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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Method: Tangent Line Approximation

Unit 4: Application of Derivatives Linearization = Tangent Line Approximation = Local Linear Approximation Method: Tangent Line Approximation

The process for finding the linearization equation, tangent line approximation, or the local linear approximation is nearly identical to the process you follow to find the equation of the tangent line. This makes sense because all we are really doing is using the tangent line to approximate our actual equation, f ( x ) . The one additional step of complexity in these problems is usually in not directly giving you your f ( x ) , and instead implying in the language of the problem what the f ( x ) needs to be.

Step 1: Determine the f ( x ) equation that you are trying to approximate.

Sometimes this equation will be given to you specifically, and sometimes you will need to determine it for yourself using the implied information in the question.

Step 2: Determine the x-value that is closest to the actual x-value that you care about, which you will use to create your tangent line.

This will be your x 0  or  a that you use in the process.

In some situations, you will be given that value directly. They will use language like “centered around x=__” or “about x=__” to tell you x-value they would like you use to construct the tangent line.

In other situations, you might have to decide for yourself what that x-value should be. The process of determining that implied x-value is usually to ask yourself, “What x-value could I plug into this exact same equation, f ( x ) , and find the answer to quickly, easily, and without a calculator?”

Step 3: Find the equation of the tangent line to your complicated f ( x )   at the x-value you determined in Step 2 .

You will see different instructors and different textbooks use different names for this equation and use different notation for the equation.

I assure you they are all the exact same equation, and that they are all a different way of writing the point–slope form of a line, y = m ( x – x 1 ) + y 1 .

Point–Slope Form

y = m ( x – x 1 ) + y 1

Linearization Equation

Local Linear Approximation Equation

This equation is one of the most common versions of alternate notation for the same tangent line equation.

This version rearranges and renames the pieces of the standard point–slope form of a line, but it really is the exact same equation.

You will also see the x-value that that you plugin to the equation referred to as either x 0  or  a .

L ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x – x 0 )

L ( x ) = f ( x 0 ) y 1 + f ′ ( x 0 ) m ( x – x 0 x 1 )

L ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x – x 0 )

L ( x ) = f ( a ) + f ′ ( a ) ( x – a )

Tangent Line Approximation Equation

You will also see the x-value that that you plugin to the equation referred to as either x 0  or  a .

f ̂ ( x ) =  f ′ ( x 0 ) ( x – x 0 ) +  f ( x 0 )

f ̂ ( x ) =   f ′ ( x 0 ) m ( x – x 0 x 1 ) +   f ( x 0 ) y 1

f ̂ ( x ) = f ′ ( x 0 ) ( x – x 0 ) + f ( x 0 )

f ̂ ( x ) =  f ′ ( a ) ( x – a ) +  f ( a )

 

 

Step 4: Use the linearization equation (tangent line equation) that you found in Step 3 to approximate the value you were initially asked to find.

Keep in mind that all you are actually doing through this linearization process is creating a new equation that approximates the actual equation you care about.

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