The average rate of change of a function over an interval [a, b] can be calculated using the formula:
Average Rate of Change = (f(b) – f(a)) / (b – a)
In this case, we need to find the average rate of change of the function f(x) = x² + 2x + 3 over the interval [4, 6].
First, let’s calculate f(4):
f(4) = (4)² + 2(4) + 3
f(4) = 16 + 8 + 3 = 27
Next, we calculate f(6):
f(6) = (6)² + 2(6) + 3
f(6) = 36 + 12 + 3 = 51
Now we can use these results to find the average rate of change:
Average Rate of Change = (f(6) – f(4)) / (6 – 4)
Average Rate of Change = (51 – 27) / (2)
Average Rate of Change = 24 / 2 = 12
So, the average rate of change of f(x) over the interval [4, 6] is 12.