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SECTION 3.1The Derivative and the Tangent Line Problem
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Remember what the notion of limits allows us to do . . .
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Tangency
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Instantaneous Rate of Change
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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)
The derivative of at is given by
Provided the limit exists. For all for which this limit exists, is a function of .
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Graphical Representation
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f(x)
So, what’s the point?
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f(x)
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f(x)
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f(x)
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Notation and Terminology
Terminology
differentiation, differentiable, differentiable on an open interval (a,b)
Differing Notation Representing “Derivative”
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Example 1 (#2b)Estimate the slope of the graph at the points and .
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Example 2Find the derivative by the limit process (a.k.a. the formal definition).
a.
b.
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Example 3Find an equation of the tangent line to the graph of at the given point.
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Graphs of and
𝒇𝒇 ′
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Graphs of and (cont.)
𝒇 ′ (𝒙 )=𝟐 𝒙
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Example 4Use the alternative form of the derivative.
Alternative Form of the Derivative
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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
If is differentiable at , then is continuous at .
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Example 5Describe the -values at which is differentiable.
a.
b.