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,
- \( b \) is the ending point of the interval,
- \( f(a) \) and \( f(b) \) are the function’s values at these points.
In this case, we need to evaluate the function:
\( f(x) = 3x^2 + 6x + 2 \)
First, we calculate \( f(6) \) and \( f(8) \):
Step 1: Calculate \( f(6) \)
\( f(6) = 3(6)^2 + 6(6) + 2 \)
\( = 3(36) + 36 + 2 \)
\( = 108 + 36 + 2 = 146 \)
Step 2: Calculate \( f(8) \)
\( f(8) = 3(8)^2 + 6(8) + 2 \)
\( = 3(64) + 48 + 2 \)
\( = 192 + 48 + 2 = 242 \)
Step 3: Use the average rate of change formula
Now, we can find the average rate of change over the interval from 6 to 8:
\( a = 6, b = 8 \)
\( f(b) = f(8) = 242 \)
\( f(a) = f(6) = 146 \)
\( ext{Average Rate of Change} = \frac{f(8) – f(6)}{8 – 6} = \frac{242 – 146}{2} = \frac{96}{2} = 48 \)
Conclusion:
The average rate of change of the function \( f(x) \) over the interval from \( 6 \) to \( 8 \) is \( 48 \).