What Is The Slope Of The Line Tangent To The Polar Curve $r=2 \theta$ At The Point $\theta=\frac{\pi}{2}$?A. $-\frac{\pi}{2}$ B. $-\frac{2}{\pi}$ C. 0 D. $\frac{\pi}{2}$ E. 2

Introduction

In mathematics, the study of polar curves is a crucial aspect of understanding the behavior of functions in polar coordinates. Polar curves are defined by the equation $r=f(\theta)$, where $r$ is the distance from the origin to a point on the curve, and $\theta$ is the angle between the positive x-axis and the line connecting the origin to the point. The slope of the line tangent to a polar curve at a given point is an essential concept in calculus, as it helps us understand the rate of change of the function at that point.

The Polar Curve $r=2 \theta$

The given polar curve is defined by the equation $r=2 \theta$. This equation represents a curve in the polar coordinate system, where the distance from the origin to a point on the curve is directly proportional to the angle $\theta$. To find the slope of the line tangent to this curve at the point $\theta=\frac{\pi}{2}$, we need to first find the derivative of the function $r=2 \theta$ with respect to $\theta$.

Finding the Derivative of $r=2 \theta$

To find the derivative of $r=2 \theta$, we can use the power rule of differentiation, which states that if $f(x)=x^n$, then $f'(x)=nx^{n-1}$. In this case, we have $r=2 \theta$, so we can write:

$$\frac{dr}{d\theta}=2$$

This means that the derivative of $r=2 \theta$ with respect to $\theta$ is a constant function equal to 2.

Finding the Slope of the Line Tangent to the Curve

To find the slope of the line tangent to the curve at the point $\theta=\frac{\pi}{2}$, we need to use the formula for the slope of a tangent line in polar coordinates:

$$m=\frac{dy}{dx}=\frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}$$

where $r'$ is the derivative of $r$ with respect to $\theta$, and $r$ is the value of $r$ at the point $\theta=\frac{\pi}{2}$.

Evaluating the Slope of the Line Tangent to the Curve

Now that we have the formula for the slope of the line tangent to the curve, we can evaluate it at the point $\theta=\frac{\pi}{2}$. We know that $r=2 \theta$, so we can substitute this into the formula:

$$m=\frac{2\sin\frac{\pi}{2}+2\theta\cos\frac{\pi}{2}}{2\cos\frac{\pi}{2}-2\theta\sin\frac{\pi}{2}}$$

Simplifying this expression, we get:

$$m=\frac{2(1)+2\left(\frac{\pi}{2}\right)(0)}{2(0)-2\left(\frac{\pi}{2}\right)(1)}$$

$$m=\frac{2}{-2\left(\frac{\pi}{2}\right)}$$

$$m=\frac{2}{-\pi}$$

$$m=-\frac{2}{\pi}$$

Conclusion

In conclusion, the slope of the line tangent to the polar curve $r=2 \theta$ at the point $\theta=\frac{\pi}{2}$ is $-\frac{2}{\pi}$. This result can be obtained by using the formula for the slope of a tangent line in polar coordinates and evaluating it at the given point.

Discussion

The concept of finding the slope of a tangent line to a polar curve is an essential aspect of calculus. It helps us understand the behavior of functions in polar coordinates and is used in various applications, such as physics and engineering. The result obtained in this problem can be used to analyze the behavior of the polar curve $r=2 \theta$ and its derivatives.

Final Answer

The final answer is $\boxed{-\frac{2}{\pi}}$.

Introduction

In our previous article, we explored the concept of finding the slope of the line tangent to a polar curve. We used the formula for the slope of a tangent line in polar coordinates to find the slope of the line tangent to the polar curve $r=2 \theta$ at the point $\theta=\frac{\pi}{2}$. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the significance of finding the slope of the line tangent to a polar curve?

A: Finding the slope of the line tangent to a polar curve is an essential aspect of calculus. It helps us understand the behavior of functions in polar coordinates and is used in various applications, such as physics and engineering.

Q: How do I find the slope of the line tangent to a polar curve?

A: To find the slope of the line tangent to a polar curve, you need to use the formula for the slope of a tangent line in polar coordinates:

$$m=\frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}$$

where $r'$ is the derivative of $r$ with respect to $\theta$, and $r$ is the value of $r$ at the point $\theta$.

Q: What is the formula for the slope of a tangent line in polar coordinates?

A: The formula for the slope of a tangent line in polar coordinates is:

$$m=\frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}$$

Q: How do I evaluate the slope of the line tangent to a polar curve at a given point?

A: To evaluate the slope of the line tangent to a polar curve at a given point, you need to substitute the values of $r'$, $r$, $\theta$, and $\sin\theta$ into the formula for the slope of a tangent line in polar coordinates.

Q: What is the slope of the line tangent to the polar curve $r=2 \theta$ at the point $\theta=\frac{\pi}{2}$?

A: The slope of the line tangent to the polar curve $r=2 \theta$ at the point $\theta=\frac{\pi}{2}$ is $-\frac{2}{\pi}$.

Q: Can I use the formula for the slope of a tangent line in polar coordinates to find the slope of the line tangent to any polar curve?

A: Yes, you can use the formula for the slope of a tangent line in polar coordinates to find the slope of the line tangent to any polar curve.

Q: What are some common applications of finding the slope of the line tangent to a polar curve?

A: Finding the slope of the line tangent to a polar curve has various applications in physics and engineering, such as analyzing the behavior of functions in polar coordinates and determining the rate of change of a function at a given point.

Conclusion

In conclusion, finding the slope of the line tangent to a polar curve is an essential aspect of calculus. By using the formula for the slope of a tangent line in polar coordinates, you can find the slope of the line tangent to any polar curve at a given point. We hope that this Q&A article has provided you with a better understanding of this concept.

Final Answer

The final answer is $\boxed{-\frac{2}{\pi}}$.