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Math Problems
Calculus
Relate position, velocity, speed, and acceleration using derivatives
The position of a particle moving along the x-axis is
x
(
t
)
=
cos
(
2
t
)
−
sin
(
3
t
)
x(t)=\cos(2t)-\sin(3t)
x
(
t
)
=
cos
(
2
t
)
−
sin
(
3
t
)
for time
t
≥
0
t \geq 0
t
≥
0
. When
t
=
π
t=\pi
t
=
π
, the acceleration of the particle is
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The acceleration of a particle moving along the x-axis at time
t
t
t
is given by
a
(
t
)
=
6
t
−
2
a(t)=6t-2
a
(
t
)
=
6
t
−
2
. If the position is
10
10
10
when
t
=
1
t=1
t
=
1
, then which of the following could be the function for position
x
(
t
)
=
?
x(t)= ?
x
(
t
)
=
?
\newline
(A)
9
t
2
+
1
9t^{2}+1
9
t
2
+
1
\newline
(B)
3
t
2
−
2
t
+
4
3t^{2}-2t+4
3
t
2
−
2
t
+
4
\newline
(C)
t
3
−
t
2
+
4
t
+
6
t^{3}-t^{2}+4t+6
t
3
−
t
2
+
4
t
+
6
\newline
(D)
t
3
−
t
2
+
9
t
−
20
t^{3}-t^{2}+9t-20
t
3
−
t
2
+
9
t
−
20
\newline
(E)
36
t
3
−
4
t
2
−
8
t
+
5
36t^{3}-4t^{2}-8t+5
36
t
3
−
4
t
2
−
8
t
+
5
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its position is given by
x
(
t
)
=
t
3
−
12
t
2
+
21
t
x(t)=t^{3}-12 t^{2}+21 t
x
(
t
)
=
t
3
−
12
t
2
+
21
t
. Determine the velocity of the particle at
t
=
6
t=6
t
=
6
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its velocity is given by
v
(
t
)
=
−
2
t
+
3
v(t)=-2 t+3
v
(
t
)
=
−
2
t
+
3
. Determine the acceleration of the particle at
t
=
9
t=9
t
=
9
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its position is given by
x
(
t
)
=
−
t
3
+
9
t
2
−
24
t
x(t)=-t^{3}+9 t^{2}-24 t
x
(
t
)
=
−
t
3
+
9
t
2
−
24
t
. Determine the speed of the particle at
t
=
1
t=1
t
=
1
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its position is given by
x
(
t
)
=
−
t
2
−
8
t
+
44
x(t)=-t^{2}-8 t+44
x
(
t
)
=
−
t
2
−
8
t
+
44
. Determine the acceleration of the particle at
t
=
9
t=9
t
=
9
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its velocity is given by
v
(
t
)
=
−
2
t
+
5
v(t)=-2 t+5
v
(
t
)
=
−
2
t
+
5
. Determine the acceleration of the particle at
t
=
6
t=6
t
=
6
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its velocity is given by
v
(
t
)
=
3
t
2
−
28
t
+
25
v(t)=3 t^{2}-28 t+25
v
(
t
)
=
3
t
2
−
28
t
+
25
. Determine the speed of the particle at
t
=
9
t=9
t
=
9
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its velocity is given by
v
(
t
)
=
3
t
2
+
12
t
−
36
v(t)=3 t^{2}+12 t-36
v
(
t
)
=
3
t
2
+
12
t
−
36
. Determine all values of
t
t
t
when the particle is at rest.
\newline
Answer:
t
=
t=
t
=
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its position is given by
x
(
t
)
=
−
t
2
+
8
t
−
9
x(t)=-t^{2}+8 t-9
x
(
t
)
=
−
t
2
+
8
t
−
9
. Determine the speed of the particle at
t
=
5
t=5
t
=
5
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its position is given by
x
(
t
)
=
−
t
3
+
10
t
2
−
25
t
x(t)=-t^{3}+10 t^{2}-25 t
x
(
t
)
=
−
t
3
+
10
t
2
−
25
t
. Determine the acceleration of the particle at
t
=
1
t=1
t
=
1
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its position is given by
x
(
t
)
=
t
3
−
2
t
2
−
39
t
x(t)=t^{3}-2 t^{2}-39 t
x
(
t
)
=
t
3
−
2
t
2
−
39
t
. Determine the velocity of the particle at
t
=
4
t=4
t
=
4
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its position is given by
x
(
t
)
=
t
3
−
15
t
2
+
27
t
x(t)=t^{3}-15 t^{2}+27 t
x
(
t
)
=
t
3
−
15
t
2
+
27
t
. Determine all values of
t
t
t
when the particle is at rest.
\newline
Answer:
t
=
t=
t
=
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its position is given by
x
(
t
)
=
t
3
−
3
t
2
−
9
t
x(t)=t^{3}-3 t^{2}-9 t
x
(
t
)
=
t
3
−
3
t
2
−
9
t
. Determine all values of
t
t
t
when the particle is at rest.
\newline
Answer:
t
=
t=
t
=
Get tutor help
A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its position is given by
x
(
t
)
=
t
2
+
4
t
+
15
x(t)=t^{2}+4 t+15
x
(
t
)
=
t
2
+
4
t
+
15
. Determine the speed of the particle at
t
=
4
t=4
t
=
4
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis so that at time
t
≥
0
t \geq 0
t
≥
0
its position is given by
x
(
t
)
=
−
t
3
+
8
t
2
−
20
t
x(t)=-t^{3}+8 t^{2}-20 t
x
(
t
)
=
−
t
3
+
8
t
2
−
20
t
. Determine the speed of the particle at
t
=
1
t=1
t
=
1
.
\newline
Answer:
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A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
2
)
=
4
and
v
(
2
)
=
6
x(2)=4 \quad \text { and } \quad v(2)=6
x
(
2
)
=
4
and
v
(
2
)
=
6
\newline
Which of the following expression gives the velocity of the particle when
t
=
9
?
t=9 ?
t
=
9
?
\newline
6
+
∫
9
2
a
(
t
)
d
t
6+\int_{9}^{2} a(t) d t
6
+
∫
9
2
a
(
t
)
d
t
\newline
6
+
∫
9
2
v
(
t
)
d
t
6+\int_{9}^{2} v(t) d t
6
+
∫
9
2
v
(
t
)
d
t
\newline
6
+
∫
2
9
a
(
t
)
d
t
6+\int_{2}^{9} a(t) d t
6
+
∫
2
9
a
(
t
)
d
t
\newline
6
+
∫
2
9
v
(
t
)
d
t
6+\int_{2}^{9} v(t) d t
6
+
∫
2
9
v
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
0
)
=
4
and
v
(
0
)
=
10
x(0)=4 \text { and } v(0)=10
x
(
0
)
=
4
and
v
(
0
)
=
10
\newline
Which of the following expression gives the velocity of the particle when
t
=
10
t=10
t
=
10
?
\newline
10
+
∫
10
0
x
(
t
)
d
t
10+\int_{10}^{0} x(t) d t
10
+
∫
10
0
x
(
t
)
d
t
\newline
10
+
∫
0
10
a
(
t
)
d
t
10+\int_{0}^{10} a(t) d t
10
+
∫
0
10
a
(
t
)
d
t
\newline
10
+
∫
10
0
a
(
t
)
d
t
10+\int_{10}^{0} a(t) d t
10
+
∫
10
0
a
(
t
)
d
t
\newline
10
+
∫
0
10
x
(
t
)
d
t
10+\int_{0}^{10} x(t) d t
10
+
∫
0
10
x
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
.
\newline
What is the average velocity of the particle on the interval
2
≤
t
≤
8
?
2 \leq t \leq 8 ?
2
≤
t
≤
8
?
\newline
1
6
∫
2
8
a
(
t
)
d
t
\frac{1}{6} \int_{2}^{8} a(t) d t
6
1
∫
2
8
a
(
t
)
d
t
\newline
v
(
8
)
−
v
(
2
)
6
\frac{v(8)-v(2)}{6}
6
v
(
8
)
−
v
(
2
)
\newline
a
(
8
)
−
a
(
2
)
6
\frac{a(8)-a(2)}{6}
6
a
(
8
)
−
a
(
2
)
\newline
1
6
∫
2
8
v
(
t
)
d
t
\frac{1}{6} \int_{2}^{8} v(t) d t
6
1
∫
2
8
v
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
0
)
=
5
and
v
(
0
)
=
3
x(0)=5 \quad \text { and } \quad v(0)=3
x
(
0
)
=
5
and
v
(
0
)
=
3
\newline
Which of the following expressions gives the distance traveled by the particle over the interval
0
≤
t
≤
7
0 \leq t \leq 7
0
≤
t
≤
7
?
\newline
5
+
∫
0
7
v
(
t
)
d
t
5+\int_{0}^{7} v(t) d t
5
+
∫
0
7
v
(
t
)
d
t
\newline
∣
∫
0
7
v
(
t
)
d
t
∣
\left|\int_{0}^{7} v(t) d t\right|
∣
∣
∫
0
7
v
(
t
)
d
t
∣
∣
\newline
∫
0
7
∣
v
(
t
)
∣
d
t
\int_{0}^{7}|v(t)| d t
∫
0
7
∣
v
(
t
)
∣
d
t
\newline
∫
0
7
v
(
t
)
d
t
\int_{0}^{7} v(t) d t
∫
0
7
v
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
1
)
=
7
and
v
(
1
)
=
4
x(1)=7 \quad \text { and } \quad v(1)=4
x
(
1
)
=
7
and
v
(
1
)
=
4
\newline
Which of the following expression gives the position of the particle when
t
=
9
?
t=9 ?
t
=
9
?
\newline
7
+
∫
1
9
v
(
t
)
d
t
7+\int_{1}^{9} v(t) d t
7
+
∫
1
9
v
(
t
)
d
t
\newline
1
+
∫
7
9
v
(
t
)
d
t
1+\int_{7}^{9} v(t) d t
1
+
∫
7
9
v
(
t
)
d
t
\newline
7
+
∫
9
1
v
(
t
)
d
t
7+\int_{9}^{1} v(t) d t
7
+
∫
9
1
v
(
t
)
d
t
\newline
1
+
∫
9
7
v
(
t
)
d
t
1+\int_{9}^{7} v(t) d t
1
+
∫
9
7
v
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
.
\newline
What is the average acceleration of the particle on the interval
1
≤
t
≤
10
1 \leq t \leq 10
1
≤
t
≤
10
?
\newline
1
9
∫
1
10
v
(
t
)
d
t
\frac{1}{9} \int_{1}^{10} v(t) d t
9
1
∫
1
10
v
(
t
)
d
t
\newline
a
(
10
)
−
a
(
1
)
9
\frac{a(10)-a(1)}{9}
9
a
(
10
)
−
a
(
1
)
\newline
x
(
10
)
−
x
(
1
)
9
\frac{x(10)-x(1)}{9}
9
x
(
10
)
−
x
(
1
)
\newline
1
9
∫
1
10
a
(
t
)
d
t
\frac{1}{9} \int_{1}^{10} a(t) d t
9
1
∫
1
10
a
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
2
)
=
1
and
v
(
2
)
=
7
x(2)=1 \quad \text { and } \quad v(2)=7
x
(
2
)
=
1
and
v
(
2
)
=
7
\newline
Which of the following expression gives the position of the particle when
t
=
0
t=0
t
=
0
?
\newline
1
+
∫
2
0
v
(
t
)
d
t
1+\int_{2}^{0} v(t) d t
1
+
∫
2
0
v
(
t
)
d
t
\newline
1
+
∫
0
2
v
(
t
)
d
t
1+\int_{0}^{2} v(t) d t
1
+
∫
0
2
v
(
t
)
d
t
\newline
2
+
∫
2
0
v
(
t
)
d
t
2+\int_{2}^{0} v(t) d t
2
+
∫
2
0
v
(
t
)
d
t
\newline
2
+
∫
0
2
v
(
t
)
d
t
2+\int_{0}^{2} v(t) d t
2
+
∫
0
2
v
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
0
)
=
1
and
v
(
0
)
=
3
x(0)=1 \text { and } v(0)=3
x
(
0
)
=
1
and
v
(
0
)
=
3
\newline
Which of the following expression gives the velocity of the particle when
t
=
8
?
t=8 ?
t
=
8
?
\newline
1
+
∫
8
0
v
(
t
)
d
t
1+\int_{8}^{0} v(t) d t
1
+
∫
8
0
v
(
t
)
d
t
\newline
3
+
∫
8
0
a
(
t
)
d
t
3+\int_{8}^{0} a(t) d t
3
+
∫
8
0
a
(
t
)
d
t
\newline
1
+
∫
0
8
v
(
t
)
d
t
1+\int_{0}^{8} v(t) d t
1
+
∫
0
8
v
(
t
)
d
t
\newline
3
+
∫
0
8
a
(
t
)
d
t
3+\int_{0}^{8} a(t) d t
3
+
∫
0
8
a
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
.
\newline
What is the average acceleration of the particle on the interval
0
≤
t
≤
9
?
0 \leq t \leq 9 ?
0
≤
t
≤
9
?
\newline
1
9
∫
0
9
v
(
t
)
d
t
\frac{1}{9} \int_{0}^{9} v(t) d t
9
1
∫
0
9
v
(
t
)
d
t
\newline
x
(
9
)
−
x
(
0
)
9
\frac{x(9)-x(0)}{9}
9
x
(
9
)
−
x
(
0
)
\newline
1
9
∫
0
9
a
(
t
)
d
t
\frac{1}{9} \int_{0}^{9} a(t) d t
9
1
∫
0
9
a
(
t
)
d
t
\newline
a
(
9
)
−
a
(
0
)
9
\frac{a(9)-a(0)}{9}
9
a
(
9
)
−
a
(
0
)
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
1
)
=
8
and
v
(
1
)
=
4
x(1)=8 \quad \text { and } \quad v(1)=4
x
(
1
)
=
8
and
v
(
1
)
=
4
\newline
Which of the following expression gives the position of the particle when
t
=
8
?
t=8 ?
t
=
8
?
\newline
1
+
∫
8
8
v
(
t
)
d
t
1+\int_{8}^{8} v(t) d t
1
+
∫
8
8
v
(
t
)
d
t
\newline
1
+
∫
8
8
v
(
t
)
d
t
1+\int_{8}^{8} v(t) d t
1
+
∫
8
8
v
(
t
)
d
t
\newline
8
+
∫
1
8
v
(
t
)
d
t
8+\int_{1}^{8} v(t) d t
8
+
∫
1
8
v
(
t
)
d
t
\newline
8
+
∫
8
1
v
(
t
)
d
t
8+\int_{8}^{1} v(t) d t
8
+
∫
8
1
v
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
0
)
=
6
and
v
(
0
)
=
5
x(0)=6 \quad \text { and } \quad v(0)=5
x
(
0
)
=
6
and
v
(
0
)
=
5
\newline
Which of the following expressions gives the displacement of the particle over the interval
0
≤
t
≤
6
0 \leq t \leq 6
0
≤
t
≤
6
?
\newline
∫
0
6
v
(
t
)
d
t
\int_{0}^{6} v(t) d t
∫
0
6
v
(
t
)
d
t
\newline
∫
0
6
∣
v
(
t
)
∣
d
t
\int_{0}^{6}|v(t)| d t
∫
0
6
∣
v
(
t
)
∣
d
t
\newline
6
+
∫
0
6
∣
v
(
t
)
∣
d
t
6+\int_{0}^{6}|v(t)| d t
6
+
∫
0
6
∣
v
(
t
)
∣
d
t
\newline
6
+
∫
0
6
v
(
t
)
d
t
6+\int_{0}^{6} v(t) d t
6
+
∫
0
6
v
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
5
)
=
7
and
v
(
5
)
=
4
x(5)=7 \quad \text { and } \quad v(5)=4
x
(
5
)
=
7
and
v
(
5
)
=
4
\newline
Which of the following expression gives the velocity of the particle when
t
=
0
t=0
t
=
0
?
\newline
4
+
∫
0
5
x
(
t
)
d
t
4+\int_{0}^{5} x(t) d t
4
+
∫
0
5
x
(
t
)
d
t
\newline
4
+
∫
5
0
x
(
t
)
d
t
4+\int_{5}^{0} x(t) d t
4
+
∫
5
0
x
(
t
)
d
t
\newline
4
+
∫
5
0
a
(
t
)
d
t
4+\int_{5}^{0} a(t) d t
4
+
∫
5
0
a
(
t
)
d
t
\newline
4
+
∫
0
5
a
(
t
)
d
t
4+\int_{0}^{5} a(t) d t
4
+
∫
0
5
a
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
0
)
=
4
and
v
(
0
)
=
6
x(0)=4 \quad \text { and } \quad v(0)=6
x
(
0
)
=
4
and
v
(
0
)
=
6
\newline
Which of the following expressions gives the distance traveled by the particle over the interval
0
≤
t
≤
7
0 \leq t \leq 7
0
≤
t
≤
7
?
\newline
4
+
∫
0
7
∣
v
(
t
)
∣
d
t
4+\int_{0}^{7}|v(t)| d t
4
+
∫
0
7
∣
v
(
t
)
∣
d
t
\newline
∫
0
7
∣
v
(
t
)
∣
d
t
\int_{0}^{7}|v(t)| d t
∫
0
7
∣
v
(
t
)
∣
d
t
\newline
∫
0
7
v
(
t
)
d
t
\int_{0}^{7} v(t) d t
∫
0
7
v
(
t
)
d
t
\newline
4
+
∫
0
7
v
(
t
)
d
t
4+\int_{0}^{7} v(t) d t
4
+
∫
0
7
v
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
0
)
=
7
and
v
(
0
)
=
8
x(0)=7 \quad \text { and } \quad v(0)=8
x
(
0
)
=
7
and
v
(
0
)
=
8
\newline
Which of the following expressions gives the distance traveled by the particle over the interval
0
≤
t
≤
10
0 \leq t \leq 10
0
≤
t
≤
10
?
\newline
∫
0
10
x
(
t
)
d
t
\int_{0}^{10} x(t) d t
∫
0
10
x
(
t
)
d
t
\newline
∫
0
10
∣
v
(
t
)
∣
d
t
\int_{0}^{10}|v(t)| d t
∫
0
10
∣
v
(
t
)
∣
d
t
\newline
∫
0
10
v
(
t
)
d
t
\int_{0}^{10} v(t) d t
∫
0
10
v
(
t
)
d
t
\newline
∫
0
10
∣
x
(
t
)
∣
d
t
\int_{0}^{10}|x(t)| d t
∫
0
10
∣
x
(
t
)
∣
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
1
)
=
2
and
v
(
1
)
=
5
x(1)=2 \quad \text { and } \quad v(1)=5
x
(
1
)
=
2
and
v
(
1
)
=
5
\newline
Which of the following expressions gives the distance traveled by the particle over the interval
1
≤
t
≤
6
1 \leq t \leq 6
1
≤
t
≤
6
?
\newline
∫
1
6
v
(
t
)
d
t
\int_{1}^{6} v(t) d t
∫
1
6
v
(
t
)
d
t
\newline
∫
1
6
∣
v
(
t
)
∣
d
t
\int_{1}^{6}|v(t)| d t
∫
1
6
∣
v
(
t
)
∣
d
t
\newline
∫
1
6
x
(
t
)
d
t
\int_{1}^{6} x(t) d t
∫
1
6
x
(
t
)
d
t
\newline
∫
1
6
∣
x
(
t
)
∣
d
t
\int_{1}^{6}|x(t)| d t
∫
1
6
∣
x
(
t
)
∣
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
0
)
=
5
and
v
(
0
)
=
6
x(0)=5 \quad \text { and } \quad v(0)=6
x
(
0
)
=
5
and
v
(
0
)
=
6
\newline
Which of the following expressions gives the displacement of the particle over the interval
0
≤
t
≤
7
0 \leq t \leq 7
0
≤
t
≤
7
?
\newline
∫
0
7
∣
v
(
t
)
∣
d
t
\int_{0}^{7}|v(t)| d t
∫
0
7
∣
v
(
t
)
∣
d
t
\newline
∫
0
7
x
(
t
)
d
t
\int_{0}^{7} x(t) d t
∫
0
7
x
(
t
)
d
t
\newline
∫
0
7
v
(
t
)
d
t
\int_{0}^{7} v(t) d t
∫
0
7
v
(
t
)
d
t
\newline
∫
0
7
∣
x
(
t
)
∣
d
t
\int_{0}^{7}|x(t)| d t
∫
0
7
∣
x
(
t
)
∣
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
0
)
=
7
and
v
(
0
)
=
3
x(0)=7 \quad \text { and } \quad v(0)=3
x
(
0
)
=
7
and
v
(
0
)
=
3
\newline
Which of the following expressions gives the displacement of the particle over the interval
0
≤
t
≤
7
0 \leq t \leq 7
0
≤
t
≤
7
?
\newline
∫
0
7
∣
x
(
t
)
∣
d
t
\int_{0}^{7}|x(t)| d t
∫
0
7
∣
x
(
t
)
∣
d
t
\newline
∫
0
7
v
(
t
)
d
t
\int_{0}^{7} v(t) d t
∫
0
7
v
(
t
)
d
t
\newline
∫
0
7
∣
v
(
t
)
∣
d
t
\int_{0}^{7}|v(t)| d t
∫
0
7
∣
v
(
t
)
∣
d
t
\newline
∫
0
7
x
(
t
)
d
t
\int_{0}^{7} x(t) d t
∫
0
7
x
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
7
)
=
6
and
v
(
7
)
=
3
x(7)=6 \quad \text { and } \quad v(7)=3
x
(
7
)
=
6
and
v
(
7
)
=
3
\newline
Which of the following expression gives the velocity of the particle when
t
=
8
?
t=8 ?
t
=
8
?
\newline
∫
8
7
a
(
t
)
d
t
\int_{8}^{7} a(t) d t
∫
8
7
a
(
t
)
d
t
\newline
3
+
∫
8
7
a
(
t
)
d
t
3+\int_{8}^{7} a(t) d t
3
+
∫
8
7
a
(
t
)
d
t
\newline
∫
7
8
a
(
t
)
d
t
\int_{7}^{8} a(t) d t
∫
7
8
a
(
t
)
d
t
\newline
3
+
∫
7
8
a
(
t
)
d
t
3+\int_{7}^{8} a(t) d t
3
+
∫
7
8
a
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
2
)
=
7
and
v
(
2
)
=
3
x(2)=7 \quad \text { and } \quad v(2)=3
x
(
2
)
=
7
and
v
(
2
)
=
3
\newline
Which of the following expressions gives the displacement of the particle over the interval
2
≤
t
≤
10
2 \leq t \leq 10
2
≤
t
≤
10
?
\newline
∫
2
10
∣
v
(
t
)
∣
d
t
\int_{2}^{10}|v(t)| d t
∫
2
10
∣
v
(
t
)
∣
d
t
\newline
7
+
∫
2
10
∣
v
(
t
)
∣
d
t
7+\int_{2}^{10}|v(t)| d t
7
+
∫
2
10
∣
v
(
t
)
∣
d
t
\newline
∫
2
10
v
(
t
)
d
t
\int_{2}^{10} v(t) d t
∫
2
10
v
(
t
)
d
t
\newline
7
+
∫
2
10
v
(
t
)
d
t
7+\int_{2}^{10} v(t) d t
7
+
∫
2
10
v
(
t
)
d
t
Get tutor help
A particle moves along the
x
x
x
-axis such that at any time
t
≥
0
t \geq 0
t
≥
0
its position is
x
(
t
)
x(t)
x
(
t
)
, its velocity is
v
(
t
)
v(t)
v
(
t
)
, and its acceleration is
a
(
t
)
a(t)
a
(
t
)
. You are given:
\newline
x
(
7
)
=
1
and
v
(
7
)
=
8
x(7)=1 \quad \text { and } \quad v(7)=8
x
(
7
)
=
1
and
v
(
7
)
=
8
\newline
Which of the following expression gives the position of the particle when
t
=
0
t=0
t
=
0
?
\newline
1
+
∫
7
0
v
(
t
)
d
t
1+\int_{7}^{0} v(t) d t
1
+
∫
7
0
v
(
t
)
d
t
\newline
1
+
∫
0
7
v
(
t
)
d
t
1+\int_{0}^{7} v(t) d t
1
+
∫
0
7
v
(
t
)
d
t
\newline
7
+
∫
0
7
v
(
t
)
d
t
7+\int_{0}^{7} v(t) d t
7
+
∫
0
7
v
(
t
)
d
t
\newline
7
+
∫
7
0
v
(
t
)
d
t
7+\int_{7}^{0} v(t) d t
7
+
∫
7
0
v
(
t
)
d
t
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A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
(
t
1.5
−
5
t
)
cos
(
2
t
)
v(t)=\left(t^{1.5}-5 t\right) \cos (2 t)
v
(
t
)
=
(
t
1.5
−
5
t
)
cos
(
2
t
)
. What is the average velocity 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:
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A particle travels along the
x
x
x
-axis such that its acceleration is given by
a
(
t
)
=
t
0.3
+
3
cos
(
t
−
5
)
a(t)=t^{0.3}+3 \cos (t-5)
a
(
t
)
=
t
0.3
+
3
cos
(
t
−
5
)
. If the velocity of the particle is
v
=
−
5
v=-5
v
=
−
5
when
t
=
0
t=0
t
=
0
, what is the velocity of the particle when
t
=
5
t=5
t
=
5
? You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
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The position of a bicycle riding down a straight road is measured by the differentiable function
f
f
f
, where
f
(
t
)
f(t)
f
(
t
)
is measured in feet and
t
t
t
is measured in minutes. What are the units of
f
′
′
(
t
)
f^{\prime \prime}(t)
f
′′
(
t
)
?
\newline
feet
\newline
minutes
\newline
feet / minute
\newline
minutes / foot
\newline
feet / minute
2
{ }^{2}
2
\newline
minutes / foot
2
{ }^{2}
2
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The velocity of a bicycle riding down a straight road is measured by the differentiable function
f
f
f
, where
f
(
t
)
f(t)
f
(
t
)
is measured in meters per second and
t
t
t
is measured in seconds. What are the units of
∫
1
6
f
(
t
)
d
t
\int_{1}^{6} f(t) d t
∫
1
6
f
(
t
)
d
t
?
\newline
seconds
\newline
meters
\newline
seconds / meter
\newline
meters / second
\newline
seconds
/
/
/
meter
2
^{2}
2
\newline
meters / second
2
{ }^{2}
2
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The atmospheric pressure of the air changes with height above sea level. The rate of change of the air pressure at a given height above sea level can be measured by the differentiable function
f
(
h
)
f(h)
f
(
h
)
, in psi per meter, where
h
h
h
is measured in meters. What are the units of
f
′
(
h
)
f^{\prime}(h)
f
′
(
h
)
?
\newline
meters
\newline
psi
\newline
meters / psi
\newline
psi / meter
\newline
meters
/
p
s
i
2
/ \mathrm{psi}^{2}
/
psi
2
\newline
psi
/
/
/
meter
2
^{2}
2
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The velocity of a bicycle riding down a straight road is measured by the differentiable function
f
f
f
, where
f
(
t
)
f(t)
f
(
t
)
is measured in feet per second and
t
t
t
is measured in seconds. What are the units of
∫
2
6
f
′
(
t
)
d
t
\int_{2}^{6} f^{\prime}(t) d t
∫
2
6
f
′
(
t
)
d
t
?
\newline
feet
\newline
seconds
\newline
feet / second
\newline
seconds / foot
\newline
feet / second
2
{ }^{2}
2
\newline
seconds / foot
2
{ }^{2}
2
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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
sin
(
2
t
+
3
)
v(t)=t^{0.7} \sin (2 t+3)
v
(
t
)
=
t
0.7
sin
(
2
t
+
3
)
. What is the acceleration of the particle at time
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
1.1
cos
(
t
2
−
5
)
v(t)=t^{1.1} \cos \left(t^{2}-5\right)
v
(
t
)
=
t
1.1
cos
(
t
2
−
5
)
. 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
0.9
sin
(
t
−
1
)
v(t)=t^{0.9} \sin (t-1)
v
(
t
)
=
t
0.9
sin
(
t
−
1
)
. If the position of the particle is
x
=
−
3
x=-3
x
=
−
3
when
t
=
3
t=3
t
=
3
, what is the position of the particle when
t
=
5
t=5
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
2.1
sin
(
2
t
)
v(t)=t^{2.1} \sin (2 t)
v
(
t
)
=
t
2.1
sin
(
2
t
)
. Find all times when the speed of the particle is equal to
2
2
2
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:
t
=
t=
t
=
Get tutor help
A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
(
t
2.3
+
5
)
sin
(
2
t
)
v(t)=\left(t^{2.3}+5\right) \sin (2 t)
v
(
t
)
=
(
t
2.3
+
5
)
sin
(
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
1.9
cos
(
3
t
−
1
)
v(t)=t^{1.9} \cos (3 t-1)
v
(
t
)
=
t
1.9
cos
(
3
t
−
1
)
. What is the distance traveled by the particle over 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:
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A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
(
t
1.6
+
3
)
cos
(
3
t
)
v(t)=\left(t^{1.6}+3\right) \cos (3 t)
v
(
t
)
=
(
t
1.6
+
3
)
cos
(
3
t
)
. What is the average velocity of the particle on 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:
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A particle travels along the
x
x
x
-axis such that its acceleration is given by
a
(
t
)
=
(
t
0.5
+
3
)
cos
(
3
t
)
a(t)=\left(t^{0.5}+3\right) \cos (3 t)
a
(
t
)
=
(
t
0.5
+
3
)
cos
(
3
t
)
. If the velocity of the particle is
v
=
−
2
v=-2
v
=
−
2
when
t
=
1
t=1
t
=
1
, what is the velocity 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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