What is the rate of change for f(x) = - 2 cos(4x - 3) on the interval from x = pi/4 to x = pi/2?

Solution:

The average rate of change of a function f(x) is equivalent to the slope of the secant line connecting the two points on the function on the interval [a, b].

It can be expressed as:

Average rate of change = [f(b) - f(a)] / b - a

Given f(x) = - 2 cos(4x - 3), on the interval [π/ 4, π/ 2]

f (π/2) = - 2 × cos (4 (π/2) - 3)

= - 2 × cos (2π - 3)

f (π/2) = 2 cos 3 [ ∵ cos (2π - θ) = cos θ]

f (π/4) = - 2 × cos (4(π/4) - 3)

= - 2 × cos (π - 3)

f (π/4) = - 2 cos 3 [ ∵ cos (π - θ) = - cos θ]

The rate of change = \(\dfrac{2 \cos 3 - (- 2 \cos 3)}{\dfrac{π}{2} - \dfrac{π}{4}}\)

= (4 cos 3)/ π/4 = (16 cos 3)/ π

What is the rate of change for f(x) = - 2 cos(4x - 3) on the interval from x = pi/4 to x = pi/2?

Summary: 

The rate of change for the function f(x) = - 2 cos(4x - 3) on the interval from [π/ 4, π/ 2] is (16 cos 3)/ π.