7. After boiling, the 18 liters of water was reduced to 7 liters. How many water was
1 answer:
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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
.
The correct answer is actually the last one.
The second derivative gives us information about the concavity of a function: if
then the function is concave downwards in that point, whereas if
then the function is concave upwards in that point.
This already shows why the first option is wrong - if the function was concave downwards for all x, then the second derivate would have been negative for all x, which isn't the case, because we have, for example,
Also, the second derivative gives no information about specific points of the function. Suppose, in fact, that passes through the origin, so
. Now translate the function upwards, for example. we have that
doesn't pass through the origin, but the second derivative is always
. So, the second option is wrong as well.
Now, about the last two. The answer you chose would be correct if the exercise was about the first derivative . In fact, the first derivative gives information about the increasing or decreasing behaviour of the function - positive and negative derivative, respectively. So, if the first derivative is negative before a certain point and positive after that point. It means that the function is decreasing before that point, and increasing after. So, that point is a relative minimum.
But in this exercise we're dealing with second derivative, so we don't have information about the increasing/decreasing behaviour. Instead, we know that the second derivative is negative before zero - which means that the function is concave downwards before zero - and positive after zero - which means that the function is concave upwards after zero.
A point where the function changes its concavity is called a point of inflection, which is the correct answer.