To determine over which interval the exponential function [tex]\(f(x) = e^x\)[/tex] and the linear function [tex]\(g(x) = x\)[/tex] are approximately the same, we need to compare the differences between these two functions over several specified intervals.
We are given the following intervals to consider:
1. From 0.25 to 0.5
2. From 0.5 to 0.75
3. From 0.75 to 1.0
4. From 1.25 to 1.5
Let's analyze the differences over these intervals.
The average differences between [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] over the given intervals are:
1. From 0.25 to 0.5: [tex]\(1.0838600827835139\)[/tex]
2. From 0.5 to 0.75: [tex]\(1.248213425685215\)[/tex]
3. From 0.75 to 1.0: [tex]\(1.5302536494605297\)[/tex]
4. From 1.25 to 1.5: [tex]\(2.5905928532946856\)[/tex]
We need to identify the interval where the average difference between the exponential and linear functions is the smallest. Based on the provided averages, the differences are as follows:
- The interval from 0.25 to 0.5 has the smallest average difference: [tex]\(1.0838600827835139\)[/tex].
Thus, we conclude that the interval over which the exponential function [tex]\(e^x\)[/tex] and the linear function [tex]\(x\)[/tex] are approximately the same is from [tex]\(0.25\)[/tex] to [tex]\(0.5\)[/tex].
