To find the average rate of change of the function over the interval [tex]\( 40 \leq x \leq 55 \)[/tex], we need to consider the values of the function at the endpoints of the interval. The average rate of change of a function [tex]\( f \)[/tex] over the interval [tex]\([a, b]\)[/tex] is given by the formula:
[tex]\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \][/tex]
Here, [tex]\( a = 40 \)[/tex] and [tex]\( b = 55 \)[/tex]. From the given table, we have:
[tex]\[ f(40) = 17 \][/tex]
[tex]\[ f(55) = 11 \][/tex]
Let's substitute these values into the formula:
1. Calculate the difference in the function values:
[tex]\[ f(55) - f(40) = 11 - 17 = -6 \][/tex]
2. Calculate the difference in the [tex]\( x \)[/tex] values:
[tex]\[ 55 - 40 = 15 \][/tex]
3. Now, compute the average rate of change:
[tex]\[ \frac{f(55) - f(40)}{55 - 40} = \frac{-6}{15} \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{-6}{15} = -0.4 \][/tex]
So, the average rate of change of the function over the interval [tex]\( 40 \leq x \leq 55 \)[/tex] is [tex]\(-0.4\)[/tex].
