The rate of change of the function f(x) = 0.1x² over the interval 1 ≤ x ≤ 4 is 0.5
The rate of change of a function represents how much the function is changing with respect to its input variable. To determine the rate of change of the function f(x) = 0.1x² over the interval 1 ≤ x ≤ 4, we need to find the average rate of change.
The average rate of change is calculated by finding the difference in the function values at the endpoints of the interval and dividing it by the difference in the input values.
First, we evaluate the function at the endpoints of the interval:
At x = 1: f(1) = 0.1(1)² = 0.1(1) = 0.1
At x = 4: f(4) = 0.1(4)² = 0.1(16) = 1.6
Next, we find the difference in the function values: 1.6 - 0.1 = 1.5
Then, we find the difference in the input values: 4 - 1 = 3
Finally, we divide the difference in the function values by the difference in the input values to find the average rate of change:
Average rate of change = (1.6 - 0.1) / (4 - 1) = 1.5 / 3 = 0.5
Therefore, the rate of change of the function f(x) = 0.1x² over the interval 1 ≤ x ≤ 4 is 0.5. This means that on average, for each unit increase in x within this interval, the function increases by 0.5