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Putnam
Putnam 1939
B5
B5
Part of
Putnam 1939
Problems
(1)
Putnam 1939 B5
Source:
8/20/2021
Do either
(
1
)
(1)
(
1
)
or
(
2
)
(2)
(
2
)
:
(
1
)
(1)
(
1
)
Prove that
∫
1
k
[
x
]
f
′
(
x
)
d
x
=
[
k
]
f
(
k
)
−
∑
1
[
k
]
f
(
n
)
,
\int_{1}^{k} [x] f'(x) dx = [k] f(k) - \sum_{1}{[k]} f(n),
∫
1
k
[
x
]
f
′
(
x
)
d
x
=
[
k
]
f
(
k
)
−
∑
1
[
k
]
f
(
n
)
,
where
k
>
1
,
k > 1,
k
>
1
,
and
[
z
]
[z]
[
z
]
denotes the greatest integer
≤
z
.
\leq z.
≤
z
.
Find a similar expression for:
∫
1
k
[
x
2
]
f
′
(
x
)
d
x
.
\int_{1}^{k} [x^2] f'(x) dx.
∫
1
k
[
x
2
]
f
′
(
x
)
d
x
.
(
2
)
(2)
(
2
)
A particle moves freely in a straight line except for a resistive force proportional to its speed. Its speed falls from
1
,
000
f
t
s
1,000 \dfrac{ft}{s}
1
,
000
s
f
t
to
900
f
t
s
900 \dfrac{ft}{s}
900
s
f
t
over
1200
f
t
.
1200 ft.
1200
f
t
.
Find the time taken to the nearest
0.01
s
.
0.01 s.
0.01
s
.
[No calculators or log tables allowed!]
Putnam