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Problem 9
2021 CMC (Non-Math Major) Problem 9
2021 CMC (Non-Math Major) Problem 9
Source: 2021 CMC (Non-Math Major) Problem 9
November 25, 2022
calculus
Problem Statement
Let
f
(
x
)
f(x)
f
(
x
)
be a twice continuously differentiable function on closed interval
[
a
,
b
]
[a,b]
[
a
,
b
]
Prove that
lim
n
→
∞
n
2
[
∫
a
b
f
(
x
)
d
x
−
b
−
a
n
∑
k
=
1
n
f
(
a
+
2
k
−
1
2
n
(
b
−
a
)
)
]
=
(
b
−
a
)
2
24
[
f
′
(
b
)
−
f
′
(
a
)
]
\lim_{n \to \infty} n^2[\int_{a}^{b}f(x)dx-\frac{b-a}{n}\sum_{k=1}^{n}f(a+\frac{2k-1}{2n}(b-a))]=\frac{(b-a)^2}{24}[f'(b)-f'(a)]
lim
n
→
∞
n
2
[
∫
a
b
f
(
x
)
d
x
−
n
b
−
a
∑
k
=
1
n
f
(
a
+
2
n
2
k
−
1
(
b
−
a
))]
=
24
(
b
−
a
)
2
[
f
′
(
b
)
−
f
′
(
a
)]
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