Calculus Breakdown: Analyzing A Quadratic Function's Continuity And Variation

Hey guys! Let's dive into a classic calculus problem. We're going to dissect a quadratic function, exploring its continuity and how it behaves. This is a super fundamental concept, and understanding it lays the groundwork for tackling more complex problems down the road. So, grab your pens and let's get started! We will start by going over the key concepts we need to master to be able to solve this math problem. The first thing we will cover is continuity, which, in simple terms, asks whether you can draw the graph of a function without lifting your pen. If you can, then it's continuous! The next one is variation, which is all about understanding how a function changes – whether it's increasing, decreasing, or staying constant – as the input (usually 'x') changes. Let's get to it. Here's the function we will be analyzing. Let's assume that $f(x) = x^2 + 2x - 3$, and this function is defined on the interval $I = [-1, + infty[$. Now, let's break down the two main parts of the question. First, we need to show that the function is continuous on the interval $I$. Second, we need to create a variation table for this function.

Demystifying Continuity: A Deep Dive

Okay, so let's tackle the first part: proving the continuity of the function $f(x) = x^2 + 2x - 3$ on the interval $I = [-1, + infty[$. Continuity is a fundamental concept in calculus, and it essentially means that a function doesn't have any abrupt jumps, holes, or breaks within a given interval. In simpler terms, you can draw the graph of a continuous function without lifting your pen. For a function to be continuous at a point, three conditions must be met:

1. The function must be defined at that point (i.e., the point must be in the function's domain).
2. The limit of the function as x approaches that point must exist.
3. The limit of the function as x approaches that point must be equal to the value of the function at that point. Since $f(x)$ is a polynomial function, it's continuous everywhere. Polynomial functions are, by definition, continuous across their entire domain. The function $f(x) = x^2 + 2x - 3$ is a quadratic function (a polynomial of degree 2). Its graph is a parabola. These parabolas are smooth curves without any breaks or gaps, meaning the function is continuous for all real numbers. Since the interval $I = [-1, + infty[$ is a subset of the real numbers, the function is also continuous on this interval. To make it easier to understand, we can analyze the function from a different perspective. The function is made up of basic continuous functions: $x^2$, $2x$, and the constant $-3$. The sum of continuous functions is also continuous. Therefore, $f(x)$ is continuous on the given interval. Remember, a continuous function means there are no sudden jumps or breaks in the graph. A good way to visualize this is to imagine drawing the function on a graph without lifting your pencil. Because $f(x)$ is a quadratic polynomial, its graph will be a smooth parabola, which can be drawn without lifting your pencil. So, to wrap up part a, we've successfully demonstrated that the function is continuous on the specified interval. Understanding continuity is super important. It's a core concept that helps us understand a function's behavior, and it's used in lots of other calculus concepts, like derivatives and integrals. That means we can now confidently move on to the next part.

Unveiling the Function's Behavior: The Variation Table

Alright, now that we have demonstrated continuity, it's time to analyze the function's behavior: that is, to create a table of variations. The variation table helps us understand where the function is increasing, decreasing, and where it might have any turning points (like maximum or minimum values). Let's start with the first thing. To create a variation table, we need to find the derivative of the function. The derivative of a function, denoted as $f'(x)$, tells us the rate of change of the function at any given point. The derivative of $f(x) = x^2 + 2x - 3$ is found using the power rule: $f'(x) = 2x + 2$. The next step is to find the critical points of the function. Critical points are points where the derivative is equal to zero or undefined. In our case, we need to solve the equation $2x + 2 = 0$ to find the critical point. So, $2x = -2$, and therefore, $x = -1$. The critical point is $x = -1$. Now, we have to create a variation table. Here's how we create a variation table. We'll have rows for the values of $x$, the sign of the derivative $f'(x)$, and the behavior of the function $f(x)$.

We can now proceed to evaluate the table.

• x: The first row represents the values of $x$ within the domain. Here, our domain is $I = [-1, + infty[$. We'll include the critical point we found, which is -1.
• f'(x): In the second row, we will enter the sign of the derivative $f'(x)$. We already know that the derivative is $2x + 2$. Let's examine the sign of $f'(x)$ in different intervals. If $x < -1$, then $f'(x) < 0$, because $2x + 2 < 0$. If $x > -1$, then $f'(x) > 0$, because $2x + 2 > 0$. At the critical point $x = -1$, the derivative $f'(-1) = 0$.
• f(x): The last row describes the behavior of the function $f(x)$. If $f'(x) < 0$, then $f(x)$ is decreasing. If $f'(x) > 0$, then $f(x)$ is increasing. At the critical point where $f'(x) = 0$, the function has a local minimum or maximum (in our case, a minimum). Let's calculate the minimum point. We found that the critical point is $x = -1$. So, we will have $f(-1) = (-1)^2 + 2(-1) - 3 = 1 - 2 - 3 = -4$. So, the minimum point is at (-1,-4). This confirms the function is decreasing until $x = -1$ and increasing after $x = -1$, as predicted. The variation table perfectly captures how the function's behavior shifts at the critical point. This whole process of creating the variation table provides us with a complete picture of how our quadratic function is changing across the interval. It is really useful because it helps us visualize and understand key aspects of the function.

Conclusion: Putting It All Together

We've successfully navigated the analysis of our quadratic function! We proved its continuity, which means there were no sudden breaks or jumps. Then, we used the derivative to build a variation table, revealing where the function increases, decreases, and its turning points. The skills we used, like finding derivatives, critical points, and interpreting the sign of the derivative, are absolutely central to calculus. Keep practicing, and you'll become a master of analyzing and understanding all sorts of functions! Keep up the awesome work, and you'll conquer calculus in no time. Guys, I hope this breakdown helped! Don't hesitate to ask if you have more questions. Keep practicing, and you will get there!