**GLOSSARY**
**family of curves**

**Family of curves: y = cx**^{2}*x*

*y*

*O*

*y= f (x)*

*A*

*C*

*B*

a b

**Extreme Value Theorem**

*x*
*y*

*O*

*y= cx*^{2}

*constant, or parameter, that appears in the equation. For example,*
the equation y = cx^{2}represents the family of parabolas with vertex
*at the origin and axis along the y-axis. The individual curves of a*
family share common properties that can be deduced from the
equation of the family.

first derivative *See*DERIVATIVE.

*First Derivative Test Let c be a critical number of the function y = f(x) on an*
open interval (a, b); that is, the derivative f′(x) is either 0 or

undefined at x = c. Then, if f′ changes sign from positive to negative
*at c,ƒ has a relative maximum at c; if f*′ changes sign from negative to
*positive at c,ƒ has a relative minimum at c; and if f*′ does not change
*sign at c,ƒ has neither a relative maximum nor a relative minimum at*
*c. Examples are y = –x*^{2}, y = x^{2/3}, and y = x^{3}, respectively.

*See also*CRITICAL NUMBER; MAXIMUM, RELATIVE; MINIMUM,

RELATIVE.

first-order differential equation A differential equation in which the
highest derivative of the unknown function is the first derivative: an
example is xy′ + y^{2}= sin x.

focus *Of an ellipse, See*ELLIPSE.
*Of a hyperbola, See*HYPERBOLA.
*Of a parabola, See*PARABOLA.

Fourier series An infinite series of sine and cosine terms of the form
a0/2 +_{n=1}

### Σ

^{∞}

^{(a}

^{n}

^{cos nx + b}

^{n}

^{sin nx)}

= a0/2 + a1cos x + b1sin x + a2cos 2x + b2sin 2x + . . . Under certain conditions, a function f(x), regarded as a periodic function over the interval [–π, π], can be represented by a Fourier series. The coefficients are found from the following formulas:

**GLOSSARY** **first derivative – Fourier series**

**GLOSSARY** **first derivative – Fourier series**

*x*
**has a relative minimum at**
**x = 0; (c) ƒ has neither a**
**maximum nor a minimum**
**at x = 0**

an= (1/π) –π

### ∫

^{π}f(x) cos nx dx, n = 0, 1, 2, . . . bn= (1/π) –π

### ∫

^{π}f(x) sin nx dx, n = 1, 2, . . . (note that the first formula applies also for n = 0, giving us the coefficient a0).

For example, the function f(x) = x, regarded as a periodic function
over [–π, π], is represented by the series 2[(sin x)/1 – (sin 2x)/2 +
(sin 3x)/3 – + . . .], which has only sine terms. Fourier series are used
in physics to describe vibration and wave phenomena. The series is
named after its discoverer, JEAN-BAPTISTE-JOSEPH FOURIER.
*function(s) Algebraic: See*ALGEBRAIC FUNCTIONS.

Average value of: The average value of a function y = f(x) over an interval [a, b] is defined as 1 . For example, the

b a f x dx

a b

−

### ∫

^{( )}

**GLOSSARY**
**function(s)**

**GLOSSARY**
**function(s)**

0

*-*π π *x*

*y*

2π -2π

*(a)*

0 *x*

*y*

π
*-*π

*(b)* *S*_{1}

*S*_{2}
*S*_{3}
*S*_{4}

**Fourier series: (a) the**
**function f(x) = x,**
**f(x + 2π) = f(x); (b) the**
**first four partial sums of**
**the Fourier series of f(x)**

average value of y = x^{2}over the interval [1, 2] is = 7/3.

Geometrically, the average value of f(x) is the height of the rectangle with base [a, b] whose area is equal to the area under the graph of f(x) from x = a to x = b.

*Composite: See*COMPOSITE FUNCTION.
*Constant: See*CONSTANT FUNCTION.
*Continuous: See*CONTINUITY.
*Cubic: See*CUBIC FUNCTION.

*Decreasing: A function f(x) is decreasing on a interval if, given*
any two numbers x_{1}and x_{2}in the interval, f(x_{1}) > f(x_{2}) whenever

**a function; (b) a function;**

**(c) not a function; (d) not**

Definition of: A function ƒ is a rule of correspondence, or

*“mapping,” between two sets D and R, such that every element of D*
*corresponds to (“is mapped on”) exactly one element of R. The set D*
*is called the domain of ƒ, and the set R the range. We can think of the*
*function as a set of ordered pairs {(x, y)} in which each x is an*
*element of the domain and each y an element of the range, and no*
*two y correspond to the same x. We also require that every element of*
*D must appear as the first element in one pair (in other words, that no*
*element of D is “left out”).*

For example, if D = {1, 2, 3} and R = {3, 7, π}, the set of ordered
pairs {(1, 3), (2, 7), (3, π)} is a function, as is the set {(1, 3), (2, 7),
*(3, 7)} (the same y can come from two different x), but the set {(1, 3),*
(2, 7), (1, π)} is not; neither is the set {(1, 3), (2, 7)}, because not all
*elements of D are being used. The rule of correspondence can be*
entirely arbitrary, as in the example just given, or it can be an
empirical rule (for example, the set of hourly temperature readings at
some location over a 24-hour period). In calculus, however, a function
*is usually given by a formula that tells us how to obtain each y from*
*x. For example, the formula y = 2x + 1 tells us to take any number x,*
double it, and then add 1.

*See also*DOMAIN; FUNCTION NOTATION; ONTO FUNCTION; RANGE.
*Derivative of, See*DERIVATIVE.

*Domain of, See*DOMAIN.

*Elementary, See*ELEMENTARY FUNCTIONS.
*Even, See*EVEN FUNCTION.

*Exponential, See*EXPONENTIAL FUNCTION.

*Extreme value of, See*EXTREME VALUE OF A FUNCTION.
*Graph of: The set of all points (x, y) for which y is a given*
*function of x.*

*Greatest integer, See*GREATEST INTEGER FUNCTION.
*Hyperbolic, See*HYPERBOLIC FUNCTIONS.

Increasing: A function f(x) is increasing on a interval if, given any
two numbers x1and x2in the interval, f(x1) < f(x2) whenever x1< x2.
*A function f(x) is increasing at a point c in its domain if its derivative*
is positive at x = c; that is, if f′(c) > 0.

Integrable: A function is integrable over an interval [a, b] if the
definite integral _{a}

### ∫

^{b}f(x) dx exists (i.e., is a real number). Any function that is continuous on [a, b] is integrable there.

*Inverse, See*INVERSE FUNCTION.
*Limit of, See*LIMIT OF A FUNCTION.
*Linear, See*LINEAR FUNCTION.

*Logarithmic, See*LOGARITHMIC FUNCTION.

**GLOSSARY**

Maximum on an interval: The largest value a function can attain on that interval.

Minimum on an interval: The smallest value a function can attain on that interval.

*Notation: the symbol y = f(x) (read: “y is a function of x”). This*
*is usually followed by a formula that expresses y explicitly as a*
*function of x. For example, the formula y = f(x) = 2x + 1 tells us to*
take a number from the domain (in this case, all real numbers),
double it, and add 1. Thus we have f(3) = 2 · (3) + 1 = 7,
f(0) = 2 · (0) + 1 = 1, f(a) = 2a + 1, f(2a) = 2(2a) + 1 = 4a + 1,
f(x + 1) = 2(x + 1) + 1 = 2x + 3, and so on.

Another way of looking at the functional notation is f( ) = 2( ) + 1, where the blank space is to be filled with the desired value or

*expression, taken from the domain of ƒ. (Note: the parentheses in f(x)*
*do not indicate multiplication.) We can think of x as an “input” to the*
*function, and of y as the corresponding “output.” The function itself is*
*denoted by ƒ (of course, any other letter would do, such as g or h).*

*Odd, See*ODD FUNCTION.

*One-to-one, See*ONE-TO-ONE FUNCTION.
*Onto, See*ONTO FUNCTION.

*Periodic, See*PERIODIC FUNCTION.
*Polynomial, See*POLYNOMIAL FUNCTION.
*Quadratic, See*QUADRATIC FUNCTION.
*Range of, See*RANGE.

*Rational, See*RATIONAL FUNCTION.

*Transcendental, See*TRANSCENDENTAL FUNCTION.
*Trigonometric, See*TRIGONOMETRIC FUNCTIONS.
*Zero of, See*ZERO OF A FUNCTION.

Fundamental Theorem of Calculus First form: Let F(x) be an antiderivative
of f(x), that is, F′(x) = f(x). Then _{a}

### ∫

^{b}f(x) dx = F(b) – F(a)

The last expression is also written as F(x)

### |

a b. Second form:d/dx _{a}

### ∫

^{x}f(t) dt = f(x)

*Here we regard the integral as a function of its upper limit x, so we*
*use t for the variable of integration to distinguish it from x.*

*See also*ANTIDERIVATIVE; AREA FUNCTION.

future value The amount of money, or balance, in a bank account at the end of
a specified time period since the money was deposited. If the amount
*deposited (the principal) is P, the annual interest rate is r (expressed as*
**GLOSSARY** **Fundamental Theorem of Calculus – future value**

**GLOSSARY** **Fundamental Theorem of Calculus – future value**

*a decimal), and the money is compounded n times a year for t years,*
*then the future value A is given by the formula A = P(1 + r/n)*^{nt}.

For example if P = $100, r = 5%, and n = 12 (that is, the money is
compounded monthly), then the future value after t = 5 years is A =
100(1 + 0.05/12)^{12·5}*= $128.34. If the bank is using continuous*
compounding, the formula is A = Pe^{rt}, where e ~ 2.78 is the base of
natural logarithms. For the data given above, the future value for
continuous compounding will be A = 100e^{0.05·5}= $128.40.

*See also*PRESENT VALUE.

generalized harmonic series *See p-*SERIES.

generalized power rule *See*POWER RULE, GENERALIZED.

general solution of a differential equation A solution that contains arbitrary
constants. For example, the general solution of the equation y″ + y = 0
*is y = A cos x + B sin x, where A and B are arbitrary constants.*

*See also*PARTICULAR SOLUTION OF A DIFFERENTIAL EQUATION.
*geometric mean Of n positive numbers a*1, a2, . . ., anis the expression

. For example, the geometric mean of the numbers
1, 2, 3, and 5 is = ^{4}√—

30 2.34.

geometric progression A progression, or sequence, of numbers in which
each number is obtained from its predecessor by multiplication by a
*constant number, called the quotient of the progression. Examples are*
1, 2, 4, 8, 16, . . ., 2^{n – 1}, . . . (here the initial term is 1 and the quotient
is 2), 1, 1/2, 1/4, 1/8, 1/16, . . ., 1/2^{n – 1}, . . . (initial term 1, quotient
1/2), and 1, –1, 1, –1, . . ., (–1)^{n – 1}, . . . (initial term 1, quotient –1).

*Generally, if the initial term is a and the quotient q, we can write the*
progression as a, aq, aq^{2}, . . ., aq^{n – 1}, . . ., where aq^{n – 1}*is the nth*
term (note that the first term can be written as aq^{0}, which explains
*the n – 1 in the nth term). A geometric progression can be finite or*
infinite. If finite, it ends after the term aq^{n – 1}, in which case we drop
the three dots behind it.

geometric series The sum of the terms of a geometric progression. For
example, 1 + 2 + 4 + 8 + . . . + 2^{n – 1}. A geometric series can be finite
or infinite. If finite, we can write it as a + aq + aq^{2}+ . . . + aq^{n – 1},
*where a is the initial term, q the quotient, and n the number of terms*
(note that the first term can be written as aq^{0}, which explains the
*n – 1 in the nth term). In this case the sum of the series can be found*
from the formula S = a(1 – q^{n})/(1 – q). For the example given above,
the sum is S = 1 · (1 – 2^{n})/(1 – 2) = 2^{n}– 1. If the series is infinite and
if |^{q}|*< 1 (that is, q is between –1 and 1), then the series converges to*

4√(1 · 2 · 3 · 5)

n√^{(a}^{1}^{a}^{2}^{. . . a}^{n}^{)}

**GLOSSARY**