3 3 Increasing and Decreasing Functions Homework p
- Slides: 45
3. 3: Increasing and Decreasing Functions Homework: p. 183 5, 9, 15, 25, 39, 47, 57 -69 odd, 73, 83 Standards Learning Objectives: EK 2. 2 A 2: Key • Determine intervals on features of function and their which a function is derivatives can be increasing or identified and decreasing related to their graphical, • Apply the First numerical and Derivative Test to find ANALYTICAL
Lesson Structure 3. 3 3. 4 1 2 3 4 Con Con Con TE TE SLE
Concept 1: Increasing and Decreasing Function(s)
Concept 1: (extended) – Guidelines for Finding Intervals Let f be continuous on the interval (a, b). To find the open intervals on which f is increasing of decreasing, use the following steps: 1. Locate the critical numbers (by finding the derivative) of f in (a, b) and use these numbers to test intervals 2. Determine the sign of f’(x) at one test value in each interval 3. Based on the sign of f’(x) determine if the function increases or decreases
Example 1: Intervals for which f is increasing or decreasing •
Example 1: Intervals for which f is increasing or decreasing •
Example 1: Intervals for Which f is increasing or decreasing •
Example 1: Intervals for Which f is increasing or decreasing •
Example 1: Intervals for Which f is increasing or decreasing •
Example 1: Intervals for Which f is increasing or decreasing • Minimum Maximum Minimum
Example 1: Intervals for which f is increasing or decreasing • Minimum Maximum Minimum
Student Led Example 1: Intervals for which f is increasing or decreasing • m o d t n ra d do re
Student Led Example 1: Intervals for which f is increasing or decreasing •
SLE 1: Graph
Concept 2: The First Derivative Test
Concept 2: The First Derivative Test
Concept 2: The First Derivative Test
Concept 2: The First Derivative Test
3. 4: Concavity and the Second Derivative Test Homework: p. 192 9, 15, 21, 25, 29*, 33 -41 odd, 75, Learning 77 Standards Objectives: EK 2. 2 A 1: First and second derivatives of a function can provide information about the function an its graph including intervals of increase or decrease, local and • Determine intervals on which a function is concave up or concave down • Find any points of inflection of the graph of a function • Apply the Second
Concept 1: Defining Concavity
Concept 1: Defining Concavity
Concept 2: Test for Concavity
Concept 3: Finding Points of Inflection
Concept 4: The Second Derivative Test
We’re going to try and accomplish all our objectives in one problem. But first, some concepts Wha’?
1. Find all critical points of a function 2. Determine intervals for which a function is increasing or decreasing 3. Locate relative extrema and concavity 4. Find points of inflection using the second derivative test 5. Graph the function 6. Do it all without technology
Example 1: Objective 1 - Find all critical points of a function •
Example 1: Objective 2 – Determine Increasing/Decreasing Intervals •
Example 1: Objective 3 – LOCATE RELATIVE EXTREMA and Concavity •
Example 1: Objective 3 – Locate Relative Extrema and CONCAVITY •
Example 1: Objective 4 – Find Points of Inflection •
Example 1: Objective 5 - Graphing
Example 1: Objective 5 - Graphing
Example 1: Objective 5 - Graphing
Example 1: Objective 5 - Graphing
Example 1: Objective 1 - Find all critical points of a function •
Example 1: Objective 2 – Determine Increasing/Decreasing Intervals •
Example 1: Objective 3 – LOCATE RELATIVE EXTREMA and Concavity •
Example 1: Objective 3 – Locate Relative Extrema and CONCAVITY •
Example 1: Objective 4 – Find Points of Inflection SKIP
Example 1: Objective 5 - Graphing
Example 1: Objective 5 - Graphing
Example 1: Objective 5 - Graphing
Example 1: Objective 5 - Graphing