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Math Problems
Calculus
Linear approximation
Find the linearization of
f
(
x
)
=
5
x
3
−
3
x
2
f(x) = 5x^3 - 3x^2
f
(
x
)
=
5
x
3
−
3
x
2
at
x
=
2
x = 2
x
=
2
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
______
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Find the linearization of
f
(
x
)
=
55
x
f(x) = 55x
f
(
x
)
=
55
x
at
x
=
1
x = 1
x
=
1
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
_
_
\_\_
__
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Find the linearization of
f
(
x
)
=
x
5
f(x) = x^5
f
(
x
)
=
x
5
at
x
=
2
x = 2
x
=
2
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
______
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Find the linearization of
f
(
x
)
=
3
x
4
+
x
2
f(x) = 3x^4 + x^2
f
(
x
)
=
3
x
4
+
x
2
at
x
=
−
1
x = -1
x
=
−
1
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
______
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Find the linearization of
f
(
x
)
=
4
x
f(x) = 4\sqrt{x}
f
(
x
)
=
4
x
at
x
=
16
x = 16
x
=
16
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
______
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Find the linearization of
f
(
x
)
=
x
f(x) = \sqrt{x}
f
(
x
)
=
x
at
x
=
9
x = 9
x
=
9
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
______
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Find the linearization of
f
(
x
)
=
x
2
3
f(x) = x^{\frac{2}{3}}
f
(
x
)
=
x
3
2
at
x
=
27
x = 27
x
=
27
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
______
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Find the linearization of
f
(
x
)
=
x
3
f(x) = \sqrt[3]{x}
f
(
x
)
=
3
x
at
x
=
1
x = 1
x
=
1
. Write an exact answer.
\newline
L(x) = ______
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Find the linearization of
f
(
x
)
=
x
4
−
x
3
f(x) = x^4 - x^3
f
(
x
)
=
x
4
−
x
3
at
x
=
2
x = 2
x
=
2
. Write an exact answer.
\newline
L(x) = ______
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Find the linearization of
f
(
x
)
=
2
x
3
−
3
x
2
f(x) = 2x^3 - 3x^2
f
(
x
)
=
2
x
3
−
3
x
2
at
x
=
2
x = 2
x
=
2
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
_
_
_
_
_
\_\_\_\_\_
_____
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Find the linearization of
f
(
x
)
=
x
2
−
4
x
+
5
f(x) = x^2 - 4x + 5
f
(
x
)
=
x
2
−
4
x
+
5
at
x
=
3
x = 3
x
=
3
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
______
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Find the linearization of
f
(
x
)
=
x
1
4
f(x) = x^{\frac{1}{4}}
f
(
x
)
=
x
4
1
at
x
=
16
x = 16
x
=
16
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
______
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Find the linearization of
f
(
x
)
=
1
x
2
f(x) = \frac{1}{x^2}
f
(
x
)
=
x
2
1
at
x
=
2
x = 2
x
=
2
. Write an exact answer.
\newline
L(x) = ______
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Find the linearization of
f
(
x
)
=
2
x
f(x) = \frac{2}{x}
f
(
x
)
=
x
2
at
x
=
1
x = 1
x
=
1
. Write an exact answer.
\newline
L
(
x
)
=
L(x) =
L
(
x
)
=
______
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Let
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
be a differentiable function such that
d
y
d
x
=
x
y
\frac{d y}{d x}=\frac{x}{y}
d
x
d
y
=
y
x
and
f
(
8
)
=
2
f(8)=2
f
(
8
)
=
2
. What is the approximation of
f
(
8.1
)
f(8.1)
f
(
8.1
)
using the line tangent to the graph of
f
f
f
at
x
=
8
x=8
x
=
8
?
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Find the equation of the tangent line to the graph of
y
(
x
)
=
4
cot
6
(
x
)
y(x)=4 \cot ^{6}(x)
y
(
x
)
=
4
cot
6
(
x
)
at
x
=
π
4
x=\frac{\pi}{4}
x
=
4
π
.
\newline
Equation of the tangent line is
y
=
y=
y
=
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Question
\newline
Find the equation of the line tangent to the graph of
f
(
x
)
=
x
−
1
f(x)=\sqrt{x-1}
f
(
x
)
=
x
−
1
at
x
=
5
x=5
x
=
5
.
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g
(
n
)
=
−
11
⋅
4
n
g(n)=-11 \cdot 4^{n}
g
(
n
)
=
−
11
⋅
4
n
\newline
Complete the recursive formula of
g
(
n
)
g(n)
g
(
n
)
.
\newline
g
(
1
)
=
_
_
_
_
_
g(1)=\_\_\_\_\_
g
(
1
)
=
_____
\newline
g
(
n
)
=
g
(
n
−
1
)
.
_
_
_
_
_
g(n)=g(n-1) \text {. } \_\_\_\_\_
g
(
n
)
=
g
(
n
−
1
)
.
_____
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Let
f
f
f
be an exponential function of the form
f
(
x
)
=
a
b
x
f(x)=a b^{x}
f
(
x
)
=
a
b
x
. If
f
(
0
)
=
4
f(0)=4
f
(
0
)
=
4
and
f
(
3
)
=
108
f(3)=108
f
(
3
)
=
108
, what is the equation of
f
f
f
?
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Solve for the exact value of
x
x
x
.
\newline
6
ln
(
4
x
+
3
)
+
19
=
37
6 \ln (4 x+3)+19=37
6
ln
(
4
x
+
3
)
+
19
=
37
\newline
Answer:
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{
f
(
1
)
=
37
f
(
n
)
=
f
(
n
−
1
)
⋅
0.3
\begin{cases} f(1)=37 \\\\ f(n)=f(n-1)\cdot 0.3 \end{cases}
⎩
⎨
⎧
f
(
1
)
=
37
f
(
n
)
=
f
(
n
−
1
)
⋅
0.3
\newline
Find an explicit formula for f(n).
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The graph of
f
(
x
)
=
10
(
0.75
)
x
f(x)=10(0.75)^x
f
(
x
)
=
10
(
0.75
)
x
for
x
≥
0
x\geq0
x
≥
0
is shown. What is the initial value of the function?
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Given
f
(
x
)
=
−
x
2
−
12
f(x)=-x^{2}-12
f
(
x
)
=
−
x
2
−
12
, find
f
(
4
)
f(4)
f
(
4
)
\newline
Answer:
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3
3
3
. Find the tangent line to
f
(
x
)
=
f(x)=
f
(
x
)
=
(
2
−
4
x
2
)
5
\left(2-4 x^{2}\right)^{5}
(
2
−
4
x
2
)
5
at
x
=
1
x=1
x
=
1
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For the following equation, evaluate
f
′
(
3
)
f^{\prime}(3)
f
′
(
3
)
.
\newline
f
(
x
)
=
−
5
x
+
1
f(x)=-5 x+1
f
(
x
)
=
−
5
x
+
1
\newline
Answer:
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For the function
f
(
x
)
=
8
x
2
−
2
x
+
11
f(x)=8 x^{2}-2 x+11
f
(
x
)
=
8
x
2
−
2
x
+
11
, find the slope of the tangent line at
x
=
11
x=11
x
=
11
.
\newline
Answer:
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For the function
f
(
x
)
=
−
10
x
2
−
10
x
−
8
f(x)=-10 x^{2}-10 x-8
f
(
x
)
=
−
10
x
2
−
10
x
−
8
, find the slope of the tangent line at
x
=
−
10
x=-10
x
=
−
10
.
\newline
Answer:
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Let
x
2
+
y
2
=
25
x^{2}+y^{2}=25
x
2
+
y
2
=
25
.
\newline
What is the value of
d
2
y
d
x
2
\frac{d^{2} y}{d x^{2}}
d
x
2
d
2
y
at the point
(
4
,
3
)
(4,3)
(
4
,
3
)
?
\newline
Give an exact number.
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