The 5-unit moving average of the function g(x)=16−x^3

To compute the 5-unit moving average of the function \( g(x) = 16 - x^3 \), we need to average the values of the function over 5 consecutive units.

A moving average of a function is typically defined as:

\[
\text{Moving Average}(x) = \frac{1}{n} \sum_{i=0}^{n-1} g(x - i)
\]

Here, \( n = 5 \) (since it's a 5-unit moving average) and \( g(x) = 16 - x^3 \).

So, the 5-unit moving average of \( g(x) \) is:

\[
\text{Moving Average}(x) = \frac{1}{5} \left[ g(x) + g(x-1) + g(x-2) + g(x-3) + g(x-4) \right]
\]

Substitute \( g(x) = 16 - x^3 \) into this equation:

\[
\text{Moving Average}(x) = \frac{1}{5} \left[ (16 - x^3) + (16 - (x-1)^3) + (16 - (x-2)^3) + (16 - (x-3)^3) + (16 - (x-4)^3) \right]
\]

Now expand and simplify each cubic expression:

\[
\text{Moving Average}(x) = \frac{1}{5} \left[ 16 - x^3 + 16 - (x-1)^3 + 16 - (x-2)^3 + 16 - (x-3)^3 + 16 - (x-4)^3 \right]
\]

This is the general formula for the 5-unit moving average of \( g(x) \). From here, you can calculate specific values at given points of \( x \).

Would you like me to simplify this further or compute it at specific points?

---

### Related Questions:
1. How do you calculate moving averages for different window sizes (e.g., 3-unit moving average)?
2. How does the behavior of the moving average change with different functions?
3. Can moving averages be applied to non-polynomial functions?
4. How does the moving average relate to smoothing data?
5. How would this moving average look graphically when plotted?

**Tip**: Moving averages are commonly used to smooth out fluctuations in data or functions over a specific range.

Learn how to calculate the 5-unit moving average of the function g(x) = 16 − x^3. This problem explores key calculus and algebra concepts, showing how to apply the moving average formula to a cubic function. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation