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Let’s check out your problem:
Find the derivative of
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
at
x
=
8
x = 8
x
=
8
.
\newline
Write your answer as an integer or a
fractions
" target="_blank" class="backlink">fraction. Simplify any fractions.
\newline
____
\newline
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Math Problems
Precalculus
Find values of derivatives using limits
Full solution
Q.
Find the derivative of
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
at
x
=
8
x = 8
x
=
8
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
Identify Function and Point:
Identify the function and the point where the derivative is needed.
\newline
Function:
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
\newline
Point:
x
=
8
x = 8
x
=
8
Calculate Derivative of
f
(
x
)
f(x)
f
(
x
)
:
Calculate the derivative of
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
.
\newline
The derivative of
x
x
x
with respect to
x
x
x
is
1
1
1
.
Evaluate Derivative at
x
=
8
x = 8
x
=
8
:
Evaluate the derivative at
x
=
8
x = 8
x
=
8
. Since the derivative of
x
x
x
is
1
1
1
, it remains
1
1
1
at any point, including
x
=
8
x = 8
x
=
8
.
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Consider the curve given by the equation
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y
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5
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y
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50
x y^{2}+5 x y=50
x
y
2
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y
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50
. It can be shown that
d
y
d
x
=
−
y
(
y
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5
)
x
(
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y
+
5
)
.
\frac{d y}{d x}=\frac{-y(y+5)}{x(2 y+5)} \text {. }
d
x
d
y
=
x
(
2
y
+
5
)
−
y
(
y
+
5
)
.
\newline
Write the equation of the vertical line that is tangent to the curve.
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What is the value of
d
d
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x
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d
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Question
The rate of change
d
P
d
t
\frac{d P}{d t}
d
t
d
P
of the number of algae in a tank is modeled by the following differential equation:
\newline
d
P
d
t
=
2317
10614
P
(
1
−
P
662
)
\frac{d P}{d t}=\frac{2317}{10614} P\left(1-\frac{P}{662}\right)
d
t
d
P
=
10614
2317
P
(
1
−
662
P
)
\newline
At
t
=
0
t=0
t
=
0
, the number of algae in the tank is
174
174
174
and is increasing at a rate of
28
28
28
algae per minute. At what value of
P
P
P
is
P
(
t
)
P(t)
P
(
t
)
growing the fastest?
\newline
Answer:
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Question
The rate of change
d
P
d
t
\frac{d P}{d t}
d
t
d
P
of the number of bacteria in a tank is modeled by the following differential equation:
\newline
d
P
d
t
=
2
9849
P
(
598
−
P
)
\frac{d P}{d t}=\frac{2}{9849} P(598-P)
d
t
d
P
=
9849
2
P
(
598
−
P
)
\newline
At
t
=
0
t=0
t
=
0
, the number of bacteria in the tank is
196
196
196
and is increasing at a rate of
16
16
16
bacteria per minute. At what value of
P
P
P
does the graph of
P
(
t
)
P(t)
P
(
t
)
have an inflection point?
\newline
Answer:
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Posted 1 year ago
Question
The rate of change
d
P
d
t
\frac{d P}{d t}
d
t
d
P
of the number of people infected by a disease is modeled by the following differential equation:
\newline
d
P
d
t
=
45
125404
P
(
800
−
P
)
\frac{d P}{d t}=\frac{45}{125404} P(800-P)
d
t
d
P
=
125404
45
P
(
800
−
P
)
\newline
At
t
=
0
t=0
t
=
0
, the number of people infected by the disease is
214
214
214
and is increasing at a rate of
45
45
45
people per hour. What is the limiting value for the total number of people infected by the disease as time increases?
\newline
Answer:
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Posted 1 year ago
Question
The exponential function
f
f
f
is graphed in the
x
y
x y
x
y
-plane. As
x
x
x
increases by
1
,
y
1, y
1
,
y
increases by a factor of
3
3
3
. Which of the following could be
f
f
f
?
\newline
Choose
1
1
1
answer:
\newline
(A)
f
(
x
)
=
(
1
3
)
x
f(x)=\left(\frac{1}{3}\right)^{x}
f
(
x
)
=
(
3
1
)
x
\newline
(B)
f
(
x
)
=
(
1
3
)
x
+
3
f(x)=\left(\frac{1}{3}\right)^{x}+3
f
(
x
)
=
(
3
1
)
x
+
3
\newline
(C)
f
(
x
)
=
3
x
+
2
f(x)=3^{x}+2
f
(
x
)
=
3
x
+
2
\newline
(D)
f
(
x
)
=
2
(
3
)
x
f(x)=2(3)^{x}
f
(
x
)
=
2
(
3
)
x
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Posted 7 months ago
Question
Which of the following is a correct interpretation of the expression
\newline
−
4
+
13
-4+13
−
4
+
13
?
\newline
Choose
1
1
1
answer:
\newline
(A) The number that is
4
4
4
to the left of
−
13
-13
−
13
on the number line
\newline
(B) The number that is
4
4
4
to the right of
−
13
-13
−
13
on the number line
\newline
(C) The number that is
13
13
13
to the left of
−
4
-4
−
4
on the number line
\newline
(D) The number that is
13
13
13
to the right of
−
4
-4
−
4
on the number line
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Posted 1 year ago
Question
The function
f
f
f
is defined by
f
(
x
)
=
x
3
+
2
sin
(
3
x
−
3
)
f(x)=x^{3}+2 \sin (3 x-3)
f
(
x
)
=
x
3
+
2
sin
(
3
x
−
3
)
. Use a calculator to write the equation of the line tangent to the graph of
f
f
f
when
x
=
0.5
x=0.5
x
=
0.5
. You should round all decimals to
3
3
3
places.
\newline
Answer:
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Posted 1 year ago
Question
The function
f
f
f
is defined by
f
(
x
)
=
x
2
−
2
x
+
3
cos
(
x
2
−
x
)
f(x)=x^{2}-2 x+3 \cos \left(x^{2}-x\right)
f
(
x
)
=
x
2
−
2
x
+
3
cos
(
x
2
−
x
)
. Use a calculator to write the equation of the line tangent to the graph of
f
f
f
when
x
=
−
2.5
x=-2.5
x
=
−
2.5
. You should round all decimals to
3
3
3
places.
\newline
Answer:
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Posted 1 year ago
Question
The function
f
f
f
is defined by
f
(
x
)
=
x
2
+
x
−
2
sin
(
2
x
)
f(x)=x^{2}+x-2 \sin (2 x)
f
(
x
)
=
x
2
+
x
−
2
sin
(
2
x
)
. Use a calculator to write the equation of the line tangent to the graph of
f
f
f
when
x
=
3
x=3
x
=
3
. You should round all decimals to
3
3
3
places.
\newline
Answer:
Get tutor help
Posted 1 year ago
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