Problem 13 - Applied Calculus

Question

Find the average value of each function over the given interval. \(f(x)=3\) on [10,50]

Step-by-Step Solution

Answer

The average value of the function is 3.

Step 1: Understand the Problem

We need to find the average value of the constant function \(f(x) = 3\) over the interval \([10, 50]\). A constant function means that the function value is the same at every point in the interval.

Step 2: Use the Average Value Formula

The average value of a function \(f(x)\) over an interval \([a, b]\) is given by the formula: \[ \text{Average Value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]. Here, \(a = 10\) and \(b = 50\), and \(f(x) = 3\).

Step 3: Set Up the Integral

For the constant function \(f(x) = 3\), the integral becomes \( \int_{10}^{50} 3 \, dx \).

Step 4: Compute the Integral

Calculating the integral of a constant, \( \int 3 \, dx = 3x + C\), over the interval \([10, 50]\), gives \( \left[ 3x \right]_{10}^{50} = 3(50) - 3(10)\).

Step 5: Evaluate the Integral

Calculate \(3(50) - 3(10)\) which results in \(150 - 30 = 120\).

Step 6: Find the Average Value

Substitute back into the average value formula: \[ \text{Average Value} = \frac{1}{50 - 10} \cdot 120 = \frac{1}{40} \cdot 120 = 3 \].

Key Concepts

Constant Function

A constant function is one of the simplest types of functions in mathematics. This function maintains the same output for any input in its domain. When we consider a constant function like \( f(x) = 3 \), it means that no matter what value of \( x \) you plug in, the output will always be 3. This idea makes calculations, such as finding the average or integrating, straightforward.

• A constant function does not change, which means its graph is a horizontal line.
• The slope of a constant function is zero, ensuring the function value remains the same at all points.
• In terms of real-life applications, constant functions can model situations where a value remains steady over time, such as a fixed rate or cost.

Definite Integral

The definite integral is a fundamental concept in calculus that allows us to calculate the accumulated value of a function over a specific interval. For example, when dealing with the function \( f(x) = 3 \), the definite integral computes the total quantity of the constant function over that specified range.
To integrate a constant function like \( f(x) = 3 \) from 10 to 50, you perform the integral calculation:

• For any constant \( k \) over an interval \([a, b]\), the integral is given by \( \int_{a}^{b} k \, dx = k(b-a) \).
• This simplifies the task as you multiply the constant by the length of the interval.

The definite integral allows not just for calculating areas under curves but helps in finding more practical solutions, such as average values, as needed in this exercise.

Average Value Formula

The average value of a function over an interval gives us a single value representing the entire function's behavior within that range. It's like boiling down all the values a function takes to one representative value. In mathematical terms, the average value of a function \( f(x) \) over an interval \([a, b]\) is computed using this vital formula:

• \[ \text{Average Value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]
• This formula divides the integral, representing the total accumulation, by the interval width \( b-a \), which balances the function's values over that span.
• For our specific function \( f(x) = 3 \), the constant nature simplifies the process, making the average value equal to the function's value (since there's no variation).

Utilizing this formula is critical in both pure mathematics and practical scenarios, where understanding an underlying average can illuminate trends or simplify complex analyses.