When decreases in one variable are accompanied by decreases in another variable, the variables are described as?
1 answer:
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Answer:
Step-by-step explanation:
First open the brackets
Then collect the like terms
Solve the like terms
The simplified form is
Answer:
The Taylor series is .
The radius of convergence is .
Step-by-step explanation:
<em>The Taylor expansion.</em>
Recall that as we want the Taylor series centered at its expression is given in powers of
. With this in mind we need to do some transformations with the goal to obtain the asked Taylor series from the Taylor expansion of
.
Then,
Now, in order to make a more compact notation write . Thus, the above expression becomes
Notice that, if x is very close from 3, then y is very close from 0. Then, we can use the Taylor expansion of the logarithm. Hence,
Now, substitute in the previous equality. Thus,
<em>Radius of convergence.</em>
We find the radius of convergence with the Cauchy-Hadamard formula:
Where stands for the coefficients of the Taylor series and
for the radius of convergence.
In this case the coefficients of the Taylor series are
and in consequence . Then,
Applying the properties of roots
Hence,
Recall that
So, as we get that
.
Data: (Cylinder)
h (height) = 8 cm
r (radius) = 5 cm
Adopting:
V (volume) = ?
Solving:(<span>Cylinder volume)
</span>
<span>Note: Now, let's find the volume of a hemisphere.
</span>
Data: (hemisphere volume)
V (volume) = ?
r (radius) = 5 cm
Adopting:
If: We know that the volume of a sphere is
, but we have a hemisphere, so the formula will be half the volume of the hemisphere 
Formula: (<span>Volume of the hemisphere)
</span>
Solving:
<span>Now, to find the total volume of the figure, add the values: (cylinder volume + hemisphere volume)
</span>
Volume of the figure = cylinder volume + hemisphere volume
Volume of the figure = 628 cm³ + 261.6 cm³
h (height) = 8 cm
r (radius) = 5 cm
Adopting:
V (volume) = ?
Solving:(<span>Cylinder volume)
</span>
<span>Note: Now, let's find the volume of a hemisphere.
</span>
Data: (hemisphere volume)
V (volume) = ?
r (radius) = 5 cm
Adopting:
If: We know that the volume of a sphere is
Formula: (<span>Volume of the hemisphere)
</span>
Solving:
<span>Now, to find the total volume of the figure, add the values: (cylinder volume + hemisphere volume)
</span>
Volume of the figure = cylinder volume + hemisphere volume
Volume of the figure = 628 cm³ + 261.6 cm³