Calculate Average Rate Of Change For Y=4x^2 Over [0, 7/4]

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Hey guys! Let's dive into a fun math problem where we need to find the average rate of change for the function y = 4x² over the interval from x = 0 to x = 7/4. This might sound a bit complicated, but don't worry, we'll break it down step by step so it’s super easy to understand. We will cover what the average rate of change means and how to calculate it. So, grab your calculators and let’s get started!

Understanding Average Rate of Change

Before we jump into the calculation, let’s quickly talk about what the average rate of change actually means. Think of it like this: imagine you're driving a car. The average speed you travel over a certain distance is the total distance you covered divided by the total time it took. Similarly, the average rate of change of a function tells us how much the function's value changes, on average, for each unit change in x over a specific interval.

Average Rate of Change Explained. In mathematical terms, the average rate of change of a function y = f(x) over an interval [a, b] is given by the formula:

(f(b) - f(a)) / (b - a)

This formula is essentially calculating the slope of the line that connects the points (a, f(a)) and (b, f(b)) on the graph of the function. This line is called the secant line. So, when we find the average rate of change, we’re finding the slope of this secant line. Make sure you understand that this is an average – the function might be changing at different rates at different points within the interval, but this gives us the overall change over the entire interval.

Why is this useful? Well, the average rate of change can give us a simplified view of how a function behaves over an interval. It’s used in various real-world applications, such as:

  • Physics: Calculating average velocity or acceleration.
  • Economics: Determining average cost or revenue changes.
  • Biology: Modeling population growth rates.

So, you see, understanding the average rate of change is not just a math exercise; it’s a tool that helps us understand and model changes in various situations. Now that we have a good grasp of what it is, let’s apply it to our specific problem.

Setting Up the Problem

Okay, let’s get back to our problem. We have the function y = 4x², and we want to find the average rate of change between x = 0 and x = 7/4. To do this, we'll use the formula we just discussed:

(f(b) - f(a)) / (b - a)

In our case:

  • f(x) = 4x²
  • a = 0
  • b = 7/4

So, we need to find f(a) and f(b), which means we need to plug in x = 0 and x = 7/4 into our function y = 4x². This is a straightforward process, but it's crucial to be careful with our calculations to avoid errors. Once we have these values, we can plug them into the formula and calculate the average rate of change.

This initial setup is like planning a route before a journey. We know where we are starting (x = 0), where we want to go (x = 7/4), and the “road” we are traveling (y = 4x²). Now, we just need to follow the plan and do the calculations. Think of it as finding the elevation change between two points on a mountain – we need to know the elevation at both points before we can find the difference and, therefore, the average slope. Let’s move on to the next step, where we’ll actually calculate the values of f(0) and f(7/4). Remember, precision is key here, so let’s take our time and make sure we get it right!

Calculating f(a) and f(b)

Alright, let's roll up our sleeves and calculate f(a) and f(b). Remember, f(x) = 4x², a = 0, and b = 7/4. First, let's find f(a), which is f(0). We simply plug in x = 0 into the function:

f(0) = 4 * (0)² = 4 * 0 = 0

So, f(0) = 0. That was pretty straightforward, right? Now, let's tackle f(b), which is f(7/4). We plug in x = 7/4 into the function:

f(7/4) = 4 * (7/4)²

Here, we need to square 7/4 first:

(7/4)² = (7/4) * (7/4) = 49/16

Now, multiply by 4:

4 * (49/16) = 49/4

So, f(7/4) = 49/4. Great! We've calculated both f(0) and f(7/4). These are the “elevations” at our starting and ending points. Now, we have all the pieces we need to plug into the average rate of change formula. Think of this step as measuring the height at the beginning and the end of our interval – we know how high we started and how high we ended. Next, we'll use these values to find the overall change in height relative to the distance traveled, which will give us the average slope.

Remember, keeping track of each step and double-checking our calculations helps prevent errors. Math can be like a puzzle – each piece needs to fit perfectly. We’ve found two crucial pieces, and now we’re ready to put them together. Let’s move on to the final calculation where we’ll use these values in the average rate of change formula. We're almost there, guys – keep up the great work!

Applying the Average Rate of Change Formula

Okay, we've done the groundwork, and now it's time for the grand finale: plugging our values into the average rate of change formula. Remember, the formula is:

(f(b) - f(a)) / (b - a)

We found that:

  • f(a) = f(0) = 0
  • f(b) = f(7/4) = 49/4
  • a = 0
  • b = 7/4

Now, let's plug these values into the formula:

((49/4) - 0) / (7/4 - 0)

This simplifies to:

(49/4) / (7/4)

To divide fractions, we multiply by the reciprocal of the divisor:

(49/4) * (4/7)

The 4s cancel out:

49/7

And finally:

49/7 = 7

So, the average rate of change for the function y = 4x² over the interval from x = 0 to x = 7/4 is 7. That's it! We've solved the problem. This final calculation is like putting the last piece of a puzzle in place – everything clicks, and we see the whole picture. The average rate of change of 7 tells us that, on average, the function’s value increases by 7 units for every 1 unit increase in x over the given interval.

Conclusion and Answer

Woo-hoo! We made it! We've successfully calculated the average rate of change for the function y = 4x² over the interval [0, 7/4]. By following our step-by-step approach, we found that the average rate of change is 7. Remember, the key to solving these problems is to break them down into smaller, manageable steps. First, we understood what the average rate of change means. Then, we set up the problem by identifying our function and interval. Next, we calculated the function values at the interval endpoints. Finally, we plugged everything into the formula and simplified to get our answer.

So, the correct answer is A. 7. Great job, everyone! I hope this explanation has made the concept of average rate of change much clearer for you guys. Math can be challenging, but with a little bit of patience and a systematic approach, you can tackle any problem. Keep practicing, and you'll become a math whiz in no time!

Now, go forth and conquer more math problems, knowing you've got the tools and the knowledge to succeed. You’ve seen how to calculate the average rate of change, and you understand its significance. Whether it's in physics, economics, or any other field, you're now equipped to analyze how things change over time. So, keep that calculator handy, stay curious, and never stop exploring the fascinating world of mathematics! Until next time, keep up the amazing work!