Lesson 3: 5.3 Increasing and Decreasing Intervals
Duration of Days: 2
Lesson Objective
Use the sign of the first derivative to determine if a function f is increasing or decreasing on a given interval.
Construct a sign chart (number line) to organize critical points and the resulting intervals.
Formally justify the behavior of a function: "f(x) is increasing because f'(x) > 0.
Identify relative extrema by observing where the sign of the first derivative changes (The First Derivative Test).
If we know the slope is positive for an entire interval, what must be true about the y-values of the function?
Can a function be increasing even if the derivative is zero at a single point? (e.g., f(x) = x^3).
How do we use "test values" to determine the behavior of an entire interval?
Why is it insufficient to just say "the graph is going up" on an AP Exam?
Monotonic
Strictly Increasing / Strictly Decreasing
Sign Chart
First Derivative Test
Relative Extrema
Justification
Standard FUN-1.B: Explain the relationship between the sign of the derivative and the monotonic behavior of the function.
Mathematical Practice 3.D: Provide reasons or rationales for conclusions.
Description:
This lesson teaches students to solve the inequality f'(x) > 0 and f'(x) < 0. Students find critical points, place them on a number line, and test values in each "zone" to see if the derivative is positive or negative.
Purpose:
This is the primary way students describe the "shape" of a function algebraically. On the AP Exam, students are frequently asked to "Find the intervals where f is increasing. Justify your answer." This lesson provides the exact logical structure required for those points.
DOK 1 (Recall): Stating that f'(x) > 0 implies f(x) is increasing.
DOK 2 (Skill/Concept): Creating a sign chart for a polynomial or rational function.
DOK 3 (Strategic Thinking): Identifying a relative maximum by noting that f' changes from positive to negative at a critical point.
Scaffolding (Support):The Directional Arrows: Encourage students to draw literal upward or downward arrows under their sign charts. It makes the "peaks" and "valleys" visually obvious.
Template for Justification: Provide a sentence starter: "Since f'(x) < 0 on the interval (a, b), the function f(x) is decreasing on that interval.
Extension (Challenge):Working Backwards: Give students a sign chart for f' and ask them to sketch a possible graph of f(x).
Trigonometric Functions: Have students find increasing/decreasing intervals for trig functions on the interval [0, 2pi].
AP College Board Assessments