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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Identifier: Optimization

Unit 4: Application of Derivatives Optimization Identifier: Optimization

The first main identifier is that you are dealing with a word problem, and not just a general equation that they give you. From there you will see the problem asking you to optimize, maximize, minimize, find the minimum of, or find the max of. Basically, words that are asking you to find the largest or smallest amount.

Note : Many people get these problems confused with what are referred to as Related Rates problems (which we will discuss in the not-too-distant future). The key when trying to distinguish between the two types of problems is to note that optimization problems are asking you for a max or min amount of something. It is a fixed value they are wanting as an answer. A related rates problem will ask you about how something is changing. The language will be about how quickly or slowly something is changing. These problems are describing movement.

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