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.