AP Calculus Unit 5 Lesson 9: Connecting f, f', and f'' graphically

In this engaging and visual lesson, we take a deep dive into the relationships between a function 𝑓, its first derivative 𝑓′ , and its second derivative 𝑓′′. Through step-by-step visual explanations you'll gain a deeper understanding of how these derivatives reveal key characteristics of a function's behavior, both analytically and graphically.
What You'll Learn:
Understanding the First Derivative: Learn how 𝑓′ reveals whether a function is increasing or decreasing, identifies critical points for relative maxima/minima, and provides graphical insights into the slope of 𝑓.
The Power of the Second Derivative: Explore how 𝑓′′ explains concavity (up or down), identifies inflection points where curvature changes, and helps justify extrema and analyze function behavior.
Connecting Graphs of 𝑓, 𝑓′, and 𝑓′′: Develop strategies to match function, first derivative, and second derivative graphs while interpreting changes in slope, curvature, and critical points.
This lesson is packed with illustrations, problem-solving, and guided explanations to help students master the interplay between 𝑓, 𝑓′, and 𝑓′′. Whether you're preparing for an exam, teaching calculus, or simply refining your skills, this video offers a clear and accessible exploration of these fundamental concepts.