Stage 7 Lesson Hub

This Lesson uses the derivative to determine maxima, minima, concavity, inflection points and related qualitative information about functions and their graphs. Applications are found in Stage 8.

What's Here 1. The Shape of a Function The Sign of the Derivative and the First Derivative Test Applications to Curve Sketching Concavity and the Second Derivative 2. Extreme Values Critical Points The "Wizard of Oz" Testing Critical Points The Second Derivative Test Full Curve Sketching 3. The Mean Value Theorem Consequences of the MVT Rolle's Theorem and a Proof References: The Field Guide to Functions.

What To Do This lesson is short and straightforward. To see what kind of problems you will be expected to do, look at the Onward & Upward quiz and the Practice problems. Expected Study Time You should expect to spend approximately three hours working through this Lesson in order to prepare for the Practice area and "Onward and Upward" activities of this Stage. Objectives: Find the critical points and extreme values of a function. Determine the monotonicity and concavity properties of a function. Apply this knowledge to curve sketching. Use the First Derivative Test, the Second Derivative Test, and the Mean Value Theorem.