Let f*f* be the function defined by f, of, x, equals, cosine, left parenthesis, start fraction, pi, divided by, 2, end fraction, x, right parenthesis, .*f*(*x*)=cos(2*π*​*x*). What is the average value of f*f* on the interval open square bracket, one half, comma, start fraction, 7, divided by, 2, end fraction, close square bracket[21​,27​] written in simplest form?

We are tasked with finding the average value of the function \( f(x) = \cos(2\pi x) \) on the interval \( \left[\frac{1}{2}, \frac{7}{2}\right] \).

### Step 1: Formula for the Average Value of a Function
The average value of a function \( f(x) \) over the interval \( [a, b] \) is given by:

\[
\text{Average value of } f(x) = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx
\]

Here, \( f(x) = \cos(2\pi x) \), \( a = \frac{1}{2} \), and \( b = \frac{7}{2} \).

### Step 2: Set Up the Integral
We need to compute the integral:

\[
\int_{\frac{1}{2}}^{\frac{7}{2}} \cos(2\pi x) \, dx
\]

### Step 3: Calculate the Integral
To integrate \( \cos(2\pi x) \), we use the formula for the integral of cosine:

\[
\int \cos(kx) \, dx = \frac{1}{k} \sin(kx)
\]

In this case, \( k = 2\pi \), so:

\[
\int \cos(2\pi x) \, dx = \frac{1}{2\pi} \sin(2\pi x)
\]

Now, we evaluate the definite integral:

\[
\int_{\frac{1}{2}}^{\frac{7}{2}} \cos(2\pi x) \, dx = \left[ \frac{1}{2\pi} \sin(2\pi x) \right]_{\frac{1}{2}}^{\frac{7}{2}}
\]

### Step 4: Evaluate the Definite Integral
Evaluate the sine at the limits of integration:

- At \( x = \frac{7}{2} \), we have \( 2\pi \cdot \frac{7}{2} = 7\pi \), so \( \sin(7\pi) = 0 \).
- At \( x = \frac{1}{2} \), we have \( 2\pi \cdot \frac{1}{2} = \pi \), so \( \sin(\pi) = 0 \).

Thus, the integral evaluates to:

\[
\frac{1}{2\pi} \left( 0 - 0 \right) = 0
\]

### Step 5: Average Value
Since the integral is zero, the average value of \( f(x) = \cos(2\pi x) \) over the interval \( \left[\frac{1}{2}, \frac{7}{2}\right] \) is:

\[
\text{Average value} = \frac{1}{\frac{7}{2} - \frac{1}{2}} \times 0 = \frac{1}{3} \times 0 = 0
\]

### Final Answer:
The average value of \( f(x) = \cos(2\pi x) \) on the interval \( \left[\frac{1}{2}, \frac{7}{2}\right] \) is \( \boxed{0} \).

Would you like further details or explanations on any part of the solution?

Here are some related questions:
1. How does the periodicity of the cosine function affect its average value over an interval?
2. What would the average value of \( \cos(x) \) be on the interval \( [0, 2\pi] \)?
3. How can you compute the average value of a function involving a sine or cosine wave over multiple periods?
4. How do you compute the integral of trigonometric functions in general?
5. What are some practical applications of calculating the average value of a function?

**Tip**: When working with trigonometric integrals, always check the periodicity of the function, as this can often simplify the computation of averages over certain intervals.

Let f(x) = cos(2πx). What is the average value of f(x) on the interval [1/2, 7/2] written in simplest form?

Learn how to calculate the average value of the function f(x) = cos(2πx) on the interval [1/2, 7/2] with a detailed solution. This problem explores concepts in trigonometry and integral calculus. Suitable for students in Grades 11-12.

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