To find the average rate of change of [tex]\(g(x)\)[/tex] from [tex]\(x=0\)[/tex] to [tex]\(x=7\)[/tex], we need to follow a few steps.
1. Identify the values of [tex]\(g(x)\)[/tex] at [tex]\(x=0\)[/tex] and [tex]\(x=7\)[/tex]:
- From the table, at [tex]\(x=0\)[/tex], [tex]\(g(0) = 1\)[/tex].
- At [tex]\(x=7\)[/tex], [tex]\(g(7) = 5\)[/tex].
2. Calculate the change in [tex]\(g(x)\)[/tex]:
- Change in [tex]\(g(x)\)[/tex] = [tex]\(g(7) - g(0)\)[/tex].
- Substituting the values, Change in [tex]\(g(x)\)[/tex] = [tex]\(5 - 1 = 4\)[/tex].
3. Determine the change in [tex]\(x\)[/tex]:
- The change in [tex]\(x\)[/tex] from [tex]\(x=0\)[/tex] to [tex]\(x=7\)[/tex] is [tex]\(7 - 0 = 7\)[/tex].
4. Calculate the average rate of change:
- The average rate of change of [tex]\(g\)[/tex] is given by the formula: [tex]\(\frac{\text{Change in } g(x)}{\text{Change in } x}\)[/tex].
- Substituting the changes:
[tex]\[
\frac{\text{Change in } g(x)}{\text{Change in } x} = \frac{4}{7}
\][/tex]
So, the average rate of change of [tex]\(g(x)\)[/tex] from [tex]\(x=0\)[/tex] to [tex]\(x=7\)[/tex] is approximately [tex]\(0.5714285714285714\)[/tex].
