Average Rate Of Change Of F(x) = X² + 3x Over [-5, 0]
Hey there, math enthusiasts! Today, we're diving into the fascinating world of functions and their rates of change. Specifically, we're going to tackle the function f(x) = x² + 3x and figure out its average rate of change over the interval [-5, 0]. Sounds intriguing, right? Let's get started!
Understanding the Average Rate of Change
Before we jump into the calculations, it's crucial to grasp what the average rate of change actually represents. In simple terms, it's the slope of the line connecting two points on a curve. Imagine you're driving a car, and you want to know your average speed over a certain time period. That's essentially what the average rate of change tells us – the average amount the function's output changes for each unit change in its input over a specific interval.
The Formula for Success
The magic formula we'll be using is:
Average Rate of Change = (f(b) - f(a)) / (b - a)
Where:
- f(x) is our function
- [a, b] is the interval we're interested in
- f(a) is the function's value at the beginning of the interval
- f(b) is the function's value at the end of the interval
This formula, guys, is your key to unlocking the mysteries of how functions behave. It's like a secret code that reveals the average trend of the function within a given range.
Visualizing the Concept
Think of it this way: if you were to draw a line connecting the points (a, f(a)) and (b, f(b)) on the graph of the function, the average rate of change would be the slope of that line. A positive average rate of change means the function is generally increasing over the interval, while a negative one indicates a decreasing trend. A zero average rate of change? That means the function's output is, on average, staying the same.
Calculating the Average Rate of Change for f(x) = x² + 3x
Alright, let's put our newfound knowledge to the test! We have our function, f(x) = x² + 3x, and our interval, [-5, 0]. Now, it's time to crunch some numbers.
Step 1: Find f(a) and f(b)
First, we need to figure out the function's values at the endpoints of our interval. In this case, a = -5 and b = 0.
Let's start with f(a) = f(-5):
- f(-5) = (-5)² + 3(-5)
- f(-5) = 25 - 15
- f(-5) = 10
So, f(-5) = 10. That's the function's value at the start of our interval. Now, let's find f(b) = f(0):
- f(0) = (0)² + 3(0)
- f(0) = 0 + 0
- f(0) = 0
Therefore, f(0) = 0. This tells us the function's value at the end of our interval.
Step 2: Plug the Values into the Formula
Now that we have f(a) and f(b), we can plug them into our average rate of change formula:
Average Rate of Change = (f(b) - f(a)) / (b - a)
Average Rate of Change = (0 - 10) / (0 - (-5))
Step 3: Simplify and Solve
Let's simplify the expression:
Average Rate of Change = -10 / 5
Average Rate of Change = -2
Wait a minute! It seems there was a slight error in the options provided in the original question. Based on our calculations, the average rate of change is actually -2, not -3. It's always a good practice, guys, to double-check your work and trust your calculations!
Analyzing the Result
So, what does this -2 average rate of change tell us? It means that, on average, for every one-unit increase in x over the interval [-5, 0], the function's value decreases by 2 units. This indicates a downward trend in the function's behavior within this interval.
Connecting the Dots: Slope and the Average Rate of Change
Remember how we talked about the average rate of change being the slope of the line connecting two points on the curve? In this case, if we were to draw a line between the points (-5, 10) and (0, 0) on the graph of f(x) = x² + 3x, that line would have a slope of -2. This visual representation can really solidify your understanding of the concept.
Why is the Average Rate of Change Important?
You might be wondering,
