Glossary: A Quick Guide to the Mathematical
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• 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 thanone point, as when you have two mountain peaks of exactly the sameheight.) Not to be confused with a local maximum, which is to the absolutemaximum as the police chief is to the army chief-of-staff. The absolutemaximum 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 onyour politics) “politician” for “chief-of-staff”.
On a graph, it is a point that is having a really bad day. As low as it canget.
• absolute value Drop the negative sign if there is one. Otherwise, just
• 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 backwardinto the seat when somebody really puts the pedal to the metal. Sincethe rate of change of a function is its derivative, the acceleration is thederivative of the velocity function. Since the velocity function is itself thederivative of the function giving your position, the acceleration functionis 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 youshouldn’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 lacedoily you used to see on your grandma’s couch - made obsolete by plasticslipcovers). 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 theintroduction of indefinite integrals, and then never used again, havingbeen 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 parentswho 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. Youmaneuver your way next to them and casually strike up a conversation. Making good time, you get closer and closer, till you’re practically knock-ing knees. In calculus, you just keep getting closer. In the real world, youstart explaining your love of partial fractions, they excuse themselves toget 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 planethat the graph of the function approaches, getting closer and closer as wetravel along the line. Functions that have had one too many may weaveback and forth across an asymptote, but still, the further out you go, thecloser 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 ofvarious types of carbon in the rock. Since carbon-12 does not decay overtime, and carbon-14 does decay over time, they can tell by the ratio ofcarbon-14 to carbon-12 how old the rocks are. What is this topic doingin a calculus book? The rate of decay of the carbon-14 and any otherradioactive substance is exponential. That is to say, the amount at timet is given by f (t) = Coe−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, andeach point is given by specifying two numbers (7, 4), which means go out 7units in the x-direction and then 4 units in the y-direction. Why the funnyname? 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 coordi-
nates. 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 versionstates
• 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 teacherhas shown you before. It’s best demonstrated by example. If we want tocomplete the square on x2 + 8x + 10, we write it as:
x2+8x+10 = x2+8x+(8/2)2+10−(8/2)2 = x2+8x+(8/2)2−6 = (x+4)2−6
Why would we want to complete the square on a quantity?
example, suppose that you want to graph x2 + 8x + 10 + y2 = 0. Bycompleting the square, this becomes:
(x + 4)2 + y2 = 6. This is the equation of a circle of radius 6 centered atthe point (-4, 0).
• completing the square dance. “We’re not playing music anymore, so
• complex number. A number that neglected to ‘get real’. Currently in
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 negativenumber, but what they really mean is that you can’t take the square rootof a negative number and expect to get a real number. No, you get acomplex number instead. Given a complex number of the form a + bi, a iscalled the real part and bi is called the imaginary part. Normally doesn’tcome up in a first calculus course.
• composition of functions. Applying one function to another. For in-
stance sin ( x) is the composition of sin x with
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 acup (up.cup., there’s a mnemonic device). In order to tell whether afunction is concave up or down, one uses the infamous second derivativetest, which is discussed in detail elsewhere in this book.
variable, which has no single value. When you say, “My spouse was myconstant supporter,” you mean that he or she never wavered, despite yourconviction for arms dealing and tax evasion, your decision to come out ofthe closet, and your involvement in the Perot for President campaign.
• continuity. You know, no big surprises. Everything keeps going for-
ward on an even keel. Here’s the technical definition: A function f (x) iscontinuous at a point a if
holds for all values of a where f (x) is defined. Now for a less technicaldefinition: 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 oncontinuity for more details.
• critical point. The point that was made when you weren’t paying at-
tention. Also, a value of x that makes the derivative f (x) of a functioneither equal to 0 or nonexistent. It comes up either in graphing functions,telling you where the critical changes in the graph occur, or in appliedmax/min problems, where it tells where the potential maxima or minimaare occurring.
• definite integral. The definite integral of a function f (x) over an interval
a ≤ x ≤ b 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 dictionarylike “apothecary”. You should be reading about it in this book. But, ifyou insist on a nutshell definition, the derivative of f (x) is the rate ofchange of f (x). Geometrically, it also represents the slope of the tangentline to the graph of the function y = f (x) at the point (x, f (x)), but that’sa 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 differentiableeverywhere, 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 differentialis also a small change in a variable. For instance, dy is a small changein the value of y. Although dy often occurs as part of the symbol for thederivative and as part of the symbol for the integral, and although in thoseother guises, dy plays a role very similar to the one intended when we callit a differential, it is best to just think of the differential dy as a very smallchange in y.
• differential equation. An equation that involves derivatives, as in
These equations govern most of the physical world, so treat them withrespect.
• 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 isjust the set of all values for x that it makes sense to plug into f (x). For
• double integration. Calm down. It’s okay. If you are looking up this
word, then that means some jerk from multivariable calculus has said toyou, “If you think integration’s hard, wait until you hit double integra-tion.” First of all, it’s a lie. Double integration isn’t that hard. Andsecondly, you don’t need to worry about it for quite a while yet. Back tothe 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 evertried to write a sentence without it? It comes up all over the place. Infact it’s the most commonly occurring letter in the whole alphabet! Samething in calculus. One tantalizing tidbit is that it is the only number youcould 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 acircle 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, notso easy to do.
• exponential growth. Exponential growth is “VERY VERY FAST GROWTH”.
When people say exponential growth, they are trying to impress the hellout of you. A function experiences exponential growth if it is at least asbig as a function of the form CKx, where C > 0 and K > 1. For example,the function 2x experiences exponential growth. Notice that the function2x doubles in value each time x increases by one. So although 21 is only2, 210 is already 1024, and 220 is 1,048,576. That is “VERY VERY FASTGROWTH”.
• 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 aminimum is a point where a high or low extreme occurs.
• factorial. n! = n × (n − 1) × .3 × 2 × 1, that is to say, the factorial of an
integer n is simply the product of all of the integers from 1 to n. A goodquestion to stump your professor with is, “ How do you take the factorialof 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 twotypes of function - social functions and mathematical functions. Thoughcompletely different, they use much of the same terminology.
Social functions are also called mixers or gatherings. Usually they involveparties hosted by dormitory floors (or assistant deans) with kegs of beer(or little cucumber sandwiches). The location where the function takesplace is known as the domain of the function. The place where the food iscooked is known as the range of the function. A function that lasts untilmorning is said to be continuous. One that is broken up by the police andresumed the next day is called discontinuous. The phone number of thedreamboat you met is called the value of the function. It often winds upin the range.
The same terminology is used by mathematicians to describe what theycall a function. The main difference is that when a mathematician has afunction, everyone gets exactly one value! No one leaves with two numbersat 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 inthe number 3 for x and it spits out the number 9. It’s domain is the setof values that are legal to put in, and its range is the set of possible valuesit 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.
F (b) − F (a), 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 functionevaluated at a. The other part states that
Both parts show that derivatives and integrals are intimately related, andwe 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 ex-
pression for the absolute extremum, maximum or minimum. It comes
from the fact that this extremum is the most extreme extremum on theglobe. For instance, the coldest place on earth, emotionally speaking, isWashington 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 picturessay 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 ofpoints (x, y) in the Cartesian plane that satisfy the equation y = f (x). Most important property? Any vertical line can intersect the graph atmost once, since a vertical line is defined by a particular value of x. Butfor 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 thismaterial, just because it is expendable and they run out of time. But yourcourse isn’t skipping it. That’s okay, because they are actually a cinch todeal with. The hyperbolic sine of x, denoted sinh x (pronounced like cinch
cosh (rhymes with posh) is defined to be cosh x =
each is the other’s derivative. All of the other hyperbolic trig functionsare defined in terms of these two in exactly the same way the other trigfunctions are defined in terms of the sine and cosine functions.
• hypocycloid. Just kidding. We don’t know, some kind of cycloid, prob-
ably. 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) isf (x). Not to be confused with the definite integral, which gives a numberas 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 func-tions are integrable.
• integrand. The function inside the integral that is being integrated.
• integrandstand. A stand upon which to put the function inside the
An inverse trigonometric function is the function that reverses the effectof the original trig function, kind of like the democrats and the republi-cans, when they are taking turns being elected to power. The one undoeswhatever the other had accomplished while in office. The inverse functionfor sin is denoted arcsin. So if y = sin x, then x = arcsin y. The notationarcsin is used, rather than sin−1, so as to prevent people from confusing
• 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 cannotbe written as a fraction of two integers. Classic examples include π, e
2. Every irrational number has a decimal representation that is
non-repeating. There are tons of these numbers, actually more of thesethan there are of the rational numbers. (Mathematicians say there areuncountably many of these.) Of course, you would be completely justifiedin noting that there are infinitely many rational numbers, so how couldthere 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, butthat’s a little too far afield for us. Good question to ask the professorthough.
• 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 approachas you plug values into a function, and the values get closer and closer toa 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 straightthing between any two points. The equation of a line has two generalforms, the point-slope form y − y0 = m(x − x0), where (x0, y0) is a pointon the line and m is the slope of the line) and the slope-y-intercept formy = mx+b, where b is the y-coordinate where the line intersects the y-axisand m is the slope).
• linear equation. An equation that represents a line. Looks something
like 3x + 2y = 4. No x2 or sin x or even an xy. It can always be put in ageneral form Ax + By + C = 0, where A, B and C are constants that arepossibly zero. (Note that both of the equations for lines in the previousdefinition 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 anextremum, namely a maximum or a minimum. Possibly if you expanded
your vision you would see a larger or smaller value somewhere far away onthe graph, but comparing this point only to its near neighbors, it comesout on top (in the case of local maximum)or on the bottom (in the case oflocal minimum). The local maximum is a big fish in a small pond. A localminimum may be able to find someone even lower than him if he wandersout 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 fixednumber 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 atheoretical mathematician and they are incapable of separating their the-oretical world from the world of the classroom. “Map” is another wordused for a function. It could be used as in “ This is a map from the realsto the reals,” translation,“ This is a function that takes a real number andturns it into another real number.”
• maxima. Plural of “maximum”. One of the few examples of an applica-
tion of Latin. Who says the language is dead? It’s just resting. See “localmaximum” and “absolute maximum” for more details.
• minima. Look, we just explained maxima. Do we have to do everything
• natural logarithm. The natural logarithm, usually denoted ln, is the
logarithm to the base e. Also, a method of birth control used at treefarms.
• negative. Pessimistic or depressed. Often treatable with Prozac.
• origin. The point in space with all coordinates equal to 0. Thought to
• orthogonal. Fancy math word for perpendicular. When you say, “He’s
orthogonal to the rest of the world,” you mean he’s perpendicular to ev-eryone 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 whenyou have a curve passing through the origin in the shape of an upwardopening cup. The prefix para comes from the Greek and means at or tothe side of, as in paralegal: at or to the side of a lawyer, paranormal: ator to the side of normal and Paraguay: at or to the side of Guay, a tinycountry most people don’t know about. Parabolas can also be obtainedby slicing a right circular cone by a plane that is parallel to a line in thecone passing through the vertex. How about that for a useful fact?
• π. 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 π is used for the circumference of a unitdiameter circle because it was thought up by some ancient greeks. Wellit was, but the letter π was not used for this number until a few hundredyears ago, and introduced by an Englishman at that. His motivation isunclear. Some suspect that it is because perimeter starts with a p. Othersknow that the English like a good pie for lunch.
What some of those ancient Greeks did give us was the sorry method ofmeasuring angles using 360 degrees. The origin of the number 360 is alsomurky, though it is suspected to have something to do with the fact that apizza 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 agood way of slicing the π, so to speak.
For most purposes π is about 3.14. It took thousands of years before πwas known to ten decimal places. Today, mathematicians have calculatedπ to over three billion decimal places. Fortunately, most professors willnot ask you to memorize more than the first hundred thousand.
• π a la mode. The number 3.14159. with a big scoop of vanilla ice cream
• 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 theslightest 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, iff (t) = t2, in units of feet and seconds, then at time t=0, you are at theorigin, at time t=1 second you are 1 foot to the right of the origin and attime t= 2 seconds, you are 4 feet to the right of the origin. Of course attime t= 52 hours, you have travelled farther than the speed of light wouldallow, breaking one of the most basic laws of the physical universe. Cool.
(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 factorthe left-hand side. There are two solutions, which may be equal.
• 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. Whatmathematicians cook on is the range in the kitchen. Fancy poultry on amenu 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(asmeasured by your speedometer) is your rate of change. Another name forthe rate of change of a function IS the derivative of that function.
• rational function. A function that makes a lot of sense.
• rational number. A number that has both feet on the ground. It’s all
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 is337/122. Each rational number has a decimal representation that is eitherterminating (consisting of finitely many decimal places) or repeating. In-terestingly enough, there are a lot more irrational numbers than rationalnumbers. 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
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 afunction 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 bya line. That is a secant line. Why isn’t it just called a line? History. Mostoften secant lines come up in references to tangent lines, where you take asequence of secant lines, fixing one of the points on the curve and movingthe second point along the curve toward the first point. The sequence ofsecant lines that you obtain approach the so-called tangent line. Since thetangent line has such a fancy name, people felt bad for the secant line andgave 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), youjust 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 giventime, 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 sexualharassment.
• Theorems and proofs. A theorem is a claim on some subject, such as,
“The derivative of sin x is cos x.” A proof is a detailed, logical, completelyconvincing argument showing why it’s true. Learning the difference be-tween what does and what does not constitute a proof is one of the mostimportant things you can get out of a calculus course, though there isseldom time for a detailed discussion of this issue in a crowded curricu-lum. Good proofs should convince all reasonable people. Of course thereare 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 thatthey cannot understand the legal meaning of “proof beyond a reasonabledoubt”. If you want to avoid jury duty, point out that you learned abouttheorems and proofs in calculus. For similar reasons, lawyers are kept outof calculus classes because of a belief by mathematicians that they cannotunderstand the mathematical meaning of “proof”. (If you are a lawyer orfuture lawyer: please don’t sue us). Mainly because mathematicians, likelawyers, 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 sin2 x +cos2 x = 1, although there are also many less significant ones runningaround underfoot. Note that the classic immediately gives you otherssuch as tan2 x + 1 = sec2 x by dividing through by cos2 x.
• 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 awhole 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 thatfor a runner to get from A to B, he or she must first traverse half thedistance, and then half the remaining distance and then half the remainingdistance ad infinitum. Since, clearly, the runner cannot perform infinitelymany steps in a finite amount of time, motion is an impossibility, and istherefore an illusion. So, all the world is just a dream. Roll over and goback to sleep.

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