Determining the Average Rate of Change of a Function over an Interval | Calculation and Explanation

The function f is given by f(x)=1+3cosx. What is the average rate of change of f over the interval [0,π] ?

To find the average rate of change of a function over an interval, we need to calculate the difference in the values of the function at the endpoints of the interval and divide it by the difference in the inputs

To find the average rate of change of a function over an interval, we need to calculate the difference in the values of the function at the endpoints of the interval and divide it by the difference in the inputs.

In this case, the function is given by f(x) = 1 + 3cos(x), and the interval we are interested in is [0, π]. Therefore, we need to find the values of f(π) and f(0) and calculate the difference between them.

To find f(π), we substitute π into the function:
f(π) = 1 + 3cos(π)
Since cosine of π is -1, we have:
f(π) = 1 + 3(-1) = 1 – 3 = -2

To find f(0), we substitute 0 into the function:
f(0) = 1 + 3cos(0)
Since cosine of 0 is 1, we have:
f(0) = 1 + 3(1) = 1 + 3 = 4

Now, we calculate the difference in the function values:
f(π) – f(0) = -2 – 4 = -6

Next, we calculate the difference in the inputs:
π – 0 = π

Finally, we find the average rate of change by dividing the difference in the function values by the difference in the inputs:
Average rate of change = (f(π) – f(0)) / (π – 0) = -6 / π

Therefore, the average rate of change of the function f over the interval [0,π] is -6 / π.

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