Unit 5: Analytical Applications of Differentiation

Students will be able to:
The "First Derivative Test": Students will know that when f' changes sign, the original function f has reached a peak (maximum) or a valley (minimum).
The "Second Derivative Test": Students will understand Concavity. They’ll know that f'' tells us if a graph is "cupping up" (like a smile) or "cupping down" (like a frown).
Point of Inflection: Students will identify the exact moment a curve changes its "bend" (from concave up to concave down).
Mean Value Theorem (MVT): Students will grasp one of the most important theorems in calculus—that if you travel at an average speed of 60 mph, there must have been at least one moment where your speedometer read exactly 60 mph.
Optimization: This is the "peak" of the unit. Students will use derivatives to find the "best" way to do something—like the dimensions of a box that hold the most volume using the least material.

5.1 Using the Mean Value Theorem
5.2 Extreme Value Theorem, Global Versus Local
Extrema, and Critical Points
5.3 Determining Intervals on Which a Function is
Increasing or Decreasing
5.4 Using the First Derivative Test to Determine Relative
Local Extrema
5.5 Using the Candidates Test to Determine Absolute
(Global) Extrema
5.6 Determining Concavity of Functions over Their
Domains
5.7 Using the Second Derivative Test to Determine
Extrema
5.8 Sketching Graphs of Functions and Their Derivatives
5.9 Connecting a Function, Its First Derivative, and Its
Second Derivative
5.10 Introduction to Optimization Problems
5.11 Solving Optimization Problems
5.12 Exploring Behaviors of Implicit Relations

College Board AP Classroom Assessments