the function ( f ) is given by ( f(x)=1 + 3cos x ). what is the average rate of change of ( f ) over the…

the function ( f ) is given by ( f(x)=1 + 3cos x ). what is the average rate of change of ( f ) over the interval ( 0,pi )?\na ( -\frac{6}{pi} )\nb ( -\frac{2}{pi} )\nc ( \frac{2}{pi} )\nd 1

the function ( f ) is given by ( f(x)=1 + 3cos x ). what is the average rate of change of ( f ) over the interval ( 0,pi )?\na ( -\frac{6}{pi} )\nb ( -\frac{2}{pi} )\nc ( \frac{2}{pi} )\nd 1

Answer

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). Here, (a = 0), (b=\pi), and (f(x)=1 + 3\cos x).

Step2: Calculate (f(0)) and (f(\pi))

  • For (x = 0): (f(0)=1+3\cos(0)). Since (\cos(0)=1), then (f(0)=1 + 3\times1=4).
  • For (x=\pi): (f(\pi)=1+3\cos(\pi)). Since (\cos(\pi)=- 1), then (f(\pi)=1+3\times(-1)=1 - 3=-2).

Step3: Substitute into the average - rate - of - change formula

(\frac{f(\pi)-f(0)}{\pi - 0}=\frac{-2 - 4}{\pi}=\frac{-6}{\pi})

Answer:

A. (-\frac{6}{\pi})