Bot GPT 4-o mini To calculate the average rate of change of the function \( f(x) = 3x^2 \) over the interval \( 1 \leq x \leq 5 \), we can use the formula:
\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = 1 \) and \( b = 5 \).
First, we need to calculate \( f(1) \) and \( f(5) \):
\[ f(1) = 3(1^2) = 3(1) = 3 \]
\[ f(5) = 3(5^2) = 3(25) = 75 \]
Now we can plug these values into our formula:
\[ \text{Average rate of change} = \frac{f(5) - f(1)}{5 - 1} = \frac{75 - 3}{5 - 1} \]
Calculating the difference in the numerator:
\[ 75 - 3 = 72 \]
Calculating the difference in the denominator:
\[ 5 - 1 = 4 \]
Now we can substitute these values back into the formula:
\[ \text{Average rate of change} = \frac{72}{4} = 18 \]
It appears that none of the answer choices reflect the average rate of change we calculated. However, upon reviewing, the calculations are correct:
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Calculate \( f(1) \): \[ f(1) = 3(1^2) = 3 \]
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Calculate \( f(5) \): \[ f(5) = 3(5^2) = 75 \]
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Determine the average rate of change: \[ \text{Average rate of change} = \frac{75 - 3}{5 - 1} = \frac{72}{4} = 18 \]
Since the given answer options do not include 18, it might be worth verifying the problem statement or the options given.
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