" GMm
- dx = ( G, M and m are constant)
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Answers
Answer:
f
G
,
M
, and
m
are constants and
F
=
G
M
m
r
2
a. Calculate the differential of
F
,
d
F
.
b. If
r
increases by
4
%
(meaning
d
r
=
.04
r
) use the differential to estimate the percentage change in
F
.
Differentiation:
Differential of any variable
y
is given by
d
y
.
Differentiation Formulas:
1.
d
(
x
n
)
=
n
x
n
−
1
d
x
.
2.
d
(
ln
x
)
=
d
x
x
.
3.
d
(
constant
)
=
0
Answer and Explanation:
Given
F
=
G
M
m
r
2
,where
r
is the only variable.
a.
Differentiating the equation:
d
F
=
d
(
G
M
m
r
2
)
d
F
=
−
G
M
m
r
3
d
r
.
b.
Given that
r
is increased by
4
%
.
⇒
d
r
r
=
4
100
=
0.04.
Taking logarithm and differentiating:
ln
F
=
ln
(
G
M
m
)
−
2
ln
r
d
F
F
=
0
−
2
d
r
r
d
F
F
=
−
2
(
0.04
)
d
F
F
=
−
0.08
d
F
F
∗
100
=
−
8
%
⇒
F
decreases by
8
%
.
Explanation: