Math103

Front 1

Integer

Back 1

whole number (positive or negative)

*every integer is a rational number as it can be divided by 1.

Front 2

Rational Number

Back 2

expressed as a quotient a/b of two integers, where b does not equal 0.

ex. 2/3, 6 1/2

Front 3

Irrational Number

Back 3

cannot be expressed as the quotient of two numbers.

i.e. square root of 2 or pie

Front 4

Absolute Value

Back 4

any number (positive or negative) expressed as a positive number.

|-3| = 3

Front 5

Exponents - Product Law

Back 5

When multiplying, add exponents

Front 6

Exponents - Quotient Law

Back 6

When dividing, subtract exponents

Front 7

Exponents - Power Law

Back 7

When an exponent is multiplied to the power of another exponent, multiply exponents.

i.e. (a^r)^s = a^rxs

Front 8

Factoring

Back 8

ab + ac = a (b+c)

Front 9

Factoring Two Squares

Back 9

a^2 - b^2 = (a + b)(a + b)

Front 10

Quadratic Equation

Back 10

for ax^2 + bx + c = 0

x = (-b +- Vb^2 - 4ac)/2a

Front 11

absolute maximum

Back 11

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.

Front 12

absolute minimum

Back 12

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.

Front 13

absolute value

Back 13

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

Front 14

acceleration

Back 14

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).

Front 15

antiderivative

Back 15

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'.

Front 16

antidifferentiation

Back 16

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'.

Front 17

asymptote

Back 17

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.

Front 18

Cartesian coordinates

Back 18

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.

Front 19

Cartesian plane

Back 19

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

Front 20

chain rule

Back 20

``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

df/dx = df/du x du/dx

Front 21

completing the square

Back 21

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).

Front 22

complex number

Back 22

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.

Front 23

composition of functions

Back 23

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

Front 24

concavity

Back 24

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.

Front 25

constant

Back 25

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.

Front 26

continuity

Back 26

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

lim f(x) = f(a)
x->a

Moreover, a function is continuous if

lim f(x) = f(a)
x->a

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.

Front 27

critical point

Back 27

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.

Front 28

definite integral

Back 28

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.

Front 29

derivative

Back 29

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.

Front 30

differentiable function

Back 30

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.

Front 31

differential

Back 31

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.

Front 32

differential equation

Back 32

An equation that involves derivatives, as in

6 (dy/dx) + y - x = 25

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

Front 33

domain of a function

Back 33

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 .

Front 34

e

Back 34

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

(d/dx)e^x = e^x

Front 35

function

Back 35

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.

Front 36

Fundamental Theorem of Calculus

Back 36

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. b|a f(x) dx = 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:

b|a f'(x)dx = f(b) - f(a)


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

(d/dx) x|a f(t) dt = f(x)

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.

Front 37

limit

Back 37

the number you approach as you plug values into a function, and the values get closer and closer to a given number

Front 38

linear equation

Back 38

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.

Front 39

polynomial

Back 39

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

Front 40

position function

Back 40

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.