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Calculus
Relate position, velocity, speed, and acceleration using derivatives
A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
t
2.1
sin
(
2
t
+
5
)
v(t)=t^{2.1} \sin (2 t+5)
v
(
t
)
=
t
2.1
sin
(
2
t
+
5
)
. Find all times when the speed of the particle is equal to
1
1
1
on the interval
0
≤
t
≤
3
0 \leq t \leq 3
0
≤
t
≤
3
. You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
t
=
t=
t
=
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A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
(
t
2.1
+
5
)
cos
(
2
t
)
v(t)=\left(t^{2.1}+5\right) \cos (2 t)
v
(
t
)
=
(
t
2.1
+
5
)
cos
(
2
t
)
. What is the distance traveled by the particle over the interval
0
≤
t
≤
5
0 \leq t \leq 5
0
≤
t
≤
5
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
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A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
(
t
0.7
+
t
)
cos
(
2
t
)
v(t)=\left(t^{0.7}+t\right) \cos (2 t)
v
(
t
)
=
(
t
0.7
+
t
)
cos
(
2
t
)
. If the position of the particle is
x
=
−
2
x=-2
x
=
−
2
when
t
=
2.5
t=2.5
t
=
2.5
what is the position of the particle when
t
=
1
t=1
t
=
1
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
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A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
t
0.5
sin
(
t
−
4
)
v(t)=t^{0.5} \sin (t-4)
v
(
t
)
=
t
0.5
sin
(
t
−
4
)
. What is the distance traveled by the particle over the interval
0
≤
t
≤
6
0 \leq t \leq 6
0
≤
t
≤
6
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
Get tutor help
A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
(
t
0.7
+
4
)
sin
(
2
t
)
v(t)=\left(t^{0.7}+4\right) \sin (2 t)
v
(
t
)
=
(
t
0.7
+
4
)
sin
(
2
t
)
. What is the distance traveled by the particle over the interval
0
≤
t
≤
6
0 \leq t \leq 6
0
≤
t
≤
6
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
Get tutor help
A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
t
0.4
cos
(
t
−
4
)
v(t)=t^{0.4} \cos (t-4)
v
(
t
)
=
t
0.4
cos
(
t
−
4
)
. What is the average acceleration of the particle on the interval
0
≤
t
≤
5
0 \leq t \leq 5
0
≤
t
≤
5
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
Get tutor help
A particle travels along the
x
x
x
-axis such that its acceleration is given by
a
(
t
)
=
t
0.5
sin
(
2
t
)
a(t)=t^{0.5} \sin (2 t)
a
(
t
)
=
t
0.5
sin
(
2
t
)
. If the velocity of the particle is
v
=
−
3
v=-3
v
=
−
3
when
t
=
1
t=1
t
=
1
, what is the velocity of the particle when
t
=
4
t=4
t
=
4
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
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A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
t
2.3
−
3
−
2
cos
(
3
t
)
v(t)=t^{2.3}-3-2 \cos (3 t)
v
(
t
)
=
t
2.3
−
3
−
2
cos
(
3
t
)
. What is the average acceleration of the particle on the interval
0
≤
t
≤
3
0 \leq t \leq 3
0
≤
t
≤
3
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
Get tutor help
A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
(
t
0.3
+
5
)
cos
(
2
t
)
v(t)=\left(t^{0.3}+5\right) \cos (2 t)
v
(
t
)
=
(
t
0.3
+
5
)
cos
(
2
t
)
. What is the average velocity of the particle on the interval
0
≤
t
≤
4
0 \leq t \leq 4
0
≤
t
≤
4
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
Get tutor help
A particle travels along the
x
x
x
-axis such that its position is given by
x
(
t
)
=
t
1.9
sin
(
t
2
)
x(t)=t^{1.9} \sin \left(t^{2}\right)
x
(
t
)
=
t
1.9
sin
(
t
2
)
. What is the velocity of the particle at time
t
=
0.5
t=0.5
t
=
0.5
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
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A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
t
0.3
−
4
cos
(
2
t
)
v(t)=t^{0.3}-4 \cos (2 t)
v
(
t
)
=
t
0.3
−
4
cos
(
2
t
)
. What is the acceleration of the particle at time
t
=
5
?
t=5 ?
t
=
5
?
You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
Get tutor help
A particle travels along the
x
x
x
-axis such that its position is given by
x
(
t
)
=
t
1.7
sin
(
3
t
)
x(t)=t^{1.7} \sin (3 t)
x
(
t
)
=
t
1.7
sin
(
3
t
)
. What is the velocity of the particle at time
t
=
3.5
t=3.5
t
=
3.5
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
Get tutor help
A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
t
2.1
sin
(
3
t
+
2
)
v(t)=t^{2.1} \sin (3 t+2)
v
(
t
)
=
t
2.1
sin
(
3
t
+
2
)
. What is the distance traveled by the particle over the interval
0
≤
t
≤
5
0 \leq t \leq 5
0
≤
t
≤
5
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
Get tutor help
A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
t
1.1
−
2
−
5
cos
(
3
t
)
v(t)=t^{1.1}-2-5 \cos (3 t)
v
(
t
)
=
t
1.1
−
2
−
5
cos
(
3
t
)
. What is the acceleration of the particle at time
t
=
1
t=1
t
=
1
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
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