Question

Find the average value of the function f(x)=−4sin(x) on the interval [π2,3π2] and determine a number c in this interval for which f(x) is equal to the average value.

Answer 1

Answer:

Step-by-step explanation:

The average value theorem sets:

if f (x) is continuous in [a, b] and derivable in (a, b) there is a c Є (a, b) such that

$\frac{f(b)-f(a)}{b-a}=f'(c)$ , where

f(a)=f(π/2)=-4*sin(π/2) = -4*1= -4

f(b)=(3π/2)=-4*sin(3π/2) = -4*-1 = 4

$\frac{4-(-4)}{(3\pi/2)-(\pi/2)}=f'(c)$

$\frac{8}{\pi }=f'(c)$

$f'(x)=-4cos(x)$ ⇒

$f'(c)=-4cos(c)=\frac{8}{\pi }\\c=acos(\frac{-2}{\pi })$

c≅130