Mathematics is a language. You need to know the terms and notation in order to successfully master the concepts. Out of the hundreds of key words you might find in your textbook, I’ve complied a list of the top ten AP Calculus vocabulary words, which I hope will give you a good foundation for further study.

The words on this list (with one exception) pertain to both the AB and BC versions of the AP Calculus test.

And so, without further ado, here’s your top ten!

## Top Ten AP Calculus Vocabulary Words

### 1. Limit

The **limit** of a function *y* = *f*(*x*) (as *x* approaches a number *a*) is something like the ultimate *trend* of the values of *f* (near *x* = *a*).

The limit notation, , stands for the phrase: “The values of *f* get closer and closer to *y* = *L* as the *x*-values approach *a*.”

It’s essential to understand both what a limit is conceptually as well as how to find limits graphically, analytically, and by algebraic manipulation.

For more about limits, check out: What is the Limit of a Function?

### 2. Continuous

We say that a function *f* is **continuous** at a *x* = *a* if the value of *f*(*a*) matches the predicted value coming from the limit as *x* → *a*. That is,

In more intuitive terms, this means that there is no break or hole in the graph at *x* = *a*.

We also say that a function is **continuous on an interval** if it’s continuous at each individual point in that interval. Some teachers say that a function is continuous on an interval if “you can draw the graph without lifting your pencil” on that interval.

Take a look at the following brief review for more information: AP Calculus Exam Review: Limits and Continuity.

### 3. Derivative

The **derivative** of a function *f* is a (typically different) function *f* ' that measures the *rate of change* of the *y*-values of *f* with respect to change in the *x*-value. Another notation for the derivative is *dy*/*dx*.

The derivative value *f* '(*a*) also measures the *slope of the tangent line* to the curve at the point (*a*, *f*(*a*)).

There are a number of formulas that go along with this definition. First and foremost, the *limit definition* of the derivative is based finding the slope of the tangent line using a limit. It’s valid for all functions but is hard to work with, especially for complicated functions.

Other formulas, such as the Power Rule, Product and Quotient Rules, and Chain Rule, serve to find derivatives of all kinds of functions.

For more about derivatives and the various derivative formulas, check out Calculus Review: Derivative Rules.

### 4. Velocity (and Acceleration)

Suppose an object moves along a straight line over time, and suppose the function *s*(*t*) measures the position of the object at any time *t*. Then the **velocity** of the object, as a function of time, is the *derivative* of position. That is, *v*(*t*) = *s* '(*t*).

The reason for this is that velocity is the *rate of change* of position (see item 3 above).

The rate of change of velocity is called **acceleration**. And so acceleration is the derivative of velocity. In other words, acceleration is the *second derivative* of position.

### 5. Chain Rule

Among all of the derivative rules you have encountered, perhaps the most useful and yet most misunderstood is the chain rule. The **chain rule** is a formula for finding the derivative of a *composition* of functions.

What makes this rule so important is that so many important functions are actually compositions of two or more basic functions.

You can find more details about the chain rule, along with worked out examples, by clicking AP Calculus Review: Chain Rule.

### 6. Logarithm

The **logarithm** of a number is equal to the exponent on a given base that would give that number. For example, the logarithm in base 2 of the number 32 is 5. *Why?* Because 32 = 2^{5}.

But that’s algebra not calculus, right? How did *logarithm* make it to the Top Ten AP Calculus Vocabulary Words?

Well there are two big reasons.

First, there is a particular number called *e* that goes into defining the **natural logarithm**. The number *e* is roughly 2.7, but it’s not crucial to memorize the constant itself.

Instead, you’ll need to be aware of the properties of the natural logarithm and its close cousin, the natural exponential function, *e** ^{x}*.

Second, logarithms play a huge role in certain derivative formulas. The technique of **logarithmic differentiation** uses the natural logarithm, ln *x*, to break down powers, products, and/or quotients into simpler operations so that it’s easier to apply your derivative formulas.

Be sure to brush up on your logarithms before the test. Here’s an article to help you do just that: AP Calculus Review: Properties of Exponents and Logarithms.

### 7. Slope Field

A **slope field** is a certain kind of visualization of a *differential equation*.

Suppose you have a differential equation of the form *dy*/*dx* = …, where the dots represent an expression involving both *x* and *y*. The slope field is found by plugging in sample points (*x*, *y*) and drawing short segments of slope equal to the value of *dy*/*dx* that you get at each point.

For example, below is the slope field of *dy*/*dx* = *x* + *y*.

While it may seem at first that slope fields are hard to grasp, they are actually quite intuitive. I like to think of the slope field as a map of the currents in a river. Then any particular function that solves the differential equation is really just a path of a boat through the water along those currents.

### 8. Integral

Integrals form a huge part of the AP Calculus exam. In fact you might say they are an… (wait for it)…. *integral* part of the test!

Integration is the reverse operation of differentiation, but of course there’s much more to the story.

For a good review of integrals, check out: AP Calculus Exam Review: Integrals

### 9. Series (BC only)

A **series** is the sum of a *sequence* of numbers.

When studying series you have to be aware of terms like **convergent**, **divergent**, **integral test**, **conditionally convergent**, **alternating**, etc.

Moreover, once you have series of *numbers* mastered, then they throw series of *functions* at you! Those are called **Taylor series** or **Maclaurin series**.

If you want a detailed review of this topic, take a look at this review of sequences and series.

### 10. Theorem

Last but not least, let’s talk about theorems in mathematics. A **theorem** is simply a true mathematical statement.

For the AP Calculus exams, you’ll need to understand and be able to apply various theorems, including:

- Extreme Value Theorem (EVT)
- Mean Value Theorem (MVT)
- Rolle’s Theorem
- The Fundamental Theorem of Calculus (FTC)

Check out AP Calculus Review: Reasoning with Theorems and Definitions for more.

## Summary

Here’s that list of AP Calculus vocabulary one more time. Happy studying!

- Limit
- Continuous
- Derivative
- Velocity (and Acceleration)
- Chain Rule
- Logarithm
- Slope Field
- Integral
- Series
- Theorem

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##### 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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