Problem 18 - Calculus Early Transcendentals

Question

Find the points at which the following polar curves have a horizontal or a vertical tangent line. $$r=3+6 \sin \theta$$

Step-by-Step Solution

Answer

Answer: To find the points where horizontal tangent lines occur on the polar curve $$r = 3 + 6\sin{\theta}$$, substitute the values of θ = π/2 + kπ, where k is an integer (step 2), into the polar-to-Cartesian conversion formulas (step 5) to find the Cartesian coordinates (x, y) of these points: $$x = \left(3 + 6\cos\left(k\pi\right)\right)(-\sin\left(k\pi\right))$$ $$y = \left(3 + 6\cos\left(k\pi\right)\right)\cos\left(k\pi\right)$$ By plugging in different integer values for k, we can find the specific tangent points on the curve.

Step 1: Find the derivative of r with respect to θ

We are given the polar curve: $$r = 3 + 6\sin{\theta}$$. Differentiate r with respect to θ: $$\frac{dr}{d\theta} = 6\cos{\theta}$$

Step 2: Find θ for horizontal tangents

A horizontal tangent occurs when $$\frac{dr}{d\theta} = 0$$. Using the derivative from Step 1: $$6\cos{\theta} = 0$$ Divide by 6: $$\cos{\theta} = 0$$ A cosine emerges zero at θ = π/2 + kπ, where k is an integer. In this case: $$\theta = \frac{\pi}{2}\ + k\pi$$ for k an integer

Step 3: Find the polar coordinates for horizontal tangents

Now that we have the values of θ for horizontal tangents, substitute θ into the equation of the curve to find the corresponding r: $$r = 3 + 6\sin\left(\frac{\pi}{2} + k\pi\right)$$ Remember that sine function has the following property: $$\sin\left(\frac{\pi}{2} + x\right) = \cos{x}$$ Therefore, we have: $$r = 3 + 6\cos\left(k\pi\right)$$

Step 4: Find θ for vertical tangents

A vertical tangent occurs when $$\frac{dr}{d\theta}$$ is infinite. Looking back at the derivative from Step 1, it's clear that this happens when cosine function is zero: $$\cos\theta = 0$$ But this is the same condition as for horizontal tangents, so there are no distinct vertical tangents.

Step 5: Convert polar to Cartesian coordinates

Now we need to convert the polar coordinates of the tangent points (r, θ) to Cartesian coordinates (x, y). The conversion formulas are: $$x = r\cos{\theta}$$ $$y = r\sin{\theta}$$ Substitute the values from Step 3: $$x = \left(3 + 6\cos\left(k\pi\right)\right)\cos\left(\frac{\pi}{2} + k\pi\right)$$ $$y = \left(3 + 6\cos\left(k\pi\right)\right)\sin\left(\frac{\pi}{2} + k\pi\right)$$ Through properties of sine and cosine, we have: $$x = \left(3 + 6\cos\left(k\pi\right)\right)(-\sin\left(k\pi\right))$$ $$y = \left(3 + 6\cos\left(k\pi\right)\right)\cos\left(k\pi\right)$$ Practically, we can find the tangent points for a set value of k (e.g., k = -2, -1, 0, 1, 2). Each point given by (x, y) will represent the location of a horizontal tangent on the polar curve.

Key Concepts

Derivatives in Calculus

Derivatives are essential tools in calculus used to understand how functions change. When dealing with polar curves, the focus is on finding how the radius, denoted as \( r \), changes with respect to the angle \( \theta \). This relationship is expressed by the derivative \( \frac{dr}{d\theta} \).

For the polar curve \( r = 3 + 6\sin{\theta} \), we computed the derivative as \( \frac{dr}{d\theta} = 6\cos{\theta} \). This tells us how the radius changes as \( \theta \) varies. By setting \( \frac{dr}{d\theta} = 0 \), we identify points where the curve's slope is zero, leading us to horizontal tangents. Understanding derivatives in this way helps in recognizing critical points where a curve's nature changes, such as transitioning from increasing to decreasing.

• First, identify your function, like \( r(\theta) \).
• Take the derivative with respect to \( \theta \).
• Use derivative values to find special conditions, like tangents.

In polar curves, these steps help in sketching the nature and behavior of the curve.

Tangent Lines

Tangent lines to curves give valuable insights into the behavior of the curve at specific points. For polar curves, identifying where these tangents are horizontal or vertical is a common task.

A horizontal tangent line occurs when the derivative \( \frac{dr}{d\theta} \) is zero, as it implies no change in \( r \) at that point, meaning the curve is flat in the \( \theta \) direction. For \( r = 3 + 6\sin{\theta} \), this happens when \( 6\cos{\theta} = 0 \) or \( \cos{\theta} = 0 \). This fascinating occurrence shows that, no matter how the curve is twisting or turning, at those points, it's moving perfectly horizontally.

• Horizontal tangents: Set \( \frac{dr}{d\theta} = 0 \) to find.
• Vertical tangents: Find where \( \frac{dx}{d\theta} \) or \( \frac{dy}{d\theta} \) is infinite.

Understanding where and why these tangent points occur helps visualize how a curve behaves, useful in applications from graphics to engineering.

Polar to Cartesian Conversion

Transitioning from polar to Cartesian coordinates is key in many applications since many tools and systems work in Cartesian grids. This conversion transforms points defined by angles and radii into \( x \) and \( y \) coordinates, making them easier to visualize and manipulate.

The conversion uses two primary formulas:

• \( x = r \cos{\theta} \)
• \( y = r \sin{\theta} \)

For the polar equation \( r = 3 + 6\cos(k\pi) \), applying these formulas for each selected \( \theta \) results in Cartesian points. These points illustrate physical locations where horizontal tangents from the original polar equation occur.

Converting coordinates helps in evaluating curves with explicit Cartesian geometry and is invaluable when calculating distances, areas, or when further analyzing the shapes of curves.