Problem 4 - Calculus Early Transcendentals

Question

Explain how to find the average value of a function on an interval \([a, b]\) and why this definition is analogous to the definition of the average of a set of numbers.

Step-by-Step Solution

Answer

Answer: The average value of a function provides us with a single representative value that has the same area under the curve as the original function within the specified interval. It can be calculated using the following formula: Average value of f(x) on [a, b] = (1 /(b-a)) ∫(a to b) f(x) dx. It involves determining the function f(x) and the interval [a, b], calculating the integral of the function over the interval, ∫(a to b) f(x) dx, and then dividing the integral by the width of the interval, (b - a).

Step 1: Introduce the concept of average value of a function

The average value of a function provides us with a single representative value that, if the function were a continuous line, would have the same area under the curve as the original function within the specified interval. Mathematically, the average value of a function \(f(x)\) on the interval \([a, b]\) is represented as follows: Average value of \(f(x)\) on \([a, b] = \frac{1}{b-a} \int_a^b f(x) dx\)

Step 2: Show the step-by-step process to find the average value

To find the average value of a function on the interval \([a, b]\), follow the steps below: 1. Determine the function \(f(x)\) and the interval \([a, b]\). 2. Calculate the integral of the function, \(\int_a^b f(x) dx\). 3. Divide the integral by the width of the interval, which is \((b - a)\), i.e. \(\frac{1}{b-a} \int_a^b f(x) dx\).

Step 3: Explain the analogy between average value of a function and average of a set of numbers

The definition of the average value of a function is analogous to the definition of the average of a set of numbers because both involve taking a weighted sum and dividing it by a measure of the size of the set. For a set of numbers, the average is calculated as follows: Average of \(n\) numbers = \(\frac{\text{sum of the numbers}}{\text{number of elements (n)}}\) For the average value of a function, we consider the function values on the interval as a continuous set, and the integral essentially represents a weighted sum of these values over the interval \([a, b]\). By dividing the sum (integral) by the size of the interval (\(b - a\)), we obtain an average value. In both cases, the sum is divided by a measure of the size of the set (number of elements for discrete data, length of the interval for continuous data), which is why these definitions are analogous.