EXPECTED VALUE of a random variable is also called
its mean.
See also MEAN VALUE; STATISTICS: DESCRIPTIVE.
mean value Let f be 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 f is
denoted
–
f and 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 f represents, for example, the air temperature at the general post office in Adelaide, Australia,
recorded over a 24-hour period, then
–
f represents the
average temperature at downtown Adelaide that day.
mean-value theorem (Lagrange’s mean-value theorem) French mathematician JOSEPH-LOUIS LAGRANGE
(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
a and 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 theorem 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 c between a and b such that
Note that the quantity
is the SLOPE (rise
over run) of the line segment connecting the two endpoints. 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 tangent line at that point. Summing, that is integrating,
over all values and dividing by the length of the interval 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 consequences:
1. If the derivative of a function is always positive,
then the function is increasing.
This means that if a and b are two numbers with a < b,
then we have f(a) < f(b). Since, for some number c, we
have
and the quantities b – a and
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 a and b we
have that f(a) equals f(b). This follows from the meanvalue theorem, since for some value c we 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 derivative, then the two functions differ by a constant,
that is, f(x) = g(x)+C for 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) = C for
some constant value C.
′ =
−
−
f c
f b f a
b a
( )
( ) ( )
′
−
=
− ( )
−
∫ f x dx
b a
f b f a
b a
a
b ( )
( )
f b f a
b a
( ) ( )
−
−
′ =
−
−
f c
f b f a
b a
( )
( ) ( )
f b a
f x dx
a
b
= − ∫
1
( )
334 mean value
its mean.
See also MEAN VALUE; STATISTICS: DESCRIPTIVE.
mean value Let f be 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 f is
denoted
–
f and 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 f represents, for example, the air temperature at the general post office in Adelaide, Australia,
recorded over a 24-hour period, then
–
f represents the
average temperature at downtown Adelaide that day.
mean-value theorem (Lagrange’s mean-value theorem) French mathematician JOSEPH-LOUIS LAGRANGE
(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
a and 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 theorem 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 c between a and b such that
Note that the quantity
is the SLOPE (rise
over run) of the line segment connecting the two endpoints. 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 tangent line at that point. Summing, that is integrating,
over all values and dividing by the length of the interval 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 consequences:
1. If the derivative of a function is always positive,
then the function is increasing.
This means that if a and b are two numbers with a < b,
then we have f(a) < f(b). Since, for some number c, we
have
and the quantities b – a and
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 a and b we
have that f(a) equals f(b). This follows from the meanvalue theorem, since for some value c we 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 derivative, then the two functions differ by a constant,
that is, f(x) = g(x)+C for 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) = C for
some constant value C.
′ =
−
−
f c
f b f a
b a
( )
( ) ( )
′
−
=
− ( )
−
∫ f x dx
b a
f b f a
b a
a
b ( )
( )
f b f a
b a
( ) ( )
−
−
′ =
−
−
f c
f b f a
b a
( )
( ) ( )
f b a
f x dx
a
b
= − ∫
1
( )
334 mean value