Find the mean, median and mode of f(x) = sin x[0,pi/2]

To find the **mean**, **median**, and **mode** of the function \( f(x) = \sin(x) \) on the interval \( \left[ 0, \frac{\pi}{2} \right] \), we'll follow these steps:

### 1. Mean
The **mean** of a function over an interval is given by:

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

For \( f(x) = \sin(x) \) on the interval \( \left[ 0, \frac{\pi}{2} \right] \), we have:

\[
\text{Mean} = \frac{1}{\frac{\pi}{2} - 0} \int_{0}^{\frac{\pi}{2}} \sin(x) \, dx = \frac{2}{\pi} \int_{0}^{\frac{\pi}{2}} \sin(x) \, dx
\]

We know that the integral of \( \sin(x) \) is \( -\cos(x) \), so:

\[
\int_{0}^{\frac{\pi}{2}} \sin(x) \, dx = \left[ -\cos(x) \right]_{0}^{\frac{\pi}{2}} = -\cos\left( \frac{\pi}{2} \right) + \cos(0) = 0 + 1 = 1
\]

Thus, the mean is:

\[
\text{Mean} = \frac{2}{\pi} \times 1 = \frac{2}{\pi}
\]

### 2. Median
The **median** is the value \( m \) such that half of the total area under the curve lies to the left of \( m \). Mathematically, we find \( m \) such that:

\[
\int_0^m \sin(x) \, dx = \frac{1}{2} \int_0^{\frac{\pi}{2}} \sin(x) \, dx
\]

We already computed that the total area under the curve is 1, so we want:

\[
\int_0^m \sin(x) \, dx = \frac{1}{2}
\]

Solving this:

\[
-\cos(m) + \cos(0) = \frac{1}{2} \implies -\cos(m) + 1 = \frac{1}{2} \implies \cos(m) = \frac{1}{2}
\]

So:

\[
m = \cos^{-1}\left( \frac{1}{2} \right) = \frac{\pi}{3}
\]

Thus, the median is \( \frac{\pi}{3} \).

### 3. Mode
The **mode** is the value of \( x \) that maximizes the function. Since \( \sin(x) \) is increasing on \( \left[ 0, \frac{\pi}{2} \right] \), the maximum value occurs at \( x = \frac{\pi}{2} \), so the mode is:

\[
\text{Mode} = \frac{\pi}{2}
\]

### Summary:
- **Mean**: \( \frac{2}{\pi} \)
- **Median**: \( \frac{\pi}{3} \)
- **Mode**: \( \frac{\pi}{2} \)

Would you like further clarification or more details?

Here are 5 related questions for you to consider:
1. How would the mean change if the interval was \( \left[0, \pi \right] \)?
2. What is the effect of modifying the function to \( f(x) = \cos(x) \) on the same interval?
3. Can you find the variance of \( f(x) = \sin(x) \) on the interval \( \left[ 0, \frac{\pi}{2} \right] \)?
4. What is the geometric interpretation of the median in this context?
5. How does the shape of the \( \sin(x) \) function affect its mode?

**Tip**: When calculating means over an interval, always remember to divide by the length of the interval after integrating.

Find the mean, median and mode of f(x) = sin(x) on the interval [0, pi/2]

Explore the detailed solution to finding the mean, median, and mode of the trigonometric function f(x) = sin(x) on the interval [0, pi/2]. This problem uses integral calculus and function analysis, ideal for students in undergraduate-level calculus courses.

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