Hey guys! Today, we're diving into a classic calculus problem: finding the instantaneous rate of change of a function at a specific point. In simpler terms, we want to know how quickly the function f(x) = -3x² + 2x - 9 is changing when x is exactly -2. This is a fundamental concept in calculus, and understanding it opens the door to many cool applications in physics, engineering, economics, and more. So, let's break it down step-by-step, making sure everyone's on board.
Understanding Instantaneous Rate of Change
Before we jump into the calculations, let's get crystal clear on what we mean by "instantaneous rate of change." You might already be familiar with the idea of average rate of change. Think of it like this: if you drive 100 miles in 2 hours, your average speed is 50 miles per hour. But, that doesn't tell you how fast you were going at any specific moment during the drive. Maybe you were stuck in traffic for a while, then sped up on the highway. The instantaneous rate of change is like looking at your speedometer at one precise instant – it tells you exactly how fast you're going right now.
In calculus terms, the instantaneous rate of change of a function at a point is the slope of the tangent line to the function's graph at that point. Imagine drawing a line that just barely touches the curve of the function at x = -2. The steepness of that line is what we're after. The instantaneous rate of change, also known as the derivative, measures how a function's output changes with respect to its input at a specific point. It's crucial for understanding the behavior of functions, particularly their increasing or decreasing trends and concavity. In practical terms, it helps us analyze rates of change in various real-world phenomena, such as velocity in physics, reaction rates in chemistry, and marginal cost in economics. To calculate it, we need to use the concept of a limit. The limit allows us to zoom in closer and closer to the point of interest until we essentially have the slope of a line touching the curve at just that single point. This might sound a bit abstract, but don't worry, we'll make it concrete with the example below.
The Derivative: Our Key Tool
The magic tool we use to find the instantaneous rate of change is called the derivative. The derivative of a function, often written as f'(x) (read as "f prime of x"), gives us a formula for the slope of the tangent line at any point x. Once we have the derivative, we can simply plug in x = -2 to find the instantaneous rate of change at that specific point. So, the first step is to find the derivative of our function, f(x) = -3x² + 2x - 9. There are a couple of ways to do this, but the most common method is using the power rule.
Applying the Power Rule
The power rule is a handy shortcut for finding the derivative of terms that look like x raised to some power (like x², x³, etc.). It states that if we have a term axⁿ, its derivative is nax^(n-1). In plain English, you multiply the coefficient (a) by the exponent (n), and then reduce the exponent by 1. Let's apply this to our function:
- Term -3x²: Here, a = -3 and n = 2. So, the derivative of this term is 2 * (-3) * x^(2-1) = -6x.
- Term 2x: This is the same as 2x¹, so a = 2 and n = 1. The derivative is 1 * 2 * x^(1-1) = 2x⁰ = 2 (since anything to the power of 0 is 1).
- Term -9: This is a constant term. The derivative of any constant is always 0.
Now, we put it all together. The derivative of f(x) = -3x² + 2x - 9 is f'(x) = -6x + 2 + 0, which simplifies to f'(x) = -6x + 2. Awesome! We've got our derivative. This formula now allows us to calculate the instantaneous rate of change at any x-value we want.
Calculating the Instantaneous Rate of Change at x = -2
We've found the derivative, f'(x) = -6x + 2. Now, the final step is super easy: we simply plug in x = -2 into this formula. This will give us the slope of the tangent line at that specific point, which is the instantaneous rate of change we're looking for.
So, f'(-2) = -6 * (-2) + 2 = 12 + 2 = 14. That's it! The instantaneous rate of change of f(x) = -3x² + 2x - 9 at x = -2 is 14. This means that at the point where x is -2, the function is increasing quite rapidly – for a small change in x, the function's value is changing 14 times as much.
Interpreting the Result
Okay, so we got 14. But what does this number mean? Remember, the instantaneous rate of change is the slope of the tangent line. A positive slope means the function is increasing at that point. In our case, the function is increasing quite steeply since the slope is 14. If we were to graph the function and draw a tangent line at x = -2, it would be a line going upwards from left to right, with a relatively steep incline. This provides a visual way to understand the instantaneous behavior of the function at a particular point, which is a fundamental concept in calculus and its applications.
Alternative Method: Using the Limit Definition of the Derivative
While the power rule is a quick way to find derivatives, it's important to understand where it comes from. The power rule is derived from the limit definition of the derivative, which is a more fundamental concept. This method is a bit more involved, but it gives you a deeper understanding of what the derivative really represents.
The limit definition of the derivative is:
f'(x) = lim (h->0) [f(x + h) - f(x)] / h
This formula essentially calculates the slope of a secant line (a line that intersects the curve at two points) as the distance between the two points (h) gets infinitesimally small, approaching zero. This limit gives us the slope of the tangent line at a single point.
Applying the Limit Definition to Our Function
Let's apply this to our function, f(x) = -3x² + 2x - 9. This will take a little bit of algebra, so bear with me!
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Find f(x + h): We substitute (x + h) for x in our function:
f(x + h) = -3(x + h)² + 2(x + h) - 9
Expand this: f(x + h) = -3(x² + 2xh + h²) + 2x + 2h - 9
Distribute the -3: f(x + h) = -3x² - 6xh - 3h² + 2x + 2h - 9
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Find f(x + h) - f(x): Now, we subtract the original function from f(x + h):
f(x + h) - f(x) = (-3x² - 6xh - 3h² + 2x + 2h - 9) - (-3x² + 2x - 9)
Notice how the -3x², 2x, and -9 terms cancel out, leaving us with:
f(x + h) - f(x) = -6xh - 3h² + 2h
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Divide by h: Next, we divide the result by h:
[f(x + h) - f(x)] / h = (-6xh - 3h² + 2h) / h
We can factor out an h from the numerator: h(-6x - 3h + 2) / h
Now, cancel the h terms: -6x - 3h + 2
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Take the limit as h approaches 0: Finally, we take the limit as h approaches 0:
lim (h->0) (-6x - 3h + 2) = -6x - 3(0) + 2 = -6x + 2
Boom! We got the same derivative, f'(x) = -6x + 2, using the limit definition. This confirms our result from the power rule and gives us a deeper understanding of how derivatives are calculated.
Calculating at x = -2 (Again!)
Just like before, we plug in x = -2 into our derivative: f'(-2) = -6(-2) + 2 = 12 + 2 = 14. We arrive at the same answer, 14, which reinforces the consistency of calculus methods.
Key Takeaways and Why This Matters
Finding the instantaneous rate of change might seem like an abstract mathematical exercise, but it has profound implications in various fields. Understanding how a function changes at a specific point is crucial for:
- Physics: Determining the instantaneous velocity and acceleration of an object.
- Engineering: Optimizing designs and predicting the behavior of systems.
- Economics: Analyzing marginal cost and revenue to make informed business decisions.
- Computer Science: Developing algorithms that adapt to changing conditions.
The Bigger Picture
By understanding derivatives and instantaneous rates of change, you gain a powerful tool for analyzing and predicting change in the world around you. It's a fundamental concept in calculus that unlocks a deeper understanding of how things work.
Conclusion
So, there you have it! We've successfully found the instantaneous rate of change of f(x) = -3x² + 2x - 9 at x = -2, using both the power rule and the limit definition of the derivative. The answer, 14, tells us how rapidly the function is changing at that specific point. This concept is a cornerstone of calculus and has wide-ranging applications in various fields. Keep practicing, and you'll become a pro at finding instantaneous rates of change in no time! Remember, the instantaneous rate of change provides critical insights into the behavior of functions, allowing us to understand and predict how systems evolve over time. Whether you're analyzing the trajectory of a rocket or the growth of a population, the principles of calculus, and specifically the derivative, provide a powerful framework for making sense of the world.