Problem 68 - Calculus Early Transcendentals

Question

\(63-68\) Find the points on the given curve where the tangent line is horizontal or vertical. $$r^{2}=\sin 2 \theta$$

Step-by-Step Solution

Answer

Find critical \(\theta\) where derivatives equal 0 for horizontal/vertical tangents.

Step 1: Understanding the Problem

We are given the polar equation \( r^{2} = \sin 2\theta \) and need to find points where the tangent line to this curve is horizontal or vertical.

Step 2: Polar to Cartesian Conversion

Recall that in polar coordinates, a point \((r, \theta)\) can be converted to Cartesian coordinates using \(x = r \cos\theta\) and \(y = r \sin\theta\). However, since we're working directly in polar form to find horizontal and vertical tangents, let's focus on these conditions.

Step 3: Condition for Horizontal Tangent

A tangent line is horizontal if \( \frac{dy}{d\theta} = 0 \) and \( \frac{dx}{d\theta} eq 0 \). Calculate \( \frac{dy}{d\theta} = r \cos \theta + 2r \sin \theta \cdot \frac{dr}{d\theta} \) and set it to zero. Similarly, \( \frac{dx}{d\theta} = -r \sin \theta + 2r \cos \theta \cdot \frac{dr}{d\theta} \). Use these expressions to solve for the condition.

Step 4: Condition for Vertical Tangent

A tangent line is vertical if \( \frac{dx}{d\theta} = 0 \) and \( \frac{dy}{d\theta} eq 0 \). Use the expressions for \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \) from the previous step and set \( \frac{dx}{d\theta} = 0 \). This will help us find the condition for vertical tangents.

Step 5: Solving for \(r\) and \(\theta\) Conditions

We use the given equation \( r^2 = \sin 2\theta \) to help find derivations \( \frac{dr}{d\theta} \). The derivative \( \frac{dr}{d\theta} = \frac{1}{2r} \frac{d}{d\theta}(\sin 2\theta) = \frac{1}{2r} \cdot 2\cos 2\theta = \frac{\cos 2\theta}{r} \). Substitute \( \frac{dr}{d\theta} \) into the tangent conditions in previous steps to find critical \(\theta\) values.

Step 6: Calculating Horizontal and Vertical Points

Solve equations \( \cos \theta + 2\sin \theta \cdot \frac{\cos 2\theta}{r} = 0 \) for horizontal tangents and \( -\sin \theta + 2\cos \theta \cdot \frac{\cos 2\theta}{r} = 0 \) for vertical tangents. Use the equation \( r^2 = \sin 2\theta \) to find specific \(r\) and \(\theta\). You'll uncover solutions such as \( (r = \sqrt{\sin 2\theta}, \theta = \frac{\pi}{4}) \) and others depending on periodic properties of sine and cosine.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to describe the position of points in a plane. Instead of using traditional Cartesian coordinates (x, y), polar coordinates define a point based on a radius and an angle.

The key components in polar coordinates are:

• **Radius** (r): Distance from the origin to the point.
• **Angle** (\( \theta \)): Measured from the positive x-axis to the point in a counter-clockwise direction.

These coordinates provide a more intuitive way to handle problems involving symmetry or circular paths. They are particularly useful when dealing with curves like spirals or cardioids, which can become complex in Cartesian coordinates. By converting between Cartesian and polar coordinates, you can apply different calculus techniques that are sometimes more straightforward in one system than the other.

Tangent Lines

Tangent lines are straight lines that touch a curve at exactly one point without crossing it. This concept is essential in calculus as it helps to find the slope of a curve at any given point.

In polar coordinates, understanding tangent lines requires considering how changes in \( \theta \) affect both \( r \) and the position on the curve.

• **Horizontal Tangents**: Occur when the slope is zero.
• **Vertical Tangents**: Occur when the slope is undefined.

These lines are closely related to derivatives because the slope at any point is determined by differentiating the equations defining the curve. For polar equations, this means working with \( \frac{dr}{d\theta} \), \( \frac{dx}{d\theta} \), and \( \frac{dy}{d\theta} \). Interacting with these derivatives allows you to find where the tangent lines are horizontal or vertical.

Differentiation

Differentiation in calculus is the process of finding the derivative of a function. In the context of polar coordinates, it involves determining how the radius \( r \) and angle \( \theta \) change with respect to one another.

When we find derivatives in polar coordinates, we generally need to:

• Differentiate \( r \) with respect to \( \theta \) to obtain \( \frac{dr}{d\theta} \).
• Compute Cartesian derivatives by transforming polar equations.
• Use derivatives to understand the behavior of curves, such as finding tangent lines.

For instance, in solving equations that determine tangent lines in polar form, you need \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \), which are derived from polar relationships. These derivatives are crucial for defining the slope of a curve at a point, which is essential for sketching graphs and analyzing curve properties.

Horizontal and Vertical Tangents

Horizontal and vertical tangents are special cases of tangent lines, where the curve has specific slopes. Determining them in polar coordinates involves setting the correct derivative equal to zero.

For a curve given in polar form, horizontal and vertical tangents are identified as follows:

• **Horizontal Tangents**: Occur when \( \frac{dy}{d\theta} = 0 \) and \( \frac{dx}{d\theta} eq 0 \).
• **Vertical Tangents**: Occur when \( \frac{dx}{d\theta} = 0 \) and \( \frac{dy}{d\theta} eq 0 \).

To find these critical points, you work with the polar equation and its derivatives, solving for \( \theta \) and \( r \) using known identities and trigonometric relationships. These intersections often reveal key characteristics about the symmetry and nature of the curve itself.