Conquer AP Calculus: The Fundamental Theorem of Calculus, key formulas, and more

You didn’t sign up for AP Calculus AB or BC just to struggle through fundamental concepts and end up with a failing grade: you enrolled to excel. Whether your goal is earning valuable college credit, boosting your GPA, or proving you can master challenging calculus theorems, we’re here to help every step of the way.

We’ve gathered all the essential formulas and calculus theorems you’ll need to confidently approach AP Calculus questions, in both AB and BC courses, so you can focus on memorizing what matters most. We strip away the noise, break down complex topics, and guide you in building practical, lasting confidence for test day. Let’s break down what’s on the AP Calculus exam, from key concepts to strategies for tackling every section.

Overview of the exam format

Before we dive into specifics, let’s go over the structure of the test. Both the AP Calc AB and BC exams are 3 hours and 15 minutes long, so you’ll need to be ready for an extended testing experience. Be sure to bring an approved calculator, a reliable watch, and some snacks for your break to maintain your energy throughout.

During the structured format of either exam, you will work through 45 multiple-choice questions and 6 free-response questions. You will have 105 minutes to answer all the multiple-choice questions, followed by 90 minutes dedicated to the free-response section. Keeping this timing in mind will help you pace yourself effectively on exam day.

The content covered on both the AP Calc AB exam and the AP Calc BC exam can generally be organized into five main sections:

• Area 1: Limits
• Area 2: Derivatives
• Area 3: Integrals and Differential Equations
• Area 4: Polar Coordinates, Parametric Equations, and Vector-Valued Functions
• Area 5: Infinite Series

Each content area contains indispensable knowledge, rules, and processes to master ahead of your exams.

Importance of formulas and theorems

Many students wonder if an equation sheet is given during AP Calculus exams. Unfortunately, you won’t be given a reference sheet during the actual test, so it’s crucial you know the formulas (and how to apply them) going into it. Luckily, the exams rely more on conceptual understanding than shallow memorization. So if you are practicing applying your knowledge across the five sections, you should have good exposure to everything you need to know. Creating a “cheat sheet” of core terms on your own before the exam and using the first minute of testing time to write down everything you remember on scratch paper can also be a helpful strategy.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) and the Second FTC are the most important calculus theorems connecting differentiation and integration in both AP Calculus AB and BC. In simple terms, the FTC states that the derivative of a function is the reverse of its integral, and vice versa. This established principle makes it easy to calculate exact areas under curves by finding the “antiderivative,” and to calculate both the rate and actual value of change of a function. The Second FTC posits that the definite integral of a function f(a) and f(b) equals the difference of its antiderivatives at those points. Basically, the FTC is like a mathematical trick that enables the calculation of unknown values by reversing (deriving or integrating) an expression.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is another core concept that all AP students should be familiar with. It guarantees the existence of values within an interval, which becomes particularly important when using intervals to determine the absolute extrema of a specific expression. IVT allows us to assume that a point on a function’s curve can be calculated, and that the function is continuous between the two interval points given.

Other key formulas in AP Calculus

The FTC and IVT are just the beginning: there are tens of important formulas, principles, and notations to be familiar with ahead of the exam. The exhaustive lists below break down important topics for both AP Calculus AB and AP Calculus BC, with helpful links to visual examples.

AP Calculus (general):

• Average Rate of Change (AROC): Also known as the slope. For non-linear functions, the slope changes across different parts of the graph.
• Limits and Continuity: Explore properties of limits at a point and limits at infinity, which describe a function as it gets closer to a certain value.
• Squeeze Theorem: A key calculus theorem used to evaluate tricky or unclear limits.
• Definition of Continuity: Understanding continuity is foundational for AP Calculus AB and BC.
• Limit Definitions of the Derivative: Connecting limits to differentiation in AP Calculus AB and BC.
• Definition of Differentiability & Differentiation Rules: Core for mastering calculus theorems and derivative applications.
• Derivative Rules (Basic Rules, Trigonometric Functions): Must-know strategies for deriving functions, along with refreshers from trigonometry, which are crucial for solving geometric problems. Also covered are:
  • Product Rule in Calculus: Essential for differentiating product functions.
  • Quotient Rule: Key for handling division of functions.
  • Chain Rule: Fundamental for composite functions.
• Derivatives of Inverse Functions: Allows you to effectively reverse a derived expression.
• PVA (Derivatives, Integrals): Position, velocity, and acceleration applications in calculus.
• L’Hospital’s Rule: A powerful tool for evaluating indeterminate limits.
• Mean Value Theorem (MVT) and Rolle’s Theorem: Central AP Calculus theorems for understanding function behavior.
• Extreme Value Theorem (EVT): Guarantees absolute extrema on closed intervals in calculus.
• First Derivative Test and Second Derivative Test: Analyze critical points using calculus theorems.
• Riemann Sums: Approximating area under curves, foundational for the FTC.
• Limit of a Right Riemann Sum: Linking sums to definite integrals.
• Integral Rules: Basic Integration Rules, Trigonometric Integrals, Properties of Definite Integrals, BC-only: Improper Integrals
• Average Value of a Function: Calculated using the FTC.
• Total Distance Traveled with integrals: Key practical application of calculus theorems.
• Area Between Curves (In Terms of x, In Terms of y): Using FTC and Second FTC to determine values.
• Volume (Disk Method, Washer Method): Finding volume using revolving regions.

BC only:

• Integration by Parts: Advanced technique featured on the AP Calculus BC exam.
• Euler’s Method: Numerical approximation method for differential equations.
• Exponential Growth and Decay: Applications of calculus in modeling real-world phenomena.
• Logistic Growth and Decay Models: Advanced concept important for both calculus and statistical analysis.
• Arc Length Formula: Advanced integration topic on the AP Calculus BC test.
• Parametric Equations: Slope, Speed, and Arc Length
• Derivatives of Vector-Valued Functions: Important calculus theorem for parametric motion problems.
• Total Distance with Vector Functions: Advanced application for AP Calculus BC students.
• Polar to Rectangular Coordinates Conversion: Integrates geometric concepts.
• Slope of a Polar Curve: Unique calculus theorem application in AP Calculus BC.
• Sum of a Geometric Series: Series convergence, a topic exclusive to the BC exam.
• Convergence Tests for Series: Determine series behavior using advanced calculus theorems.
• Taylor and Maclaurin Polynomials: Power series expansion techniques for AP Calculus BC.
• Known Power Series Representations: Recognize standard series, an important skill for AP Calculus BC.

There’s a lot of ground to cover, but mastering these key calculus theorems is critical to succeed in both AP Calculus AB and AP Calculus BC. Review each concept carefully to maximize your exam performance.

Other topics to know

The theorems and formulas above don’t exist in isolation: they build off of each other and intersect in many ways. Bridge different concepts together by covering these additional and important topics:

• Connection between derivatives and integrals
• Techniques for Integration
• Integration by substitution
• Applications of Derivatives
• Optimization Problems
• Related Rates Problems
• Series and Sequences
• Convergence and Divergence

Proficiency in one area of calculus is often contingent upon mastery of another: if you don’t understand how to derive an equation, for example, you’ll have a hard time with optimization problems. That’s why it’s important to start with the basics (limits, derivatives, and integrals), brush up on algebra rules, and make connections between different topics. Take the time to actively take notes as you complete your coursework and assignments. While much of the AP exam will be digital, practicing writing out expressions by hand helps with retention, getting more comfortable with complex notations, and preparing for free-response questions.

Conclusion

The AP Calculus AB and AP Calculus BC exams are true marathons, with over three hours of challenging problems each grounded in key calculus theorems. The intensity builds quickly, and tackling these advanced questions without a formula sheet can make the exam feel even more demanding.

Despite this, the rigorous nature of AP Calculus doesn’t center on memorization. Instead, both AP Calculus AB and BC require a deep understanding of how central concepts, such as the connection between derivatives and integrals, are linked together. Success depends on recognizing when to apply foundational calculus theorems and employing well-practiced methods in the right context. This strategy-first approach gives prepared students a real advantage.

Effective preparation involves working through a diverse set of AP Calculus practice problems and applying formulas and techniques, rather than just memorizing them. The goal is for the core ideas of calculus to become second nature well before exam day. With this focused approach, both AP Calculus AB and BC shift from survival mode to confident execution: let your diligent preparation carry you forward.