First and Second Derivative Tests
Students analyze the first and second derivatives to determine intervals of increase, decrease, and concavity. They use these tests to locate local extrema and points of inflection.
About This Topic
Students analyze the first and second derivatives to determine intervals of increase, decrease, and concavity. They use these tests to locate local extrema and points of inflection.
Key Questions
- How does the first derivative indicate increasing or decreasing behavior?
- What does the second derivative tell us about concavity?
- How do we identify points of inflection?
Planning templates for Calculus
5E Model
The 5E Model structures lessons through five phases (Engage, Explore, Explain, Elaborate, and Evaluate), guiding students from curiosity to deep understanding through inquiry-based learning.
Unit PlannerMath Unit
Plan a multi-week math unit with conceptual coherence: from building number sense and procedural fluency to applying skills in context and developing mathematical reasoning across a connected sequence of lessons.
RubricMath Rubric
Build a math rubric that assesses problem-solving, mathematical reasoning, and communication alongside procedural accuracy, giving students feedback on how they think, not just whether they got the right answer.
More in Applications of Differentiation
Extreme Value Theorem and Optimization
Students use derivatives to find absolute and relative extrema of functions. They apply these concepts to solve applied optimization problems in various contexts.
2 methodologies
The Mean Value Theorem
This topic covers the Mean Value Theorem and its implications for the behavior of functions. Students connect average rates of change to instantaneous rates of change.
2 methodologies
Curve Sketching and Kinematics
Students synthesize derivative tests to sketch accurate graphs of functions. They also apply differentiation to particle motion, analyzing position, velocity, and acceleration.
2 methodologies