Unit 4: Application of Derivatives

Application of Derivatives Overview
Evaluating a Derivative (Finding the Slope)
5 Topics
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
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
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
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 2nd 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
Identifier: Physics Relationships
Method: Physics Relationships
Example 1: Physics Relationships
Mean Value Theorem
4 Topics
Identifier: Mean Value Theorem
Method: Mean Value Theorem
Example 1: Mechanical
Example 2: Conceptual
1st Derivative Test: Local Extrema & Increasing/Decreasing Intervals
5 Topics
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
Identifier: Global Extrema
Method: Global Extrema
Example 1: Global Extrema
2nd Derivative Test Option 1: Intervals of Concavity and Inflection Points
3 Topics
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
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
Identifier: Curve Sketching
Method: Curve Sketching
Example 1: Curve Sketching
Optimization
3 Topics
Identifier: Optimization
Method: Optimization
Example 1: Optimization
L’Hospital’s Rule
3 Topics
Identifier: L’Hospital’s Rule
Method: L’Hospital’s Rule
Example 1: L’Hospital’s Rule
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1st Derivative Test: Local Extrema & Increasing/Decreasing Intervals

Unit 4: Application of Derivatives 1st Derivative Test: Local Extrema & Increasing/Decreasing Intervals

There is no more fundamental application of the derivative than running a first derivative test. You will do a ton of first derivative test problems. The great news about this style of problem is that the steps for running the process are always the exact same. Once you get the steps down, you just repeat the process only with a new starting equation, f ( x ) .

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