Average Rate Of Change: Which Statement Is True?
Hey guys! Let's dive into a math problem that involves the average rate of change. It's a concept that pops up a lot in calculus and even in everyday life when we talk about things like speed or growth. We're going to break down a specific question about a function g(x) and figure out which statement about its average rate of change is definitely true. So, buckle up, and let's get started!
Understanding Average Rate of Change
First, let's make sure we're all on the same page about what average rate of change actually means. In simple terms, it's the measure of how much a function's output changes, on average, over a certain interval of its input. Think of it like this: if you're driving a car, your average speed is the total distance you traveled divided by the total time it took. Similarly, for a function, the average rate of change between two points is the change in the function's value divided by the change in the input value.
Mathematically, the average rate of change of a function f(x) between two points x = a and x = b is given by the formula:
(f(b) - f(a)) / (b - a)
This formula essentially calculates the slope of the secant line connecting the points (a, f(a)) and (b, f(b)) on the graph of the function. The secant line is just a straight line that intersects the curve at these two points. So, the average rate of change gives us an idea of the function's overall trend between those two points.
To really grasp this, let's think about a few examples. Imagine a function that represents the height of a plant over time. The average rate of change between two weeks would tell us how much the plant grew, on average, per week during that period. Or, consider a function that represents the temperature of a room over time. The average rate of change would tell us how much the temperature changed, on average, per hour.
Understanding this concept is crucial for tackling the problem we're about to discuss. We need to be comfortable with the formula and what it represents graphically and in real-world scenarios. So, keep this definition in mind as we move on to the specific question about g(x).
Analyzing the Given Information
Now, let's focus on the problem at hand. We're given that the average rate of change of the function g(x) between x = 4 and x = 7 is 5/6. This is the key piece of information we need to work with. Our goal is to figure out which statement must be true based on this fact.
Let's break down what this information tells us. We know the interval we're considering is from x = 4 to x = 7. We also know the average rate of change over this interval is 5/6. This means that, on average, the function's output is changing at a rate of 5/6 units for every 1 unit change in the input within this interval.
To make this more concrete, let's plug the given values into our average rate of change formula:
(g(7) - g(4)) / (7 - 4) = 5/6
This equation is the mathematical representation of the information we've been given. It tells us that the difference in the function's value at x = 7 and x = 4, divided by the difference in the input values (which is 7 - 4 = 3), is equal to 5/6.
This equation is crucial because it allows us to relate the function's values at the two endpoints of the interval to the average rate of change. We can use this equation to analyze the given statements and determine which one must be true. It's like having a secret code that connects the dots between the function's behavior and the information we're given. So, let's keep this equation in mind as we examine the possible answers.
Evaluating the Statements
Okay, now we're at the heart of the problem: evaluating the given statements. We need to carefully consider each statement and determine whether it must be true based on the information we have about the average rate of change of g(x). Remember, just because a statement could be true doesn't mean it must be true. We're looking for the statement that is guaranteed to be correct given the information we have.
Let's look at the first statement:
A. g(7) - g(4) = 5/6
This statement looks tempting at first glance, as it involves the difference in the function's values at x = 7 and x = 4. However, it's crucial to remember the average rate of change formula. We know that the average rate of change is (g(7) - g(4)) / (7 - 4), which equals 5/6. This statement is missing the crucial division by (7-4). To see if this statement is true, let's go back to our equation:
(g(7) - g(4)) / (7 - 4) = 5/6
We know that (7 - 4) = 3, so we can rewrite the equation as:
(g(7) - g(4)) / 3 = 5/6
To isolate (g(7) - g(4)), we need to multiply both sides of the equation by 3:
g(7) - g(4) = (5/6) * 3
g(7) - g(4) = 15/6
g(7) - g(4) = 5/2
So, we see that g(7) - g(4) is actually equal to 5/2, not 5/6. Therefore, statement A is incorrect.
Now, let's consider the second statement:
B. x(7-4) / (7-4) = 5/6
This statement looks a bit strange. It includes the variable x, which doesn't seem to fit in the context of the average rate of change formula. Also, we can immediately simplify the left side of the equation: (7-4) / (7-4) is simply equal to 1, so the left side becomes just x. This means the statement is saying x = 5/6, which has nothing to do with the average rate of change of g(x) between x=4 and x=7. Therefore, statement B is also incorrect.
Identifying the Correct Statement
Since statements A and B are incorrect, let's take a closer look at our work and see if we can deduce the correct statement from the information we have. We know the average rate of change formula is the key:
(g(7) - g(4)) / (7 - 4) = 5/6
We've already used this to show that g(7) - g(4) = 5/2. However, the core of the given information is the equation itself. This equation must be true based on the problem statement. It directly represents the average rate of change of g(x) between x=4 and x=7.
Therefore, the statement that must be true is the one that accurately reflects this equation. After analyzing the initial options, if there were an option that directly stated or was equivalent to:
(g(7) - g(4)) / (7 - 4) = 5/6
That would be the correct answer. Sometimes, the correct answer might be a slight algebraic manipulation of this equation, but it would fundamentally represent the same relationship.
Key Takeaways and Practice
So, guys, we've walked through this problem step-by-step, from understanding the definition of average rate of change to analyzing the given statements and identifying the one that must be true. This type of question really tests your understanding of the concept and your ability to apply the formula correctly.
Here are a few key takeaways to keep in mind:
- Know the formula: The average rate of change formula, (f(b) - f(a)) / (b - a), is your best friend in these problems. Memorize it and understand what each part represents.
- Don't jump to conclusions: Just because a statement looks similar to the formula doesn't mean it's correct. Carefully evaluate each statement and make sure it aligns with the given information.
- Manipulate the equation: Sometimes, the correct statement might be a rearranged version of the average rate of change formula. Practice manipulating the equation to isolate different variables.
- Think about the meaning: Remember that the average rate of change represents the slope of the secant line. Visualizing this can help you understand the concept better.
To really solidify your understanding, try practicing similar problems. Look for questions that give you the average rate of change and ask you to find a relationship between the function's values at different points. The more you practice, the more comfortable you'll become with these types of problems. Keep up the great work, and you'll be mastering average rate of change in no time!