The Curve and Its Properties

Oct 13, 2025 by magicalceylontours.com 32 views

Let a Curve be Defined as $\sin y = x + y$

Determine Which of the Following Statements is/are True

In mathematics, curves are defined by equations that relate the variables x and y. These equations can be used to determine various properties of the curve, such as its shape, orientation, and behavior. In this article, we will explore the properties of a curve defined by the equation $\sin y = x + y$ and determine which of the following statements is/are true.

The Curve and Its Properties

The given equation is $\sin y = x + y$ . To understand the properties of this curve, we need to analyze its components. The sine function is a periodic function that oscillates between -1 and 1. The variable x represents the horizontal axis, while the variable y represents the vertical axis. The equation $\sin y = x + y$ implies that the value of y is equal to the sine of y plus the value of x.

Statement I: The Curve has a Vertical Tangent when $\cos y - 1 = 0$

To determine if the curve has a vertical tangent when $\cos y - 1 = 0$ , we need to analyze the equation $\cos y - 1 = 0$ . This equation implies that the value of $\cos y$ is equal to 1. Since the cosine function is equal to 1 at $y = 0$ , we can conclude that the curve has a vertical tangent when $y = 0$ .

However, we need to verify if this is true for the given equation $\sin y = x + y$ . To do this, we can differentiate the equation with respect to x. Using the chain rule, we get:

\frac{dy}{dx} = \frac{1}{\cos y}

Since $\cos y - 1 = 0$ , we have $\cos y = 1$ . Substituting this value into the equation, we get:

\frac{dy}{dx} = \frac{1}{1} = 1

This implies that the curve has a slope of 1 when $y = 0$ . However, this does not necessarily mean that the curve has a vertical tangent. A vertical tangent occurs when the slope of the curve is undefined, which is not the case here.

Therefore, we can conclude that statement I is false.

Statement II: The Curve has No Horizontal Tangents because $\cos y - 1 \neq 0$

To determine if the curve has no horizontal tangents because $\cos y - 1 \neq 0$ , we need to analyze the equation $\cos y - 1 \neq 0$ . This equation implies that the value of $\cos y$ is not equal to 1. Since the cosine function is not equal to 1 for any value of y, we can conclude that the curve has no horizontal tangents.

However, we need to verify if this is true for the given equation $\sin y = x + y$ . To do this, we can differentiate the equation with respect to x. Using the chain rule, we get:

\frac{dy}{dx} = \frac{1}{\cos y}

Since $\cos y - 1 \neq 0$ , we have $\cos y \neq 1$ . This implies that the curve has a slope of $\frac{1}{\cos y}$ , which is not equal to 0. Therefore, the curve has no horizontal tangents.

Therefore, we can conclude that statement II is true.

Conclusion

In conclusion, we have analyzed the properties of a curve defined by the equation $\sin y = x + y$ and determined which of the following statements is/are true. We found that statement I is false, while statement II is true. The curve has no horizontal tangents because $\cos y - 1 \neq 0$ . This implies that the curve has a slope of $\frac{1}{\cos y}$ , which is not equal to 0.

References

[1] Calculus, James Stewart, 7th edition
[2] Differential Equations, Lawrence Perko, 2nd edition

Additional Resources

[1] Khan Academy: Calculus
[2] MIT OpenCourseWare: Calculus

Discussion

Do you have any questions about the properties of the curve defined by the equation $\sin y = x + y$ ?
Can you think of any other properties of the curve that we did not discuss in this article?
How can we use the equation $\sin y = x + y$ to model real-world phenomena?
Q&A: Let a Curve be Defined as $\sin y = x + y$

Determine Which of the Following Statements is/are True

In our previous article, we analyzed the properties of a curve defined by the equation $\sin y = x + y$ and determined which of the following statements is/are true. We found that statement I is false, while statement II is true. The curve has no horizontal tangents because $\cos y - 1 \neq 0$ . This implies that the curve has a slope of $\frac{1}{\cos y}$ , which is not equal to 0.

In this article, we will answer some frequently asked questions about the curve defined by the equation $\sin y = x + y$ .

Q: What is the significance of the equation $\sin y = x + y$ ?

A: The equation $\sin y = x + y$ is a mathematical representation of a curve that has a unique property. The curve has no horizontal tangents because $\cos y - 1 \neq 0$ . This implies that the curve has a slope of $\frac{1}{\cos y}$ , which is not equal to 0.

Q: Can you explain the concept of a vertical tangent?

A: A vertical tangent is a point on a curve where the slope of the curve is undefined. This occurs when the denominator of the slope is equal to 0. In the case of the curve defined by the equation $\sin y = x + y$ , the slope is given by $\frac{dy}{dx} = \frac{1}{\cos y}$ . Since $\cos y - 1 \neq 0$ , the slope is never equal to 0, and therefore, the curve has no vertical tangents.

Q: How can we use the equation $\sin y = x + y$ to model real-world phenomena?

A: The equation $\sin y = x + y$ can be used to model various real-world phenomena, such as the motion of a pendulum or the vibration of a spring. The curve defined by the equation can be used to represent the position of the pendulum or the spring as a function of time.

Q: Can you provide an example of a real-world application of the curve defined by the equation $\sin y = x + y$ ?

A: One example of a real-world application of the curve defined by the equation $\sin y = x + y$ is in the design of a suspension bridge. The curve can be used to represent the shape of the bridge as a function of the distance from the center of the bridge. The curve can also be used to model the motion of the bridge as it is subjected to wind and other external forces.

Q: How can we determine the slope of the curve defined by the equation $\sin y = x + y$ ?

A: The slope of the curve defined by the equation $\sin y = x + y$ can be determined by differentiating the equation with respect to x. Using the chain rule, we get:

\frac{dy}{dx} = \frac{1}{\cos y}

This implies that the slope of the curve is given by $\frac{1}{\cos y}$ .

Q: Can you explain the concept of a horizontal tangent?

A: A horizontal tangent is a point on a curve where the slope of the curve is equal to 0. This occurs when the denominator of the slope is equal to 0. In the case of the curve defined by the equation $\sin y = x + y$ , the slope is given by $\frac{dy}{dx} = \frac{1}{\cos y}$ . Since $\cos y - 1 \neq 0$ , the slope is never equal to 0, and therefore, the curve has no horizontal tangents.

Q: How can we use the equation $\sin y = x + y$ to model the motion of a pendulum?

A: The equation $\sin y = x + y$ can be used to model the motion of a pendulum by representing the position of the pendulum as a function of time. The curve defined by the equation can be used to represent the shape of the pendulum as it swings back and forth.

Q: Can you provide an example of a mathematical model that uses the equation $\sin y = x + y$ ?

A: One example of a mathematical model that uses the equation $\sin y = x + y$ is the model of a simple harmonic oscillator. The equation can be used to represent the position of the oscillator as a function of time, and the curve defined by the equation can be used to model the motion of the oscillator.

Conclusion

In conclusion, we have answered some frequently asked questions about the curve defined by the equation $\sin y = x + y$ . We have explained the significance of the equation, the concept of a vertical tangent, and the concept of a horizontal tangent. We have also provided examples of real-world applications of the curve and mathematical models that use the equation.