let be the function given by . what is the average value of on the closed interval ?

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30-03-2025

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Let's explore how to determine the average value of a function over a specified interval. This concept is crucial in various fields, including calculus and statistics. We'll use the example of the function f(x) = x² + 1 on the closed interval [0, 2].

Understanding Average Value of a Function

The average value of a function, f(x), on a closed interval [a, b] is given by the following formula:

(1/b-a) ∫_a^b f(x) dx

This formula essentially calculates the definite integral of the function over the interval and then divides by the length of the interval. The definite integral represents the area under the curve, and dividing by the interval's length gives us the average height of the function over that span.

Step-by-Step Calculation for f(x) = x² + 1 on [0, 2]

Let's apply this to our specific function, f(x) = x² + 1, on the interval [0, 2].

Step 1: Find the definite integral:

First, we need to compute the definite integral of f(x) from 0 to 2:

∫₀² (x² + 1) dx = [x³/3 + x]₀²

Step 2: Evaluate the integral at the limits:

Plugging in the upper and lower limits of integration, we get:

[ (2³/3 + 2) - (0³/3 + 0) ] = (8/3 + 2) = 14/3

Step 3: Divide by the interval length:

The length of the interval [0, 2] is 2 - 0 = 2. Now, we divide the result from Step 2 by this length:

(14/3) / 2 = 7/3

Therefore, the average value of f(x) = x² + 1 on the closed interval [0, 2] is 7/3.

Visualizing the Average Value

Imagine the graph of f(x) = x² + 1. The average value of 7/3 represents the height of a rectangle with a width equal to the interval length (2) and an area equal to the area under the curve of f(x) from 0 to 2.

Applications and Further Exploration

This method of finding the average value of a function has broad applications in various fields:

• Physics: Calculating average velocity or acceleration.
• Engineering: Determining average stress or strain on a structure.
• Economics: Finding average cost or revenue over a period.

Understanding how to calculate and interpret the average value of a function is a fundamental concept in calculus with significant practical applications. Further exploration could involve examining more complex functions or intervals, and exploring the connection between the average value and the Mean Value Theorem for Integrals.