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Math Problems
Calculus
Find derivatives using the chain rule I
Determine the number of real solutions for the quadratic equation
2
x
2
−
3
x
+
1
=
0
2x^2 - 3x + 1 = 0
2
x
2
−
3
x
+
1
=
0
using the discriminant.
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Simplify:
g
(
x
)
=
1
−
∣
x
∣
2
−
∣
x
∣
g(x)=\frac{1-|x|}{2-|x|}
g
(
x
)
=
2
−
∣
x
∣
1
−
∣
x
∣
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Differentiate:
y
=
(
x
−
1
)
2
y=(x-1)^{2}
y
=
(
x
−
1
)
2
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lim
x
→
+
∞
x
ln
(
x
)
x
2
+
1
=
?
\lim _{x \rightarrow+\infty} \frac{\sqrt{x} \ln (x)}{x^{2}+1}=?
lim
x
→
+
∞
x
2
+
1
x
l
n
(
x
)
=
?
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Find the derivative of
y
=
tan
8
x
+
3
(
x
)
y=\tan ^{8 x+3}(x)
y
=
tan
8
x
+
3
(
x
)
.
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In the following equation, what is the value of
c
c
c
?
\newline
8
c
=
(
8
−
4
)
5
8^c = (8^{-4})^5
8
c
=
(
8
−
4
)
5
\newline
c
=
c =
c
=
____
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Find the derivative of
f
(
x
)
f(x)
f
(
x
)
. Where
f
(
x
)
=
e
(
x
+
1
)
f(x)=e^{(x+1)}
f
(
x
)
=
e
(
x
+
1
)
f
′
(
x
)
=
?
f'(x) = ?
f
′
(
x
)
=
?
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Solve.
\newline
x
2
=
2
x
+
1
x^{2}=2 x+1
x
2
=
2
x
+
1
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y
=
(
1
−
x
)
2
(
2
x
+
3
)
y=(1-x)^2(2x+3)
y
=
(
1
−
x
)
2
(
2
x
+
3
)
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0
=
−
(
y
2
−
2
y
+
x
)
0=-\left(y^{2}-2 y+x\right)
0
=
−
(
y
2
−
2
y
+
x
)
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y
=
(
−
x
2
−
16
x
)
−
30
y=\left(-x^{2}-16 x\right)-30
y
=
(
−
x
2
−
16
x
)
−
30
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y
=
−
(
x
+
7
)
2
−
9
y=-(x+7)^2-9
y
=
−
(
x
+
7
)
2
−
9
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Solve the equation:
y
=
x
2
+
8
x
y=x^{2}+8x
y
=
x
2
+
8
x
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What is the discriminant of the quadratic equation
−
x
2
−
x
−
2
=
0
-x^{2}-x-2=0
−
x
2
−
x
−
2
=
0
?
\newline
−
9
-9
−
9
\newline
−
7
-7
−
7
\newline
9
9
9
\newline
7
7
7
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f
(
x
)
=
−
(
x
+
2
)
2
+
16
f(x) = -(x+2)^2 + 16
f
(
x
)
=
−
(
x
+
2
)
2
+
16
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A(x)=(
8
8
8
+
2
2
2
x)^
2
2
2
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Find the inverse of each function.
\newline
f
(
x
)
=
−
2
−
3
x
2
f(x)=\frac{-2-3 x}{2}
f
(
x
)
=
2
−
2
−
3
x
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Find the derivative of
\newline
f
(
x
)
f(x)
f
(
x
)
.
\newline
f
(
x
)
=
e
x
x
f(x)=\frac{e^{x}}{x}
f
(
x
)
=
x
e
x
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Find the derivative
f
(
x
)
=
e
(
x
+
1
)
f(x)=e^{(x+1)}
f
(
x
)
=
e
(
x
+
1
)
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Evaluate.
\newline
−
3
3
⋅
(
1
375
)
1
3
=
\sqrt[3]{-3} \cdot\left(\frac{1}{375}\right)^{\frac{1}{3}}=
3
−
3
⋅
(
375
1
)
3
1
=
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Find the sum.
\newline
∑
k
=
1
38
(
6
k
−
105
)
=
\sum_{k=1}^{38}(6 k-105)=
k
=
1
∑
38
(
6
k
−
105
)
=
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Find
lim
x
→
3
f
(
x
)
\lim _{x \rightarrow 3} f(x)
lim
x
→
3
f
(
x
)
for
f
(
x
)
=
x
2
−
4
11
−
2
x
f(x)=\frac{x^{2}-4}{11-2 x}
f
(
x
)
=
11
−
2
x
x
2
−
4
.
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Find
lim
x
→
0
g
(
x
)
\lim _{x \rightarrow 0} g(x)
lim
x
→
0
g
(
x
)
for
g
(
x
)
=
4
x
2
−
15
4
x
+
3
g(x)=\frac{4 x^{2}-15}{4 x+3}
g
(
x
)
=
4
x
+
3
4
x
2
−
15
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∑
j
=
0
2
(
j
2
)
=
\sum_{j=0}^{2}\left(j^{2}\right)=
∑
j
=
0
2
(
j
2
)
=
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∑
k
=
1
3
(
k
−
4
)
=
\sum_{k=1}^{3}(k-4)=
∑
k
=
1
3
(
k
−
4
)
=
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∑
k
=
0
1
(
2
−
k
)
=
\sum_{k=0}^{1}(2-k)=
∑
k
=
0
1
(
2
−
k
)
=
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∑
n
=
0
2
(
n
)
=
\sum_{n=0}^{2}(n)=
∑
n
=
0
2
(
n
)
=
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∑
n
=
0
2
(
−
n
)
=
\sum_{n=0}^{2}(-n)=
∑
n
=
0
2
(
−
n
)
=
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∑
k
=
0
1
(
2
k
−
1
)
=
\sum_{k=0}^{1}(2 k-1)=
∑
k
=
0
1
(
2
k
−
1
)
=
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∑
x
=
1
2
(
6
x
)
=
\sum_{x=1}^{2}(6 x)=
∑
x
=
1
2
(
6
x
)
=
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∑
y
=
1
2
(
1
−
y
)
=
\sum_{y=1}^{2}(1-y)=
∑
y
=
1
2
(
1
−
y
)
=
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∑
t
=
0
1
(
4
−
t
)
=
\sum_{t=0}^{1}(4-t)=
∑
t
=
0
1
(
4
−
t
)
=
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∑
n
=
1
3
(
n
−
1
)
=
\sum_{n=1}^{3}(n-1)=
∑
n
=
1
3
(
n
−
1
)
=
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∑
x
=
1
3
(
4
x
)
=
\sum_{x=1}^{3}(4 x)=
∑
x
=
1
3
(
4
x
)
=
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∑
j
=
1
3
(
3
j
)
=
\sum_{j=1}^{3}(3 j)=
∑
j
=
1
3
(
3
j
)
=
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Polynomial function
g
g
g
is defined as
g
(
x
)
=
x
3
−
a
x
2
−
17
x
+
12
g(x)=x^{3}-a x^{2}-17 x+12
g
(
x
)
=
x
3
−
a
x
2
−
17
x
+
12
, where
a
a
a
is a constant. If
x
+
4
x+4
x
+
4
is a factor of the polynomial, then what is the value of
a
a
a
?
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Solve the following equation for
x
x
x
.
\newline
x
2
−
x
−
2
=
x
−
2
x
=
□
\begin{array}{l} \sqrt{x^{2}-x-2}=x-2 \\ x=\square \end{array}
x
2
−
x
−
2
=
x
−
2
x
=
□
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If
3
a
=
3
4
5
3^{a}=\sqrt[5]{3^{4}}
3
a
=
5
3
4
, what is the value of
a
a
a
?
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The equation
2
c
⋅
b
c
=
1
0
c
2^{c} \cdot b^{c}=10^{c}
2
c
⋅
b
c
=
1
0
c
is true for all values of
c
c
c
. What is the value of
b
b
b
?
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If
3
a
=
3
2
5
3^{a}=\sqrt[5]{3^{2}}
3
a
=
5
3
2
, what is the value of
a
a
a
?
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The equation
(
x
3
+
y
)
⋅
x
5
=
x
20
\left(x^{3+y}\right) \cdot x^{5}=x^{20}
(
x
3
+
y
)
⋅
x
5
=
x
20
is true for all values of
x
x
x
. What is the value of
y
y
y
?
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The equation
(
z
3
+
t
)
4
=
z
20
\left(z^{3+t}\right)^{4}=z^{20}
(
z
3
+
t
)
4
=
z
20
is true for all values of
z
z
z
. What is the value of
t
t
t
?
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Evaluate the expression
2
(
5
)
x
2(5)^{x}
2
(
5
)
x
for
x
=
2
x=2
x
=
2
.
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Evaluate the expression
2
x
−
x
2
2^{x}-x^{2}
2
x
−
x
2
for
x
=
5
x=5
x
=
5
.
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Evaluate the expression
10
(
3
)
x
10(3)^{x}
10
(
3
)
x
for
x
=
2
x=2
x
=
2
.
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Evaluate the expression
6
2
+
x
−
x
2
6^{2}+x-x^{2}
6
2
+
x
−
x
2
for
x
=
3
x=3
x
=
3
.
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Evaluate the expression
4
x
−
3
x
4^{x}-3^{x}
4
x
−
3
x
for
x
=
2
x=2
x
=
2
.
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Evaluate the expression
7
2
x
2
−
2
\frac{7^{2}}{x^{2}-2}
x
2
−
2
7
2
for
x
=
3
x=3
x
=
3
.
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Evaluate the expression
4
x
2
x
\frac{4^{x}}{2^{x}}
2
x
4
x
for
x
=
3
x=3
x
=
3
.
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What is the inverse of the function
\newline
g
(
x
)
=
−
2
(
x
−
4
)
?
g(x)=-2(x-4)?
g
(
x
)
=
−
2
(
x
−
4
)?
\newline
g
−
1
(
x
)
=
g^{-1}(x)=
g
−
1
(
x
)
=
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