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Over which interval are the exponential and linear functions approximately the same?

A. from 0.25 to 0.5
B. from 0.5 to 0.75
C. from 0.75 to 1.0
D. from 1.25 to 1.5

Sagot :

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].
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