Question
Find the average rate of change of f(x)=3x2 over the interval 3,5. Write your answer as an integer, fraction, or decimal rounded to the nearest tenth. Simplify any fractions.
Answers
GPT-4o mini
To find the average rate of change of the function \( f(x) = 3x^2 \) over the interval \( [3, 5] \), we can use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 3 \) and \( b = 5 \).
First, we compute \( f(3) \) and \( f(5) \):
1. Calculate \( f(3) \):
\[
f(3) = 3(3^2) = 3 \cdot 9 = 27
\]
2. Calculate \( f(5) \):
\[
f(5) = 3(5^2) = 3 \cdot 25 = 75
\]
Now substitute these values into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(5) - f(3)}{5 - 3} = \frac{75 - 27}{5 - 3} = \frac{48}{2} = 24
\]
Thus, the average rate of change of \( f(x) = 3x^2 \) over the interval \( [3, 5] \) is
\[
\boxed{24}
\]
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 3 \) and \( b = 5 \).
First, we compute \( f(3) \) and \( f(5) \):
1. Calculate \( f(3) \):
\[
f(3) = 3(3^2) = 3 \cdot 9 = 27
\]
2. Calculate \( f(5) \):
\[
f(5) = 3(5^2) = 3 \cdot 25 = 75
\]
Now substitute these values into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(5) - f(3)}{5 - 3} = \frac{75 - 27}{5 - 3} = \frac{48}{2} = 24
\]
Thus, the average rate of change of \( f(x) = 3x^2 \) over the interval \( [3, 5] \) is
\[
\boxed{24}
\]