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AP® Calculus AB-BC

The Best AP® Calculus AB-BC Review: Topic Summaries, Examples, and Free Practice

Welcome to Albert’s collection of science topic reviews for teaching and reviewing AP® Calculus AB-BC. Teachers and students can explore our easy-to-follow guides below for use at home or in the classroom.

Overview and Scoring

What is on the AP® Calculus exam? How is it scored?

Explore the structure and topics of the AP® Calculus AB and BC exams. Understand scoring criteria, key concepts, and strategies to maximize your score.

Subject icon for The Best AP® Calculus AB Review Guide for 2025

The Best AP® Calculus AB Review Guide for 2025

Subject icon for AP® Calculus AB FAQ: Everything You Need to Know for 2024

AP® Calculus AB FAQ: Everything You Need to Know for 2024

Subject icon for Is AP® Calculus Hard?

Is AP® Calculus Hard?

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AP® Calculus AB Score Calculator

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AP® Calculus BC Score Calculator

Study and Test Prep

How can I ace the AP® Calculus AB or BC exam? What are the must-know strategies and study tips?

These articles offer key advice for mastering the AP® Calculus AB and BC exams. Learn effective study techniques, review strategies, and exam tips to strengthen your understanding and boost your score.

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One Month AP® Calculus Study Guide

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How to Study for AP® Calculus

Subject icon for The Ultimate List of AP® Calculus Tips

The Ultimate List of AP® Calculus Tips

Unit 1: Limits and Continuity

How to solve limits to infinity? How to do average rate of change? How to tell if a function is continuos

Build a rock‑solid limits foundation for the AP® Calculus AB/BC exam in one place. These bite‑size reviews cover every limits and continuity skill—average rate of change, formal limit laws, rational and infinite limits, discontinuities, asymptotes, and the IVT—paired with fast practice to sharpen speed and accuracy.

1.1 Introducing Calculus: Can Change Occur at an Instant?

1.2 Defining Limits and Using Limit Notation

1.3 Estimating Limit Values from Graphs

1.4 Estimating Limit Values from Tables

1.5 Determining Limits Using Algebraic Properties of Limits

1.6 Determining Limits Using Algebraic Manipulation

1.7 Selecting Procedures for Determining Limits

1.8 Determining Limits Using the Squeeze Theorem

1.9 Connecting Multiple Representations of Limits

1.10 Exploring Types of Discontinuities

1.11 Defining Continuity at a Point

1.12 Confirming Continuity over an Interval

1.13 Removing Discontinuities

1.14 Connecting Infinite Limits and Vertical Asymptotes

1.15 Connecting Limits at Infinity and Horizontal Asymptotes

1.16 Working with the Intermediate Value Theorem (IVT)

Unit 2: Differentiation: Definition and Fundamental Properties

How to find the instantaneous rate of change? What is the quotient rule? When does a derivative not exist?

Master derivatives fast: start with the difference quotient, notation, and slope estimates, learn when derivatives fail, then apply the power, product, quotient, and trig rules to any polynomial or trig function. These bite‑size AP® Calculus AB/BC guides pair clear rules with quick practice so you can differentiate accurately under test pressure.

2.1 Defining Average and Instantaneous Rates of Change at a Point

2.2 Defining the Derivative of a Function and Using Derivative Notation

2.3 Estimating Derivatives of a Function at a Point

2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

2.5 Applying the Power Rule

2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

2.7 Derivatives of cos(x), sin(x), e^x, and ln(x)

2.8 The Product Rule

2.9 The Quotient Rule

2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

When to use the chain rule? What is implicit differentiation? How to find the derivative of an inverse function?

Expand your AP® Calculus derivative arsenal with concise guides on the Chain Rule, implicit differentiation, and derivatives of inverse and inverse trig functions—plus higher‑order derivatives. Bite‑size explanations and targeted practice problems ensure you can apply every rule quickly and accurately on exam day.

3.1 The Chain Rule

3.2 Implicit Differentiation

3.3 Differentiating Inverse Functions

3.4 Differentiating Inverse Trigonometric Functions

3.5 Selecting Procedures for Calculating Derivatives

3.6 Calculating Higher-Order Derivatives

Unit 4: Contextual Applications of Differentiation

How to find instantaneous velocity? When is a particle at rest? What is an indeterminate form?

Turn derivative skills into real‑world power: learn to interpret rate of change, analyze particle motion, and solve classic related‑rates scenarios like expanding cones. Wrap up with linearization for quick approximations and L’Hospital’s Rule for indeterminate forms, giving you the applied‑calculus edge for AP® Calculus AB/BC.

4.1 Interpreting the Meaning of the Derivative in Context

4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration

4.3 Rates of Change in Applied Contexts Other Than Motion

4.4 Introduction to Related Rates

4.5 Solving Related Rates Problems

4.6 Approximating Values of a Function Using Local Linearity and Linearization

4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms

Unit 5: Analytical Applications of Differentiation

What is the mean value theorem? How to find relative extrema? How to find concavity?

Tackle AP® Calculus AB/BC graph analysis and optimization in one hub: use the Mean and Extreme Value Theorems plus first‑ and second‑derivative tests to pinpoint intervals, extrema, and concavity. Finish by sketching graphs, probing implicit relations, and solving classic optimization problems with confidence.

5.1 Using the Mean Value Theorem

5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points

5.3 Determining Intervals on Which a Function Is Increasing or Decreasing

5.4 Using the First Derivative Test to Determine Relative (Local) Extrema

5.5 Using the Candidates Test to Determine Absolute (Global) Extrema

5.6 Determining Concavity of Functions over Their Domains

5.7 Using the Second Derivative Test to Determine Extrema

5.8 Sketching Graphs of Functions and Their Derivatives

5.9 Connecting a Function, Its First Derivative, and Its Second Derivative

5.10 Introduction to Optimization Problems

5.11 Solving Optimization Problems

5.12 Exploring Behaviors of Implicit Relations

Unit 6: Integration and Accumulation of Change

How to do Riemann sums? What is the fundamental theorem of calculus? How to do u substitution?

Start with Riemann‑sum approximations and the Fundamental Theorem of Calculus to see how accumulation really works, then dive into definite‑integral properties and accumulation functions. Quick guides to u‑substitution, completing the square, integration by parts, partial fractions, and improper integrals give you every technique needed to ace AP® Calculus AB/BC integration problems.

6.1 Exploring Accumulations of Change

6.2 Approximating Areas with Riemann Sums

6.3 Riemann Sums, Summation Notation, and Definite Integral Notation

6.4 The Fundamental Theorem of Calculus and Accumulation Functions

6.5 Interpreting the Behavior of Accumulation Functions Involving Area

6.6 Applying Properties of Definite Integrals

6.7 The Fundamental Theorem of Calculus and Definite Integrals

6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

6.9 Integrating Using Substitution

6.10 Integrating Functions Using Long Division and Completing the Square

6.11 Integrating Using Integration by Parts (BC)

6.12 Using Linear Partial Fractions (BC)

6.13 Evaluating Improper Integrals (BC)

6.14 Selecting Techniques for Antidifferentiation

Unit 7: Differential Equations

How to solve differential equations? How to sketch a slope field? What is Euler's method?

Master AP® Calculus AB/BC differential equations in one stop: see how slope fields, Euler’s Method, and separation of variables unlock exact and approximate solutions, from exponential to logistic growth models. Quick guides on verifying solutions, finding particulars, and interpreting slope‑field behavior give you the tools to tackle every DE question the exam throws your way.

7.1 Modeling Situations with Differential Equations

7.2 Verifying Solutions for Differential Equations

7.3 Sketching Slope Fields

7.4 Reasoning Using Slope Fields

7.5 Approximating Solutions Using Euler’s Method (BC)

7.6 Finding General Solutions Using Separation of Variables

7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

7.8 Exponential Models with Differential Equations

7.9 Logistic Models with Differential Equations (BC)

Unit 8: Applications of Integration

How to find average value of a function? How to find area between two curves? When to disk vs washer method?

Conquer geometric applications of integrals for AP® Calculus AB/BC: learn to find average value, net change, and displacement, then master area between curves and shaded regions. Step‑by‑step guides to cross‑sections, disc‑and‑washer volumes, and arc length equip you to solve any solid‑of‑revolution or area problem the exam throws your way.

8.1 Finding the Average Value of a Function on an Interval

8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals

8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts

8.4 Finding the Area Between Curves Expressed as Functions of x

8.5 Finding the Area Between Curves Expressed as Functions of y

8.6 Finding the Area Between Curves That Intersect at More Than Two Points

8.7 Volumes with Cross Sections: Squares and Rectangles

8.8 Volumes with Cross Sections: Triangles and Semicircles

8.9 Volume with Disc Method: Revolving Around the x- or y-Axis

8.10 Volume with Disc Method: Revolving Around Other Axes

8.11 Volume with Washer Method: Revolving Around the x- or y-Axis

8.12 Volume with Washer Method: Revolving Around Other Axes

8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC)

Unit 9: Parametric Equations, Polar Coordinates, and Vector- Valued Functions (BC)

How to find second derivative of parametric equations? How to find arc length? What is a vector valued function?

Sharpen skills with parametric, vector‑valued, and polar functions as you compute first and second derivatives, arc length, and areas between polar curves. From integrating vectors to spotting when a particle rests, these concise guides equip you for every advanced curve question on test day.

9.1 Defining and Differentiating Parametric Equations

9.2 Second Derivatives of Parametric Equations

9.3 Finding Arc Lengths of Curves Given by Parametric Equations

9.4 Defining and Differentiating Vector-Valued Functions

9.5 Integrating Vector-Valued Functions

9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions

9.7 Defining Polar Coordinates and Differentiating in Polar Form

9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

9.9 Finding the Area of the Region Bounded by Two Polar Curves

Unit 10: Infinite Sequences and Series (BC)

How to find the sum of an infinite series? What is the alternating series test? What is the taylor polynomial?

Lock in series mastery: learn every convergence test—geometric, p‑series, comparison, ratio, integral, alternating, and more—while distinguishing absolute vs conditional cases and bounding errors. Then harness power, Taylor, and Maclaurin series to represent functions and nail interval of convergence questions on exam day.

10.1 Defining Convergent and Divergent Infinite Series

10.2 Working with Geometric Series

10.3 The nth Term Test for Divergence

10.4 Integral Test for Convergence

10.5 Harmonic Series and p-Series

10.6 Comparison Tests for Convergence

10.7 Alternating Series Test for Convergence

10.8 Ratio Test for Convergence

10.9 Determining Absolute or Conditional Convergence

10.10 Alternating Series Error Bound

10.11 Finding Taylor Polynomial Approximations of Functions

10.12 Lagrange Error Bound

10.13 Radius and Interval of Convergence of Power Series

10.14 Finding Taylor or Maclaurin Series for a Function

10.15 Representing Functions as Power Series

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