every intermediate height from the base to the apex. A
rigorous proof of the theorem, however, relies on the
notion of the “completeness” of the real numbers
(meaning that no points, like the square root of 11, are
missing from the real line). This is a subtle property, one
that was not properly understood until the late 1800s
with the construction of a D
EDEKIND CUT
.
The intermediate-value theorem is useful for locat-
ing roots of equations. For example, consider the func-
tion f(x) = x3+2 x– 5. It is continuous on the interval
[1,2] and satisfies f(1) < 0 and f(2) > 0. It follows then
that the equation x3+ 2x– 5 = 0 has a root some-
where between 1 and 2. By working with smaller and
smaller intervals, one can often use this method to
determine the location of a root with a good degree of
precision. The
BISECTION METHOD
, for example,
employs this technique.
The intermediate-value theorem has a number of
amusing consequences:
In theory, it is always possible to slice a pan-
cake, no matter how irregular its shape,
exactly in half with a single straight-line cut.
Hold a knife to the left of the pancake so that 100 per-
cent of the cake lies to its right. Now slide the knife, in
parallel, across the pancake until it lies on the other
side of the cake. At this location, zero percent of the
pancake lies to the right of the knife. By the intermedi-
ate-value theorem, there must be some intermediate
location for the knife that yields the value of 50 percent
lying to its right. That is, there is, in theory, a knife
position that cuts the pancake exactly in half.
Note that this result does not depend on the angle
we initially hold the knife—vertically, horizontally, or
diagonally. We have in fact shown that it is always pos-
sible to slice a pancake in half with a knife held at any
previously set angle.
In theory, it is always possible to simultaneously
slice two pancakes each exactly in half with a
single straight-line cut, no matter the shapes of
the pancakes nor their location on the table.
This result is known as the two-pancake theorem.
The previous result assures us we are always able to
slice the first pancake exactly in half, pointing the knife
at any angle we care to choose. The concern is that the
knife might or might not cut the second pancake. Sup-
pose we find a direction, deem this angle zero degrees,
that slices the first pancake in half, but misses the sec-
ond pancake entirely, with 100 percent of the second
pancake lying to the right of the knife, say. Turn the
knife 180°. We now have a knife cut (same line, oppo-
site direction) that slices the first pancake exactly in
half, with the second pancake lying entirely to its left,
that is, zero percent to the right. By the intermediate-
value theorem, there must be an intermediate angle
between zero degrees and 180°that slices the first pan-
cake exactly in half, and has 50 percent of the second
pancake lying to its right—that is, one that simultane-
ously slices the second pancake in half as well.
At any instant, there are two points on the
Earth’s equator directly opposite each other
with exactly the same air temperature.
For each position θdegrees longitude on the equa-
tor, let f(θ) be the air temperature at this position
minus the air temperature at the opposite point of the
equator, at θ+ 180°:
f(θ) = temp(θ) – temp(θ+ 180°)
Notice that f(θ+ 180°) equals the same value, but
opposite in sign, to f(θ):
f(θ+ 180°) = temp(θ+ 180°) – temp(θ) = –f(θ)
Thus the function f(θ) moves from positive to negative
values. By the intermediate-value theorem, there must
be an intermediate location where the function is zero.
This is the desired position on the Earth’s equator.
See also
EXTREME
-
VALUE THEOREM
;
FIXED POINT
;
HAM
-
SANDWICH THEOREM
;
SPHERE
.
interpolation The process of estimating the value of
a function between two values already known is called
interpolation. For example, if the temperature of a cup
of tea was initially 200°F, and two minutes later it was
180°F, one might guess that its temperature at the one-
minute mark was 190°F, based in the assumption that
the temperature decreases steadily over time. This rep-
resents that simplest method of interpolation, called
linear interpolation: we presuppose that the variation
of the function can be described as a straight line pass-
ing through the known values.
interpolation 277