First, let's understand the table provided. The table gives values of [tex]\( D = f(t) \)[/tex], which represent the total US debt in billions of dollars [tex]\( t \)[/tex] years after 2000.
To calculate the average rate of change of the US debt between the years corresponding to [tex]\( t = 1 \)[/tex] and [tex]\( t = 5 \)[/tex], we need to evaluate the expression [tex]\(\frac{f(5) - f(1)}{5 - 1}\)[/tex].
Let's break this down into steps:
1. Identify [tex]\( f(5) \)[/tex] and [tex]\( f(1) \)[/tex] from the table:
- [tex]\( f(1) = 5817.1 \)[/tex] billion dollars.
- [tex]\( f(5) = 7960.4 \)[/tex] billion dollars.
2. Substitute these values into the expression [tex]\(\frac{f(5) - f(1)}{5 - 1}\)[/tex]:
[tex]\[
\frac{f(5) - f(1)}{5 - 1} = \frac{7960.4 - 5817.1}{5 - 1}
\][/tex]
3. Calculate the difference in the numerator:
[tex]\[
7960.4 - 5817.1 = 2143.3
\][/tex]
4. Calculate the difference in the denominator:
[tex]\[
5 - 1 = 4
\][/tex]
5. Compute the average rate of change:
[tex]\[
\frac{2143.3}{4} = 535.825
\][/tex]
6. Round the answer to one decimal place:
[tex]\[
535.825 \approx 535.8
\][/tex]
Therefore, the average rate of change of the US debt from [tex]\( t = 1 \)[/tex] to [tex]\( t = 5 \)[/tex] is [tex]\( \boxed{535.8} \)[/tex] billion dollars per year.
This tells us that between the years corresponding to [tex]\( t = 1 \)[/tex] (which is year 2001) and [tex]\( t = 5 \)[/tex] (which is year 2005), the US debt increased on average by approximately [tex]\( 535.8 \)[/tex] billion dollars per year.
