3. f(x) = 1/4 cos 1/8 x\namplitude:\ndomain:\nperiod:\nrange:\nfrequency:\nfind the average rate of change…

Explanation:

Step1: Find the amplitude

For the function $y = A\cos(Bx)$, the amplitude is $|A|$. Here $A=\frac{1}{4}$, so the amplitude is $\left|\frac{1}{4}\right|=\frac{1}{4}$.

Step2: Find the period

The period of the function $y = A\cos(Bx)$ is $T=\frac{2\pi}{|B|}$. Here $B = \frac{1}{8}$, so $T=\frac{2\pi}{\frac{1}{8}}=16\pi$.

Step3: Find the frequency

The frequency $f=\frac{1}{T}$. Since $T = 16\pi$, then $f=\frac{1}{16\pi}$.

Step4: Find the domain

The domain of the cosine - type function $y=\frac{1}{4}\cos\frac{1}{8}x$ is all real numbers, i.e., $(-\infty,\infty)$.

Step5: Find the range

The range of the cosine function $y = \cos t$ is $[- 1,1]$. For $y=\frac{1}{4}\cos\frac{1}{8}x$, when $\cos\frac{1}{8}x=-1$, $y =-\frac{1}{4}$; when $\cos\frac{1}{8}x = 1$, $y=\frac{1}{4}$. So the range is $\left[-\frac{1}{4},\frac{1}{4}\right]$.

Step6: Find the 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 $a = 0$, $b=\pi$, $f(x)=\frac{1}{4}\cos\frac{1}{8}x$. Then $f(0)=\frac{1}{4}\cos(0)=\frac{1}{4}$, $f(\pi)=\frac{1}{4}\cos\frac{\pi}{8}$. The average rate of change is $\frac{\frac{1}{4}\cos\frac{\pi}{8}-\frac{1}{4}}{\pi - 0}=\frac{\frac{1}{4}(\cos\frac{\pi}{8}-1)}{\pi}=\frac{\cos\frac{\pi}{8}-1}{4\pi}$.

Answer:

amplitude: $\frac{1}{4}$ period: $16\pi$ frequency: $\frac{1}{16\pi}$ domain: $(-\infty,\infty)$ range: $\left[-\frac{1}{4},\frac{1}{4}\right]$ average rate of change over $[0,\pi]$: $\frac{\cos\frac{\pi}{8}-1}{4\pi}$