Respuesta :
Answer:
[tex]\frac{16}{\pi}[/tex]
Step-by-step explanation:
The average rate of a function [tex]f[/tex] on interval [tex]x=a[/tex] to [tex]x=b[/tex] is:
[tex]\frac{f(b)-f(a)}{b-a}[/tex].
[tex]a=0[/tex] and [tex]b=\frac{\pi}{4}[/tex] here for this problem where [tex]f(x)=-2\cos(4x)-3[/tex].
So we will need to evaluate [tex]f[/tex] for [tex]x=0 \text{ and } b=\frac{\pi}{4}[/tex].
To find [tex]f(0)[/tex] we will replace [tex]x[/tex] in [tex]f(x)=-2\cos(4x)-3[/tex] with [tex]0[/tex]:
[tex]f(0)=-2\cos(4(0))-3[/tex]
[tex]f(0)=-2\cos(0)-3[/tex]
[tex]f(0)=-2(1)-3[/tex]
[tex]f(0)=-2-3[/tex]
[tex]f(0)=-5[/tex]
To find [tex]f(\frac{\pi}{4})[/tex] we will replace [tex]x[/tex] in [tex]f(x)=-2\cos(4x)-3[/tex] with [tex]\frac{\pi}{4}[/tex]:
[tex]f(\frac{\pi}{4})=-2\cos(4(\frac{\pi}{4}))-3[/tex]
[tex]f(\frac{\pi}{4})=-2\cos(\pi)-3[/tex]
[tex]f(\frac{\pi}{4})=-2(-1)-3[/tex]
[tex]f(\frac{\pi}{4})=2-3[/tex]
[tex]f(\frac{\pi}{4})=-1[/tex]
So now we need to compute the change in [tex]f[/tex] over the change in [tex]x[/tex]:
[tex]\frac{-1-(-5)}{\frac{\pi}{4}-0}[/tex]
[tex]\frac{-1+5}{\frac{\pi}{4}}[/tex]
Dividing by [tex]\frac{\pi}{4}[/tex] is the same as multiplying by [tex]\frac{4}{\pi}[/tex]:
[tex](-1+5)\cdot \frac{4}{\pi}[/tex]
[tex]4 \cdot \frac{4}{\pi}[/tex]
[tex]\frac{16}{\pi}[/tex]
