Use derivatives to analyze the shape of a graph. Apply the Mean Value Theorem, find critical points and extrema with the first and second derivative tests, determine concavity and inflection points, sketch functions and their derivatives, and solve optimization problems.
All 12 topics from the College Board CED, in order.
Topic 5.1
Using the Mean Value Theorem
The MVT guarantees a point where instantaneous rate equals average rate.
Topic 5.2
Extreme Value Theorem & Critical Points
Global vs. local extrema and where they can occur.
Topic 5.3
Increasing & Decreasing Intervals
Using the sign of f′ to find where f rises or falls.
Topic 5.4
First Derivative Test
Classifying relative extrema from sign changes of f′.
Topic 5.5
Candidates Test for Absolute Extrema
Finding global max/min on a closed interval.
Topic 5.6
Determining Concavity
Using f″ to find concave up/down and inflection points.
Topic 5.7
Second Derivative Test
Classifying extrema using the sign of f″.
Topic 5.8
Sketching f and Its Derivatives
Reading graphs of f, f′, and f″ against each other.
Topic 5.9
Connecting f, f′, and f″
Translating features among a function and its derivatives.
Topic 5.10
Introduction to Optimization
Framing real-world max/min problems.
Topic 5.11
Solving Optimization Problems
Full procedure for optimizing a quantity.
Topic 5.12
Behaviors of Implicit Relations
Analyzing extrema and concavity of implicit curves.
About Unit 5
Unit 5 uses derivatives to analyze the shape and behavior of a function’s graph. It opens with the Mean Value Theorem — on a closed interval where f is continuous and differentiable, there is a point where the instantaneous rate equals the average rate. Then you find critical points (where f′ = 0 or is undefined) and use the sign of f′ to determine where f is increasing or decreasing.
The first derivative test classifies relative extrema from sign changes of f′; the candidates test finds absolute extrema on a closed interval; and the sign of f″ gives concavity and inflection points, feeding the second derivative test. You’ll sketch and connect the graphs of f, f′, and f″, and finish with optimization — turning a real-world max/min problem into a function to optimize — and the behavior of implicit relations.
On the exam this unit is 15–18% for AB and 8–11% for BC — the largest AB unit — and takes about 15 class periods (AB). The four mathematical practices below run through every topic: