What Is The Average Rate Of Change Of The Function G(x) = 5/(x-1) + 2 Over The Interval [-4, 3]?
The average rate of change of a function is a fundamental concept in calculus and is essential for understanding how a function's output changes with respect to its input over a given interval. In this article, we will explore how to calculate the average rate of change for the function g(x) = 5/(x-1) + 2 over the interval [-4, 3]. This involves understanding the definition of average rate of change, applying it to the given function, and performing the necessary calculations. This concept is crucial not only in mathematics but also in various fields like physics, economics, and engineering, where understanding rates of change is vital for modeling and analyzing real-world phenomena.
Defining Average Rate of Change
The average rate of change of a function f(x) over an interval [a, b] is defined as the change in the function's value divided by the change in the input value. Mathematically, it is expressed as:
(Average Rate of Change = (f(b) - f(a)) / (b - a))
This formula represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)) on the graph of the function. The average rate of change gives us a measure of how the function's output changes on average for each unit change in the input over the specified interval. It's a crucial concept for understanding the behavior of functions and their applications in various fields.
Applying the Concept to g(x) = 5/(x-1) + 2
To find the average rate of change for the function g(x) = 5/(x-1) + 2 over the interval [-4, 3], we need to apply the formula mentioned above. First, we will identify the values of a and b, which are the endpoints of the interval. In this case, a = -4 and b = 3. Next, we will calculate the function values at these points, g(-4) and g(3). These values will be used to determine the change in the function's output. Once we have these values, we can plug them into the formula for the average rate of change and simplify to get the final answer. This process will illustrate how the average rate of change provides a summary of the function's behavior over the given interval.
Calculating g(-4)
To calculate g(-4), we substitute -4 for x in the function g(x) = 5/(x-1) + 2:
g(-4) = 5/(-4 - 1) + 2
Simplifying the denominator inside the fraction:
g(-4) = 5/(-5) + 2
Now, we perform the division:
g(-4) = -1 + 2
Finally, we add the numbers:
g(-4) = 1
So, the value of the function g(x) at x = -4 is 1. This value is a crucial component in calculating the average rate of change over the interval [-4, 3]. Understanding how to correctly substitute and simplify expressions is fundamental in evaluating functions at specific points.
Calculating g(3)
Next, we calculate g(3) by substituting 3 for x in the function g(x) = 5/(x-1) + 2:
g(3) = 5/(3 - 1) + 2
Simplifying the denominator:
g(3) = 5/2 + 2
To add these numbers, we need a common denominator. We can rewrite 2 as 4/2:
g(3) = 5/2 + 4/2
Now, we add the fractions:
g(3) = 9/2
So, the value of the function g(x) at x = 3 is 9/2. This value, along with g(-4), will be used to calculate the average rate of change over the interval [-4, 3]. Correctly evaluating functions at different points is essential for understanding their behavior and calculating rates of change.
Applying the Average Rate of Change Formula
Now that we have calculated g(-4) = 1 and g(3) = 9/2, we can apply the average rate of change formula:
(Average Rate of Change = (g(b) - g(a)) / (b - a))
In our case, a = -4, b = 3, g(a) = 1, and g(b) = 9/2. Substituting these values into the formula:
(Average Rate of Change = (9/2 - 1) / (3 - (-4)))
First, we simplify the numerator:
9/2 - 1 = 9/2 - 2/2 = 7/2
Next, we simplify the denominator:
3 - (-4) = 3 + 4 = 7
Now, we can rewrite the average rate of change expression:
(Average Rate of Change = (7/2) / 7)
To divide by 7, we multiply by its reciprocal (1/7):
(Average Rate of Change = (7/2) * (1/7))
Multiplying the fractions:
(Average Rate of Change = 7/(2 * 7))
Simplifying the fraction by canceling out the 7s:
(Average Rate of Change = 1/2)
Final Answer and Conclusion
Thus, the average rate of change of the function g(x) = 5/(x-1) + 2 over the interval [-4, 3] is 1/2. This value represents the average change in the function's output for each unit change in the input over this interval. Understanding the concept of average rate of change is crucial for analyzing the behavior of functions and their applications in various fields. By calculating the function values at the endpoints of the interval and applying the formula, we can determine how the function changes on average over the interval. The correct answer in this case is not among the provided options, highlighting the importance of careful calculation and verification.
Importance of Average Rate of Change
The average rate of change is a vital concept in calculus and has numerous applications in various fields. It provides a simple yet powerful way to understand how a function's output changes with respect to its input over an interval. In fields like physics, it can represent average velocity; in economics, it can represent the average cost or revenue change; and in engineering, it can represent the average change in a system's output. The average rate of change lays the foundation for understanding more complex concepts like instantaneous rate of change and derivatives, which are central to calculus. By mastering the average rate of change, students and professionals can gain deeper insights into the behavior of functions and their applications in real-world scenarios.
Potential Errors and How to Avoid Them
When calculating the average rate of change, several potential errors can occur, which can lead to incorrect results. One common mistake is incorrectly evaluating the function at the endpoints of the interval. This can be avoided by carefully substituting the values into the function and simplifying the expression step by step. Another common error is mishandling the arithmetic operations, especially when dealing with fractions or negative numbers. To prevent this, it is essential to double-check each calculation and use parentheses to maintain the correct order of operations. Additionally, errors can arise from incorrect substitution into the average rate of change formula itself. Ensuring that the function values and interval endpoints are placed in the correct positions in the formula is crucial. By being aware of these potential pitfalls and taking steps to avoid them, one can ensure accurate calculations of the average rate of change.
Applications and Further Exploration
The concept of the average rate of change extends beyond theoretical calculations and has practical applications in various fields. In physics, it can represent average velocity or acceleration over a time interval. In economics, it can describe the average change in cost or revenue for each unit produced or sold. In biology, it can represent the average growth rate of a population over a certain period. Further exploration of this concept leads to understanding instantaneous rate of change, which is the foundation of differential calculus. By examining the limit of the average rate of change as the interval approaches zero, we can find the derivative of a function, which provides valuable information about the function's behavior at a specific point. Understanding the applications and further extensions of the average rate of change enriches one's comprehension of calculus and its utility in real-world problem-solving.