Question
what is the average rate of change of the function f(x) = x^2 - 4x - 7 over the [2,5]
Answers
GPT-4o mini
To find the average rate of change of the function \( f(x) = x^2 - 4x - 7 \) over the interval \([2, 5]\), we use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 2 \) and \( b = 5 \).
First, we calculate \( f(2) \) and \( f(5) \):
1. Calculate \( f(2) \):
\[
f(2) = (2)^2 - 4(2) - 7 = 4 - 8 - 7 = -11
\]
2. Calculate \( f(5) \):
\[
f(5) = (5)^2 - 4(5) - 7 = 25 - 20 - 7 = -2
\]
Now, substitute these values into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(5) - f(2)}{5 - 2} = \frac{-2 - (-11)}{5 - 2} = \frac{-2 + 11}{3} = \frac{9}{3} = 3
\]
Thus, the average rate of change of the function \( f(x) = x^2 - 4x - 7 \) over the interval \([2, 5]\) is \( 3 \).
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 2 \) and \( b = 5 \).
First, we calculate \( f(2) \) and \( f(5) \):
1. Calculate \( f(2) \):
\[
f(2) = (2)^2 - 4(2) - 7 = 4 - 8 - 7 = -11
\]
2. Calculate \( f(5) \):
\[
f(5) = (5)^2 - 4(5) - 7 = 25 - 20 - 7 = -2
\]
Now, substitute these values into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(5) - f(2)}{5 - 2} = \frac{-2 - (-11)}{5 - 2} = \frac{-2 + 11}{3} = \frac{9}{3} = 3
\]
Thus, the average rate of change of the function \( f(x) = x^2 - 4x - 7 \) over the interval \([2, 5]\) is \( 3 \).