- 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
- 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
- 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
- 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
(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
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
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
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 .
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
- definite integral. The definite integral of a function f(x) over
is a number, sometimes thought of as the area
under a graph. Not to be confused with the indefinite integral, which gives a
- derivative. Hey, this is the most important idea in all of
You shouldn't be looking up the definition as if it's some word in the
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
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
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
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
- 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
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.
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
t or some other letter) and it spits out a new real number. For instance,
= 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
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
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
(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
- 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
A function is integrable if its integral exists. Most of the standard functions
- 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 ,
is used, rather than
so as to prevent people from confusing the inverse trig function
- 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
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
- 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
3x + 2y = 4. No x2 or
or even an xy.
It can always be put in a general form
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
- 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
- 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 ,
the logarithm to the base e. Also, a method of birth control used at tree
- 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
- . 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.
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
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
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
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.
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
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
- 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
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
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.