Iustrating the Fundamental Theorem of Calculus 87
Then, if we make this
division infinitely fine,
we wi get the exact
amount of alcohol,
won’t we?
We, that’s
true, but it’s
not realistic.
You’d have to a up
an infinite number of
infinitely fine portions.
Lk at this
expreion. Does
it remind you of
something?
Ah!
It lks like an
imitating linear
function!
I...
I s.
p x x x
3 4 3
( )
×
( )
88 Chapter 3 Let’s Integrate a Function!
Step 4—Review of the Imitating Linear Function
When the derivative of f(x) is given by f
′ 
(x), we had f(x) f
′ 
(a) (xa) + f(a)
near x = a.
Transposing f(a), we get
u
f x f a f a x a
( )
( )
( )
( )
or (Difference in f ) (Derivative of f ) × (Difference in x)
If we assume that the interval between two consecutive values of x
0
, x
1
,
x
2
, x
3
, ..., x
6
is small enough, x
1
is close to x
0
, x
2
is close to x
1
, and so on.
Now, let’s introduce a new function, q(x), whose derivative is p(x). This
means q
′ 
(x) = p(x).
Using u for this q(x), we get
(Difference in q) (Derivative of q) × (Difference in x)
q x q x p x x x
1 0 0 1 0
( )
( )
( )
( )
q x q x p x x x
2 1 1 2 1
( )
( )
( )
( )
The sum of the right sides of these expressions is the same as the sum
of the left sides.
Some terms in the expressions for the sum cancel each other out.
q x q x p x x x
q x q x p x x x
q x q x
1 0 0 1 0
2 1 1 2 1
3 2
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ))
( )
( )
( )
( )
( )
( )
( )
( )
( )
p x x x
q x q x p x x x
q x q x p x x
2 3 2
4 3 3 4 3
5 4 4 5
( )
+
( )
( )
( )
( )
x
q x q x p x x x
4
6 5 5 6 5
q x q x
6 0
( )
( )
The sum
Substituting x
6
= 9 and x
0
= 0, we get
The approximate amount of alcohol = the sum × 20
q x q x
6 0
20
( )
( )
{ }
×
q q9 0 20
( )
( )
{ }
×
So we nd to find
function q(x) that
satisfies
q
′ 
(x) = p(x).
Iustrating the Fundamental Theorem of Calculus 89
We have just
obtained the
foowing
relationship of
expreions
shown in the
diagram.
But if we increase
the number of
points
x
0
, x
1
, x
2
, x
3
,
and so on, until it
becomes infinite,
we can say that
relationship u
changes from
aroximation
to“equality.”
But, since the sum
of the expreions
have bn imitating
the constant value
q(9) − q(0) ,
we get the
relationship
shownhere.
*
Step 5—Approximation
Exact Value
* We wi obtain this relationship
more rigorously on page 94.
The approximate amount of alcohol
(÷ 20) given by the stepwise function:
(Constant)
The exact amount
of alcohol (÷ 20)
The exact amount
of alcohol (÷ 20)
=
=
=
The sum of
for an infinite number of x
i
p x x x
i i i
( )
( )
+1
q q9 0
( )
( )
p x x x p x x x
0 1 0 1 2 1
( )
( )
+
( )
( )
+ ...
q q9 0
( )
( )

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