Final answer:
The average rate of change of cos x from 0 to 4π/3 is found using the formula [f(b) - f(a)] / (b - a), where f is cos x, a = 0, and b = 4π/3. The calculation yields an average rate of change of -1/2, making option (b) the correct answer.
Step-by-step explanation:
The question asks us to find the average rate of change of cos x from 0 to 4π/3. To do this, we can use the formula for the average rate of change, which is given by Δy/Δx, where Δy is the change in the function values and Δx is the change in x-values over the interval. In terms of a function f(x), this can be written as [f(b) - f(a)] / (b - a), where a and b are the endpoints of the interval.
Applying this to the cosine function, we have a = 0 and b = 4π/3. The cosine of 0 is 1, and the cosine of 4π/3 is -0.5. Thus, the change in the function's value (Δy) is -0.5 - 1 = -1.5. The change in x (Δx) is 4π/3 - 0 = 4π/3.
Therefore, the average rate of change is (-1.5) / (4π/3). When we simplify this expression, we get the average rate of change as -1/2. Thus, the answer to the question is (b) -1/2