Calc AB · U5
Analytical Applications of Differentiation
Mean Value Theorem, extrema, monotonicity, concavity, derivative graphs, optimization, and implicit-relation behavior.
Test yourself
A focused, distraction-free run through 60 multiple-choice questions. Hints if you need them; your score at the end.
Difficulty guide
The label describes the problem, not your ability.
1/5 - Foundational
Warm-up setup; checks one basic idea.
2/5 - Routine exam skill
Normal syllabus skill; usually one main rule or interpretation.
3/5 - Standard multi-step
Exam-style problem with multiple steps or a common trap.
4/5 - Hard exam skill
High-end exam problem; combines methods carefully.
5/5 - Challenge
Stretch problem beyond normal exam pressure.
Topic 5.1
Using the Mean Value Theorem
Verifying hypotheses and interpreting guaranteed derivative values
Topic 5.2
Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
Identifying candidate points and distinguishing guaranteed absolute extrema from local behavior
Topic 5.3
Determining Intervals on Which a Function Is Increasing or Decreasing
Using the sign of the first derivative to describe monotonic behavior
Topic 5.4
Using the First Derivative Test to Determine Relative Extrema
Classifying local extrema from derivative sign changes
Topic 5.5
Using the Candidates Test to Determine Absolute Extrema
Comparing endpoint and critical-point values on closed intervals
Topic 5.6
Determining Concavity of Functions over Their Domains
Using the second derivative to determine concavity and inflection points
Topic 5.7
Using the Second Derivative Test to Determine Extrema
Classifying critical points using second derivative information
Topic 5.8
Sketching Graphs of Functions and Their Derivatives
Using graphs of derivatives to infer features of original functions
Topic 5.9
Connecting a Function, Its First Derivative, and Its Second Derivative
Combining sign, slope, and concavity information across representations
Topic 5.10
Introduction to Optimization Problems
Building objective functions, constraints, and feasible domains
Topic 5.11
Solving Optimization Problems
Differentiating objective functions and confirming optimal values
Topic 5.12
Exploring Behaviors of Implicit Relations
Using implicit derivatives to analyze slopes, tangent behavior, and extrema on relations