{"id":10097,"date":"2017-05-30T14:20:01","date_gmt":"2017-05-30T21:20:01","guid":{"rendered":"https:\/\/magoosh.com\/hs\/?p=10097"},"modified":"2018-10-24T03:43:46","modified_gmt":"2018-10-24T10:43:46","slug":"ap-calculus-vocabulary-words","status":"publish","type":"post","link":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-vocabulary-words\/","title":{"rendered":"Top 10 AP Calculus Vocabulary Words to Know"},"content":{"rendered":"

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.<\/p>\n

The words on this list (with one exception) pertain to both the AB and BC versions of the AP Calculus test.<\/p>\n

And so, without further ado, here’s your top ten!<\/p>\n

Top Ten AP Calculus Vocabulary Words<\/h2>\n

1. Limit<\/h3>\n

The limit<\/strong> of a function y<\/em> = f<\/em>(x<\/em>) (as x<\/em> approaches a number a<\/em>) is something like the ultimate trend<\/em> of the values of f<\/em> (near x<\/em> = a<\/em>). <\/p>\n

The limit notation, \"Limit, stands for the phrase: “The values of f<\/em> get closer and closer to y<\/em> = L<\/em> as the x<\/em>-values approach a<\/em>.”<\/p>\n

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

For more about limits, check out: What is the Limit of a Function?<\/a><\/p>\n

2. Continuous<\/h3>\n

We say that a function f<\/em> is continuous<\/strong> at a x<\/em> = a<\/em> if the value of f<\/em>(a<\/em>) matches the predicted value coming from the limit as x<\/em> → a<\/em>. That is,<\/p>\n

\"the<\/div>\n

In more intuitive terms, this means that there is no break or hole in the graph at x<\/em> = a<\/em>. <\/p>\n

We also say that a function is continuous on an interval<\/strong> 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.<\/p>\n

Take a look at the following brief review for more information: AP Calculus Exam Review: Limits and Continuity<\/a>.<\/p>\n

3. Derivative<\/h3>\n

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

The derivative value f<\/em> '(a<\/em>) also measures the slope of the tangent line<\/em> to the curve at the point (a<\/em>, f<\/em>(a<\/em>)).<\/p>\n

\n
\"three
Three different tangent lines for a curve y<\/em> = f<\/em>(x<\/em>).<\/figcaption><\/figure>\n<\/div>\n

There are a number of formulas that go along with this definition. First and foremost, the limit definition<\/em> 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.<\/p>\n

\"Limit<\/p>\n

Other formulas, such as the Power Rule, Product and Quotient Rules, and Chain Rule, serve to find derivatives of all kinds of functions. <\/p>\n

For more about derivatives and the various derivative formulas, check out Calculus Review: Derivative Rules<\/a>.<\/p>\n

4. Velocity (and Acceleration)<\/h3>\n

Suppose an object moves along a straight line over time, and suppose the function s<\/em>(t<\/em>) measures the position of the object at any time t<\/em>. Then the velocity<\/strong> of the object, as a function of time, is the derivative<\/em> of position. That is, v<\/em>(t<\/em>) = s<\/em> '(t<\/em>).<\/p>\n

The reason for this is that velocity is the rate of change<\/em> of position (see item 3 above).<\/p>\n

The rate of change of velocity is called acceleration<\/strong>. And so acceleration is the derivative of velocity. In other words, acceleration is the second derivative<\/em> of position.<\/p>\n

\"position,<\/p>\n

5. Chain Rule<\/h3>\n

Among all of the derivative rules you have encountered, perhaps the most useful and yet most misunderstood is the chain rule. The chain rule<\/strong> is a formula for finding the derivative of a composition<\/em> of functions. <\/p>\n

\"Statement<\/p>\n

What makes this rule so important is that so many important functions are actually compositions of two or more basic functions.<\/p>\n

You can find more details about the chain rule, along with worked out examples, by clicking AP Calculus Review: Chain Rule<\/a>.<\/p>\n

6. Logarithm<\/h3>\n

The logarithm<\/strong> 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?<\/em> Because 32 = 25<\/sup>.<\/p>\n

But that’s algebra not calculus, right? How did logarithm<\/em> make it to the Top Ten AP Calculus Vocabulary Words? <\/p>\n

\n
\"Calculus
How did all this algebra get into calculus???<\/figcaption><\/figure>\n<\/div>\n

Well there are two big reasons.<\/p>\n

First, there is a particular number called e<\/em> that goes into defining the natural logarithm<\/strong>. The number e<\/em> is roughly 2.7, but it’s not crucial to memorize the constant itself. <\/p>\n

Instead, you’ll need to be aware of the properties of the natural logarithm and its close cousin, the natural exponential function, e<\/em>x<\/sup><\/em>. <\/p>\n

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

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<\/a>.<\/p>\n

7. Slope Field<\/h3>\n

A slope field<\/strong> is a certain kind of visualization of a differential equation<\/em>.<\/p>\n

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

For example, below is the slope field of dy<\/em>\/dx<\/em> = x<\/em> + y<\/em>.<\/p>\n

\n
\"Slope
Slope field for dy\/dx = x+y<\/figcaption><\/figure>\n<\/div>\n

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.<\/p>\n

8. Integral<\/h3>\n

Integrals form a huge part of the AP Calculus exam. In fact you might say they are an… (wait for it)…. integral<\/em> part of the test!<\/p>\n

\n
\"Improper
One important application of the integral is in determining area under or between curves.<\/figcaption><\/figure>\n<\/div>\n

Integration is the reverse operation of differentiation, but of course there’s much more to the story.<\/p>\n

For a good review of integrals, check out: AP Calculus Exam Review: Integrals<\/a><\/p>\n

9. Series (BC only)<\/h3>\n

A series<\/strong> is the sum of a sequence<\/em> of numbers.<\/p>\n

\"Series<\/p>\n

When studying series you have to be aware of terms like convergent<\/strong>, divergent<\/strong>, integral test<\/strong>, conditionally convergent<\/strong>, alternating<\/strong>, etc.<\/p>\n

Moreover, once you have series of numbers<\/em> mastered, then they throw series of functions<\/em> at you! Those are called Taylor series<\/strong> or Maclaurin series<\/strong>.<\/p>\n

If you want a detailed review of this topic, take a look at this review of sequences and series<\/a>.<\/p>\n

10. Theorem<\/h3>\n

Last but not least, let’s talk about theorems in mathematics. A theorem<\/strong> is simply a true mathematical statement.<\/p>\n

For the AP Calculus exams, you’ll need to understand and be able to apply various theorems, including:<\/p>\n