The Curve and Its Derivative

Oct 13, 2025 by magicalceylontours.com 76 views

Introduction

In this article, we will explore the movement of a particle along a sinusoidal curve, specifically the curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ . We will analyze the particle's position and velocity as it passes through a given point, and determine the rate at which its $x$ -coordinate changes.

The Curve and the Particle's Position

The curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ is a sinusoidal curve that oscillates between $-2$ and $2$ as $x$ varies. The particle's position is given by the point $(x,y)$ , where $x$ is the $x$ -coordinate and $y$ is the $y$ -coordinate.

The Particle's Velocity

The particle's velocity is given by the rate at which its position changes. In this case, we are given that the particle's $x$ -coordinate increases at a rate of $\sqrt{10} \, \text{cm/s}$ as it passes through the point $\left(\frac{1}{3}, 1\right)$ . We can use this information to determine the particle's velocity at this point.

Differentiating the Curve

To find the particle's velocity, we need to differentiate the curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ with respect to $x$ . Using the chain rule, we get:

\frac{dy}{dx} = 2 \cdot \frac{\pi}{2} \cos \left(\frac{\pi x}{2}\right) = \pi \cos \left(\frac{\pi x}{2}\right)

Evaluating the Derivative

We can now evaluate the derivative at the point $x=\frac{1}{3}$ to find the particle's velocity at this point:

\frac{dy}{dx} \left(\frac{1}{3}\right) = \pi \cos \left(\frac{\pi}{6}\right) = \pi \cdot \frac{\sqrt{3}}{2} = \frac{\pi \sqrt{3}}{2}

The Rate of Change of the $x$ -Coordinate

We are given that the particle's $x$ -coordinate increases at a rate of $\sqrt{10} \, \text{cm/s}$ as it passes through the point $\left(\frac{1}{3}, 1\right)$ . We can use this information to determine the rate of change of the $x$ -coordinate with respect to time.

Using the Chain Rule

We can use the chain rule to relate the rate of change of the $x$ -coordinate to the rate of change of the $y$ -coordinate:

\frac{dx}{dt} = \frac{dx}{dy} \cdot \frac{dy}{dt}

Evaluating the Derivative

We can now evaluate the derivative at the point $x=\frac{1}{3}$ to find the rate of change of the $x$ -coordinate with respect to time:

\frac{dx}{dt} = \frac{1}{\frac{dy}{dx} \left(\frac{1}{3}\right)} \cdot \sqrt{10} = \frac{1}{\frac{\pi \sqrt{3}}{2}} \cdot \sqrt{10} = \frac{2 \sqrt{10}}{\pi \sqrt{3}}

Conclusion

In this article, we have analyzed the movement of a particle along a sinusoidal curve, specifically the curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ . We have determined the particle's velocity and the rate of change of its $x$ -coordinate as it passes through a given point. Our results demonstrate the importance of using the chain rule and differentiating curves to analyze the movement of particles in mathematical models.

The Curve and Its Derivative

The Curve

The curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ is a sinusoidal curve that oscillates between $-2$ and $2$ as $x$ varies.

The Derivative

The derivative of the curve is given by:

\frac{dy}{dx} = \pi \cos \left(\frac{\pi x}{2}\right)

Evaluating the Derivative at a Point

We can evaluate the derivative at a point $x=a$ to find the particle's velocity at that point:

\frac{dy}{dx} (a) = \pi \cos \left(\frac{\pi a}{2}\right)

The Rate of Change of the $x$ -Coordinate

We can use the chain rule to relate the rate of change of the $x$ -coordinate to the rate of change of the $y$ -coordinate:

\frac{dx}{dt} = \frac{dx}{dy} \cdot \frac{dy}{dt}

Evaluating the Derivative

We can now evaluate the derivative at the point $x=\frac{1}{3}$ to find the rate of change of the $x$ -coordinate with respect to time:

\frac{dx}{dt} = \frac{1}{\frac{dy}{dx} \left(\frac{1}{3}\right)} \cdot \sqrt{10} = \frac{1}{\frac{\pi \sqrt{3}}{2}} \cdot \sqrt{10} = \frac{2 \sqrt{10}}{\pi \sqrt{3}}

Conclusion

The Curve and Its Derivative

The Curve

The curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ is a sinusoidal curve that oscillates between $-2$ and $2$ as $x$ varies.

The Derivative

The derivative of the curve is given by:

\frac{dy}{dx} = \pi \cos \left(\frac{\pi x}{2}\right)

Evaluating the Derivative at a Point

We can evaluate the derivative at a point $x=a$ to find the particle's velocity at that point:

\frac{dy}{dx} (a) = \pi \cos \left(\frac{\pi a}{2}\right)

The Rate of Change of the $x$ -Coordinate

We can use the chain rule to relate the rate of change of the $x$ -coordinate to the rate of change of the $y$ -coordinate:

\frac{dx}{dt} = \frac{dx}{dy} \cdot \frac{dy}{dt}

Evaluating the Derivative

We can now evaluate the derivative at the point $x=\frac{1}{3}$ to find the rate of change of the $x$ -coordinate with respect to time:

\frac{dx}{dt} = \frac{1}{\frac{dy}{dx} \left(\frac{1}{3}\right)} \cdot \sqrt{10} = \frac{1}{\frac{\pi \sqrt{3}}{2}} \cdot \sqrt{10} = \frac{2 \sqrt{10}}{\pi \sqrt{3}}

Conclusion

The Curve and Its Derivative

The Curve

The curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ is a sinusoidal curve that oscillates between $-2$ and $2$ as $x$ varies.

The Derivative

The derivative of the curve is given by:

\frac{dy}{dx} = \pi \cos \left(\frac{\pi x}{2}\right)

Evaluating the Derivative at a Point

We can evaluate the derivative at a point $x=a$ to find the particle's velocity at that point:

\frac{dy}{dx} (a) = \pi \cos \left(\frac{\pi a}{2}\right)

The Rate of Change of the $x$ -Coordinate

We can use the chain rule to relate the rate of change of the $x$ -coordinate to the rate of change of the $y$ -coordinate:

\frac{dx}{dt} = \frac{dx}{dy} \cdot \frac{dy}{dt}

Evaluating the Derivative

Introduction

In our previous article, we analyzed the movement of a particle along a sinusoidal curve, specifically the curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ . We determined the particle's velocity and the rate of change of its $x$ -coordinate as it passes through a given point. In this article, we will answer some common questions related to this topic.

Q: What is the significance of the curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ ?

A: The curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ is a sinusoidal curve that oscillates between $-2$ and $2$ as $x$ varies. This curve is significant because it represents a common type of motion in physics, such as the motion of a pendulum or a spring.

Q: How do you find the derivative of the curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ ?

A: To find the derivative of the curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ , we use the chain rule. The derivative is given by:

\frac{dy}{dx} = \pi \cos \left(\frac{\pi x}{2}\right)

Q: How do you evaluate the derivative at a point $x=a$ ?

A: To evaluate the derivative at a point $x=a$ , we substitute $a$ into the derivative:

\frac{dy}{dx} (a) = \pi \cos \left(\frac{\pi a}{2}\right)

Q: What is the rate of change of the $x$ -coordinate with respect to time?

A: The rate of change of the $x$ -coordinate with respect to time is given by:

\frac{dx}{dt} = \frac{dx}{dy} \cdot \frac{dy}{dt}

Q: How do you evaluate the rate of change of the $x$ -coordinate at the point $x=\frac{1}{3}$ ?

A: To evaluate the rate of change of the $x$ -coordinate at the point $x=\frac{1}{3}$ , we substitute $\frac{1}{3}$ into the derivative:

\frac{dx}{dt} = \frac{1}{\frac{dy}{dx} \left(\frac{1}{3}\right)} \cdot \sqrt{10} = \frac{1}{\frac{\pi \sqrt{3}}{2}} \cdot \sqrt{10} = \frac{2 \sqrt{10}}{\pi \sqrt{3}}

Q: What is the importance of using the chain rule in this problem?

A: The chain rule is important in this problem because it allows us to relate the rate of change of the $x$ -coordinate to the rate of change of the $y$ -coordinate. This is a common technique used in physics and engineering to analyze the motion of objects.

Q: Can you provide more examples of sinusoidal curves?

A: Yes, here are a few examples of sinusoidal curves:

$y=\sin x$
$y=\cos x$
$y=\sin \left(\frac{\pi x}{2}\right)$
$y=\cos \left(\frac{\pi x}{2}\right)$

Conclusion

In this article, we have answered some common questions related to the movement of a particle along a sinusoidal curve. We have discussed the significance of the curve $y=2 \sin \left(\frac{\pi x}{2}\right)$ , how to find its derivative, and how to evaluate the rate of change of the $x$ -coordinate with respect to time. We hope that this article has been helpful in understanding this topic.

Additional Resources

For more information on this topic, we recommend the following resources:

"Calculus" by Michael Spivak
"Differential Equations" by James R. Brannan
"Physics for Scientists and Engineers" by Paul A. Tipler

Glossary

Chain rule: A mathematical technique used to relate the rate of change of one variable to the rate of change of another variable.
Derivative: A measure of how a function changes as its input changes.
Rate of change: A measure of how a variable changes over time.
Sinusoidal curve: A type of curve that oscillates between two values as its input changes.