Fourteen lessons make up Unit 5 - Analytical Applications of Differentiation of the AP Calculus AB module. For each lesson, scholars watch a short video explaining a process and then complete provided practice problems to gain mastery. The unit begins with the Mean Value Theorem. Pupils first see the definition of the theorem and hear an explanation of it in simple terms. The helpful video then compares the Mean Value Theorem with the Intermediate Value Theorem. Subsequent lessons teach learners to use derivatives to find out how to find critical points from equations of functions that may be extrema, how to use the derivative and critical points to find where a function is either increasing or decreasing and how to find a critical point and how intervals of increasing and decreasing work together to identify local extrema. Other topics include finding the absolute maximums and minimums of a function on an interval by testing critical points and determining whether an interval is concave up or concave down using graphs and the second derivative. For the mid-unit review, scholars apply what they have learned to use the Mean Value Theorem, find extrema given the equation or graph of the derivative, and find when a function decreases. In the second part of the unit, learners find extrema and intervals of concavity of the function, sketch graphs of a function's derivative, and interpret intercepts and critical points of each graph to gain insight into the other. They determine whether a function is increasing or decreasing and its concavity and see the importance of setting up a one-variable equation. They use their knowledge of derivatives to find solutions. For the Unit 5 review, pupils review all the topics presented in Analytical Applications of Differentiation resources.
