Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

In order to graph a polar equation such as \(r=f(t)\) using a parametric equation grapher, you must replace this equation by \(x=f(t) \cos t\) and \(y=f(t) \sin t .\) These equations can be obtained by multiplying \(r=f(t)\) by \(\cos t\) and \(\sin t\), respectively. Confirm the discussions of conics in the text by graphing \(r=4 e /(1+e \cos t)\) for \(e=0.1,0.5,0.9,1,1.1\) and \(1.3\) on \([-\pi, \pi]\).

Step-by-Step Solution

Answer

Parametric equations x(t) and y(t) for r=4e/(1+e cos t) result in different conics based on the eccentricity e; ellipses for e<1, parabolas for e=1, and hyperbolas for e>1.

Step 1: Convert Polar to Parametric Equations

The given polar equation is \(r = 4e/(1 + e\cos t)\). To graph using a parametric grapher, convert it to parametric form with \(x = r\cos t\) and \(y = r\sin t\). Start by expressing \(r\) as a function of \(t\). Then, substitute \(r\) in the equations to get \(x = (4e/(1 + e\cos t))\cos t\) and \(y = (4e/(1 + e\cos t))\sin t\).

Step 2: Choose Values for Eccentricity

You need to graph the equations using different eccentricity values, \(e = 0.1, 0.5, 0.9, 1, 1.1,\) and \(1.3\). For each value, substitute \(e\) into both equations for \(x(t)\) and \(y(t)\).

Step 3: Graph for Each Eccentricity on the Interval

Using a graphing tool, plot the parametric equations \(x = (4e/(1 + e\cos t))\cos t\) and \(y = (4e/(1 + e\cos t))\sin t\) over the interval \([-\pi, \pi]\). For each \(e\) value, this will show how the conic sections change shape. Note the shapes: ellipse for \(e < 1\), parabola for \(e = 1\), and hyperbola for \(e > 1\).

Step 4: Analyze the Graphs

Observe the graphs for each \(e\) value. When \(e = 0.1, 0.5, 0.9\), the graph forms an ellipse, indicating the orbit is closed. For \(e = 1\), the graph is a parabola, indicating it is an open conic. When \(e = 1.1, 1.3\), the graph forms a hyperbola, indicating the paths diverge.

Key Concepts

Parametric Equations

Parametric equations are a way to express mathematical curves using parameters. Rather than defining the curve solely in terms of x and y coordinates, parametric equations use a separate variable—usually denoted as t or sometimes θ—to express both coordinates. This approach might initially seem complex, but it offers incredible versatility in describing different shapes and paths. A parametric equation takes the form of two linked equations: \(x(t)\) and \(y(t)\), where 't' is the parameter. For instance, in our exercise, the polar equation \(r = f(t)\) converts directly to parametric equations \(x(t) = f(t)\cos t\) and \(y(t) = f(t)\sin t\). This transformation allows a graphing tool to plot complex shapes more efficiently. Keep in mind that parametric equations often provide more explicit information about the direction and speed of the plotted curve.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. They are fundamental in mathematics and can represent many natural phenomena and mechanical systems. The primary types of conic sections include ellipses, parabolas, and hyperbolas. - **Ellipse**: Formed when the plane cuts all around the cone, leading to a closed curve. This is seen when the eccentricity (e) is less than 1. - **Parabola**: Occurs when the plane is parallel to the edge of the cone, resulting in an open curve where e equals 1. - **Hyperbola**: Develops when the plane intersects the cone at a somewhat steeper angle, resulting in two open, diverging curves when e is greater than 1. In the exercise, by varying the eccentricity values within the polar equation, one can observe transitions between these conic sections, illustrating how the shape of the path changes based on this parameter.

Eccentricity

Eccentricity is a number that characterizes the shape of conic sections. It defines how much the conic section deviates from being a circle. Utilized in both polar and parametric equations, eccentricity is crucial in determining the geometry of an orbit or path. - For ellipses, the eccentricity is between 0 and 1; the closer to 0, the more circular the ellipse.- When the eccentricity is precisely 1, the conic section is a parabola.- If the eccentricity value exceeds 1, the shape becomes a hyperbola.In polar graphing, particularly when using parametric equations, adjusting the eccentricity provides a vivid portrayal of how conic sections morph from one another, allowing us to see ellipses disappear into parabolas, or stretch into hyperbolas. Adjusting 'e' in equations such as our exercise "\(r=4e/(1+e\cos t)\)" reveals how conic paths are formed depending on this critical parameter.

Transformation from Polar to Cartesian

Transforming equations from polar to Cartesian coordinates bridges two ways of representing points on a plane. While polar coordinates use the radius and angle \((r,θ)\), Cartesian coordinates express points using \((x, y)\). This transformation is crucial for graphing using tools that work better with Cartesian input.For conversion, the formulas are rather straightforward:

• \(x = r \cos(θ)\)
• \(y = r \sin(θ)\)

Using these relations, any polar equation can be converted to its Cartesian form, as shown in our example where the polar equation is expressed as parametric equations for plotting in the Cartesian plane.This transformation process not only highlights different perspectives on the same mathematical concept but also aids in visualizing and interpreting complex curves and paths. By understanding and applying these conversions, you add flexibility to your mathematical toolkit, accessing insights from both coordinate systems.