Question

what is the average rate of change of the function f(x) = x^2 - 4 - 7 over the [2,5]

Answer 1

To find the average rate of change of the function \( f(x) = x^2 - 4 - 7 \) over the interval \([2, 5]\), we first need to simplify the function:

\[
f(x) = x^2 - 11
\]

The average rate of change of a function \( f \) over the interval \([a, b]\) is given by the formula:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

In this case, \( a = 2 \) and \( b = 5 \). We will calculate \( f(2) \) and \( f(5) \):

1. Calculate \( f(2) \):

\[
f(2) = 2^2 - 11 = 4 - 11 = -7
\]

2. Calculate \( f(5) \):

\[
f(5) = 5^2 - 11 = 25 - 11 = 14
\]

Now we can plug these values into the average rate of change formula:

\[
\text{Average Rate of Change} = \frac{f(5) - f(2)}{5 - 2} = \frac{14 - (-7)}{5 - 2} = \frac{14 + 7}{3} = \frac{21}{3} = 7
\]

Thus, the average rate of change of the function \( f(x) = x^2 - 4 - 7 \) over the interval \([2, 5]\) is **7**.