Finding the Average Value of a Function on a Given Interval

Introduction

As a high school mathematics teacher, one of the important concepts I teach my students is how to find the average value of a function on a given interval. This skill is essential for understanding and applying various mathematical concepts, such as integration, probability, and statistical analysis.

Understanding the Problem

In this article, we will focus on finding the average value of the function f(x) = x + 1 on the interval [0, 15].

The average value of a function f(x) on an interval [a, b] is defined as the integral of the function over the interval divided by the length of the interval. Mathematically, this can be expressed as:

Average value = (∫(a,b) f(x) dx) / (b - a)

where ∫(a,b) f(x) dx represents the integral of the function f(x) over the interval [a, b].

Step-by-Step Solution

1. Identify the function and the interval: The function is f(x) = x + 1, and the interval is [0, 15].

2. Compute the integral of the function over the interval: To find the integral of f(x) = x + 1 over the interval [0, 15], we can use the fundamental theorem of calculus:

   ∫(0,15) f(x) dx = ∫(0,15) (x + 1) dx = (x^2/2 + x)|_0^15 = (225/2 + 15) - (0/2 + 0) = 127.5

3. Compute the length of the interval: The length of the interval [0, 15] is 15 - 0 = 15.

4. Calculate the average value: Applying the formula for the average value, we get:

   Average value = (∫(0,15) f(x) dx) / (b - a) = 127.5 / 15 = 8.5

Conclusion

In this article, we have learned how to find the average value of a function on a given interval. By following the step-by-step solution, we were able to determine that the average value of the function f(x) = x + 1 on the interval [0, 15] is 8.5.

This skill is essential for various mathematical applications, and it is important for high school students to understand and practice it. By mastering this concept, students can develop a deeper understanding of mathematical functions and their properties, which will benefit them in their future mathematical studies and real-world problem-solving.