# Glossary: A Quick Guide to the Mathematical Jargon

• absolute maximum.  The all-time, one-and-only, single, absolute and total maximum value of a function over a specified domain of the function. (Although it is the unique maximum value, it could occur at more than one point, as when you have two mountain peaks of exactly the same height.) Not to be confused with a local maximum, which is to the absolute maximum as the police chief is to the army chief-of-staff. The absolute maximum is sometimes also called the global maximum.

• absolute minimum.  Same definition as for the absolute maximum, only substitute the word minimum'' everywhere the word maximum occurs. Also substitute lawyer'' for police chief'' and (optionally, depending on your politics) politician'' for chief-of-staff''.

On a graph , it is a point that is having a really bad day. As low as it can get.

• absolute value  Drop the negative sign if there is one. Otherwise, just leave the number alone.

• acceleration.  Acceleration is the rate of change of the velocity. It causes that funny feeling in the pit of your stomach as you are mushed backward into the seat when somebody really puts the pedal to the metal. Since the rate of change of a function is its derivative, the acceleration is the derivative of the velocity function. Since the velocity function is itself the derivative of the function giving your position, the acceleration function is the second derivative of the function giving your position.(In mathese, a= dv/dt = d2s/dt2, where s is the position function).

• algebra.  Hold it. If you don't know what algebra is (a bunch of letters like x and y and a bunch of rules for playing around with them), then you shouldn't be taking calculus. Return to GO, do not collect \$200.

• antiderivative.  You guessed it. This is the opposite of the derivative. Doesn't deserve the negative connotations associated to some of the other anti' words like antichrist', antisocial' or anti-macassar' (that little lace doily you used to see on your grandma's couch - made obsolete by plastic slipcovers). The antiderivative of a function f(x) is another function F(x) whose derivative is f(x). Also called the indefinite integral of f(x). The antiderivative' terminology is traditionally usually used just before the introduction of indefinite integrals , and then never used again, having been forever replaced with the term indefinite integral'.

• antidifferentiation.  The process of taking an antiderivative. Also a strong aversion to distinguishing between different people, as with parents who insist on calling all five of their children Frank'.

• asymptote.  An asymptote is like one of those people you meet at a party who is devastatingly attractive and you just want to get close. You maneuver your way next to them and casually strike up a conversation. Making good time, you get closer and closer, till you're practically knocking knees. In calculus, you just keep getting closer. In the real world, you start explaining your love of partial fractions, they excuse themselves to get a drink, and you see them driving away through the window. An asymptote for the graph of a function is a line sitting in the x-y plane that the graph of the function approaches, getting closer and closer as we travel along the line. Functions that have had one too many may weave back and forth across an asymptote, but still, the further out you go, the closer they get.

• callipygian. Appears near calculus' in the dictionary. Check it out.

• carbon dating.  This is the essence of the social life of geologists. They get together, crush a bunch of rocks, and then determine the amounts of various types of carbon in the rock. Since carbon-12 does not decay over time, and carbon-14 does decay over time, they can tell by the ratio of carbon-14 to carbon-12 how old the rocks are. What is this topic doing in a calculus book? The rate of decay  of the carbon-14  and any other radioactive substance is exponential. That is to say, the amount at time t is given by f(t) = Co e-kt. A great source of problems and examples.

• Cartesian coordinates.  These are just the standard coordinates in the plane. You know, the ones where you have an x-axis and a y-axis, and each point is given by specifying two numbers (7, 4), which means go out 7 units in the x-direction and then 4 units in the y-direction. Why the funny name? They are named after the French mathematician Rene Descartes, whose Latin name was Cartesius.

• Cartesian plane.  That's a plane upon which we have Cartesian coordinates. It also describes the entire Air Force of the country of Cartesia.

• chain rule.  Never allow yourself to be chained up by someone whose body is covered by more tattoos than latex.'' The mathematical version states

(f(g(x))'=f'(g(x))g'(x)

or

• completing the square.  Here's a phrase that gets thrown around a lot and is the kind of thing that every teacher assumes some other teacher has shown you before. It's best demonstrated by example. If we want to complete the square on x2 + 8x +10, we write it as:

Why would we want to complete the square on a quantity? For one example, suppose that you want to graph x2+8x +10 +y2= 0. By completing the square, this becomes:

(x+4)2 +y2 = 6. This is the equation of a circle of radius 6 centered at the point (-4, 0).

• completing the square dance. We're not playing music anymore, so swing your butts right out the door.''

• complex number.  A number that neglected to get real'. Currently in therapy.

It's also one of those numbers like 7+ 6i, where i is the number . We know, everybody says you can't take the square root of a negative number, but what they really mean is that you can't take the square root of a negative number and expect to get a real number. No, you get a complex number instead. Given a complex number of the form a + bi, a is called the real part and bi is called the imaginary part. Normally doesn't come up in a first calculus course.

• composition of functions.  Applying one function to another. For instance is the composition of with . If successful, the two functions are then performed by an orchestra.

• concavity.  A part of the graph of a function is said to be concave down if it looks like part of a frown, and concave up if it looks like part of a cup (up...cup..., there's a mnemonic device). In order to tell whether a function is concave up or down, one uses the infamous second derivative test, which is discussed in detail elsewhere in this book.

• constant.  A fixed number, like 3 or . To be distinguished from a variable, which has no single value. When you say, My spouse was my constant supporter," you mean that he or she never wavered, despite your conviction for arms dealing and tax evasion, your decision to come out of the closet, and your involvement in the Perot for President campaign.

• continuity.  You know, no big surprises. Everything keeps going forward on an even keel. Here's the technical definition: A function f(x) is continuous at a point a if

Moreover, a function is continuous if

holds for all values of a where f(x) is defined. Now for a less technical definition: a function is continuous everywhere if you can draw the entire graph of the function without lifting your pencil from the page. (Okay, you can lift your pencil long enough to draw the axes.) See the section on continuity for more details.

• critical point.  The point that was made when you weren't paying attention. Also, a value of x that makes the derivative f'(x) of a function either equal to 0 or nonexistent. It comes up either in graphing functions, telling you where the critical changes in the graph occur, or in applied max/min problems, where it tells where the potential maxima or minima are occurring.

• definite integral.  The definite integral of a function f(x) over an interval is a number, sometimes thought of as the area under a graph. Not to be confused with the indefinite integral, which gives a function.

• derivative.  Hey, this is the most important idea in all of calculus. You shouldn't be looking up the definition as if it's some word in the dictionary like apothecary''. You should be reading about it in this book. But, if you insist on a nutshell definition, the derivative of f(x) is the rate of change of f(x). Geometrically, it also represents the slope of the tangent line to the graph of the function y=f(x) at the point (x,f(x)), but that's a mouthful.

• dictionary table tennis. See lexicon ping-pong.

• differentiable function.  A function is differentiable at a point if its derivative exists at that point. For instance, f(x)= x2 is differentiable everywhere, whereas g(x) = |x| is differentiable everywhere except x= 0. Why don't we say derivativeable''? Because it sounds ridiculous.

• differential.  It has something to do with the transmission of your car, but it's way too complicated for us to understand. Oh, yeah, a differential is also a small change in a variable. For instance, dy is a small change in the value of y. Although dy often occurs as part of the symbol for the derivative and as part of the symbol for the integral, and although in those other guises, dy plays a role very similar to the one intended when we call it a differential, it is best to just think of the differential dy as a very small change in y.

• differential equation.  An equation that involves derivatives, as in

These equations govern most of the physical world, so treat them with respect.

• domain of a function.  A dog's domain is all of the land that he can traverse in a day, starting with one full bladder. A function's domain is just the set of all values for x that it makes sense to plug into f(x). For instance, the domain of is all .

• double integration.  Calm down. It's okay. If you are looking up this word, then that means some jerk from multivariable calculus has said to you, If you think integration's hard, wait until you hit double integration.'' First of all, it's a lie. Double integration isn't that hard. And secondly, you don't need to worry about it for quite a while yet. Back to the good stuff.

• e.  e is one of those numbers that is so important, it gets it's very own name. In fact, e = 2.71828.... Why is it so important? Have you ever tried to write a sentence without it? It comes up all over the place. In fact it's the most commonly occurring letter in the whole alphabet! Same thing in calculus. One tantalizing tidbit is that it is the only number you could pick such that

• ellipse. Step on a circle until it squawks, and you've got an ellipse. It's a bit longer than it is wide. The general formula is like the formula for a circle only with a few extra a's and b's thrown in, as in x2/a2 + y2/b2 = 1.

• exponent.  That little number that appears as a superscript next to another number or function. Also called the power. If you are divorced, this is not what people are referring to when they say, How's your ex' "?

• exponential function.  This is the function f(x) = ex. It's most famous property? It is its own derivative. That's like being your own mother, not so easy to do.

• exponential growth.  Exponential growth is VERY VERY FAST GROWTH''. When people say exponential growth, they are trying to impress the hell out of you. A function experiences exponential growth if it is at least as big as a function of the form CKx, where C > 0 and K > 1. For example, the function 2x experiences exponential growth. Notice that the function 2x doubles in value each time x increases by one. So although 21 is only 2, 210 is already 1024, and 220 is 1,048,576. That is VERY VERY FAST GROWTH''.

• extrema (or extremum).  (extrema is the plural form.) Just a word for either a maximum or a minimum. Let's face it: A maximum or a minimum is a point where a high or low extreme occurs.

• factorial.  , that is to say, the factorial of an integer n is simply the product of all of the integers from 1 to n. A good question to stump your professor with is,  How do you take the factorial of a number like 3/2 or -2?''

• function.  Functions are something that just about everyone encounters during the four years of college, even the TV majors and the drama types who avoid any class where the word calculate is used. There are two types of function - social functions and mathematical functions. Though completely different, they use much of the same terminology.

Social functions are also called mixers or gatherings. Usually they involve parties hosted by dormitory floors (or assistant deans) with kegs of beer (or little cucumber sandwiches). The location where the function takes place is known as the domain of the function. The place where the food is cooked is known as the range of the function. A function that lasts until morning is said to be continuous. One that is broken up by the police and resumed the next day is called discontinuous. The phone number of the dreamboat you met is called the value of the function. It often winds up in the range.

The same terminology is used by mathematicians to describe what they call a function. The main difference is that when a mathematician has a function, everyone gets exactly one value! No one leaves with two numbers at a mathematical function, and no one leaves with none.

A mathematical function is a machine where you put in a real number (often denoted by a variable x, but sometimes by t or some other letter) and it spits out a new real number. For instance, f(x) = x2. You put in the number 3 for x and it spits out the number 9. It's domain is the set of values that are legal to put in, and its range is the set of possible values it can spit out.

• Fundamental Theorem of Calculus.  This theorem is usually stated in two parts. One part states that finding areas under curves can be done by taking antiderivatives and plugging in the limits. , where F(x) is any function whose derivative is f(x), F'(x) = f(x).This can also be stated as:

If you integrate the derivative of a function over an interval from a to b, you just get the original function evaluated at b minus the original function evaluated at a. The other part states that

Both parts show that derivatives and integrals are intimately related, and we don't just mean on a first name basis. If it weren't for this theorem, calculus courses would be half as long as they are.

• global (extremum, maximum, minimum, warming).  Another expression for the absolute extremum, maximum or minimum. It comes from the fact that this extremum is the most extreme extremum on the globe. For instance, the coldest place on earth, emotionally speaking, is Washington D.C. It is a global extremum.

• graph.  A pictorial representation of a function formed by plotting f(x) in the y direction on the x-y plane. Very useful, because while pictures say a thousand words, a graph gives an infinite number of function values.

Given a function f(x), the graph of the function is simply the set of points (x,y) in the Cartesian plane that satisfy the equation y=f(x). Most important property? Any vertical line can intersect the graph at most once, since a vertical line is defined by a particular value of x. But for a particular value of x, there is only one value of y such that y = f(x).

• hyperbolic trigonometric functions.  Well, this means you are in a slightly more heavy duty calculus course. Most calculus courses skip this material, just because it is expendable and they run out of time. But your course isn't skipping it. That's okay, because they are actually a cinch to deal with. The hyperbolic sine of x, denoted (pronounced like cinch - and it is) is defined to be while the hyperbolic cosine (rhymes with posh) is defined to be . Note that each is the other's derivative. All of the other hyperbolic trig functions are defined in terms of these two in exactly the same way the other trig functions are defined in terms of the sine and cosine functions.

• hypocycloid. Just kidding. We don't know, some kind of cycloid, probably. Shouldn't you be working some problems?

• indefinite integral.  The indefinite integral of a function f(x)is another function F(x) with the distinguishing feature that the derivative of F(x) is f(x). Not to be confused with the definite integral, which gives a number as an answer. Also called the antiderivative of f(x).

• integer.  ...-3,-2, -1, 0, 1, 2, 3,.... (How's that for a short definition?)

• integrable.  If people have integrity you can count on them. If people have integrability you can take their antiderivatives. Same holds for functions.

A function is integrable if its integral exists. Most of the standard functions are integrable.

• integrand.  The function inside the integral that is being integrated. Found between the and the dx.

• integrandstand. A stand upon which to put the function inside the integral that is being integrated.

• inverse trigonometric function.

An inverse trigonometric function is the function that reverses the effect of the original trig function, kind of like the democrats and the republicans, when they are taking turns being elected to power. The one undoes whatever the other had accomplished while in office. The inverse function for sin is denoted . So if , then . The notation is used, rather than , so as to prevent people from confusing the inverse trig function with .

• irrational number.  A number that is a few apples short of a picnic.

Also, a real number that is not rational, which is to say that it cannot be written as a fraction of two integers. Classic examples include , e and . Every irrational number has a decimal representation that is non-repeating. There are tons of these numbers, actually more of these than there are of the rational numbers. (Mathematicians say there are uncountably many of these.) Of course, you would be completely justified in noting that there are infinitely many rational numbers, so how could there possibly be more of these than there are of the rational numbers? That would bring us to the topic of the different kinds of infinity, but that's a little too far afield for us. Good question to ask the professor though.

• lexicon ping-pong. See dictionary table tennis.

• limit.  That bound you cannot exceed, as in Limit of three trips to the salad bar per customer." In calculus, a limit is the number you approach as you plug values into a function, and the values get closer and closer to a given number.

• line.  What we hope you waited in to get a copy of this book.

There's not a whole lot of question about what a line is. It's that straight thing between any two points. The equation of a line has two general forms, the point-slope form y-y0 = m(x-x0), where (x0,y0) is a point on the line and m is the slope of the line) and the slope-y-intercept form y=mx+b, where b is the y-coordinate where the line intersects the y-axis and m is the slope).

• linear equation.  An equation that represents a line. Looks something like 3x + 2y = 4. No x2 or or even an xy. It can always be put in a general form Ax + By + C = 0, where A, B and C are constants that are possibly zero. (Note that both of the equations for lines in the previous definition can be put in this form.)

• local (extremum, maximum, minimum).  If you were near sighted, this point would look like one where the graph of our function has an extremum, namely a maximum or a minimum. Possibly if you expanded your vision you would see a larger or smaller value somewhere far away on the graph, but comparing this point only to its near neighbors, it comes out on top (in the case of local maximum)or on the bottom (in the case of local minimum). The local maximum is a big fish in a small pond. A local minimum may be able to find someone even lower than him if he wanders out of his neighborhood.

• logarithm.  The beat of trees being cut down in the Pacific Northwest. Also a mathematical function that is the inverse of bx for some fixed number called the base' of the logarithm.

• map.  Danger! Danger! If your professor uses this word to refer to a function, then you are in serious trouble. That means that they are a theoretical mathematician and they are incapable of separating their theoretical world from the world of the classroom. Map'' is another word used for a function. It could be used as in  This is a map from the reals to the reals,'' translation, This is a function that takes a real number and turns it into another real number.''

• maxima.  Plural of maximum''. One of the few examples of an application of Latin. Who says the language is dead? It's just resting. See local maximum'' and absolute maximum'' for more details.

• minima.  Look, we just explained maxima. Do we have to do everything around here?

• natural logarithm.  The natural logarithm, usually denoted , is the logarithm to the base e. Also, a method of birth control used at tree farms.

• negative.  Pessimistic or depressed. Often treatable with Prozac.

• origin.  The point in space with all coordinates equal to 0. Thought to be somewhere in Iowa.

• orthogonal.  Fancy math word for perpendicular. When you say, He's orthogonal to the rest of the world," you mean he's perpendicular to everyone else, living in a slice of his mind that the rest of us don't have.

• parabola.  A certain type of curve. An equation of the form y=Ax2 + Bx + C will always give a parabola. Most common example is y=x2 when you have a curve passing through the origin in the shape of an upward opening cup. The prefix para comes from the Greek and means at or to the side of, as in paralegal: at or to the side of a lawyer, paranormal: at or to the side of normal and Paraguay: at or to the side of Guay, a tiny country most people don't know about. Parabolas can also be obtained by slicing a right circular cone by a plane that is parallel to a line in the cone passing through the vertex. How about that for a useful fact?

• .  That leader in the Number Hall of Fame, = 3.14159.... It can be defined to be one half of the circumference of a circle of radius 1.

You probably think that the letter is used for the circumference of a unit diameter circle because it was thought up by some ancient greeks. Well it was, but the letter was not used for this number until a few hundred years ago, and introduced by an Englishman at that. His motivation is unclear. Some suspect that it is because perimeter starts with a p. Others know that the English like a good pie for lunch.

What some of those ancient Greeks did give us was the sorry method of measuring angles using 360 degrees. The origin of the number 360 is also murky, though it is suspected to have something to do with the fact that a pizza can be nicely divided into 6 slices. For calculations involving angles, it is much easier to work with a set of angles called radians. They give a good way of slicing the , so to speak.

For most purposes is about 3.14. It took thousands of years before was known to ten decimal places. Today, mathematicians have calculated to over three billion decimal places. Fortunately, most professors will not ask you to memorize more than the first hundred thousand.

• a la mode. The number 3.14159... with a big scoop of vanilla ice cream on top. Best when warmed.

• polynomial.  You know, functions like x2- 7x + 3 or 2y15 - 4y3 + 3y -6. They do not contain any square roots or trig functions or anything the slightest bit weird. In their general form, they look like f(x)= anxn + an-1xn-1 + an-2xn-2 + ... + a2x2+a1x+ a0.

• position function.  This is a function that depends on time and tells you what your position is along a number line as time varies. For instance, if f(t) = t2, in units of feet and seconds, then at time t=0, you are at the origin, at time t=1 second you are 1 foot to the right of the origin and at time t= 2 seconds, you are 4 feet to the right of the origin. Of course at time t= 52 hours, you have travelled farther than the speed of light would allow, breaking one of the most basic laws of the physical universe. Cool.

• power rule.  Power corrupts. Absolute power corrupts absolutely.'' (Where by definition, absolute power equals power if power is positive, and otherwise the negative of power).

In calculus, the derivative of xn equals nxn-1.

• quadratic formula.  That amazing formula for finding all values of x that satisfy the equation ax2 +bx+c=0. Works even if you can't factor the left-hand side. There are two solutions, which may be equal.

or

Also slices and dices.

• range.  How far you can throw a ball is the range of your pitching arm. The set of values a function can take is the range of the function. What mathematicians cook on is the range in the kitchen. Fancy poultry on a menu is free range chicken. Home, home on the range...never mind.

• rate of change.  The rate of change is the speed at which a function is changing. If the function is measuring your position, then your speed(as measured by your speedometer) is your rate of change. Another name for the rate of change of a function IS the derivative of that function.

• rational function.  A function that makes a lot of sense.

Also the ratio of two polynomials, like .

• rational number.  A number that has both feet on the ground. It's all there.

A rational number is a number of the form a/b where a and b are integers. For instance, a few famous ones include 1/2 and 3/4. A less famous one is 337/122. Each rational number has a decimal representation that is either terminating (consisting of finitely many decimal places) or repeating. Interestingly enough, there are a lot more irrational numbers than rational numbers. Another example of math imitating life.

• real number.  Real numbers are the ones we usually deal with, including the integers, the fractions and the irrational numbers like e and that occur between the fractions. Each has a decimal representation. To be distinguished from imaginary numbers which involve .

• second derivative.  The derivative that comes after the first derivative and before the third derivative. Obtained by taking the derivative of a function twice in a row.

• secant line.  A jargon term for a line through two points in a curve. Take a curve, any curve. Then take two points on the curve. Connect them by a line. That is a secant line. Why isn't it just called a line? History. Most often secant lines come up in references to tangent lines, where you take a sequence of secant lines, fixing one of the points on the curve and moving the second point along the curve toward the first point. The sequence of secant lines that you obtain approach the so-called tangent line. Since the tangent line has such a fancy name, people felt bad for the secant line and gave it a fancy name, too.

• sine and cosine.   Two things that mathematicians ask each other at parties. What's your sine'' and what's your cosine''.

• speed.  This is the absolute value of your velocity. It's used when you don't care whether you backed into the wall at 30 mph (where velocity is -30) or drove into the wall forward at 30 mph (where velocity is 30), you just want to tell people you hit the wall going 30.

• speedometer. That little gizmo in your car that tells you how fast you are going. It is essentially a velocity function. If you look at it at a given time, it tells you your velocity (rate of change of your position function) at that time, assuming you are not backing up.

• tangent line.  A line that rubs up against and kisses'' a curve at a point, having the slope of the graph at that point. Now facing charges for sexual harassment.

• Theorems and proofs.   A theorem is a claim on some subject, such as, The derivative of is ." A proof is a detailed, logical, completely convincing argument showing why it's true. Learning the difference between what does and what does not constitute a proof is one of the most important things you can get out of a calculus course, though there is seldom time for a detailed discussion of this issue in a crowded curriculum. Good proofs should convince all reasonable people. Of course there are always those people you meet at parties who will say Wait a minute, what if a Martian was hypnotizing me while I heard the argument?" or Isn't truth all relative anyway? Why is one truth better than another?" Fortunately, they rarely get second invitations.

Mathematicians are often kept off juries because of a belief by lawyers that they cannot understand the legal meaning of proof beyond a reasonable doubt". If you want to avoid jury duty, point out that you learned about theorems and proofs in calculus. For similar reasons, lawyers are kept out of calculus classes because of a belief by mathematicians that they cannot understand the mathematical meaning of `proof''. (If you are a lawyer or future lawyer: please don't sue us). Mainly because mathematicians, like lawyers, fall into the use of jargon, theorems are also called corollaries, lemmas and propositions.

• trigonometric identity.  Any simple trig equation that relates various trig functions. The most famous and important is the classic , although there are also many less significant ones running around underfoot. Note that the classic immediately gives you others such as by dividing through by .

• velocity.  This is the rate of change of position (only differing from speed in that it can be negative if you are moving left along the number line.) It is obtained by taking the derivative of your position function.

• variable.  The single word used most often by nervous meterologists. In math, a quantity that can vary. Often represented by a letter like x or y, since it does not have a specific fixed value, but rather, can take on a whole set of different values.

• Zeno.  The last entry in any dictionary of calculus terms. He was also a Greek philosopher best known for Zeno's Paradox. He pointed out that for a runner to get from A to B, he or she must first traverse half the distance, and then half the remaining distance and then half the remaining distance ad infinitum. Since, clearly, the runner cannot perform infinitely many steps in a finite amount of time, motion is an impossibility, and is therefore an illusion. So, all the world is just a dream. Roll over and go back to sleep.