182 exponential series

Exterior angles

N

dy

––

dx

dy

––

dx

1

–

y

illustrate the essential differences between the cases

b> 1 and 0 < b< 1.

Many types of growth and decay occur at a rate

that involves exponential variation. For example, a

colony of bacteria might reproduce at a rate that dou-

bles the size of the colony every 24 hours. If initially

500 organisms are present, then after 1 day the culture

grows to a size of 1,000 organisms, after 2 days to a

size of 2,000 organisms, and by the end of day three

there are 4,000 organisms. In general, the population

size by the end of day Nis given by 500 ×2N. Any for-

mula of the form ABxwith b> 1 and Aconstant is said

to represent exponential growth. This formula is a sim-

ple example of a

POPULATION MODEL

.

The analogous formula with 0 < b< 1 represents

exponential decay. The decay of radioactive material is

a typical example of this. For example, if 50 percent of

the radioactivity produced by a nuclear explosion dis-

appears after 5 days, then after 10 days only 25 percent

of radioactivity remains, and 12.5 percent remains

after 15 days. The percentage of radioactivity present

after Ndays is given by the formula . The level of

radioactivity decreases but will never reach zero.

The

DERIVATIVE

of an exponential function f(x) = bx

can be computed via

IMPLICIT DIFFERENTIATION

after

first taking a

LOGARITHM

. Precisely, if y= bx, then

lny=x·lnb. Differentiation yields · = lnb, and so

=y· lnb=bx· lnb. That is:

This formula for the derivative is greatly simplified

if one works with base b= e, where eis defined to be

the number such that ln e= 1. We have:

The derivative of exis ex. Consequently, the

graph of the exponential function f(x)= exhas

the property that the slope of the graph at any

point is the same as the value of the function at

that point.

The function f(x) = exis sometimes called the expo-

nential function. Because the derivative of this function

is particularly simple, it is not surprising that the num-

ber e is ubiquitous throughout studies in

CALCULUS

.

exponential series The T

AYLOR SERIES

of the func-

tion f(x) = ex, given by

is called the exponential series. The

RATIO TEST

from the

study of

CONVERGENT SERIES

shows that this series con-

verges for all values of x. L

EONHARD

E

ULER

(1707–83)

made use of this series to prove his famous formula eix =

cosx+ isinx, today called E

ULER

’

S FORMULA

.

expression Any meaningful combination of symbols

that represent numbers, operations on numbers, or

other mathematical entities is called an expression. For

example, 2 + xand ab+care expressions. One could

argue that x+ y= 2 is an expression, although mathe-

maticians may prefer to call it an equation. Similarly,

3

√

–

8 could be called an expression even though it is

equivalent to a single number. In

FORMAL LOGIC

, com-

pound statements are sometimes called expressions.

For example,

¬

(p(q∨r)) is an expression.

The word express is often used in the sense “to

transform.” For example, the product (x– y)(x+ y) can

be expressed equivalently as the

DIFFERENCE OF TWO

SQUARES

: x2– y2.

exterior angle An

ANGLE

contained between one side

of a

POLYGON

and the extension of an adjacent side is

called an exterior angle of the polygon. Since two edges

of a polygon meet at a vertex, there are two exterior

angles at each vertex. One can easily see, however, that

these two angles are equal in value.

The sum of exterior angles in a convex polygon is

360°, since these angles correspond to one full turn.

This result is also true for concave polygons if one

deems angles that turn to the left as positive and ones

that turn to the right as negative.

The

EXTERIOR

-

ANGLE THEOREM

, first proved by

the geometer E

UCLID

(ca. 300–260

B

.

C

.

E

.), states that

∨