Unit 4: Application of Derivatives

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Application of Derivatives Overview

Evaluating a Derivative (Finding the Slope) 5 Topics

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Identifier: Evaluating a Derivative

Method 1: By Hand

Example 1: By Hand

Method 2: Using the Calculator

Example 2: Using the Calculator

Finding the Equation of the Tangent Line 3 Topics

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Identifier: Equation of a Tangent Line

Method: Equation of a Tangent Line

Example 1: Equation of a Tangent Line

Linearization = Tangent Line Approximation = Local Linear Approximation 4 Topics

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Identifier: Tangent Line Approximation

Method: Tangent Line Approximation

Example 1: Tangent Line Approximation

Example 2: Tangent Line Approximation

Reading and Sketching the Graphs of f(x), f′ (x), f″(x) 8 Topics

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f(x) = y-values of f(x)

f′(x) = slope of f(x)

f″(x) = concavity of f(x)

Identifier: Reading Derivatives Off Graphs

Method: Reading Derivatives Off Graphs

Example 1: Graph the 1st Derivative, f′(x), Given the Graph of f(x).

Example 2: Graph the 1st Derivative, f″(x), Given the Graph of f′(x).

Example 3: Reading Local Extrema and Intervals of Increasing and Decreasing from f′(x).

Physics Relationships 3 Topics

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Identifier: Physics Relationships

Method: Physics Relationships

Example 1: Physics Relationships

Mean Value Theorem 4 Topics

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Identifier: Mean Value Theorem

Method: Mean Value Theorem

Example 1: Mechanical

Example 2: Conceptual

1st Derivative Test: Local Extrema & Increasing/Decreasing Intervals 5 Topics

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Identifier: 1st Derivative Test

Method: 1st Derivative Test

Example 1: f′(x) = 0

Example 2: f′(x) = 0 on Closded Interval [a,b]

Example 3: f′(x) = 0 and f′(x) = DNE

Global Extrema (Absolute Max and Absolute Min) 3 Topics

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Identifier: Global Extrema

Method: Global Extrema

Example 1: Global Extrema

2nd Derivative Test Option 1: Intervals of Concavity and Inflection Points 3 Topics

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Identifier: 2nd Derivative Test

Method: 2nd Derivative Test

Example f″(x) = 0 IS an Inflection Point.

2nd Derivative Test Option 2: Finding Local Max and Local Min 3 Topics

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Identifier: 2nd Derivative Test

Method: 2nd Derivative Test

Example 1: 2nd Derivative Test

Curve Sketching Using the 1st Derivative and 2nd Derivative Tests 3 Topics

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Identifier: Curve Sketching

Method: Curve Sketching

Example 1: Curve Sketching

Optimization 3 Topics

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Identifier: Optimization

Method: Optimization

Example 1: Optimization

L’Hospital’s Rule 3 Topics

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Identifier: L’Hospital’s Rule

Method: L’Hospital’s Rule

Example 1: L’Hospital’s Rule

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