(09.03 lc) what is the rate of change for f(x) = -2 cos 4x - 3 on the interval from x = 0 to x = π/4? o a…

Explanation:

Step1: Recall rate - of - change formula

The rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 0$, $b=\frac{\pi}{4}$, and $f(x)=-2\cos(4x)-3$.

Step2: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$: $f(0)=-2\cos(4\times0)-3=-2\cos(0)-3=-2\times1 - 3=-5$.

Step3: Calculate $f(\frac{\pi}{4})$

Substitute $x=\frac{\pi}{4}$ into $f(x)$: $f(\frac{\pi}{4})=-2\cos(4\times\frac{\pi}{4})-3=-2\cos(\pi)-3=-2\times(-1)-3=-1$.

Step4: Calculate the rate of change

Use the rate - of - change formula: $\frac{f(\frac{\pi}{4})-f(0)}{\frac{\pi}{4}-0}=\frac{-1-(-5)}{\frac{\pi}{4}}=\frac{-1 + 5}{\frac{\pi}{4}}=\frac{4}{\frac{\pi}{4}}=\frac{16}{\pi}$.

Answer:

C. $\frac{16}{\pi}$