5.2 Increasing and Decreasing Functions and First Derivatives

In this lesson, you will learn to classify a function as increasing or decreasing over an interval. You will also learn to identify local extrema using the first derivative test.

Lesson Objectives

Upon completion of the lesson 5.2, you will be able to:

• • Find intervals of increase or decrease for a function by applying the test for intervals of increase or decrease (using the first derivative) that utilize the sign of the first derivative of a function to determine the interval(s) where the function is increasing or decreasing.

  • Apply the First Derivative Test to find where (at what x-values) the function has a local (or relative) maximum or local (or relative) minimum value.

Lesson Content

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

• 1. YouTube video: Intervals of Increase and Decrease

• 1. YouTube video: First Derivative Test

• 1. Critical Numbers and the First Derivative Test [University of Houston]

  2. YouTube videos: Increasing & Decreasing Functions and the 1st Derivative Test - calculus

  3. Part 3.3a1

• 1. Part 3.3a2:

• 1. Part 3.3b:

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

• 1. Section 3.4: Increasing and Decreasing Functions [McGraw-Hill]

  2. The Shape of a Graph - Part 1 [Paul's Online Math Notes] Note: Skip Example 5.

1. The First Derivative Test [Pheng Kim Ving]

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. Practice problems: Increasing and Decreasing Functions [University of Houston] (solutions and answers provided)

  2. Practice problems: Local Extreme Values [University of Houston] (Critical Numbers)

1. The Shape of a Graph - Part 1 [Paul's Online Math Notes]

   1. Note: Do problems #1 - 10.

2. Drill - First Derivative Test [University of Tennessee]

• 1. Note: Skip the 4th , 5th and 9th problems.

Additional Resources

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

• 1. Chapter 3: Graphing and Optimization [UC San Diego]

  2. Read pages 54 - 58 for The First-Derivative test for increasing and decreasing functions.

  3. Read until the end of example 4 on p. 58.

  4. YouTube video: Determining where a Function is Increasing and Decreasing using the First Derivative

1. The First Derivative: Maxima and Minima [Universita di Genova]