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 the function f(x) = 1 + 3cos(x) over the interval [0, π], we can use the formula for average rate of change:
Average rate of change = (f(a) – f(b)) / (a – b)
Where a and b represent the two endpoints of the interval
To find the average rate of change of the function f(x) = 1 + 3cos(x) over the interval [0, π], we can use the formula for average rate of change:
Average rate of change = (f(a) – f(b)) / (a – b)
Where a and b represent the two endpoints of the interval.
In this case, a = 0 and b = π. Let’s substitute these values into the formula:
Average rate of change = (f(0) – f(π)) / (0 – π)
Now, let’s find the values of f(0) and f(π).
f(0) = 1 + 3cos(0)
= 1 + 3(1)
= 1 + 3
= 4
f(π) = 1 + 3cos(π)
= 1 + 3(-1)
= 1 – 3
= -2
Substituting these values back into the formula for average rate of change:
Average rate of change = (4 – (-2)) / (0 – π)
Simplifying further:
Average rate of change = (4 + 2) / (-π)
= 6 / (-π)
Therefore, the average rate of change of f over the interval [0, π] is 6 / (-π).
Note that the negative sign indicates a decreasing trend in the function over the given interval.
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