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Math Problems
Precalculus
Find trigonometric ratios of special angles
Write inequalities to represent the situations below.
\newline
The temperature inside the lab refrigerator is no more than
45
∘
F
45{ }^{\circ} \mathrm{F}
45
∘
F
.
\newline
Use
t
\mathbf{t}
t
to represent the temperature (in
∘
F
{ }^{\circ} \mathrm{F}
∘
F
) of the refrigerator.
\newline
□
\square
□
\newline
The cruising speed of the bullet train will be at least
170
170
170
miles per hour. Use
s
\mathbf{s}
s
to represent the train's cruising speed (in miles per hour).
\newline
□
\square
□
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4
(
1
−
t
)
=
5
t
2
4(1-t)=5t^{2}
4
(
1
−
t
)
=
5
t
2
\newline
Let
t
=
x
t=x
t
=
x
and
t
=
y
t=y
t
=
y
be the solutions to the given equation.
\newline
What is the value of
−
x
y
-xy
−
x
y
?
\newline
□
\square
□
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z
=
1
−
2
i
z=1-2 i
z
=
1
−
2
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta = \square ^{\circ}
θ
=
□
∘
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The curve
y
2
sin
x
−
2
x
=
9
−
π
y^{2} \sin x-2 x=9-\pi
y
2
sin
x
−
2
x
=
9
−
π
passes through
(
π
/
2
,
3
)
(\pi / 2,3)
(
π
/2
,
3
)
. Use local linearization to estimate the value of
y
y
y
at
x
=
1.67
x=1.67
x
=
1.67
. Use
1
1
1
.
57
57
57
for
π
/
2
\pi / 2
π
/2
. Round to
2
2
2
d.p.
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In the circle shown, an arc is subtended by a central angle measuring
5
π
6
\frac{5 \pi}{6}
6
5
π
radians. What fraction of the circumference is this arc?
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Using implicit differentiation, find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
x
y
=
7
+
x
2
y
2
\sqrt{x y}=7+x^{2} y^{2}
x
y
=
7
+
x
2
y
2
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Write the log equation as an exponential equation. You do not need to solve for
x
\mathrm{x}
x
.
\newline
ln
(
6
)
=
x
−
4
\ln (6)=x-4
ln
(
6
)
=
x
−
4
\newline
Answer:
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Using implicit differentiation, find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
x
y
=
−
4
+
x
y
3
\sqrt{x y}=-4+x y^{3}
x
y
=
−
4
+
x
y
3
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The system has no solution.
\newline
The system has a unique solution:
\newline
(
x
,
y
)
=
(
□
,
□
)
(x,y)=(\square,\square)
(
x
,
y
)
=
(
□
,
□
)
\newline
\begin{cases}x+2y=8\-x-2y=8\end{cases}
\newline
The system has infinitely many solutions.
\newline
They must satisfy the following equation:
\newline
y=
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The derivative of the function
f
f
f
is defined by
f
′
(
x
)
=
x
2
−
3
x
+
3
sin
(
2
x
+
1
)
f^{\prime}(x)=x^{2}-3 x+3 \sin (2 x+1)
f
′
(
x
)
=
x
2
−
3
x
+
3
sin
(
2
x
+
1
)
. Find the
x
x
x
values, if any, in the interval
−
2.5
<
x
<
2.5
-2.5<x<2.5
−
2.5
<
x
<
2.5
where the function
f
f
f
has a relative minimum. You may use a calculator and round all values to
3
3
3
decimal places.
\newline
Answer:
x
=
x=
x
=
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The derivative of the function
f
f
f
is defined by
f
′
(
x
)
=
x
2
+
2
x
−
3
sin
(
3
x
)
f^{\prime}(x)=x^{2}+2 x-3 \sin (3 x)
f
′
(
x
)
=
x
2
+
2
x
−
3
sin
(
3
x
)
. Find the
x
x
x
values, if any, in the interval
−
2.5
<
x
<
3
-2.5<x<3
−
2.5
<
x
<
3
where the function
f
f
f
has a relative minimum. You may use a calculator and round all values to
3
3
3
decimal places.
\newline
Answer:
x
=
x=
x
=
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Given the function
f
(
x
)
=
−
4
x
f(x)=-\frac{4}{\sqrt{x}}
f
(
x
)
=
−
x
4
, find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
. Express your answer in radical form without using negative exponents, simplifying all fractions.
\newline
Answer:
f
′
(
x
)
=
f^{\prime}(x)=
f
′
(
x
)
=
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Given the function
y
=
1
6
x
y=\frac{1}{6 \sqrt{x}}
y
=
6
x
1
, find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
. Express your answer in radical form without using negative exponents, simplifying all fractions.
\newline
Answer:
d
y
d
x
=
\frac{d y}{d x}=
d
x
d
y
=
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Given the function
f
(
x
)
=
−
3
x
2
f(x)=-\frac{3 \sqrt{x}}{2}
f
(
x
)
=
−
2
3
x
, find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
. Express your answer in radical form without using negative exponents, simplifying all fractions.
\newline
Answer:
f
′
(
x
)
=
f^{\prime}(x)=
f
′
(
x
)
=
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Given the function
y
=
3
x
3
y=3 \sqrt{x^{3}}
y
=
3
x
3
, find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
. Express your answer in radical form without using negative exponents, simplifying all fractions.
\newline
Answer:
d
y
d
x
=
\frac{d y}{d x}=
d
x
d
y
=
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Let
f
f
f
be the function defined by
f
(
x
)
=
x
2
f(x)=x^{2}
f
(
x
)
=
x
2
. If four subintervals of equal length are used, what is the value of the left Riemann sum approximation for
∫
1
2
x
2
d
x
\int_{1}^{2} x^{2} d x
∫
1
2
x
2
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Find the average value of the function
f
(
x
)
=
8
x
−
3
f(x)=\frac{8}{x-3}
f
(
x
)
=
x
−
3
8
from
x
=
5
x=5
x
=
5
to
x
=
7
x=7
x
=
7
. Write your answer as the logarithm of a single number in simplest form.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
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Find the average value of the function
f
(
x
)
=
4
6
−
x
f(x)=\frac{4}{6-x}
f
(
x
)
=
6
−
x
4
from
x
=
7
x=7
x
=
7
to
x
=
9
x=9
x
=
9
. Write your answer as the logarithm of a single number in simplest form.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
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Find the average value of the function
f
(
x
)
=
8
x
−
1
f(x)=\frac{8}{x-1}
f
(
x
)
=
x
−
1
8
from
x
=
3
x=3
x
=
3
to
x
=
7
x=7
x
=
7
. Write your answer as the logarithm of a single number in simplest form.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
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Simplify
e
ln
12
−
ln
6
e^{\ln 12-\ln 6}
e
l
n
12
−
l
n
6
and write without any logarithms.
\newline
Answer:
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