General Mathematics Vol. 12, No. 1 (2004), 61–70
An intermediate point property in some of
the classical generalized formulas of
quadrature
Petrică Dicu
Abstract
In this paper we study a property of the intermediate point (see
[6]) from the classical generalized quadrature formulas of the rectangle, trapeze, Simpson and Newton.
2000 Mathematical Subject Classification: 65D32
In [6] B. Jacobson studied a property of the intermediate point which
appear in the mean-value theorem for integrals. This property has been
studied in the articles [2], [3], and [7]. In this paper we will study this
property for the classical generalized (composed) quadrature formulas of
the rectangle, trapeze, Simpson and Newton.
61
62
Petrică Dicu
1. Let us consider the generalized quadrature formula of the rectangle.
If f : [a, b] → R, f ∈ C 2 [a, b], then for any x ∈ (a, b] there is cx ∈ (a, x) such
that:
(1)
Zx
a
¶
µ
n
X
1
(x − a)3 ′′
2k − 1
f (t)dt = (x − a)
(x − a) +
f a+
f (cx ),
n
2n
24n2
k=1
(see [1], [4], [5]).
In the above conditions we have the following theorem:
Theorem 1. If f ∈ C 4 [a, b] and f ′′′ (a) 6= 0, then for the intermediate point
cx from (1), we have:
cx − a
1
= .
x→a x − a
2
lim
Proof. Let F, G : [a, b] → R defined as follows
F (x) =
Zx
a
n
(x − a) X
f
f (t)dt −
n
k=1
µ
¶
2k + 1
(x − a)3 ′′
f (a),
a+
(x − a) −
2n
24n2
G(x) = (x − a)4 .
Since F and G are four times derivable on [a, b], and G(i) (x) 6= 0, i = 1, 4
for any x ∈ (a, b), we have
F (a) = F ′ (a) = F ′′ (a) = F ′′′ (a) = 0,
G(a) = G′ (a) = G′′ (a) = G′′′ (a) = 0.
By using successive l, Hospital rule, we obtain:
F (x)
F (IV ) (x)
= lim (IV ) .
x→a G(x)
x→a G
(x)
lim
An intermediate point property...
63
Since
F
µ
¶
n
1 X
2k − 1
3 ′′′
(x) = f (x) − 4
(2k − 1) f
a+
(x − a) −
2n
2n k=1
¶
µ
n
2k − 1
(x − a) X
4 (IV )
(x − a) and
a+
(2k − 1) f
−
2n
16n5 k=1
(IV )
′′′
G(IV ) (x) = 4!, we obtain
F (x)
1 ′′′
=
f (a).
x→a G(x)
48n2
lim
(2)
On the other hand
(x − a)3 ′′
[f (cx ) − f ′′ (a)]
2
1
F (x)
f ′′ (cx ) − f ′′ (a) cx − a
24n
=
·
,
=
·
G(x)
(x − a)4
cx − a
x−a
24n2
whence
(3)
cx − a
F (x)
1 ′′′
f (a) · lim
=
.
2
x→a x − a
x→a G(x)
24n
From relations (2) and (3) we obtain
lim
1
cx − a
= ,
x→a x − a
2
lim
which is the conclusion of Theorem 1.
2.
Now we will consider the generalized quadrature formula of the
trapeze. If f : [a, b] → R, f ∈ C 2 [a, b], then for any x ∈ [a, b] there is
cx ∈ (a, x) such that:
(4)
Zx
a
"
#
¶
n−1 µ
(x − a) f (a) X
k(x − a)
f (x)
f (t)dt =
f a+
+
+
−
n
2
n
2
k=1
−
(x − a)3 ′′
f (cx ),
12n2
(see [1], [4], [5]).
We have the following result:
64
Petrică Dicu
Theorem 2. If f ∈ C 4 [a, b] and f ′′′ (a) 6= 0, then for the intermediate point
cx from the formula (4), we have
cx − a
1
= .
x→a x − a
2
lim
Proof. We consider the functions F, G : [a, b] → R defined as follows:
"
#
¶
Zx
n−1 µ
(x − a) f (a) X
f (x)
k(x − a)
F (x) = f (t)dt −
+
+
+
f a+
n
2
n
2
k=1
a
+
(x − a)3 ′′
f (a),
12n2
G(x) = (x − a)4 .
Since F and G are four times derivable on [a, b] and G(i) (x) 6= 0, i = 1, 4
for any x ∈ (a, b), we have
F (a) = F ′ (a) = F ′′ (a) = F ′′′ (a) = 0,
G(a) = G′ (a) = G′′ (a) = G′′′ (a) = 0.
By using successive l, Hospital rule, we will have:
F (x)
F (IV ) (x)
= lim (IV ) .
x→a G(x)
x→a G
(x)
lim
Since
F
(IV )
¶
µ
n−1
4 X 3
n − 2 ′′′
k(x − a)
−
f (x) − 4
(x) =
k f a+
n
n
n k=1
¶
µ
n−1
(x − a) X 4 (IV )
(x − a) (IV )
k(x − a)
−
−
f
(x) and
a+
k f
5
n
2n
n
k=1
G(IV ) (x) = 4! we obtain:
(5)
1 ′′′
F (x)
=−
f (a).
x→a G(x)
24n2
lim
An intermediate point property...
65
On the other hand, we have:
(x − a) ′′
−
[f (cx ) − f ′′ (a)]
2
F (x)
1
f ′′ (cx ) − f ′′ (a) cx − a
12n
=
·
=
−
·
G(x)
cx − a
x−a
(x − a)4
12n2
where we find:
1 ′′′
F (x)
cx − a
=−
.
f (a) · lim
2
x→a G(x)
x→a x − a
12n
(6)
lim
From the relations (5) and (6), it follows
cx − a
1
= ,
x→a x − a
2
lim
which is exactly the assertion Theorem 2.
3.
We will continue with the generalized quadrature formula of Simp-
son. If f : [a, b] → R, f ∈ C 4 [a, b] then for any x ∈ (a, b] there is cx ∈ (a, x)
such that:
(7)
Zx
a
"
¶
n−1 µ
X
k(x − a)
(x − a)
f a+
f (a) + 2
+
f (t)dt =
6n
n
k=1
#
¶
µ
n
X
1 (x − a)5 (IV )
2k − 1
(x − a) + f (x) −
f
(cx ), (see [1], [5]).
f a+
+4
2n
4 · 6! n4
k=1
For this formula it follows:
Theorem 3. If f ∈ C 6 [a, b] and f (V ) (a) 6= 0, then for the intermediate
point cx form the formula (7), we have:
1
cx − a
=
x→a x − a
2
lim
66
Petrică Dicu
Proof. Let F, G : [a, b] → R defined as follows:
F (x) =
Zx
a
"
¶
n−1 µ
X
x−a
k(x − a)
f (t)dt −
f (a) + 2
+
f a+
6n
n
k=1
#
µ
¶
n
X
2k − 1
1 (x − a)5 (IV )
f a+
+4
f
(a),
(x − a) + f (x) +
2n
4 · 6! n4
k=1
G(x) = (x − a)6 .
We have that F and G are six times derivable on [a, b], G(i) (x) 6= 0, i = 1, 5,
for any x ∈ (a, b) and F (k) (a) = 0, G(k) (a) = 0, k = 1, 5.
By using successive l, Hospital rule, we obtain
F (V I) (x)
F (x)
= lim (V I) .
lim
x→a G
x→a G(x)
(x)
Now, from
F (V I) (x) =
(x − a) (V I)
n − 1 (V )
f (x) −
f
(x)−
n
6n
¶
µ
n−1
2 X 5 (V )
k(x − a)
−
− 6
K f
a+
n
n k=1
¶
µ
n
(x − a) X 6 (V I)
k(x − a)
−
−
a+
K f
n
3n7 k=1
µ
¶
n
1 X
2k − 1
5 (V )
− 6
(2k − 1) f
a+
(x − a) −
2n
8n k=1
¶
µ
n
1 X
2k − 1
6 (V I)
(x − a) ,
− 6
(2k − 1) f
a+
2n
8n k=1
G(V I) (x) = 6!,
An intermediate point property...
67
we obtain
(8)
F (x)
1
=− 4
f (V ) (a).
x→a G(x)
8n · 6!
On the other hand, we have
lim
1
f (IV ) (cx ) − f (IV ) (a) cx − a
F (x)
=−
·
,
·
G(x)
cx − a
x−a
4 · 6!n4
whence we obtain:
(9)
F (x)
cx − a
1
f (V ) (a) · lim
=−
.
4
x→a G(x)
x→a x − a
4 · 6!n
From the relations (8) and (9) we find
lim
lim
x→a
4.
1
cx − a
= .
x−a
2
In the final part of this paper we will consider the generalized quadra-
ture formula of Newton ([1]).
If f : [a, b] → R, f ∈ C 4 [a, b] then for any x ∈ (a, b] there is cx ∈ (a, x)
such that:
Zx
(10)
a
+3
n
X
k=1
f
µ
"
¶
n−1 µ
X
(x − a)
k(x − a)
f (t)dt =
f a+
f (a) + 2
+
8n
n
k=1
#
µ
¶
¶
n
X
3k − 1
3k − 2
f a+
a+
(x − a) + 3
(x − a) + f (x) −
3n
3n
k=1
1 (x − a)5 (IV )
f
(cx ).
9 · 6! n4
The result obtained consist in following:
−
Theorem 4. If f ∈ C 6 [a, b] and f (V ) (a) 6= 0 then for the intermediate point
cx from (10) we have:
cx − a
1
= .
x→a x − a
2
lim
68
Petrică Dicu
Proof. Let F, G : [a, b] → R defined as follows:
F (x) =
Zx
a
"
¶
n−1 µ
X
(x − a)
k(x − a)
f (t)dt −
f (a) + 2
+
f a+
8n
n
k=1
#
¶
¶
µ
µ
n
n
X
X
3k − 1
3k − 2
(x − a) + 3
(x − a) + f (x) +
+3
f a+
f a+
3n
3n
k=1
k=1
+
1 (x − a)5 (IV )
f
(a),
9 · 6! n4
G(x) = (x − a)6 .
Since F and G are seven times derivable on [a, b], G(i) (x) 6= 0, i = 1, 6
for any x ∈ (a, b) and F (k) (a) = 0, G(k) (a) = 0, k = 0, 5, by using successive
l, Hospital rule, we obtain:
F (x)
F V I (x)
= lim (V I) .
x→a G(x)
x→a G
(x)
lim
We have that:
F (V I) (x) =
4n − 3 (V )
(x − a) (V I)
f (x) −
f
(x)−
4n
8n
¶
¶
µ
µ
n−1
n−1
3 X 5 (V )
(x − a) X 6 (V I)
k(x − a)
k(x − a)
− 6
−
−
k f
k f
a+
a+
n
n
2n k=1
4n7 k=1
¶
µ
n
1 X
3k − 2
5 V
(x − a) −
−
(3k − 2) f
a+
3n
108n6 k=1
¶
µ
n
(x − a) X
3k − 2
6 (V I)
−
(3k − 2) f
(x − a) −
a+
3 · 648 k=1
3n
µ
¶
n
3k − 1
1 X
5 (V )
(3k − 1) f
a+
−
(x − a) −
3n
108n6 k=1
An intermediate point property...
69
¶
µ
n
(x − a) X
3k − 1
6 (V I)
−
(3k − 1) f
(x − a) ,
a+
3 · 648 k=1
3n
G(V I) (x) = 6!.
Therefore:
1
F (x)
=−
f (V ) (a).
4
x→a G(x)
18 · 6!n
lim
(11)
On the other hand, we have
1 f (IV ) (cx ) − f (IV ) (a) cx − a
F (x)
=−
·
,
G(x)
cx − a
x−a
9 · 6!n4
whence:
(12)
F (x)
1
cx − a
=−
.
f (V ) (a) · lim
4
x→a G(x)
x→a x − a
9 · 6!n
lim
From the relations (11) and (12) we obtain
cx − a
1
= .
x→a x − a
2
lim
References
[1] D. Acu, Extremal problems in the numerical integration of the functions, Doctor thesis (Cluj - Napoca), 1980 (in Romanian).
[2] D.Acu, About the inetrmediate point in the mean-value theorems, The
annual session of scientifical comunations of the Faculty of Sciences in
Sibiu, 17 - 18 May, 2001 (in Romanian).
[3] D. Acu, P. Dicu, About some properties of the intermediate point in
certain mean-value formulas, General Mathematics, vol. 10, no. 1-2,
2002, 51 - 64.
70
Petrică Dicu
[4] A. Ghizzetti, A. Ossicini, Quadrature Formulae, Birkhäuser Verlag
Basel und Stuttgart, 1970.
[5] D. V. Ionescu, Numerical quadratures, Bucharest, Technical Editure,
1957 (in Romanian).
[6] B. Jacobson, On the mean-value theorem for integrals, The American
Mathematical Monthly, vol. 89, 1982, 300 - 301.
[7] E. C. Popa, An intermediate point property in some the mean-value
theorems, Astra Matematică, vol. 1, nr. 4, 1990, 3-7 (in Romanian).
Department of Mathematics
“Lucian Blaga” University of Sibiu
Str. Dr. I. Raţiu, nr. 5-7
550012 Sibiu, Romania.
E-mail address: petrică.dicu@ulbsibiu.ro