Question

Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. $$r^{2}=4 \cos 2 \theta ;\left(0, \pm \frac{\pi}{4}\right)$$

Short answer

Answer: The equations of the tangent lines in polar coordinates at the points (0, ±π/4) are $$r(\theta) = 0$$.

Step-by-step solution

Differentiate the equation with respect to θ

First, we need to differentiate the given polar equation with respect to θ: $$\frac{d}{d\theta}(r^2) = \frac{d}{d\theta}(4\cos2\theta)$$ Applying the chain rule on the right-hand side of the equation: $$2r\frac{dr}{d\theta} = -8\sin(2\theta)$$ Now, we can solve for $$\frac{dr}{d\theta}$$: $$\frac{dr}{d\theta} = -\frac{4\sin(2\theta)}{r}$$

Calculate the slope of the tangent line

Next, we will use the formula for the slope of a tangent line in polar coordinates, which is given by: $$m = -\frac{r^2 + (r\frac{dr}{d\theta})^2}{2r\frac{dr}{dθ}}$$ Plugging the value of $$\frac{dr}{d\theta}$$ we calculated in the previous step into the equation, we get: $$m = -\frac{r^2 + (-\frac{4\sin(2\theta)}{r})^2}{2r\cdot -\frac{4\sin(2\theta)}{r}}$$ For the points (0, ±π/4): $$r = 0$$ $$\theta = \pm\frac{\pi}{4}$$

Calculate the equations of the tangent lines in polar coordinates

To find the equation of the tangent line in polar coordinates at the origin intersection, we will use the polar form of a line, which is given by: $$r(\theta) = \frac{H\cos\theta - K\sin\theta}{\cos\alpha}$$ Since the point of tangency lies on the origin (0, ±π/4), we have H = 0 and K = 0, which means the polar equation of the tangent line simplifies to: $$r(\theta) = \frac{0\cos\theta - 0\sin\theta}{\cos\alpha}$$ Which simplifies further to simply: $$r(\theta) = 0$$ So, the equations of the tangent lines in polar coordinates at the points (0, ±π/4) are: $$r(\theta) = 0$$

Key concepts

Tangent Line in Polar Coordinates

In polar coordinates, a tangent line is a line that touches a polar curve at a single point without crossing it. This is similar to tangent lines in Cartesian coordinates but adapted for polar equations. Instead of dealing with x and y, we work with the radial distance \(r\) and the angle \(\theta\).

The concept of the tangent line to a polar curve can be complex due to the nature of polar curves. However, calculating the slope of a tangent line is critical, as it describes the line's steepness or incline at a particular point. The slope \(m\) in polar coordinates can be determined using the formula:

  \(m = \frac{r \frac{dr}{d\theta} \sin \theta + r^2 \cos \theta}{r \frac{dr}{d\theta} \cos \theta - r^2 \sin \theta}\)

When the curve intersects the origin, tangent lines in polar form are often simple. For the given curve in the exercise, the tangent line equation simplifies significantly when intersecting the origin, allowing us to write \(r(\theta) = 0\) at those points.

Understanding the geometric interpretation of tangent lines in polar coordinates provides valuable insights when analyzing polar curves.

Differentiation in Polar Coordinates

Differentiation in polar coordinates involves taking the derivative with respect to \(\theta\) to analyze how the radial distance \(r\) changes. In the exercise, the goal was to find the slope of the tangent line by starting with the differentiation of the polar equation \(r^2 = 4 \cos 2\theta\).

To differentiate this equation, first apply the basic rule of differentiation to the left side. This leads us to \(\frac{d}{d\theta}(r^2) = 2r\frac{dr}{d\theta}\).

On the right side, you need to use the chain rule, a crucial calculus tool, which simplifies the differentiation process of composite functions like \(4 \cos 2\theta\). As a result, you end up with \(-8 \sin(2\theta)\).

Combining these results, we have \(2r\frac{dr}{d\theta} = -8 \sin(2\theta)\), from which you can solve for \(\frac{dr}{d\theta}\) to further use in finding the slope or other characteristics of the polar curve.

The differentiation in polar coordinates offers an essential means of examining how polar curves behave, especially when determining important features like slopes at specific angles.

Chain Rule in Polar Differentiation

The chain rule is a fundamental differentiating tool in calculus used to differentiate composite functions. It is especially useful in polar coordinates, where reflections of angles and radial functions intertwine.

For this exercise, the chain rule was critical when differentiating \(4 \cos 2\theta\), where \(2\theta\) is the inner function, and \(\cos\) is the outer function. The chain rule states that you multiply the derivative of the outer function by the derivative of the inner function.

Here's how it was applied:

• Identify the outer function, \(\cos\), and the inner function, \(2\theta\).
• Differentiating the outer function gives \(-\sin(2\theta)\), and differentiating \(2\theta\) results in \(2\).
• Multiply them to get \(-4\sin(2\theta)\).

This chain rule application allows us to handle more complex derivative calculations and is key in finding how functions like \(\cos 2\theta\) change relative to \(\theta\).

Mastering the chain rule is vital for polar differentiation and opens up the analysis of more intricate polar operations efficiently and accurately.