Curriculum Map

AP Calculus AB

College-level calculus covering limits, derivatives, integrals, and differential equations. Aligned to the College Board AP Calculus AB Course and Exam Description. Equivalent to a first-semester college calculus course.

14 Units

Early Units (Foundation)

Building Skills

Advanced Concepts

Capstone/Synthesis

Introduction to Limits

Explore the foundational concept of limits — what it means for a function to approach a value — using graphical, numerical, and intuitive methods.

3-4 days

Algebraic Techniques for Limits

Master algebraic strategies for evaluating limits exactly — including factoring, rationalization, the squeeze theorem, and limits involving infinity.

3-4 days

Continuity, Discontinuities, and Asymptotes

Formalize the concept of continuity using limits, classify types of discontinuities, explore the Intermediate Value Theorem, and connect limits to asymptotic behavior.

3-4 days

The Derivative: Definition and Interpretation

Build the derivative from first principles — as a limit of average rates of change — and develop deep intuition for what a derivative means geometrically and physically.

4-5 days

Derivative Rules: Power, Product, Quotient, and Trig

Master the fundamental differentiation rules that make calculus computationally tractable — from the power rule to derivatives of all six trigonometric functions.

4-5 days

Chain Rule, Implicit Differentiation, and Related Rates

Master the chain rule for differentiating composite functions, extend it to implicit differentiation for relations, and apply both to solve related rates problems.

5-6 days

Derivative Applications: MVT, Extreme Values, and Monotonicity

Deploy the derivative to analyze function behavior — proving results with the Mean Value Theorem, locating extreme values with the First Derivative Test, and characterizing where functions increase or decrease.

4-5 days

Derivative Applications: Concavity, Optimization, and L'Hôpital's Rule

Use the second derivative to analyze concavity and inflection points, build a complete curve sketch, solve real-world optimization problems, and resolve indeterminate limits with L'Hôpital's Rule.

5-6 days

Antiderivatives and Indefinite Integrals

Reverse the differentiation process — find functions whose derivatives equal a given function — and use initial conditions to specify unique antiderivatives for applied problems.

3-4 days

Definite Integrals and Riemann Sums

Build the definite integral from Riemann sums, interpret it as signed area and net accumulation, and connect it to the Fundamental Theorem of Calculus Part 1.

4-5 days

FTC Part 2, U-Substitution, and Integral Properties

Apply the Fundamental Theorem of Calculus Part 2 to evaluate definite integrals exactly, master u-substitution as the chain rule in reverse, and use integral properties for strategic computation.

4-5 days

Integral Applications: Area Between Curves and Volumes

Apply definite integrals to compute areas of regions bounded by two curves and volumes of solids generated by revolving regions around an axis using the disk and washer methods.

5-6 days

Integral Applications: Accumulation, Net Change, and Average Value

Interpret the definite integral as total accumulation and net change, compute the average value of a function over an interval, and solve problems involving particle motion and contextual rates.

3-4 days

Differential Equations: Slope Fields, Separation, and Exponential Models

Solve separable differential equations analytically, visualize solution families with slope fields, and model exponential growth and decay using calculus's most powerful applied framework.

4-5 days

Learning Progression

This course is designed to be taken sequentially. Earlier units establish foundational concepts that later units build upon. While you can explore units in any order, following the numbered sequence provides the most coherent learning experience.