SECTION 3.1 The Derivative and the Tangent Line Problem.

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Presentation transcript:

SECTION 3.1 The Derivative and the Tangent Line Problem

Remember what the notion of limits allows us to do...

Tangency

Instantaneous Rate of Change

The Notion of a Derivative Derivative The instantaneous rate of change of a function. Think “slope of the tangent line.” Definition of the Derivative of a Function (p. 119)

Graphical Representation

f(x) So, what’s the point?

f(x)

Notation and Terminology Terminology differentiation, differentiable, differentiable on an open interval (a,b) Differing Notation Representing “Derivative”

Example 1 (#2b)

Example 2

Example 3

Example 4 Alternative Form of the Derivative

When is a function differentiable? Functions are not differentiable... at sharp turns (v’s in the function), when the tangent line is vertical, and where a function is discontinuous. Theorem 3.1 Differentiability Implies Continuity

Example 5