第6.5节 平均值
2013-07-18 15:34阅读：
6.5
AVERAGES
Given n
numbers
y1…,
yn, their average value is defined
as
_____________________
If all
the
y1are replaced by the
average value yave, the sum will
be unchanged,
y1+ …
+ yn =
yave + …
+ yave =
n yave
.
If f is
continuous function on a closed interval [a,
b] , what is meant by the average value
of f between
a and b (Figure
6.5.1)? Let us try to imitate the procedure for finding the average
of n numbers. Take an infinite
hyperreal number Hand divide the
interval [a ,b] into infinitesimal
subintervals of length dx=(b-a)/
H. Let
Figure
6.5.1
Us
“sample”
the value of f at
the H points a,
a+dx, a+2dx,…，a +
(H-1)dx.
Then the
average value of fshould be
infinitely close to the sum of the values of
fat a, a+ dx, ……, a+
(H-1)dx, divided by
H. Thus
Since dx
= _________ , ______________
and we have
fave≈__________________.
________________.
Taking standard parts, we
are led to
DEFINITION
Let f be continuous on
[a, b]. The average value of f between a and b
is
fave = ______________
Geometrically, the area under the
curve y=f(x)is equal to the area
under the constant curve
y=favebetween a
an b,
fave·(b-a)
= _____ f(x)dx.
EXAMPLE
1 Find the
average value of
y=___from
x=1 to x=4
( Figure 6.5.2).
Figure
6.5.2
Recall that
in Section 3.8, we defined the average slope of a
function Fbetween
a and b as the
quotient
average
slope = ________________
Using the
Fundamental Theorem of Calculus we can find the connection between
the average value of F'and the
average slope of F.
THEOREM
1
Let F be an
antiderivative of a continuous function f on an open interval
I.
Then for any a < b
in I , the average slope of F between a and b is equal to the
average value of f between a and b,
___________________________.
PROOF
By the Fundamental Theorem,
F(b) - F(a) =
_____ f (x)
dx.
THEOREM 2
(Mean Value Theorem for Integrals)
Let f be continuous on [ a, b].
Then there is a point c strictly between a and b where the value of
f is equal to its average value,
________________________
PROOF
Theorem 2 is illustrated in Figure 6.5.3. We can make f continuous
on
the whole real
line by defining f(x) = f(a)
for x and
f(x)=f(b) for x >
b.
By the Second
Fundamental Theorem of Calculus ,
fhas an antiderivative
F.
By the Mean
Value Theorem there is a point
cstrictly between a
and b
at
which F'(c)
is equal to the
avera
