The average rate of change of a function between two points is calculated using the formula:
Average Rate of Change = \( \frac{f(b) – f(a)}{b – a} \)
In this case, we are looking at the function:
f(x) = 2x + 1
And we want to find the average rate of change from \(x = 5\) to \(x = 10\). Here, \(a = 5\) and \(b = 10\).
First, we need to evaluate the function at both points:
- For \(x = 5\):
- \(f(5) = 2(5) + 1 = 10 + 1 = 11\)
- For \(x = 10\):
- \(f(10) = 2(10) + 1 = 20 + 1 = 21\)
Now we can substitute these values into our average rate of change formula:
Average Rate of Change = \( \frac{f(10) – f(5)}{10 – 5} = \frac{21 – 11}{10 – 5} = \frac{10}{5} = 2 \)
Thus, the average rate of change of the function \(f(x) = 2x + 1\) from \(x = 5\) to \(x = 10\) is 2.