Problem 7 - Calculus

Question

Sketch the curve and check for \(x, y,\) and \(r\) symmetry. \(r^{2}=4 \cos 2 \theta \quad\) (lemniscate)

Step-by-Step Solution

Answer

The lemniscate is symmetric about the x-axis and the origin.

Step 1: Recognize the curve type

The equation given is \( r^2 = 4 \cos 2\theta \), which is the standard form of a lemniscate. This particular equation describes a lemniscate centered at the origin with its lobes aligned along the polar axis.

Step 2: Convert to Cartesian coordinates

To better analyze the symmetry and sketch the curve, convert the polar equation to Cartesian coordinates. Start by using the identities \( r^2 = x^2 + y^2 \) and \( \cos 2\theta = \cos^2\theta - \sin^2\theta \). Substitute \( x = r\cos\theta \) and \( y = r\sin\theta \) to rewrite the equation as \( x^2 + y^2 = 4 \frac{x^2 - y^2}{x^2 + y^2} \). Simplify this to get: \((x^2 + y^2)^2 = 4x^2 - 4y^2\).

Step 3: Determine symmetry with respect to x-axis

To check for symmetry concerning the x-axis, replace \( \theta \) with \( -\theta \) in the polar equation. The equation \( r^2 = 4 \cos (-2\theta) = 4 \cos 2\theta \) remains unchanged, indicating symmetry about the x-axis.

Step 4: Determine symmetry with respect to y-axis

To check for symmetry concerning the y-axis, replace \( \theta \) with \( \pi - \theta \). The equation becomes \( r^2 = 4 \cos ( \pi - 2\theta) = -4 \cos 2\theta \), which doesn't match the original equation. Thus, there's no symmetry about the y-axis.

Step 5: Determine symmetry with respect to the origin

Check for symmetry concerning the origin by replacing \( \theta \) with \( \theta + \pi \). The equation becomes \( r^2 = 4 \cos (2\theta + 2\pi) = 4 \cos 2\theta \), which matches the original equation, confirming symmetry about the origin.

Step 6: Sketch the curve

The lemniscate \( r^2 = 4 \cos 2\theta \) consists of two loops, one extending along the positive x-axis and another along the negative x-axis. These loops intersect at the origin \((0,0)\), with symmetric lobes about the origin and the x-axis. The curve is plotted by capturing points for values of \( \theta \) from 0 to \( 2\pi \).

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way to represent points on a plane, primarily using two values: the radial distance from a designated origin point and the angular direction from a base line. Imagine a circle, where every point is defined by how far it is from the center and the angle formed with the horizontal axis. This system becomes extremely useful when dealing with curves and shapes like the lemniscate.

• In polar coordinates, a point is denoted as \( (r, \theta) \) where \( r \) is the distance from the origin, and \( \theta \) is the angle made with the positive x-axis.
• The lemniscate equation \( r^2 = 4 \cos 2\theta \) naturally emerges in this system, as it captures the symmetry and looping nature effectively.

This approach is ideal for visualizing and understanding the intricacies of curves that exhibit radial symmetry, like the lemniscate. Understanding polar coordinates is crucial when sketching these kinds of curves, as they provide direct insight into the curve's geometric properties.

Cartesian Coordinates

While polar coordinates are excellent for dealing with curves like the lemniscate, often it's necessary to translate these into Cartesian coordinates to examine specific properties or for solving in a different context. Cartesian coordinates use an orthogonal grid system defined by x and y axes.

• In polar-to-Cartesian conversion, the formulae \( x = r \cos \theta \) and \( y = r \sin \theta \) are used.
• For the lemniscate \( r^2 = 4 \cos 2\theta \), translating to Cartesian involves substituting these relationships, yielding \( (x^2 + y^2)^2 = 4x^2 - 4y^2 \).

This transformation aids in the analysis of the curve's behavior over the Cartesian plane. It allows for the utilization of tools like standard algebraic techniques to explore aspects such as intersections, tangents, and more, offering a more detailed investigation beyond the polar framework.

Symmetry

Symmetry plays a functional role in curve sketching, bringing mathematical beauty and simplifying calculations. For the lemniscate \( r^2 = 4 \cos 2\theta \), assessing symmetry helps to determine how the curve mirrors itself about different axes or points.

• x-axis symmetry: The equation remains unchanged when testing with \( \theta \) replaced by \( -\theta \), confirming symmetry about the x-axis.
• y-axis symmetry: Testing by replacing \( \theta \) with \( \pi - \theta \) results in a negative version, disproving symmetry about the y-axis.
• Origin symmetry: The curve reflects about the origin when \( \theta \) is replaced by \( \theta + \pi \), reaffirming this reflective property.

Understanding these symmetries is essential for efficiently sketching curves and predicting their behavior, ultimately leading to quicker graphical solutions and interpretations.

Curve Sketching

Curve sketching is about translating mathematical expressions into visual representations, employing symmetries, critical points, and behaviors. The lemniscate \( r^2 = 4 \cos 2\theta \) showcases unique sketching characteristics.

• The curve consists of two loops, positioned symmetrically on the plane.
• It intersects the origin, thanks to its symmetry properties, with the lobes extending along the x-axis.
• Considering a range of \( \theta \) from 0 to \( 2\pi \) helps capture the complete shape of the curve.

To sketch such a curve, use the known symmetries and convert between coordinate systems if needed. This process involves plotting key points and observing how the lemniscate reflects across axes, ensuring an accurate depiction. Recognizing the characteristic figure-eight shape of a lemniscate assists in intuitive curve sketching, making this mathematical art form comprehensible.