4.4 Linear Approximation and Differentials

In this lesson, you will learn how to use a tangent line and its slope to approximate values of and changes in a differentiable function.

Lesson Objectives

Upon completion of the lesson 4.4, you will be able to:

• • Find the linearization of a curve at x=a.

  • Understand that the linearization is the equation of the tangent line to the curve at x=a.

  • Use the linearization to obtain an approximation for the y-value of the curve at x=a.

  • Estimate change with linear approximation.

  • Use the notation of differentials to obtain the approximate change in a function given a small change in x.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

• 1. Linearization

• 1. Example:

• 1. Finding the Linearization at a Point/Tangent Line Approximation

• 1. Tangents and linear approximation

  2. 3.05c - Linear approximation:

• 1. 3.05d - Linear approximation:

• 1. Linear Approximation and Differentials

• 1. Differentials: Intro

• 1. Using Differentials

• 1. 3.9a Differentials - Calculus

• 1. 3.9b Differentials - Calculus

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module. Linear approximations and differentials

• 1. Linear approximations

  2. Differentials

  3. Skip example 1c from the Differentials reading assignment because we have not learned the derivative for the exponential function.

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

• 1. Linear approximations (Linearization)

  2. Note: Skip Problems #1 and #5.

  3. Differentials

  4. Note: Skip Problems #2, 3, and 4.

Additional Resources

Below are additional resources that help reinforce the content for this module.

• 1. Section 3.10: Linear approximations and differentials

  2. Note: Skip example 1.2.

  3. Chapter 15: Differentials and linear approximations

  4. Application of the derivative: Lesson 32-Differentials

  5. This module includes linear approximation (linearization), differentials, and homework with answers.

  6. Note: In the homework problems, skip Question #2.