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 f(x) over a given interval [a, b], you need to find the slope of the secant line that passes through the points (a, f(a)) and (b, f(b))
To find the average rate of change of a function f(x) over a given interval [a, b], you need to find the slope of the secant line that passes through the points (a, f(a)) and (b, f(b)).
In this case, the interval is [0, π], so a = 0 and b = π.
To find f(a), you substitute a = 0 into the function f(x):
f(0) = 1 + 3cos(0) = 1 + 3(1) = 1 + 3 = 4.
Similarly, to find f(b), you substitute b = π into the function f(x):
f(π) = 1 + 3cos(π) = 1 + 3(-1) = 1 – 3 = -2.
So, the points we need are (0, 4) and (π, -2).
The slope of the secant line passing through these two points is given by:
Average rate of change = (f(b) – f(a))/(b – a).
Plugging in the values, we get:
Average rate of change = (-2 – 4)/(π – 0).
Simplifying, we have:
Average rate of change = (-6)/(π) = -6/π.
Therefore, the average rate of change of f over the interval [0, π] is -6/π.
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