To compare the average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex] with the given average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15 \)[/tex], we need to understand the meaning of the average rate of change and how it might vary between different intervals. The average rate of change of a function [tex]\( f \)[/tex] from [tex]\( x = a \)[/tex] to [tex]\( x = b \)[/tex] is calculated as:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
Given that the average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15 \)[/tex] is approximately [tex]\(-4.667\)[/tex], we want to compare this to the average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex]. Here is a detailed step-by-step explanation:
1. Average Rate of Change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15 \)[/tex]:
The average rate of change is given as [tex]\(-4.667\)[/tex]. This means that on average, the function [tex]\( f(x) \)[/tex] decreases by 4.667 units for every 1 unit increase in [tex]\( x \)[/tex] over the interval from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15 \)[/tex].
2. Average Rate of Change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex]:
We need to consider how the function [tex]\( f(x) \)[/tex] behaves from [tex]\( x = 15 \)[/tex] to [tex]\( x = 20 \)[/tex] in addition to the given rate from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15 \)[/tex]. The average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex] will factor in the addition of the new interval.
3. Given Result:
From the answer you provided, given the context of server room problems, the total sum of changes up to a certain day or span typically accumulates regularly. Here, we acknowledge the total impact of adding not only new installations but also calculating the average effect considering the continuity.
Since the average rate of change over longer intervals generally encompasses additional data points or values that might affect the function more pronouncedly or smoothly,
The average rate of change from [tex]\( x=0 \)[/tex] to [tex]\( x=20 \)[/tex] is:
- __Decreasing more slowly__ than the average rate of change from [tex]\( x=0 \)[/tex] to [tex]\( x=15 \)[/tex].
Considering its balanced cumulative rate to the added servers number effect, this implication means the rate alteration is steady but averaged less negative over the further interval elongation.
Thus, the correct statement is:
The average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex] is decreasing less rapidly than the average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15 \)[/tex].
