Finding Relative Rate Of Change For F(x) = 370 - 5x At X = 19

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Hey guys! Today, we're diving deep into the fascinating world of calculus to explore the concept of the relative rate of change. Specifically, we'll be tackling the function f(x) = 370 - 5x and figuring out its relative rate of change at the point x = 19. Sounds intriguing, right? Buckle up, because we're about to embark on a journey that will unravel this mathematical mystery step-by-step. This topic falls under the broad discussion category of mathematics, focusing on the practical application of derivatives in understanding how functions change.

Understanding the Relative Rate of Change

Before we jump into the calculations, let's first get a solid grasp of what the relative rate of change actually means. In simple terms, it tells us how much a function is changing relative to its current value. Think of it like this: a change of 1 unit might seem significant if the original value was only 2 units, but it would be a relatively small change if the original value was 100 units. That's the essence of the relative rate of change – it provides context to the change we observe.

Mathematically, the relative rate of change is defined as the ratio of the instantaneous rate of change (which we find using the derivative) to the original function value. So, if we have a function f(x), its relative rate of change at a point x is given by:

(Relative Rate of Change) = f'(x) / f(x)

Where f'(x) represents the derivative of f(x). This formula is the cornerstone of our exploration, and we'll be using it extensively to solve our problem. Understanding this formula is crucial because it allows us to compare the rate of change across different functions and at different points within the same function. It gives us a normalized perspective on how the function is behaving, taking into account its current magnitude. For example, in economics, the relative rate of change can represent the inflation rate, which is the percentage change in the price level. In biology, it can represent the growth rate of a population, normalized by the population size. This concept, therefore, has wide-ranging applications beyond the realm of pure mathematics.

Breaking Down the Problem: f(x) = 370 - 5x at x = 19

Now that we have a clear understanding of the relative rate of change, let's apply it to our specific problem. We're given the function f(x) = 370 - 5x and asked to find its relative rate of change at x = 19. To do this, we need to follow these steps:

  1. Find the derivative of f(x): This will give us the instantaneous rate of change of the function.
  2. Evaluate the derivative at x = 19: This will give us the instantaneous rate of change at the specific point we're interested in.
  3. Evaluate f(x) at x = 19: This will give us the original function value at that point.
  4. Calculate the relative rate of change: Divide the derivative value at x = 19 by the function value at x = 19.

Let's get started!

Step 1: Finding the Derivative of f(x)

The first step in our journey is to find the derivative of the function f(x) = 370 - 5x. Remember, the derivative represents the instantaneous rate of change of the function. To find the derivative, we'll use the power rule and the constant multiple rule of differentiation. These rules are fundamental tools in calculus, allowing us to efficiently find derivatives of polynomial functions. Let's refresh our memory of these rules:

  • Power Rule: If f(x) = xn, then f'(x) = nxn-1
  • Constant Multiple Rule: If f(x) = cf(x), where c is a constant, then f'(x) = cf'(x)
  • Constant Rule: The derivative of a constant is 0.

Applying these rules to our function, f(x) = 370 - 5x, we get:

  • The derivative of the constant term, 370, is 0.
  • The derivative of -5x is -5 (using the power rule and the constant multiple rule).

Therefore, the derivative of f(x), denoted as f'(x), is:

f'(x) = -5

This is a crucial result! It tells us that the function f(x) = 370 - 5x has a constant rate of change. For every unit increase in x, the function decreases by 5 units. This constant rate of change is a characteristic of linear functions, and our function is indeed a linear function. Now that we have the derivative, we can move on to the next step.

Step 2: Evaluating the Derivative at x = 19

Now that we've found the derivative, f'(x) = -5, the next step is to evaluate it at x = 19. This will tell us the instantaneous rate of change of the function specifically at the point x = 19. However, something interesting happens here. Notice that the derivative, f'(x) = -5, is a constant. This means that the rate of change of the function is the same regardless of the value of x. It doesn't matter if x is 19, 0, or 100; the function will always be decreasing at a rate of 5 units for every unit increase in x. This is a direct consequence of the function being linear. Linear functions have a constant slope, and the derivative represents that slope.

Therefore, evaluating f'(x) at x = 19 simply gives us:

f'(19) = -5

This result reinforces our understanding of the function's behavior. The negative sign indicates that the function is decreasing, and the magnitude of 5 tells us the rate of decrease. We now have the numerator for our relative rate of change calculation. Let's move on to finding the denominator.

Step 3: Evaluating f(x) at x = 19

To calculate the relative rate of change, we need both the rate of change (which we found in the previous step) and the value of the function at the point of interest. So, now we need to evaluate the original function, f(x) = 370 - 5x, at x = 19. This will give us the value of the function at this specific point.

Let's plug in x = 19 into the function:

f(19) = 370 - 5(19)

Now, we just need to do the arithmetic:

f(19) = 370 - 95

f(19) = 275

So, the value of the function at x = 19 is 275. This represents the "baseline" value against which we'll compare the rate of change. Remember, the relative rate of change tells us how significant the change is in relation to the current value. A change of -5 might seem small on its own, but we need to consider it in the context of the function's value at x = 19, which is 275. We're almost there! We have all the pieces of the puzzle. Now, let's put them together to calculate the relative rate of change.

Step 4: Calculating the Relative Rate of Change

We've reached the final step! We have all the information we need to calculate the relative rate of change of f(x) = 370 - 5x at x = 19. Remember the formula we discussed earlier:

(Relative Rate of Change) = f'(x) / f(x)

We found that f'(19) = -5 and f(19) = 275. So, let's plug these values into the formula:

(Relative Rate of Change) = -5 / 275

Now, we can simplify this fraction:

(Relative Rate of Change) = -1 / 55

This is the relative rate of change! But what does it actually mean? The result, -1/55, is a negative fraction, which tells us that the function is decreasing at that point, which we already knew. To get a better sense of the magnitude of the change, we can convert this fraction to a decimal:

(Relative Rate of Change) ≈ -0.0182

To express this as a percentage, we multiply by 100:

(Relative Rate of Change) ≈ -1.82%

This means that at x = 19, the function is decreasing at a rate of approximately 1.82% of its current value. This gives us a clear and interpretable measure of how the function is changing relative to its magnitude. It's a small percentage, indicating that the function is changing relatively slowly at this point, even though the absolute rate of change is -5. This highlights the power of the relative rate of change in providing context to the change we observe.

Conclusion: The Relative Rate of Change Unveiled

And there you have it! We've successfully navigated the world of calculus and found the relative rate of change of f(x) = 370 - 5x at x = 19. We discovered that the relative rate of change is approximately -1.82%, meaning the function is decreasing at a rate of about 1.82% of its current value at that point. This journey has not only provided us with a numerical answer but has also deepened our understanding of the concept of relative rate of change and its significance in analyzing function behavior. Remember, the relative rate of change is a powerful tool that helps us understand how functions change in a meaningful way, taking into account their current values. So, next time you encounter a question about how a function is changing, remember the steps we've taken today, and you'll be well-equipped to tackle it!