334 mean value

EXPECTED VALUE

of a random variable is also called

its mean.

See also

MEAN VALUE

;

STATISTICS

:

DESCRIPTIVE

.

mean value Let fbe a

CONTINUOUS FUNCTION

on a

closed interval [a,b]. The height of a rectangle whose

width is (b– a) and whose area is equal to the area

under the curve above the interval [a,b] is called the

mean value of the function. The mean value of fis

denoted –

fand is given by:

Loosely speaking, if one “smoothes out” the rises and

falls of the graph of the function, without changing the

area under the graph, then the height of the resulting

level curve is –

f.

If the function frepresents, for example, the air tem-

perature at the general post office in Adelaide, Australia,

recorded over a 24-hour period, then –

frepresents the

average temperature at downtown Adelaide that day.

mean-value theorem (Lagrange’s mean-value theo-

rem) French mathematician J

OSEPH

-L

OUIS

L

AGRANGE

(1736–1813) was the first to state the following

important theorem in

CALCULUS

, today called the

mean-value theorem:

If a curve is continuous over a closed interval

[a,b], and has a tangent at every point between

aand b, then there is at least one point in this

interval at which the tangent is parallel to the

line segment that connects the endpoints (a,f (a))

and (b,f (b)).

In more stringent mathematical language, this theo-

rem reads:

If a function f(x) is continuous in the closed

interval [a,b], and differentiable in the open

interval (a,b),then there exists at least one

value cbetween aand bsuch that

Note that the quantity is the

SLOPE

(rise

over run) of the line segment connecting the two end-

points. It also equals the average slope of the curve

over the entire interval [a,b]. (To see this, note that at

any point x, the quantity f′(x) is the slope of the tan-

gent line at that point. Summing, that is integrating,

over all values and dividing by the length of the inter-

val under consideration gives the average or mean

slope of the curve: .) Thus the

mean-value theorem states that for any differentiable

function defined on an interval [a,b], there exists a

location where the actual slope of the curve equals the

average slope of the graph.

The mean-value theorem has four important conse-

quences:

1. If the derivative of a function is always positive,

then the function is increasing.

This means that if aand bare two numbers with a< b,

then we have f(a) < f(b).Since, for some number c, we

have and the quantities b– aand

f′(c) are both positive, we must have that f(b) – f(a) is

also positive.

2. If the derivative of a function is always negative,

then the function is decreasing.

This is established in a manner similar to the above.

3. If the derivative of a function is always zero, then

the function is constant in value.

We need to show that for any two values aand bwe

have that f(a) equals f(b).This follows from the mean-

value theorem, since for some value cwe have:

f(b) – f(a) = f′(c)·(b– a) = 0·(b– a) = 0

4. If two functions f(x) and g(x) have the same deriva-

tive, then the two functions differ by a constant,

that is, f(x) = g(x)+Cfor some number C.

Let h(x) = f(x) – g(x).Then the derivative of h(x) is

always zero, and so by the third result h(x) = Cfor

some constant value C.