I'll tackle the first one. The second one will be left for you as an exercise.
f(x)=cot x, [pi/6, pi/2]
To find the average rate of change over the interval, we divide the definite integral by the interval, namely:
Δf(x)/Δx
=∫f(x)dx / (π/2-π/6)
=∫cot(x)dx / (π/3)
=[ln(sin(x))] /(π/3)
=[0-ln(sin(π/6)]/(π/3)
=ln(2)/π/3
=3ln(2)/π
Find the average rate of change of the function over the given interval. You must reduce and simplify your answer, if possible.
a. f(x)=cot x, [pi/6, pi/2]
b. f(theta)= -2 + cos (theta), [-pi, 0]
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