The average rate of change of a function over a specific interval can be calculated using the formula:
Average Rate of Change = \( \frac{f(b) – f(a)}{b – a} \)
Where:
- \( a \) is the starting point of the interval (in this case, 4).
- \( b \) is the ending point of the interval (in this case, 7).
- \( f(x) \) is the function we’re working with, which is \( f(x) = 3x^2 \).
To find the average rate of change from x = 4 to x = 7, we first need to calculate the function values at these points:
- Calculate \( f(4) \):
\( f(4) = 3(4)^2 = 3(16) = 48 \) - Calculate \( f(7) \):
\( f(7) = 3(7)^2 = 3(49) = 147 \)
Now, we can plug these values into the average rate of change formula:
Average Rate of Change = \( \frac{147 – 48}{7 – 4} \)
This simplifies to:
Average Rate of Change = \( \frac{99}{3} = 33 \)
Therefore, the average rate of change of the function \( f(x) = 3x^2 \) over the interval from x = 4 to x = 7 is 33.