Let $C$ Be The Curve Defined By $xy = -4$. Which Of The Following Statements Is True Of Curve $C$ At The Point $(-2, 2)$?A. It Has A Relative Minimum Because $y^{\prime}=0$ And $y^{\prime
Understanding the Curve Defined by xy = -4
In mathematics, curves are defined by equations that relate the variables x and y. One such curve is defined by the equation xy = -4. In this article, we will explore the properties of this curve, particularly at the point (-2, 2).
The Equation xy = -4
The equation xy = -4 is a quadratic equation in two variables. To understand the curve defined by this equation, we need to rewrite it in a more familiar form. We can do this by rearranging the equation to get y = -4/x.
Properties of the Curve
The curve defined by xy = -4 is a hyperbola. A hyperbola is a type of curve that has two branches, one opening upwards and the other downwards. The curve has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.
The Point (-2, 2)
The point (-2, 2) lies on the curve defined by xy = -4. To determine the properties of the curve at this point, we need to find the derivative of the equation y = -4/x.
Finding the Derivative
To find the derivative of the equation y = -4/x, we can use the power rule of differentiation. The derivative of x^n is nx^(n-1). In this case, we have y = -4/x, which can be rewritten as y = -4x^(-1).
Using the power rule, we get:
y' = -4(-1)x^(-1-1) y' = 4x^(-2) y' = 4/x^2
Evaluating the Derivative at (-2, 2)
Now that we have the derivative of the equation y = -4/x, we can evaluate it at the point (-2, 2).
y' = 4/(-2)^2 y' = 4/4 y' = 1
The derivative of the equation y = -4/x at the point (-2, 2) is 1. This means that the curve has a relative maximum at this point.
Why is this the case?
The curve defined by xy = -4 has a relative maximum at the point (-2, 2) because the derivative of the equation y = -4/x is positive at this point. This means that the curve is increasing at this point, which is a characteristic of a relative maximum.
What does this mean?
The fact that the curve has a relative maximum at the point (-2, 2) means that this point is a local maximum of the curve. In other words, the curve is higher at this point than at any other point in its neighborhood.
Implications
The fact that the curve has a relative maximum at the point (-2, 2) has important implications for the behavior of the curve in its neighborhood. For example, it means that the curve will be increasing at this point, which can have important consequences for the behavior of the curve in its neighborhood.
In conclusion, the curve defined by xy = -4 has a relative maximum at the point (-2, 2). This is because the derivative of the equation y = -4/x is positive at this point, which means that the curve is increasing at this point. This has important implications for the behavior of the curve in its neighborhood.
References
- [1] "Calculus" by Michael Spivak
- [2] "Differential Equations" by Lawrence Perko
Additional Resources
- [1] Khan Academy: Calculus
- [2] MIT OpenCourseWare: Calculus
Final Thoughts
The curve defined by xy = -4 is a fascinating example of a hyperbola. The fact that it has a relative maximum at the point (-2, 2) has important implications for the behavior of the curve in its neighborhood. This article has provided a detailed analysis of the curve and its properties, and has highlighted the importance of understanding the behavior of curves in mathematics.
Q&A: Understanding the Curve Defined by xy = -4
In our previous article, we explored the properties of the curve defined by xy = -4, particularly at the point (-2, 2). In this article, we will answer some frequently asked questions about the curve and its properties.
Q: What is the equation of the curve?
A: The equation of the curve is xy = -4.
Q: What type of curve is defined by xy = -4?
A: The curve defined by xy = -4 is a hyperbola.
Q: What are the asymptotes of the curve?
A: The curve has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.
Q: What is the derivative of the equation y = -4/x?
A: The derivative of the equation y = -4/x is y' = 4/x^2.
Q: What is the value of the derivative at the point (-2, 2)?
A: The value of the derivative at the point (-2, 2) is 1.
Q: Why is the point (-2, 2) a relative maximum of the curve?
A: The point (-2, 2) is a relative maximum of the curve because the derivative of the equation y = -4/x is positive at this point, which means that the curve is increasing at this point.
Q: What are the implications of the curve having a relative maximum at the point (-2, 2)?
A: The fact that the curve has a relative maximum at the point (-2, 2) means that this point is a local maximum of the curve. In other words, the curve is higher at this point than at any other point in its neighborhood.
Q: How does the curve behave in its neighborhood?
A: The curve will be increasing at the point (-2, 2), which can have important consequences for the behavior of the curve in its neighborhood.
Q: What are some real-world applications of the curve defined by xy = -4?
A: The curve defined by xy = -4 has many real-world applications, including physics, engineering, and economics. For example, it can be used to model the behavior of electrical circuits, mechanical systems, and economic systems.
Q: How can I learn more about the curve defined by xy = -4?
A: There are many resources available to learn more about the curve defined by xy = -4, including textbooks, online courses, and research papers. You can also consult with a mathematics professor or tutor for personalized guidance.
In conclusion, the curve defined by xy = -4 is a fascinating example of a hyperbola. The answers to these frequently asked questions provide a deeper understanding of the curve and its properties, and highlight the importance of understanding the behavior of curves in mathematics.
References
- [1] "Calculus" by Michael Spivak
- [2] "Differential Equations" by Lawrence Perko
Additional Resources
- [1] Khan Academy: Calculus
- [2] MIT OpenCourseWare: Calculus
Final Thoughts
The curve defined by xy = -4 is a fundamental concept in mathematics, and understanding its properties is essential for many real-world applications. This Q&A article provides a comprehensive overview of the curve and its properties, and is a valuable resource for anyone interested in learning more about this fascinating topic.