Have you ever wondered how to find the average value of a function over a specific interval? It's a common problem in calculus and has practical applications in various fields. In this guide, we'll walk through the process step-by-step, using the function f(x) = x² + x - 6 over the interval [0, 12] as our example. So, buckle up, guys, and let's dive into the world of average function values!
Understanding the Average Value Theorem
Before we jump into the calculations, let's quickly review the Average Value Theorem. This theorem is the foundation for finding the average value of a function. Essentially, it states that for a continuous function f(x) on a closed interval [a, b], there exists a value c within that interval such that the function's value at c (f(c)) is equal to the average value of the function over the entire interval. In simpler terms, there's at least one point in the interval where the function's height matches its average height over the interval. This concept is crucial for understanding the significance of the average value we're about to calculate.
The average value itself represents the height of a rectangle that has the same width as the interval [a, b] and the same area as the area under the curve of f(x) between a and b. Think of it as "flattening" the curve into a rectangle with an equivalent area. This geometric interpretation helps visualize what the average value represents. The theorem assures us that such a value exists, and our calculations will pinpoint exactly what that average value is for our given function and interval. So, with this understanding in mind, let's move on to the formula and the practical steps to find the average value.
The Formula for Average Value
The formula for calculating the average value of a function f(x) over the interval [a, b] is given by:
Average Value = (1 / (b - a)) ∫[a to b] f(x) dx
Where:
- f(x) is the function
- [a, b] is the interval
- ∫[a to b] f(x) dx represents the definite integral of f(x) from a to b
This formula might look a bit intimidating at first, but let's break it down. The integral part, ∫[a to b] f(x) dx, calculates the area under the curve of f(x) between the points a and b. The term (1 / (b - a)) is simply dividing this area by the width of the interval (b - a). This division effectively gives us the average height of the function over that interval, which is the average value we're looking for. Understanding each component of the formula makes the calculation process much clearer and less daunting. So, with the formula in hand, we can now apply it to our specific example and find the average value of f(x) = x² + x - 6 over the interval [0, 12].
Applying the Formula to Our Example: f(x) = x² + x - 6 on [0, 12]
Now, let's apply the formula to our specific example. We have f(x) = x² + x - 6 and the interval [0, 12]. This means a = 0 and b = 12. Let's plug these values into the formula:
Average Value = (1 / (12 - 0)) ∫[0 to 12] (x² + x - 6) dx
First, we need to evaluate the definite integral ∫[0 to 12] (x² + x - 6) dx. Remember, integrating a polynomial involves increasing the exponent of each term by 1 and then dividing by the new exponent. Applying this to our function, we get:
∫ (x² + x - 6) dx = (x³/3) + (x²/2) - 6x + C
Where C is the constant of integration. However, since we are dealing with a definite integral, we don't need to worry about C as it will cancel out when we evaluate the integral at the limits of integration. Now, we need to evaluate this expression at x = 12 and x = 0 and subtract the results:
[(12³/3) + (12²/2) - 6(12)] - [(0³/3) + (0²/2) - 6(0)]
Simplifying this expression will give us the area under the curve. This area, when divided by the width of the interval, will give us the average value of the function. So, let's crunch those numbers and see what we get!
Evaluating the Definite Integral
Let's continue with our calculation. We have:
[(12³/3) + (12²/2) - 6(12)] - [(0³/3) + (0²/2) - 6(0)]
First, let's simplify the expression inside the first set of brackets:
(12³/3) = (1728 / 3) = 576
(12²/2) = (144 / 2) = 72
6(12) = 72
So, the first part becomes:
576 + 72 - 72 = 576
The second part, [(0³/3) + (0²/2) - 6(0)], is simply 0.
Therefore, the definite integral ∫[0 to 12] (x² + x - 6) dx = 576 - 0 = 576. This value represents the area under the curve of our function f(x) = x² + x - 6 between x = 0 and x = 12. Now that we have the area, we can plug it back into our average value formula and complete our calculation. We're almost there, guys! Just one more step to go.
Calculating the Average Value
We've calculated the definite integral ∫[0 to 12] (x² + x - 6) dx to be 576. Now, let's plug this back into our average value formula:
Average Value = (1 / (12 - 0)) ∫[0 to 12] (x² + x - 6) dx
Average Value = (1 / 12) * 576
Average Value = 48
Therefore, the average value of f(x) = x² + x - 6 on the interval [0, 12] is 48. This means that the average height of the function's curve over this interval is 48 units. We've successfully found the average value! Pat yourselves on the back, guys, we've tackled a calculus problem and come out victorious.
Conclusion
Finding the average value of a function might seem challenging at first, but by understanding the Average Value Theorem and applying the formula step-by-step, it becomes a manageable task. We successfully calculated the average value of f(x) = x² + x - 6 on the interval [0, 12] to be 48. Remember, the key is to break down the problem into smaller parts: first, evaluate the definite integral to find the area under the curve, and then divide by the width of the interval. With practice, you'll become a pro at finding average values! This concept has various applications in fields like physics, engineering, and economics, so mastering it is a valuable skill. Keep practicing, guys, and happy calculating!