Shaun Ault

AP Calculus Review: Basic Formulas & Properties

It’s important to know the basics. You have to have a solid foundation in order to build a skyscraper, right? In this short review, I’ll discuss the basic formulas and properties that you’ll need to master in order to success on the AP Calculus exam.

Sky scrapers. AP Calculus Review Basic Formulas

The tallest skyscrapers require a solid foundation in order to stand.

The Basic Formulas

First and foremost, you must be you must know arithmetic, algebra, and trigonometry inside-out. For a quick review check out the following helpful articles.

Once you have mastered those fundamental topics of mathematics, you can then build your expertise in calculus.

Calculus-Specific Formulas

There are a number of basic formulas from calculus that you need to memorize for the exam.

Moreover, if you plan to take the Calculus BC exam, then you will have to know every formula that could show up on the AB exam, plus a whole slew of additional formulas and concepts that are specific to the BC exam.

It might help to look through the following “Cram Sheets” first.

The formulas can be categorized into four Big Ideas.

  1. Limits and Continuity (AB and BC)
  2. Derivatives and Their Applications (AB and BC)
  3. Integrals and Their Applications (AB and BC)
  4. Sequences and Series (BC only)

Limits and Continuity

There are not a lot of formulas for computing limits. Instead, each limit problem may require different algebraic or trigonometric “tricks.” However, it helps to know L’Hospital’s Rule:

L'Hospital's Rule

A function f is continuous at a point x = a if:

the limit as x approaches a is equal to f(a)

The Intermediate Value Theorem (IVT) states that if a function f is continuous on a closed interval [a, b], and if L is any number between f(a) and f(b), then there must be a value x = c where a < c < b, such that f(c) = L.

Average Rate/Velocity

Although this formula does not have to do with limits directly, it’s a useful concept and very important for developing the concepts of derivatives.

Average velocity formula

Derivatives and Their Applications

Limit definition for the derivative:

Limit definition of the derivative

Make sure you know every derivative rule.

Basic Differentiation Rules

basic derivative rules

Sum and Difference Rules

Derivatives of trig, exponential, and log functions

Product and Quotient Rule

Product Rule

Quotient Rule formula

Chain Rule

Statement of the Chain Rule

Inverse Functions

Derivative formula for inverse function

Derivatives of inverse trig functions

Polar and Parametric Functions

The AP Calculus BC exam also includes polar and parametric functions and their derivatives.

The derivative of a polar function, r = f(θ):

Polar derivative

The derivative of a parametric function, x = f(t) and y = g(t):

Formula for derivative of a parametric function

Applications of Derivatives — Velocity

It’s important to know the relationship between position, velocity, and acceleration in terms of derivatives.

position, velocity, and acceleration

On the AP Calculus BC test, the position may be a vector function.

velocity, acceleration, and speed for vector position function

Mean Value Theorem and Rolle’s Theorem

There are two related theorems involving differentiable functions, the Mean Value Theorem, and Rolle’s Theorem.

Mean Value Theorem (MVT): Suppose f is a function that is continuous on [a, b] and differentiable on (a, b). Then there is at least one value x = c, where a < c < b, such that

Statement of the Mean Value Theorem

Rolle’s Theorem: Suppose f is a function that is continuous on [a, b], differentiable on (a, b), and f(a) = f(b). Then there is at least one value x = c, where a < c < b, such that f '(c) = 0.

Integrals and Their Applications

Power rule for integrals

Sum and difference rule for integrals

Constant multiple rule for integrals

Constant Function Rule

Rule for 1/x

Exponential Antiderivatives

Trigonometric Antiderivatives

On the BC test, you may have to find velocity and speed for a vector position function.

Integration Techniques

The following formulas are useful for working out integrals of more complicated functions. Think of each rule as a potential tool in your toolbox. Sometimes an integral will require multiple tools.

  • u-Substitution
    Substitution Rule
  • Integration By Parts (BC only)
    Integration by parts formula

The Fundamental Theorem of Calculus (FTC)

The Fundamental Theorem of Calculus comes in two versions.

Second Fundamental Theorem of Calculus

If F(x) is any particular antiderivative for f(x), then

Definite integral of f(x) from x=a to x=b

Average Value and Mean Value Theorem for Integrals

average value formula

Mean Value Theorem for Integrals (MVTI): Suppose f is continuous on [a, b]. Then there is at least one value x = c, where a < c < b, such that

Mean Value Theorem for integrals

Applications of Integrals

Length of curve formula

On the AP Calculus BC exam, you may also have to find the length of a parametric curve defined by x = f(t) and y = g(t).

Formula for the length of a parametric curve

Use the washer or shell method to find the volume of a solid of revolution.

Formulas for Washer/Disk and Shell Methods

Sequences and Series

One of the most important formulas involving series is the Geometric Series Formula:

Formula for the sum of a geometric series

Convergence Tests

Given a series,

Series notation,

the following tests can help to prove that the series converges or diverges.

  • p-series test. If the series has general term an = 1/np, then the series converges if p > 1 and diverges if p ≤ 1.
  • Alternating series test. If the series is alternating (i.e., the terms alternate in sign forever), then the series converges if and only if an → 0 as n → ∞. And in that case, the error bound for the nth partial sum is |an+1|.
  • Ratio test.

    Ratio test
    However, if the limit is > 1, then the series diverges. No information if the limit equals 1.

  • Root test.

    Root Test

    Just as in the ratio test, if the limit is > 1, then the series diverges. No information if the limit equals 1.

Taylor and Maclaurin Series

If a function f is differentiable to all orders, then you can build its Taylor series centered at c as follows.

Taylor series for a function f

A Taylor series centered at c = 0 is called a Maclaurin series. Below are some common Maclaurin series that are worth memorizing.

Common Maclaurin series

Get Ready!

This list provides just the foundation for your study. Memorizing a list of formulas will not guarantee you a high score on the exam. You must also understand how and when to use each formula.

Furthermore, there are tons of more specialized formulas that didn’t make it to this list. Master all of the AP topics and practice, practice, practice!

Meme - One does not simply walk into the AP Calculus exam unprepared and score a 5

Are you prepared?

About Shaun Ault

Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!


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