The function f is given by f(x)=1+3cosx. What is the average rate of change of f over the interval [0,π] ?
-6/πThe difference quotient [f(π)−f(0)]/(π−0) is the average rate of change of ff over the interval [0,π][0,π].
To find the average rate of change of a function f(x) over an interval [a,b], we use the formula:
average rate of change = (f(b) – f(a))/(b – a)
In this case, our interval is [0,π], so a=0 and b=π. Therefore, we need to find:
average rate of change = (f(π) – f(0))/(π – 0)
To find f(π), we simply plug π into the expression for f:
f(π) = 1 + 3cos(π) = 1 – 3 = -2
To find f(0), we similarly plug 0 into the expression for f:
f(0) = 1 + 3cos(0) = 1 + 3 = 4
Now we can plug these values into our formula for the average rate of change:
average rate of change = (-2 – 4)/(π – 0) = -6/π
Therefore, the average rate of change of f over the interval [0,π] is -6/π.
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