derivatives on a calculator Differentiability implies continuity Intermediate Value Theorem for derivatives … and why Graphs of differentiable functions can be approximated by their tangent lines at points where the derivative exists.

How f ′(a) Might Fail to Exist

How f ′(a) Might Fail to Exist

How f ′(a) Might Fail to Exist

How f ′(a) Might Fail to Exist

How f ′(a) Might Fail to Exist

Example How f ′(a) Might Fail to Exist

How f ′(a) Might Fail to Exist Most of the functions we encounter in calculus are differentiable wherever they are defined, which means they will not have corners, cusps, vertical tangent lines or points of discontinuity within their domains. Their graphs will be unbroken and smooth, with a well-defined slope at each point.

Differentiability Implies Local Linearity A good way to think of differentiable functions is that they are locally linear; that is, a function that is differentiable at a closely resembles its own tangent line very close to a. In the jargon of graphing calculators, differentiable curves will “straighten out” when we zoom in on them at a point of differentiability.

Differentiability Implies Local Linearity

Numerical Derivatives on a Calculator

Numerical Derivatives on a Calculator The numerical derivative of f at a, which we will denote NDER is the number The numerical derivative of f, which we will denote NDER is the function

Example Derivatives on a Calculator

Derivatives on a Calculator Because of the method used internally by the calculator, you will sometimes get a derivative value at a nondifferentiable point. This is a case of where you must be “smarter” than the calculator.

Differentiability Implies Continuity The converse of Theorem 1 is false. A continuous functions might have a corner, a cusp or a vertical tangent line, and hence not be differentiable at a given point.

Intermediate Value Theorem for Derivatives Not every function can be a derivative.