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Putnam
1947 Putnam
B2
Putnam 1947 B2
Putnam 1947 B2
Source: Putnam 1947
April 3, 2022
Putnam
approximation
differentiable function
Problem Statement
Let
f
(
x
)
f(x)
f
(
x
)
be a differentiable function defined on the interval
(
0
,
1
)
(0,1)
(
0
,
1
)
such that
∣
f
′
(
x
)
∣
≤
M
|f'(x)| \leq M
∣
f
′
(
x
)
∣
≤
M
for
0
<
x
<
1
0<x<1
0
<
x
<
1
and a positive real number
M
.
M.
M
.
Prove that
∣
∫
0
1
f
(
x
)
d
x
−
1
n
∑
k
=
1
n
f
(
k
n
)
∣
≤
M
n
.
\left| \int_{0}^{1} f(x)\; dx - \frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n} \right) \right | \leq \frac{M}{n}.
∫
0
1
f
(
x
)
d
x
−
n
1
k
=
1
∑
n
f
(
n
k
)
≤
n
M
.
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