Rate Of Change Of Q(x) = -3x + 6x² + 24 At X = 3 An Explanation

Hey guys! Let's dive into a fascinating topic in calculus: the rate of change of a function. Specifically, we're going to explore the function Q(x) = -3x + 6x² + 24 and figure out how quickly it's changing at the point x = 3. This is a classic calculus problem, and understanding it will give you a solid foundation for tackling more complex concepts. So, grab your thinking caps, and let's get started!

Delving into the Concept of Rate of Change

Before we jump into the specifics of our function, let's make sure we're all on the same page about what the rate of change actually means. In simple terms, the rate of change tells us how much a function's output changes in response to a change in its input. Think of it like this: imagine you're driving a car. Your speed is the rate of change of your position – it tells you how many miles you're covering per hour. Similarly, in mathematics, the rate of change of a function tells us how much the function's value changes as we change the input variable, x. This concept is fundamental in various fields, from physics and engineering to economics and computer science.

Now, when we talk about the rate of change at a specific point, like x = 3 in our problem, we're talking about the instantaneous rate of change. This is like looking at the speedometer in your car at a single moment – it tells you how fast you're going right now. Mathematically, the instantaneous rate of change is represented by the derivative of the function. The derivative, often denoted as Q'(x), gives us a formula for calculating the rate of change at any point x. So, to find the rate of change of Q(x) at x = 3, we first need to find its derivative, Q'(x), and then plug in x = 3. This might sound a bit technical, but don't worry, we'll break it down step by step.

The rate of change is not just a theoretical concept; it has practical applications all around us. For instance, in economics, it can represent the marginal cost of production – how much it costs to produce one additional unit. In physics, it can represent the velocity of an object – how its position changes over time. Understanding the rate of change allows us to make predictions, optimize processes, and gain insights into the behavior of systems. That's why it's such a crucial concept in mathematics and its applications. So, with this understanding in mind, let's move on to finding the derivative of our function Q(x).

Finding the Derivative of Q(x) = -3x + 6x² + 24

Okay, guys, this is where the magic of calculus comes in! To find the rate of change of Q(x) at x = 3, the first step is to determine the derivative of Q(x), denoted as Q'(x). The derivative is a powerful tool that gives us a formula for the instantaneous rate of change of a function at any point. Luckily, finding the derivative of a polynomial like Q(x) is pretty straightforward using the power rule. The power rule states that if we have a term of the form ax^n, its derivative is nax^(n-1). Let's apply this rule to each term in our function Q(x) = -3x + 6x² + 24.

First, we have the term -3x, which can be written as -3x^1. Applying the power rule, we multiply the coefficient (-3) by the exponent (1) and reduce the exponent by 1, giving us -3 * 1 * x^(1-1) = -3x^0 = -3. Remember that any number raised to the power of 0 is 1, so x^0 is simply 1. Next, we have the term 6x². Applying the power rule, we get 6 * 2 * x^(2-1) = 12x^1 = 12x. Finally, we have the constant term 24. The derivative of any constant is always 0, because a constant doesn't change as x changes. Therefore, the derivative of 24 is 0. Now, we can put it all together: Q'(x) is the sum of the derivatives of each term, which is Q'(x) = -3 + 12x + 0 = -3 + 12x.

So, we've successfully found the derivative of Q(x)! Q'(x) = -3 + 12x is a new function that tells us the rate of change of Q(x) at any value of x. This is a crucial step because now we have a formula that we can use to find the rate of change specifically at x = 3. It's like having a map that shows us the steepness of a hill at any point – we just need to find the specific point we're interested in. This derivative, Q'(x), represents the slope of the tangent line to the graph of Q(x) at any point x. This geometric interpretation is incredibly useful for visualizing the rate of change. Now that we have Q'(x), the next step is to plug in x = 3 to find the rate of change at that particular point. Let's do that now!

Calculating the Rate of Change at x = 3

Alright, guys, we're in the home stretch! We've found the derivative of Q(x), which is Q'(x) = -3 + 12x. Now, to find the rate of change of Q(x) specifically at x = 3, we simply need to substitute x = 3 into our derivative function, Q'(x). This is a straightforward calculation, but it's the key to answering our original question. By plugging in x = 3, we're essentially asking the derivative function: