Problem 16 - Calculus Early Transcendentals

Question

Find the points at which the following polar curves have a horizontal or a vertical tangent line. $$r=2+2 \sin \theta$$

Step-by-Step Solution

Answer

Solution: 1. Convert the polar equation to Cartesian coordinates: $$x = (2 + 2\sin\theta)\cos\theta$$ $$y = (2 + 2\sin\theta)\sin\theta$$ 2. Compute \(dy/d\theta\) and \(dx/d\theta\). 3. Calculate \(dy/dx\) by dividing \(dy/d\theta\) by \(dx/d\theta\). 4. Identify the conditions for horizontal tangents (\(dy/dx = 0\)) and vertical tangents (\(dy/dx\) has infinite or undefined value). 5. Find the points on the curve that satisfy these conditions, convert them back to Cartesian coordinates if needed, and state the final result.

Step 1: (Step 1: Convert polar equation to Cartesian coordinates)

First, we will convert the polar curve \(r = 2 + 2\sin\theta\) to Cartesian coordinates using the conversion formulas. $$x = r\cos\theta = (2 + 2\sin\theta)\cos\theta$$ $$y = r\sin\theta = (2 + 2\sin\theta)\sin\theta$$

Step 2: (Step 2: Compute \(dy/d\theta\) and \(dx/d\theta\))

Now, we will find the derivatives \(dy/d\theta\) and \(dx/d\theta\) using the chain rule to perform implicit differentiation with respect to \(\theta\). $$\frac{dy}{d\theta} = \frac{d}{d\theta} [(2 + 2\sin\theta)\sin\theta]$$ $$\frac{dx}{d\theta} = \frac{d}{d\theta} [(2 + 2\sin\theta)\cos\theta]$$

Step 3: (Step 3: Calculate \(dy/dx\))

Next, we will find the tangent slope \(dy/dx\) by dividing \(dy/d\theta\) by \(dx/d\theta\). $$\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}$$

Step 4: (Step 4: Find conditions for horizontal and vertical tangents)

A horizontal tangent line has a slope of 0 while a vertical tangent line has an undefined slope or infinite slope. So, we will look for points on the curve when \(dy/dx = 0\) (horizontal tangents) and when \(dy/dx\) has an infinite or undefined value (vertical tangents).

Step 5: (Step 5: Identify points with horizontal and vertical tangents)

Now that we have the conditions for horizontal and vertical tangent lines, we can find the points on the curve \(r = 2 + 2\sin\theta\) which satisfy these conditions. By using the polar equation and our previous analysis, we will identify these points and then convert them back to Cartesian coordinates if needed.

Key Concepts

Cartesian Coordinates

Understand the transformation of polar to Cartesian coordinates is essential in multiple fields of mathematics and physics. In essence, Cartesian coordinates refer to a system that defines every point in a plane by a pair of numerical coordinates. These coordinates represent the horizontal (x) and vertical (y) distances from the coordinate plane's origin.

Let's start by understanding that a polar curve, such as the one given by the equation \( r = 2 + 2\sin(\theta) \), can be depicted in a polar coordinate system, where \( r \) is the radius (distance from the origin) and \( \theta \) is the angle from the positive x-axis. However, converting this into Cartesian coordinates requires us to use the relations \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \).

For the given curve, applying these transformations:

• For the x-coordinate: \( x = (2 + 2\sin(\theta))\cos(\theta) \)
• For the y-coordinate: \( y = (2 + 2\sin(\theta))\sin(\theta) \)

Once we transform these equations, we can analyze the curve in the Cartesian plane which allows us to apply concepts such as differentiation to find slopes of tangent lines at various points.

Implicit Differentiation

Implicit differentiation is a powerful technique used to differentiate equations that are not easily solved for one variable in terms of another, which is often the case with polar equations when converted to Cartesian form.

When looking at the transformed equations from polar to Cartesian coordinates, we find ourselves with expressions for \( x \) and \( y \) that are both related to \( \theta \). Since our ultimate goal is to find tangent lines, we need the derivatives \( dx/d\theta \) and \( dy/d\theta \), and subsequently \( dy/dx \), which gives us the slope of the tangent line.

Using the chain rule, for the transformed Cartesian coordinates we get:

• \( \frac{dy}{d\theta} = \frac{d}{d\theta} [(2 + 2\sin(\theta))\sin(\theta)] \)
• \( \frac{dx}{d\theta} = \frac{d}{d\theta} [(2 + 2\sin(\theta))\cos(\theta)] \)

This step is crucial as it sets up the ability to determine when the tangent line is horizontal or vertical by looking at the slope, \( dy/dx \).

Tangent Lines

The concept of tangent lines is at the very heart of differential calculus. These lines represent the direction at which a curve is heading at any given point. For a curve described by Cartesian coordinates or a converted polar equation, we determine tangent lines using derivatives.

In the context of the given curve, we calculate the slope of the tangent line as \( dy/dx \), which is obtained by dividing \( dy/d\theta \) by \( dx/d\theta \), from the implicit differentiation we previously performed.

Remember:

• A horizontal tangent line occurs where the slope \( dy/dx = 0 \).
• A vertical tangent line occurs where the slope \( dy/dx \) is undefined or infinite.

By setting up these conditions and analyzing the derivatives, we can locate the precise points on the curve where the tangent lines are exactly horizontal or vertical. These points are particularly interesting as they can represent local maxima, minima, or inflection points on the curve, which are significant in understanding curve behavior.