Calculate Average Rate Of Change Of F(x) = X^2 - 1/x On [2, 4]
Introduction
In calculus, the average rate of change of a function over an interval represents the slope of the secant line connecting the endpoints of the function on that interval. This concept is fundamental to understanding the behavior of functions and serves as a precursor to the derivative, which gives the instantaneous rate of change. In this article, we will delve into the process of calculating the average rate of change for a given function, specifically f(x) = x^2 - 1/x over the interval [2, 4]. We'll break down the steps involved, explain the underlying principles, and provide a clear, concise solution to the problem. Understanding the average rate of change is crucial not only for calculus but also for various applications in physics, engineering, economics, and other fields where quantifying change over an interval is essential. So, let's embark on this mathematical journey and unravel the intricacies of this concept.
Understanding Average Rate of Change
The average rate of change is a fundamental concept in calculus that quantifies how a function's output changes relative to its input over a specific interval. To grasp this concept fully, let's break it down. Imagine a curve representing a function f(x) on a graph. Now, consider an interval [a, b] on the x-axis. The average rate of change over this interval is essentially the slope of the line connecting the points (a, f(a)) and (b, f(b)). This line is called the secant line. Mathematically, the average rate of change is calculated using the formula:
Average Rate of Change = (f(b) - f(a)) / (b - a)
Here, f(b) represents the function's value at the end of the interval, and f(a) represents its value at the beginning. The difference f(b) - f(a) gives us the change in the function's output, while b - a gives us the change in the input. Dividing the change in output by the change in input yields the average rate at which the function is changing over that interval. It's important to note that the average rate of change doesn't tell us how the function is behaving at any specific point within the interval; rather, it provides an overall measure of change across the entire interval. This concept is crucial for understanding the behavior of functions and forms the basis for more advanced concepts like the derivative, which gives the instantaneous rate of change at a single point. In practical terms, the average rate of change can be applied to various real-world scenarios, such as calculating the average speed of a car over a journey or the average growth rate of a population over a period.
Problem Statement: f(x) = x^2 - 1/x on [2, 4]
Now, let's focus on the specific problem at hand. We are tasked with computing the average rate of change of the function f(x) = x^2 - 1/x on the interval [2, 4]. This means we need to determine how the function's value changes, on average, as x varies from 2 to 4. To do this, we will use the formula for the average rate of change that we discussed earlier:
Average Rate of Change = (f(b) - f(a)) / (b - a)
In our case, a = 2 and b = 4. So, we need to calculate f(2) and f(4), plug these values into the formula, and simplify the result. This will give us the average rate of change of the function over the specified interval. The function f(x) = x^2 - 1/x is a combination of a quadratic term (x^2) and a reciprocal term (-1/x). This combination can lead to interesting behavior as x changes, and understanding the average rate of change will help us quantify this behavior over the interval [2, 4]. Before we proceed with the calculations, it's worth noting that the average rate of change can be positive, negative, or zero, depending on whether the function is increasing, decreasing, or remaining constant, on average, over the interval. In the following sections, we will carefully compute f(2) and f(4) and then use these values to find the average rate of change.
Step 1: Calculate f(2)
To begin, we need to evaluate the function f(x) = x^2 - 1/x at x = 2. This means we substitute 2 for x in the function's expression:
f(2) = (2)^2 - 1/(2)
First, we calculate the square of 2, which is 4:
(2)^2 = 4
Next, we calculate the reciprocal of 2:
1/(2) = 0.5
Now, we substitute these values back into the expression for f(2):
f(2) = 4 - 0.5
Finally, we perform the subtraction:
f(2) = 3.5
So, the value of the function at x = 2 is 3.5. It is often beneficial to express this as a fraction for easier calculations later on. Therefore, we convert 3.5 to its fractional form:
- 5 = 7/2
Thus, f(2) = 7/2. This value represents the function's output at the beginning of our interval [2, 4]. We will use this result in the next step when we calculate f(4). Evaluating functions at specific points is a fundamental skill in calculus and is crucial for understanding their behavior. In this case, finding f(2) is a necessary step towards calculating the average rate of change over the given interval. By carefully following the order of operations, we have successfully determined the function's value at x = 2.
Step 2: Calculate f(4)
Now that we've calculated f(2), our next step is to evaluate the function f(x) = x^2 - 1/x at x = 4. We follow the same procedure as before, substituting 4 for x in the function's expression:
f(4) = (4)^2 - 1/(4)
First, we calculate the square of 4:
(4)^2 = 16
Next, we find the reciprocal of 4:
1/(4) = 0.25
Now, we substitute these values back into the expression for f(4):
f(4) = 16 - 0.25
Finally, we perform the subtraction:
f(4) = 15.75
So, the value of the function at x = 4 is 15.75. Again, it's helpful to express this as a fraction for subsequent calculations. We convert 15.75 to its fractional form:
- 75 = 63/4
Therefore, f(4) = 63/4. This value represents the function's output at the end of our interval [2, 4]. Now that we have both f(2) and f(4), we are ready to use the average rate of change formula. Calculating f(4) involved similar steps to calculating f(2), reinforcing the importance of accurate substitution and order of operations. With f(2) and f(4) in hand, we can now proceed to the final calculation and determine the average rate of change of the function over the interval [2, 4]. This value will give us a quantitative measure of how the function's output changes as x moves from 2 to 4.
Step 3: Apply the Average Rate of Change Formula
With f(2) and f(4) calculated, we can now apply the average rate of change formula to find the solution. Recall the formula:
Average Rate of Change = (f(b) - f(a)) / (b - a)
In our case, a = 2, b = 4, f(2) = 7/2, and f(4) = 63/4. Substituting these values into the formula, we get:
Average Rate of Change = (63/4 - 7/2) / (4 - 2)
Now, we need to simplify the expression. First, let's simplify the numerator. To subtract the fractions, we need a common denominator. The least common denominator for 4 and 2 is 4. So, we rewrite 7/2 as 14/4:
Average Rate of Change = (63/4 - 14/4) / (4 - 2)
Now, we subtract the fractions in the numerator:
Average Rate of Change = (49/4) / (4 - 2)
Next, we simplify the denominator:
Average Rate of Change = (49/4) / 2
To divide by 2, we can multiply by its reciprocal, which is 1/2:
Average Rate of Change = (49/4) * (1/2)
Finally, we multiply the fractions:
Average Rate of Change = 49/8
So, the average rate of change of the function f(x) = x^2 - 1/x on the interval [2, 4] is 49/8. This fraction is already in its reduced form, so we have our final answer. This result tells us that, on average, the function's output increases by 49/8 units for every 1 unit increase in x over the interval [2, 4]. The process of applying the average rate of change formula involves careful substitution, fraction manipulation, and simplification. By following these steps, we have successfully calculated the average rate of change for the given function and interval.
Conclusion
In this article, we successfully computed the average rate of change of the function f(x) = x^2 - 1/x on the interval [2, 4]. We began by understanding the fundamental concept of the average rate of change as the slope of the secant line connecting two points on a function's graph. We then broke down the problem into manageable steps: first, we calculated f(2) and f(4), ensuring accurate substitution and simplification. Next, we applied the average rate of change formula, carefully handling fraction arithmetic to arrive at the final answer. Our calculations revealed that the average rate of change is 49/8. This value represents the average change in the function's output per unit change in the input x over the interval [2, 4]. Understanding the average rate of change is crucial for grasping the behavior of functions and forms a foundation for more advanced calculus concepts. This concept has wide-ranging applications in various fields, including physics, engineering, and economics, where quantifying change over an interval is essential. By mastering the steps involved in calculating the average rate of change, you can confidently analyze and interpret the behavior of functions in a variety of contexts. The process we followed demonstrates the importance of careful calculation, attention to detail, and a solid understanding of mathematical principles. With this knowledge, you are well-equipped to tackle similar problems and further explore the fascinating world of calculus.
