Average Rate Of Change: Which Statement Is True?

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Hey guys! Let's dive into a problem about the average rate of change of a function. Understanding the average rate of change is super important in calculus and helps us analyze how a function behaves over an interval. Today, we're going to break down a problem step by step to make sure we understand exactly what's going on. So, let's get started and make math a little less scary!

Understanding Average Rate of Change

The average rate of change of a function g(x) between two points, say x = a and x = b, is essentially the slope of the secant line connecting the points (a, g(a)) and (b, g(b)) on the graph of the function. It tells us, on average, how much the function's value changes per unit change in x over that interval. Think of it like this: if you're driving a car, the average speed between two points is the total distance you traveled divided by the total time it took. Similarly, the average rate of change of a function is the total change in the function's value divided by the total change in x. Mathematically, the average rate of change is expressed as:

(g(b) - g(a)) / (b - a)

Where:

  • g(b) is the value of the function at x = b
  • g(a) is the value of the function at x = a
  • b - a is the change in x

In simpler terms, it's the change in g(x) divided by the change in x. This concept is fundamental in calculus and is closely related to the idea of a derivative, which represents the instantaneous rate of change at a single point. Understanding average rate of change helps build a strong foundation for grasping more advanced calculus concepts. It's all about understanding how a function's output changes in relation to its input over a given interval.

Now, let's apply this concept to the specific problem we're tackling today. We're given that the average rate of change of g(x) between x = 4 and x = 7 is 5/6. This means that when x changes from 4 to 7, the function g(x) changes, on average, by 5/6 for each unit change in x. To solve this problem, we'll use the formula for the average rate of change and see which of the given statements correctly reflects this information. Keep reading to see how we break down each option and determine the correct answer!

Analyzing the Given Statements

Okay, so we know that the average rate of change of g(x) between x = 4 and x = 7 is 5/6. That means:

(g(7) - g(4)) / (7 - 4) = 5/6

Let's look at each statement and see if it matches this definition.

Statement A: g(7) - g(4) = 5/6

This statement says that the difference between g(7) and g(4) is 5/6. However, the average rate of change is the difference in the function values divided by the difference in the x-values. So, this statement is missing the division by (7 - 4). It's only considering the change in the function's value, not the rate of change relative to the change in x. Therefore, this statement is not necessarily true.

Statement B: g(7 - 4) / (7 - 4) = 5/6

This statement is a bit confusing because it calculates g(7 - 4), which is g(3), and then divides it by (7 - 4), which is 3. This is not the correct way to calculate the average rate of change. The average rate of change should involve the difference in the function values at x = 7 and x = 4, not the function value at the difference of x values. Thus, this statement is incorrect.

Statement C: (g(7) - g(4)) / (7 - 4) = 5/6

This statement perfectly matches the definition of the average rate of change. It says that the difference between g(7) and g(4), divided by the difference between 7 and 4, is equal to 5/6. This is exactly what we mean by the average rate of change of g(x) between x = 4 and x = 7. So, this statement must be true.

Statement D: g(7) / g(4) = 5/6

This statement says that the ratio of g(7) to g(4) is 5/6. This is not related to the average rate of change. The average rate of change involves the difference between the function values, not the ratio. This statement is about proportional relationship between g(7) and g(4), not the rate at which g(x) changes between x=4 and x=7. Therefore, this statement is not necessarily true.

Determining the Correct Answer

After analyzing each statement, it's clear that statement C is the one that must be true. Let's recap why:

  • Statement C, (g(7) - g(4)) / (7 - 4) = 5/6, correctly represents the formula for the average rate of change. It calculates the change in g(x) (which is g(7) - g(4)) and divides it by the change in x (which is 7 - 4). This gives us the average rate at which g(x) changes per unit change in x over the interval from x = 4 to x = 7.

The other statements are incorrect because:

  • Statement A only considers the difference in g(x) values without dividing by the change in x.
  • Statement B calculates g(3) instead of using g(7) and g(4) correctly.
  • Statement D looks at the ratio of g(7) and g(4), which is not the same as the average rate of change.

So, the correct answer is:

C. (g(7) - g(4)) / (7 - 4) = 5/6

This statement accurately reflects that the average rate of change of g(x) between x = 4 and x = 7 is 5/6.

Final Thoughts

Alright, guys, we nailed it! Understanding the average rate of change is super important for grasping calculus concepts. Remember, it's all about how much a function changes on average over an interval. By using the formula and breaking down each statement, we were able to find the one that must be true. Keep practicing, and you'll become a pro in no time! Whether you're tackling derivatives or just trying to understand how things change, this concept is your friend. Keep up the great work, and I'll catch you in the next math adventure!

And always remember, don't be afraid to ask questions. Math is a journey, and every question is a step forward. Keep exploring and keep learning!