Problem 19 - Calculus Early Transcendentals

Question

Graph the conics \(r=e /(1-e \cos \theta)\) with \(e=0.4,0.6\) . \(0.8,\) and 1.0 on a common screen. How does the value of \(e\) affect the shape of the curve?

Step-by-Step Solution

Answer

As \( e \) increases, the ellipse becomes more elongated; at \( e = 1 \), the curve is a parabola.

Step 1: Understanding the Equation

The given equation is in polar form, representing conic sections. The parameter \( e \) is the eccentricity, and it determines the type of conic section. The general form is \( r = \frac{e}{1 - e \cos \theta} \). By substituting different values of \( e \), we can observe different conic sections.

Step 2: Identify Conic Sections for Different \( e \)

The value of \( e \) determines the conic section:- \( 0 < e < 1 \) indicates an ellipse.- \( e = 1 \) indicates a parabola.For our values:- \( e = 0.4, 0.6, 0.8 \) represent ellipses.- \( e = 1.0 \) represents a parabola.

Step 3: Graphing the Conics

Graph each equation by substituting the given values of \( e \):- For \( e = 0.4 \), the equation becomes \( r = \frac{0.4}{1 - 0.4 \cos \theta} \).- For \( e = 0.6 \), the equation becomes \( r = \frac{0.6}{1 - 0.6 \cos \theta} \).- For \( e = 0.8 \), the equation becomes \( r = \frac{0.8}{1 - 0.8 \cos \theta} \).- For \( e = 1.0 \), the equation becomes \( r = \frac{1.0}{1 - 1.0 \cos \theta} \).Using a graphing tool, plot these equations to observe the conics for each eccentricity value on a common screen.

Step 4: Analyze the Effect of \( e \) on the Shapes

By looking at the graphs, note the following:- For \( e = 0.4, 0.6, \) and \( 0.8 \), the graphs will show ellipses with increasing elongation as \( e \) increases.- The ellipse becomes more stretched along the major axis as \( e \) approaches 1.- At \( e = 1.0 \), the graph becomes a parabola, indicating the focus is at the edge of the curve.

Key Concepts

Eccentricity

Eccentricity is a key concept in understanding conic sections. It is denoted by the parameter \( e \), which helps determine the shape of a conic. The equation \( r = \frac{e}{1 - e \cos \theta} \) in polar coordinates describes these sections. Eccentricity can be thought of as a measure of how much a conic section deviates from being circular. A circle has an eccentricity of 0, meaning it is perfectly round.

In our exercise, we observe different values of \( e \) such as 0.4, 0.6, 0.8, and 1.0. Here's what they indicate about the conic sections:

• \( 0 < e < 1 \): This range represents ellipses. As \( e \) increases within this range, the ellipse becomes more elongated.
• \( e = 1 \): This value denotes a parabola.

Understanding the effect of \( e \) is crucial because it directly influences the shape and type of the conic section we are dealing with. When we graph these equations, the changing shapes help us visualize how eccentricity affects conic sections.

Polar Coordinates

Polar coordinates provide a different perspective from the traditional Cartesian system. They are used to represent points on a plane using a distance from a reference point (usually the origin) and an angle from a reference direction. This system is particularly useful when dealing with curves that have rotational symmetry, such as conic sections.

In our exploration of conic sections, we use the polar form \( r = \frac{e}{1 - e \cos \theta} \), where \( r \) is the radial distance from the origin and \( \theta \) is the angle. Polar coordinates are powerful for graphing conics because they align well with their symmetrical properties.

• This form easily helps visualize the conic section's orientation and shape.
• By substituting different values for \( e \), \( \theta \), and \( r \), you can observe how the curve behaves, expanding or contracting depending on the angle \( \theta \), providing a distinct pattern not as easily seen in standard coordinates.

Using polar coordinates simplifies complex graphing tasks, especially with curves having symmetry or rotational properties.

Graphing Conic Sections

Graphing conic sections is a visual method to understand how the value of eccentricity \( e \) affects their shape. By using the polar equation \( r = \frac{e}{1 - e \cos \theta} \), you visualize the nature of ellipses and parabolas in the coordinate plane. Let's break down the significance of the exercise values.

For the given values of \( e \):

• When \( e = 0.4, 0.6, \) and \( 0.8 \), we're graphing ellipses. You will notice as you increase \( e \), the ellipse appears more elongated along its major axis.
• At \( e = 1.0 \), the graph depicts a parabola. This transformation shows that as \( e \) approaches 1, the elliptical shape shifts to open as a parabola.

Graphing these conics on a common screen helps you compare the shapes and comprehends how the increment in \( e \) from 0.4 to 1.0 transitions an ellipse into a parabola. The exercise shows that even slight changes in \( e \) have significant impacts on the conic's appearance, which is vital for understanding their geometric properties.