Unit Overview
Analytical Applications of Differentiation Overview
AP Calculus ABΒ· 5 min read π 15-18% of total AP Calculus AB exam score
1. Unit at a Glance
We build from foundational theorems that justify derivative-based analysis, up to classifying core function behaviors, connecting derivatives to graph shapes, and finally applying all tools to solve applied optimization problems. This unit shifts from calculating derivatives to using them to answer meaningful questions about functions.
The concepts you learn here are not only heavily tested on the AP exam, but also lay critical groundwork for integration and differential equations later in the course.
This unit covers the following sub-topics:
AP Calculus AB Behaviors of implicit relations
Find slopes and critical points for curves defined by implicit relations.
β β β β± 6 min
AP Calculus AB Candidates test for absolute extrema
Use the Candidates Test to locate absolute extrema on closed intervals.
β β β± 4 min
AP Calculus AB Connecting f, f', f'' qualitatively
Relate features of a function to features of its first and second derivatives.
β β β β± 5 min
AP Calculus AB Determining concavity
Use the second derivative to identify intervals of concave up and concave down behavior.
β β β± 4 min
AP Calculus AB Determining intervals where a function is increasing/decreasing
Classify where a function is increasing or decreasing using the first derivative.
β β± 4 min
AP Calculus AB Extreme Value Theorem, global vs local extrema, critical points
State the Extreme Value Theorem and distinguish local vs global extrema and critical points.
β β β± 5 min
AP Calculus AB First derivative test for relative extrema
Use the first derivative test to classify relative extrema of functions.
β β β± 4 min
AP Calculus AB Introduction to optimization problems
Learn the step-by-step framework for setting up optimization problems.
β β β β± 5 min
AP Calculus AB Mean Value Theorem (MVT)
State and apply the Mean Value Theorem to functions on open and closed intervals.
β β β β± 5 min
AP Calculus AB Second derivative test
Use the second derivative test to classify relative extrema at critical points.
β β β± 4 min
AP Calculus AB Sketching graphs of f, f', f''
Sketch accurate graphs of f, f', and f'' from information about one another.
β β β β β± 6 min
AP Calculus AB Solving optimization problems
Work through full solutions to common applied optimization problems.
β β β β β± 7 min
2. Common Pitfalls
Wrong move:
Confusing the meaning of vs when describing function behavior
Why:
Mixing up which derivative tells you increasing/decreasing vs concavity is a common AP exam point deduction
Correct move:
Always explicitly label which derivative you are using when justifying function features
Wrong move:
Forgetting to include endpoints when finding absolute extrema
Why:
Absolute extrema can occur at endpoints as well as critical points on closed intervals
Correct move:
Add interval endpoints to your candidate list when running the Candidates Test
Wrong move:
Using the second derivative test when at a critical point
Why:
The second derivative test is inconclusive when , so you cannot classify extrema from it here
Correct move:
Fall back to the first derivative test to classify extrema when the second derivative test is inconclusive
3. Quick Reference Cheatsheet
Concept | Key Fact / Formula |
|---|---|
Critical Point | where or is undefined |
Extreme Value Theorem | If is continuous on , has both an absolute max and min on |
Mean Value Theorem | If continuous on and differentiable on , then for some |
Increasing/Decreasing Test | increasing when , decreasing when |
Concavity Test | concave up when , concave down when |
First Derivative Test | changes from to at = relative max; to = relative min |
Optimization Workflow | Define variables β Write objective function β Find critical points β Test candidates for extrema |
What's Next
Begin your study of this unit with the first sub-topic on analyzing behaviors of implicit relations. After you complete all 12 sub-topics in Unit 5, you will progress to the next unit on integration, the foundational topic for the second half of AP Calculus AB.