Finding the Average Rate of Change of a Function over an Interval: Calculation and Example with f(x) = 1 + 3cos(x)

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 function values at the endpoints of the interval and divide it by the difference in the x-values

To find the average rate of change of a function over an interval, we need to calculate the difference in the function values at the endpoints of the interval and divide it by the difference in the x-values. In this case, we have the function f(x) = 1 + 3cos(x) and the interval [0, π].

First, let’s find the function values at the endpoints of the interval:
f(0) = 1 + 3cos(0) = 1 + 3(1) = 1 + 3 = 4
f(π) = 1 + 3cos(π) = 1 + 3(-1) = 1 – 3 = -2

Now, let’s find the difference in the function values:
f(π) – f(0) = -2 – 4 = -6

Next, let’s find the difference in the x-values:
π – 0 = π

Finally, we can calculate the average rate of change:
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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