Respuesta :
Answer:
Explanation:To find the average rate of change for
�
=
6
sin
(
2
�
)
r=6sin(2θ) over a given interval, we need to compute the change in
�
r with respect to
�
θ and then take the average over that interval.
Given
�
=
6
sin
(
2
�
)
r=6sin(2θ), we'll first find
�
�
�
�
dθ
dr
:
�
=
6
sin
(
2
�
)
r=6sin(2θ)
We'll use the chain rule:
�
�
�
�
=
�
�
�
(
6
sin
(
2
�
)
)
dθ
dr
=
dθ
d
(6sin(2θ))
=
6
cos
(
2
�
)
×
2
=6cos(2θ)×2
=
12
cos
(
2
�
)
=12cos(2θ)
Now, to find the average rate of change over an interval, say
[
�
1
,
�
2
]
[θ
1
,θ
2
], we use the formula:
Average rate of change
=
�
(
�
2
)
−
�
(
�
1
)
�
2
−
�
1
Average rate of change=
θ
2
−θ
1
r(θ
2
)−r(θ
1
)
But here,
�
(
�
)
=
6
sin
(
2
�
)
r(θ)=6sin(2θ), so the average rate of change becomes:
6
sin
(
2
�
2
)
−
6
sin
(
2
�
1
)
�
2
−
�
1
θ
2
−θ
1
6sin(2θ
2
)−6sin(2θ
1
)
This expression gives the average rate of change of
�
r with respect to
�
θ over the interval
[
�
1
,
�
2
]
[θ
1
,θ
2
].
To find the average rate of change for
�
=
6
sin
(
2
�
)
r=6sin(2θ) over a given interval, we need to compute the change in
�
r with respect to
�
θ and then take the average over that interval.
Given
�
=
6
sin
(
2
�
)
r=6sin(2θ), we'll first find
�
�
�
�
dθ
dr
:
�
=
6
sin
(
2
�
)
r=6sin(2θ)
We'll use the chain rule:
�
�
�
�
=
�
�
�
(
6
sin
(
2
�
)
)
dθ
dr
=
dθ
d
(6sin(2θ))
=
6
cos
(
2
�
)
×
2
=6cos(2θ)×2
=
12
cos
(
2
�
)
=12cos(2θ)
Now, to find the average rate of change over an interval, say
[
�
1
,
�
2
]
[θ
1
,θ
2
], we use the formula:
Average rate of change
=
�
(
�
2
)
−
�
(
�
1
)
�
2
−
�
1
Average rate of change=
θ
2
−θ
1
r(θ
2
)−r(θ
1
)
But here,
�
(
�
)
=
6
sin
(
2
�
)
r(θ)=6sin(2θ), so the average rate of change becomes:
6
sin
(
2
�
2
)
−
6
sin
(
2
�
1
)
�
2
−
�
1
θ
2
−θ
1
6sin(2θ
2
)−6sin(2θ
1
)
This expression gives the average rate of change of
�
r with respect to
�
θ over the interval
[
�
1
,
�
2
]
[θ
1
,θ
2
].
