Understanding Average Rate Of Change For G(x) Between X=4 And X=7

The concept of average rate of change is fundamental in calculus and is crucial for understanding how a function's output changes in relation to its input. In this article, we will delve into the average rate of change of a function, particularly focusing on the function g(x) between the points x = 4 and x = 7. We are given that the average rate of change of g(x) over this interval is 5/6. Our primary goal is to identify which statement accurately represents this information. We will dissect each provided option, ensuring a comprehensive understanding of why one statement holds true while others do not. Before we begin analyzing the options, let's clarify the core concept of average rate of change. The average rate of change of a function between two points is essentially the slope of the secant line connecting those two points on the function's graph. Mathematically, it is expressed as the change in the function's output divided by the change in its input. This gives us a measure of how much the function's value changes, on average, for each unit change in the input variable. Consider a function f(x). The average rate of change between two points x = a and x = b is given by the formula: (f(b) - f(a)) / (b - a). This formula is derived directly from the slope formula (rise over run), where f(b) - f(a) represents the change in the y-values (rise) and b - a represents the change in the x-values (run). Understanding this formula is crucial for solving problems related to average rate of change. It allows us to quantify how a function's output varies over a specific interval, which has significant applications in various fields such as physics, economics, and engineering. For instance, in physics, the average rate of change can represent the average velocity of an object over a time interval. In economics, it might represent the average change in cost or revenue per unit change in production. In our specific scenario, we are dealing with the function g(x), and we are interested in its average rate of change between x = 4 and x = 7. We are told that this average rate of change is 5/6. This means that, on average, the function g(x) changes by 5/6 units for every unit increase in x between the values of 4 and 7. This information is our starting point for evaluating the given statements and determining which one accurately captures this relationship.

Dissecting the Given Statements

Now, let's analyze each statement provided in the question to determine which one accurately reflects the given information about the average rate of change of g(x). We will systematically examine each option, highlighting its mathematical meaning and comparing it to the definition of average rate of change. Understanding why some statements are incorrect is just as important as identifying the correct one, as it reinforces our grasp of the concept. Let's start with option A. This statement asserts that g(7) - g(4) = 5/6. While this expression does involve the function values at x = 7 and x = 4, it only represents the change in the function's output over the interval. It doesn't account for the change in the input variable, which is crucial for calculating the average rate of change. The average rate of change is the change in output divided by the change in input. Therefore, this statement is incomplete and does not accurately represent the average rate of change. It only gives us the difference in the function's values at the two points, but not the rate at which this change occurs relative to the change in x. Next, we consider option B. This statement presents the expression g(7 - 4) / (7 - 4) = 5/6. This is where a critical misunderstanding of function notation can lead to error. The notation g(7 - 4) means g(3), which is the value of the function g at x = 3. This is entirely different from g(7) - g(4), which is the difference between the function's values at x = 7 and x = 4. Option B incorrectly applies the function g to the difference in the x-values, rather than calculating the difference in the function's outputs. Therefore, this statement is also incorrect. It misinterprets the function evaluation and does not align with the concept of average rate of change. The numerator should represent the difference in the function's output values, not the function evaluated at the difference in input values. Moving on to option C, we have the expression (g(7) - g(4)) / (7 - 4) = 5/6. This statement perfectly matches the definition of average rate of change. The numerator, g(7) - g(4), represents the change in the function's output (the rise), and the denominator, 7 - 4, represents the change in the input (the run). The entire expression calculates the slope of the secant line connecting the points (4, g(4)) and (7, g(7)) on the graph of g(x). Since we are given that the average rate of change between x = 4 and x = 7 is 5/6, this statement must be true. It accurately applies the formula for average rate of change and correctly relates it to the given information. Finally, let's examine option D, which states g(7) / g(4) = 5/6. This statement represents the ratio of the function's values at x = 7 and x = 4. While the ratio of function values can sometimes provide insights into a function's behavior, it does not directly represent the average rate of change. The average rate of change is concerned with the difference in function values and the corresponding difference in input values, not their ratio. Therefore, this statement is incorrect. It focuses on a different relationship between the function values and does not capture the concept of average rate of change.

The Correct Statement: C

After a detailed examination of each statement, it is clear that option C is the correct answer. This option, (g(7) - g(4)) / (7 - 4) = 5/6, precisely embodies the definition of average rate of change. It calculates the change in the function's output (g(7) - g(4)) and divides it by the corresponding change in the input (7 - 4). This result directly gives us the average rate at which the function g(x) changes between the points x = 4 and x = 7. Understanding why option C is correct requires a solid grasp of the concept of average rate of change and how it is mathematically represented. The average rate of change is a measure of the function's slope over a given interval. It tells us, on average, how much the function's output changes for each unit change in the input. This concept is fundamental in calculus and has numerous applications in various fields. Option C aligns perfectly with this definition. The numerator, g(7) - g(4), represents the change in the function's y-values, while the denominator, 7 - 4, represents the change in the x-values. Dividing these gives us the slope of the secant line connecting the points (4, g(4)) and (7, g(7)), which is precisely the average rate of change. To further solidify this understanding, let's consider a hypothetical scenario. Suppose g(4) = 2 and g(7) = 4.5. Then, the average rate of change would be (4.5 - 2) / (7 - 4) = 2.5 / 3 = 5/6, which matches the given information. This example illustrates how the formula in option C accurately calculates the average rate of change based on the function's values at the endpoints of the interval. In contrast, the other options fail to capture this fundamental relationship. Option A only considers the change in the function's output without accounting for the change in input. Option B misinterprets function notation and evaluates the function at the difference of the input values, rather than calculating the difference of the function's output values. Option D looks at the ratio of the function's values, which is not relevant to the concept of average rate of change. Therefore, option C stands out as the only statement that accurately reflects the given information about the average rate of change of g(x) between x = 4 and x = 7. Its correctness stems from its precise adherence to the definition and formula for average rate of change, making it the definitive answer to the question.

Conclusion

In conclusion, the correct statement regarding the average rate of change of g(x) between x = 4 and x = 7 being 5/6 is C. (g(7) - g(4)) / (7 - 4) = 5/6. This statement accurately applies the formula for average rate of change, which is the change in the function's output divided by the change in its input. Understanding the concept of average rate of change is crucial in mathematics and its applications, and this problem serves as a great example of how to apply the definition correctly. By dissecting each option and understanding why some are incorrect, we gain a deeper appreciation for the nuances of this important mathematical concept.