The Ultimate AP Calculus AB Course

Use these free AP Calculus AB review lessons to work on limits, derivatives, integrals, and differential equations. Start with the skill you need, read the worked explanation, then check your understanding with your class materials or official released questions.

This is a topic directory for independent review. It does not provide school credit, AP enrollment, or a complete set of exam-style tests. Some linked lessons go beyond AB; those are separated below.

Plan your AP Calculus AB review

Choose one topic you can explain and one you still find difficult. For each problem, write the method, show the calculation, and explain what the answer means. Include graphs, tables, formulas, and verbal descriptions in your practice.

The College Board organizes AB into eight units. Use this roadmap alongside the official course framework; a linked lesson may cover only part of a unit.

1. Limits and continuity: start with limits, continuity, and the Intermediate Value Theorem.
2. Derivative definitions and fundamental properties: review rates of change and the basic derivative rules.
3. Composite, implicit, and inverse functions: practice choosing and combining differentiation rules.
4. Contextual applications of differentiation: interpret rates, related rates, motion, and local approximations.
5. Analytical applications of differentiation: use derivatives to justify extrema and graph behavior.
6. Integration and accumulation: connect sums, antiderivatives, and definite integrals.
7. Differential equations: interpret slope fields and solve separable equations with initial conditions.
8. Applications of integration: connect integrals to average value, motion, area, and volume.

Free calculus lessons by topic

These links open public explanations. Check each lesson for its available examples and exercises; this directory does not promise a separate printable worksheet or answer key for every topic.

Limit and Continuity

• Limit Introduction
• Neighborhood
• Estimating Limits from Tables
• Functions with Undefined Limits
• One Sided Limits
• Limit at Infinity
• Continuity at a Point
• Continuity over an Interval
• Removing Discontinuity
• Direct Substitution
• Limit Laws
• Limit Laws Combinations
• The Squeeze Theorem
• Indeterminate and Undefined
• Infinity Cases
• Trigonometric Limits
• Rationalizing Trigonometric Functions
• Algebraic Manipulation
• Redefining Function’s Value
• Rationalizing Infinite Limits
• The Intermediate Value Theorem
• Asymptotes

Derivative Basics

• Derivative Introduction
• Average and Instantaneous Rates of Change
• The Derivative of a Function
• Derivative of Trigonometric Functions
• Power Rule
• Product Rule
• Quotient Rule
• Chain Rule
• Derivative of Radicals
• Derivative of Logarithms and Exponential Functions
• Differentiability
• Differentiating Inverse Functions
• Derivative of Reciprocal Trigonometric Functions

Applications of Derivatives

• Optimization Problems
• L’Hôpital
• Implicit Relations
• Implicit Differentiation
• Related Rates
• Higher Order Derivatives
• Extreme Value Theorem
• Second Derivatives: Minimum vs. Maximum
• Curve Sketching Using Derivatives
• Mean Value Theorem

Integrals

• What is Integral?
• Applications of Integrals
• Exponential Growth and Decay
• The Anti-Derivative
• Sigma Notation (Summation Notation)
• Riemann Sums
• Rules of Integration
• Power Rule
• Fundamental Theorem of Calculus
• Trigonometric Integrals
• Substitution Rule
• Integral of Radicals
• Exponential and Logarithmic Integrals

Applications of Integrals

• Average Value of a Function
• Accumulation Problems
• Finding Area Between Curves
• Introduction to Solids of Revolution
• Volume with Cross-section Method
• Disc Method
• Washer Method
• Modeling Particle Motion

Differential Equations

• Introduction and Applications
• Separable Differential Equations
• Slope Fields
• Modeling Change
• Simple Growth and Decay

Extra calculus: BC topics and further study

Integration by parts, linear partial fractions, improper integrals, Euler’s method, logistic models, and arc length are outside the AB requirements and appear in BC. Other links here, including Newton’s method, shell volumes, surface area, and general first-order equation methods, offer further calculus study. Save these for a class that assigns them; they are not an AB completion checklist.

• Newton’s Method
• Integration by Parts
• Improper Integrals
• Arc Length
• Shell Method
• Surface Area
• Partial Fraction Expansion
• Classification of Differential Equations
• First-Order Ordinary Differential Equations
• Linear Differential Equations
• Population Models

Prepare for the May 2027 exam

College Board has changed the multiple-choice count and timing for May 2027. The current exam takes 3 hours 10 minutes: 42 multiple-choice questions in 100 minutes and six free-response questions in 90 minutes. Each section contributes half the score.

The multiple-choice section has 29 questions without a calculator and 13 in its calculator portion. Free response has two calculator questions and four without a calculator. The exam is hybrid: students use Bluebook for multiple choice and to view free-response prompts, then handwrite free-response answers.

Use College Board’s current exam guide, its linked released questions and scoring guidance, and the calculator policy. The built-in Desmos graphing calculator is available only in calculator-required portions; check the approved list before bringing a handheld model. Older practice papers may use the previous timing.

Choose a calculus book after checking the preview

For additional explanations and practice, check Calculus for Beginners. Open its public preview and compare the contents with the topic you need before buying. It is a calculus study book; this directory does not establish that it is a complete AP AB exam preparation package. The website offers a printable PDF, with a separate Amazon paperback listing. Check each listing’s details and edition before buying.

Questions about using this course directory

Should I study every linked topic for AB?

Work through your teacher’s sequence and the official AB framework. The extra-calculus section keeps useful broader lessons accessible without presenting them as required AB topics. Course units also include skills that a short lesson may not fully address.

What should I review first?

If algebra or trigonometry slows you down, review function notation, equations, exponent and logarithm rules, and the unit circle alongside calculus. Try the trigonometry lessons when those gaps affect your work.

How does the Fundamental Theorem connect derivatives and integrals?

If f is continuous, the accumulation function A(x) = ∫_a^x f(t) dt has derivative A′(x) = f(x). If F is an antiderivative of f on the interval, then ∫_a^b f(x) dx = F(b) − F(a). For example, ∫₀² 3x² dx = [x³]₀² = 8. Keep the limits and variable of integration clear.

Does finishing these lessons earn college credit?

No. These are review resources. AP credit and placement depend on each college’s policy and the student’s exam result; check the institution you plan to attend.