Question Statement

Find the average rate of change of the function $f(x)=\cos x$ on the interval $[-\pi ,\pi ]$.

---

Background and Explanation

The average rate of change measures how much a function changes per unit change in the input variable over a specific interval, representing the slope of the secant line connecting the endpoints. For a function $f(x)$ on $[a,b]$, this is calculated as $\frac{f(b)-f(a)}{b-a}$.

---

Solution

We begin by defining the function and setting up the average rate of change formula.

Let

$y=f(x)=\cos x$

The average rate of change in $y$ is given by the difference quotient:

$\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}=\frac{\cos (x+\Delta x)-\cos x}{\Delta x}$

Given the interval $[-\pi ,\pi ]$, we identify the starting point and the change in $x$:

• Let $x=-\pi$ (the left endpoint)
• Then $\Delta x=\pi -(-\pi )=2\pi$ (the length of the interval)

Substituting these values into equation (1):

$\frac{\Delta y}{\Delta x}=\frac{\cos (-\pi +2\pi )-\cos (-\pi )}{2\pi }$

Simplifying the first argument: $-\pi +2\pi =\pi$, so we have:

$=\frac{\cos (\pi )-\cos (-\pi )}{2\pi }$

Using the even function property of cosine, where $\cos (-\theta )=\cos (\theta )$:

$=\frac{\cos (\pi )-\cos (\pi )}{2\pi }$

Since $\cos (\pi )=-1$, this becomes:

$=\frac{-1-(-1)}{2\pi }=\frac{0}{2\pi }=0$

Thus, the rate of change of the function $f(x)=\cos x$ over the interval $[-\pi ,\pi ]$ is $\boxed{0}$.

---

Key Formulas or Methods Used

• Average rate of change formula: $\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$ or equivalently $\frac{f(b)-f(a)}{b-a}$ for interval $[a,b]$
• Even function property: $\cos (-\theta )=\cos (\theta )$
• Trigonometric evaluation: $\cos (\pi )=-1$

---

Summary of Steps

1. Define the function: Identify $f(x)=\cos x$ and the interval $[-\pi ,\pi ]$
2. Set up the formula: Write the average rate of change as $\frac{f(x+\Delta x)-f(x)}{\Delta x}$
3. Identify interval values: Set $x=-\pi$ and calculate $\Delta x=2\pi$
4. Substitute: Plug the values into the formula to obtain $\frac{\cos (\pi )-\cos (-\pi )}{2\pi }$
5. Apply identity: Use $\cos (-\theta )=\cos (\theta )$ to simplify to $\frac{\cos (\pi )-\cos (\pi )}{2\pi }$
6. Evaluate: Calculate the final result $\frac{0}{2\pi }=0$