184 extrapolation



A represents 10, B represents 11, and so forth, with F

representing 15. Each decimal place is a power of a

sixteenth.) Unfortunately no analogous technique is

currently known to compute the ordinary base-10 dig-

its of πwith ease.

extrapolation The process of estimating the value

of a function outside a known range of values is called

extrapolation. For example, if the temperature of a

cup of tea was initially 200°F and then was measured

to be 100°F and 50°F 10 and 20 minutes later, respec-

tively, then one might guess that its temperature after

30 minutes would be 25°F. Methods of extrapolation

are normally far less reliable than

INTERPOLATION

, the

process of estimating function values between known

values. Scientists generally prefer to avoid making pre-

dictions based on extrapolation. Meteorologists, how-

ever, must use extrapolation techniques to make

weather predictions. Long-range forecasts are gener-

ally considered unreliable.

See also

POPULATION MODELS

.

extreme-value theorem This theorem asserts that a

CONTINUOUS FUNCTION

f(x),defined on a closed

INTERVAL

[a,b], reaches a maximum value and a mini-

mum value somewhere within that interval. That is,

there is a point cin the interval [a,b] such that f(x),

≤f(c),for all xin [a,b], and there is another point din

[a,b] such that f(x) ≥f(d) for all xin [a,b]. For exam-

ple, the extreme-value theorem tells us that, on the

interval [1,5] say, the function f(x) = 3x· cos(x2+ sinx)

does indeed adopt a largest value. It does not tell us,

however, where that maximum value occurs. A point at

which a function has a maximum or minimum value is

called an extremum.

The theorem is intuitively clear if we think of a

continuous function on a closed interval as one whose

graph consists of a single continuous piece with no

gaps, jumps, or holes: in moving a pencil from the left

end point (a,f(a)) to the right end point (b,f(b)),one

would not doubt that there must be a high point on

the curve where f(x) reaches its maximum value, and

a low point where it attains its minimum value. A rig-

orous proof of the theorem, however, relies on the

notion of the completeness of the real numbers (mean-

ing that no points are missing from the real line). This

is a subtle property, one that was not properly under-

stood until the late 1800s with the invention of a

D

EDEKIND CUT

. For example, the function f(x) = 2x–

x2= x(2 – x) has no maximum value on the interval

[0, 2] if the value 1 is excluded from our considera-

tions. Although this seems an artificial example, one

still needs to be sure that this type of problem can

never occur.

It is vital that the function under consideration be

continuous and that the interval under study be closed

for the theorem to be true. For example, the (discontin-

uous) function

does not reach a maximum value in the closed interval

[0,2]; nor does the (continuous) function f(x) = ,

defined on the open interval (0,2), since the function is

arbitrarily large for values xclose to zero.

The

INTERMEDIATE

-

VALUE THEOREM

is a compan-

ion result to the extreme-value theorem. It asserts that

not only does a continuous function on a closed inter-

val actually attain maximum and minimum values, but

it also takes on every value between those two extreme

values. R

OLLE

’

S THEOREM

and the

MEAN

-

VALUE THEO

-

REM

follow from the extreme-value theorem if we fur-

ther assume that the function under consideration is

differentiable.

See also

DIFFERENTIAL CALCULUS

;

MAXIMUM

/

MINIMUM

.

1

–

x