5. The function f is given by f(x) = 1 + 2sin(x). What is the average rate of change of f over the interval [0, pi/2]?A. 4/piB. 6/piC. 8/piD. 2
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 input 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 input values.
In this case, the function f(x) = 1 + 2sin(x), and we want to find the average rate of change over the interval [0, π/2].
First, let’s find the function values at the endpoints:
f(0) = 1 + 2sin(0) = 1 + 2(0) = 1
f(π/2) = 1 + 2sin(π/2) = 1 + 2(1) = 1 + 2 = 3
Next, let’s find the difference in the function values:
Δf = f(π/2) – f(0) = 3 – 1 = 2
Now, let’s find the difference in the input values:
Δx = π/2 – 0 = π/2
Finally, let’s calculate the average rate of change:
Average rate of change = Δf / Δx = 2 / (π/2) = 2 * 2/π = 4/π
So, the average rate of change of f over the interval [0, π/2] is 4/π.
Therefore, the correct answer is A. 4/π.
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