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Math Problems
Grade 8
Arithmetic sequences
6
6
6
. Use benchmark fractions to estimate sums and differences less than or greater than
1
1
1
. Write each expression in the correct answer space.
\newline
\begin{tabular}{|l|l|}
\newline
\hline Less Than
1
1
1
& Greater Than I \\
\newline
\hline & \\
\newline
\hline
\newline
\end{tabular}
\newline
7
8
+
5
10
1
5
8
−
5
6
10
10
−
2
3
1
2
+
2
3
5
12
+
1
4
1
1
6
+
7
8
\begin{array}{lll} \frac{7}{8}+\frac{5}{10} & 1 \frac{5}{8}-\frac{5}{6} & \frac{10}{10}-\frac{2}{3} \\ \frac{1}{2}+\frac{2}{3} & \frac{5}{12}+\frac{1}{4} & 1 \frac{1}{6}+\frac{7}{8} \end{array}
8
7
+
10
5
2
1
+
3
2
1
8
5
−
6
5
12
5
+
4
1
10
10
−
3
2
1
6
1
+
8
7
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Write an expression to describe the sequence below. Use
n
n
n
to represent the position of a term in the sequence, where
n
=
1
n = 1
n
=
1
for the first term.
–
8
–8
–8
,
–
16
–16
–16
,
–
24
–24
–24
,
–
32
–32
–32
, ...
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Which of the following are real numbers?
\newline
Multi-select Choices:
\newline
(A)
10
10
10
\newline
(B)
9
10
\frac{9}{10}
10
9
\newline
(C)
1.444
…
1.444\ldots
1.444
…
\newline
(D)
3
\sqrt{3}
3
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a. Let
f
(
x
)
=
7
sin
(
4
x
)
f(x)=7\sin(4x)
f
(
x
)
=
7
sin
(
4
x
)
. What is the period of
f
f
f
?
\newline
Preview
\newline
Enter a mathematical expression [more...]
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a. Let
f
(
x
)
=
7
sin
(
4
x
)
f(x)=7 \sin (4 x)
f
(
x
)
=
7
sin
(
4
x
)
. What is the period of
f
f
f
?
\newline
Preview
\newline
Enter a mathematical expression [more..]
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Let
f
f
f
be the function defined by
f
(
x
)
=
sin
(
π
4
x
)
f(x)=\sin \left(\frac{\pi}{4} x\right)
f
(
x
)
=
sin
(
4
π
x
)
. What is the average value of
f
f
f
on the interval
[
4
3
,
8
3
]
\left[\frac{4}{3}, \frac{8}{3}\right]
[
3
4
,
3
8
]
written in simplest form?
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5
5
5
.) An insurance company determines that
\newline
N
N
N
, the number of claims received in a week, is a random variable with
\newline
P
[
N
=
n
]
=
1
2
n
+
1
P[N=n]=\frac{1}{2^{n+1}}
P
[
N
=
n
]
=
2
n
+
1
1
where
\newline
n
≥
0
n \geq 0
n
≥
0
. The company also determines that the number of claims received in a given week is independent of the number of claims received in any other week.
\newline
Calculate the probability that exactly six claims will be received during a given two-week period.
\newline
(A)
\newline
5
256
\frac{5}{256}
256
5
\newline
(B)
\newline
7
256
\frac{7}{256}
256
7
\newline
(C)
\newline
5
128
\frac{5}{128}
128
5
\newline
(D)
\newline
7
128
\frac{7}{128}
128
7
\newline
(E)
\newline
5
64
\frac{5}{64}
64
5
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Baind
\newline
Math
1
1
1
\newline
Updated:
\newline
3
/
12
/
24
3/12/24
3/12/24
\newline
7.7
7.7
7.7
Assigned Practice
\newline
Name
\newline
Mattha-Burleigperiod
5
5
5
\newline
Mark the information in the diagram and then complete the flowchart proof.
\newline
1.
1.
1.
\newline
Given:
\newline
A
B
‾
≅
C
B
‾
\overline{AB} \cong \overline{CB}
A
B
≅
CB
and
\newline
D
A
‾
≅
D
C
‾
\overline{DA} \cong \overline{DC}
D
A
≅
D
C
\newline
Prove:
\newline
∠
B
A
D
≅
∠
B
C
D
\angle BAD \cong \angle BCD
∠
B
A
D
≅
∠
BC
D
\newline
List the three congruencies from the given statement or the diagram:
\newline
List Remaining Congruent Parts:
\newline
because...
\newline
Therefore...
\newline
(What you wanted to prove)
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Find the accumulation function
\newline
F
F
F
.
\newline
f
(
x
)
=
∫
−
π
x
(
3
+
sin
t
)
d
t
,
F
(
x
)
=
□
\begin{aligned} f(x) &= \int_{-\pi}^{x} (3 + \sin t) \, dt, \ F(x) &= \square \end{aligned}
f
(
x
)
=
∫
−
π
x
(
3
+
sin
t
)
d
t
,
F
(
x
)
=
□
\newline
(
a
)
(a)
(
a
)
\newline
F
(
−
π
)
=
F(-\pi) =
F
(
−
π
)
=
\newline
(
B
)
(B)
(
B
)
\newline
F
(
0
)
=
F(0) =
F
(
0
)
=
\newline
(
c
)
(c)
(
c
)
\newline
F
(
2
π
)
=
F(2\pi) =
F
(
2
π
)
=
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Solve for
r
r
r
.\begin{align*} \left(\frac{4}{r}\right)&=\left(\frac{5}{7}\right),\ r&= \end{align*}
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Solve for
t
t
t
.
(
7
3
)
=
(
4
t
)
,
t
=
\begin{align*} \left(\frac{7}{3}\right)&=\left(\frac{4}{t}\right), t&= \end{align*}
(
3
7
)
=
(
t
4
)
,
t
=
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Let
y
=
1
−
2
x
3
x
2
y=\frac{1-2 x}{3 x^{2}}
y
=
3
x
2
1
−
2
x
.
\newline
What is the value of
d
y
d
x
\frac{d y}{d x}
d
x
d
y
at
x
=
1
x=1
x
=
1
?
\newline
Choose
1
1
1
answer:
\newline
(A)
0
0
0
\newline
(B)
−
1
3
-\frac{1}{3}
−
3
1
\newline
(C)
1
1
1
\newline
(D)
1
9
\frac{1}{9}
9
1
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- Let
f
f
f
be a function such that
f
(
−
2
)
=
8
f(-2)=8
f
(
−
2
)
=
8
and
f
′
(
−
2
)
=
4
f^{\prime}(-2)=4
f
′
(
−
2
)
=
4
.
\newline
- Let
h
h
h
be the function
h
(
x
)
=
x
3
h(x)=x^{3}
h
(
x
)
=
x
3
.
\newline
Evaluate
d
d
x
[
f
(
x
)
h
(
x
)
]
\frac{d}{d x}\left[\frac{f(x)}{h(x)}\right]
d
x
d
[
h
(
x
)
f
(
x
)
]
at
x
=
−
2
x=-2
x
=
−
2
.
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- Let
g
g
g
be a function such that
g
(
−
1
)
=
5
g(-1)=5
g
(
−
1
)
=
5
and
g
′
(
−
1
)
=
6
g^{\prime}(-1)=6
g
′
(
−
1
)
=
6
.
\newline
- Let
h
h
h
be the function
h
(
x
)
=
2
x
2
h(x)=2 x^{2}
h
(
x
)
=
2
x
2
.
\newline
Evaluate
d
d
x
[
g
(
x
)
h
(
x
)
]
\frac{d}{d x}\left[\frac{g(x)}{h(x)}\right]
d
x
d
[
h
(
x
)
g
(
x
)
]
at
x
=
−
1
x=-1
x
=
−
1
.
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What is the area of the region between the graphs of
f
(
x
)
=
2
x
2
+
5
x
f(x)=2 x^{2}+5 x
f
(
x
)
=
2
x
2
+
5
x
and
g
(
x
)
=
−
x
2
−
6
x
+
4
g(x)=-x^{2}-6 x+4
g
(
x
)
=
−
x
2
−
6
x
+
4
from
x
=
−
4
x=-4
x
=
−
4
to
x
=
0
x=0
x
=
0
?
\newline
Choose
1
1
1
answer:
\newline
(A)
40
40
40
\newline
(B)
355
12
\frac{355}{12}
12
355
\newline
(C)
8
8
8
\newline
(D)
128
3
\frac{128}{3}
3
128
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What is the area of the region between the graphs of
f
(
x
)
=
x
+
10
f(x)=\sqrt{x+10}
f
(
x
)
=
x
+
10
and
g
(
x
)
=
x
−
2
g(x)=x-2
g
(
x
)
=
x
−
2
from
x
=
−
10
x=-10
x
=
−
10
to
x
=
6
x=6
x
=
6
?
\newline
Choose
1
1
1
answer:
\newline
(A)
64
3
\frac{64}{3}
3
64
\newline
(B)
160
160
160
\newline
(C)
320
3
\frac{320}{3}
3
320
\newline
(D)
128
128
128
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What is the area of the region between the graphs of
f
(
x
)
=
x
+
1
f(x)=\sqrt{x+1}
f
(
x
)
=
x
+
1
and
g
(
x
)
=
2
x
−
4
g(x)=2 x-4
g
(
x
)
=
2
x
−
4
from
x
=
0
x=0
x
=
0
to
x
=
3
x=3
x
=
3
?
\newline
Choose
1
1
1
answer:
\newline
(A)
5
3
\frac{5}{3}
3
5
\newline
(B)
−
3
-3
−
3
\newline
(C)
14
3
\frac{14}{3}
3
14
\newline
(D)
23
3
\frac{23}{3}
3
23
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What is the area of the region between the graphs of
f
(
x
)
=
x
2
+
12
x
f(x)=x^{2}+12 x
f
(
x
)
=
x
2
+
12
x
and
g
(
x
)
=
3
x
2
+
10
g(x)=3 x^{2}+10
g
(
x
)
=
3
x
2
+
10
from
x
=
1
x=1
x
=
1
to
x
=
4
x=4
x
=
4
?
\newline
Choose
1
1
1
answer:
\newline
(A)
64
3
\frac{64}{3}
3
64
\newline
(B)
77
77
77
\newline
(C)
45
45
45
\newline
(D)
18
18
18
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Let
g
(
x
)
=
3
x
4
+
8
x
3
+
4
g(x)=3 x^{4}+8 x^{3}+4
g
(
x
)
=
3
x
4
+
8
x
3
+
4
.
\newline
What is the absolute maximum value of
g
g
g
?
\newline
Choose
1
1
1
answer:
\newline
(A)
4
4
4
\newline
(B)
36
36
36
\newline
(C)
116
116
116
\newline
(D)
g
g
g
has no maximum value
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The differentiable functions
x
x
x
and
y
y
y
are related by the following equation:
\newline
x
y
=
4
x y=4
x
y
=
4
\newline
We are also given that
d
y
d
t
=
1
\frac{d y}{d t}=1
d
t
d
y
=
1
.
\newline
Find
d
x
d
t
\frac{d x}{d t}
d
t
d
x
when
y
=
0.5
y=0.5
y
=
0.5
.
Get tutor help
Let
h
(
x
)
=
{
x
+
26
−
5
x
+
1
for
x
≥
−
26
,
x
≠
−
1
k
for
x
=
−
1
h(x)=\left\{\begin{array}{ll}\frac{\sqrt{x+26}-5}{x+1} & \text { for } x \geq-26, x \neq-1 \\ k & \text { for } x=-1\end{array}\right.
h
(
x
)
=
{
x
+
1
x
+
26
−
5
k
for
x
≥
−
26
,
x
=
−
1
for
x
=
−
1
\newline
h
h
h
is continuous for all
x
>
−
26
x>-26
x
>
−
26
.
\newline
What is the value of
k
k
k
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
5
-\frac{1}{5}
−
5
1
\newline
(B)
1
1
1
\newline
(C)
1
10
\frac{1}{10}
10
1
\newline
(D)
0
0
0
Get tutor help
Let
f
(
x
)
=
x
−
1
−
2
x
−
5
f(x)=\frac{\sqrt{x-1}-2}{x-5}
f
(
x
)
=
x
−
5
x
−
1
−
2
when
x
≠
5
x \neq 5
x
=
5
.
\newline
f
f
f
is continuous for all
x
>
1
x>1
x
>
1
.
\newline
Find
f
(
5
)
f(5)
f
(
5
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
\frac{1}{2}
2
1
\newline
(B)
1
4
\frac{1}{4}
4
1
\newline
(C)
1
1
1
\newline
(D)
1
10
\frac{1}{10}
10
1
Get tutor help
f
(
x
)
=
{
sin
(
x
⋅
π
)
for
−
8
<
x
<
0
x
5
for
0
≤
x
≤
10
f(x)=\left\{\begin{array}{ll} \sin (x \cdot \pi) & \text { for }-8<x<0 \\ \frac{x}{5} & \text { for } 0 \leq x \leq 10 \end{array}\right.
f
(
x
)
=
{
sin
(
x
⋅
π
)
5
x
for
−
8
<
x
<
0
for
0
≤
x
≤
10
\newline
Find
lim
x
→
−
5
f
(
x
)
\lim _{x \rightarrow-5} f(x)
lim
x
→
−
5
f
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
0
0
0
\newline
(C)
1
1
1
\newline
(D) The limit doesn't exist.
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