Problem 2 - Essential Calculus

Question

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1-e \cos \theta}\)

Step-by-Step Solution

Answer

For \(e=1\), the graph is a parabola. For \(e=0.5\), the graph is an ellipse. For \(e=1.5\), the graph is a hyperbola.

Step 1: Graph for \(e=1\)

Graph the polar equation \(r=\frac{2(1)}{1-(1) \cos \theta}\) using a graphing utility. The resulting graph represents a conic section.

Step 2: Identify the conic for \(e=1\)

If \(e=1\), the plotted graph is a straight line, which means that it is a parabola.

Step 3: Graph for \(e=0.5\)

Next, graph the polar equation \(r=\frac{2(0.5)}{1-(0.5) \cos \theta}\) using a graphing utility. The resulting graph represents another type of conic section.

Step 4: Identify the conic for \(e=0.5\)

If \(e=0.5\), the graph will be an ellipse due to the fact that for \(0

Step 5: Graph for \(e=1.5\)

Finally, graph the polar equation \(r=\frac{2(1.5)}{1-(1.5) \cos \theta}\) using the graphing utility. The resulting graph represents a different kind of conic section.

Step 6: Identify the conic for \(e=1.5\)

If \(e=1.5\), the graph will be a hyperbola. This is because if \(e>1\), the conic section is a hyperbola.

Key Concepts

Conic Sections

A conic section is the curve obtained by intersecting a cone with a plane. When graphing polar equations, the value of the eccentricity e determines the type of conic section represented by the equation.

The three types of conic sections are

• Parabolas, which have an eccentricity of e = 1.
• Ellipses, with 0 < e < 1.
• Hyperbolas, where e > 1.

In the given exercise, the polar equation r = \(\frac{2 e}{1-e \cos \theta}\), with different values of e, represents different conic sections. By graphing this equation for various e, students can see the transition from one type of conic to another and better understand the properties of each section. It's essential to recognize these conic sections visually, as they are foundational in fields such as physics, engineering, and astronomy.

Polar Coordinates

Polar coordinates represent points on a plane using a distance from a reference point and an angle from a reference direction. The reference point is often called the pole (akin to the origin in the Cartesian coordinate system), and the reference direction typically aligns with the positive x-axis, which is known as the polar axis.

Any point in the plane can be represented by a pair (r, \theta), where r is the radial distance from the pole, and \theta is the angle measured in radians from the polar axis. Polar coordinates are particularly useful in situations where the relationship between two points is more straightforwardly described in terms of angles and distances, or where the systems exhibit radial symmetry.

Polar equations, such as r = \(\frac{2 e}{1-e \cos \theta}\), express the radial distance r as a function of the angle \theta and possibly other parameters, like the eccentricity e in the case of conic sections.

Transformation to Cartesian Coordinates

For better understanding or for different applications, polar coordinates can be converted to Cartesian coordinates using the equations:

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

This relation is key when transferring a problem from one coordinate system to another.

Graphing Utility

A graphing utility is a software or tool that helps plot equations onto a coordinate system. Graphing utilities can handle both Cartesian and polar coordinate systems, making them incredibly useful in visualizing complex equations, including conic sections.

When working with polar equations, graphing utilities allow students to quickly and accurately draw the curves associated with given equations. These tools often provide interactive features: by adjusting parameters, like the eccentricity e in our given exercise, users can observe real-time changes in the graph's shape, which offers a deeper conceptual understanding of the mathematical relationships involved.

Graphing utilities also aid in identifying curves by providing gridlines and points of reference on the graph. For any given equation, understanding the resulting graph's shape, symmetry, and intersections can give insights into solving related mathematical problems or applying these concepts in practical scenarios such as satellite trajectories or design of optical lenses.

Advantages in Education

For educators and students alike, graphing utilities are essential tools. They provide a visual aid that reinforces the connection between algebraic expressions and their geometric counterparts. By incorporating such tools in exercises like graphing r = \(\frac{2 e}{1-e \cos \theta}\), educators enhance learning experiences and support students in developing problem-solving skills.

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Problem 2

Other exercises in this chapter

Problem 1

Find \(d y / d x\). $$ x=t^{2}, y=5-4 t $$

View solution

Problem 1

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$ (4,3 \pi / 6) $$

View solution

Problem 2

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminatin

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Problem 2

Find \(d y / d x\). $$ x=\sqrt[3]{t}, y=4-t $$

View solution

Key Concepts

Conic Sections

A conic section is the curve obtained by intersecting a cone with a plane. When graphing polar equations, the value of the eccentricity e determines the type of conic section represented by the equation.

The three types of conic sections are

• Parabolas, which have an eccentricity of e = 1.
• Ellipses, with 0 < e < 1.
• Hyperbolas, where e > 1.

In the given exercise, the polar equation r = \(\frac{2 e}{1-e \cos \theta}\), with different values of e, represents different conic sections. By graphing this equation for various e, students can see the transition from one type of conic to another and better understand the properties of each section. It's essential to recognize these conic sections visually, as they are foundational in fields such as physics, engineering, and astronomy.

Polar Coordinates

Polar coordinates represent points on a plane using a distance from a reference point and an angle from a reference direction. The reference point is often called the pole (akin to the origin in the Cartesian coordinate system), and the reference direction typically aligns with the positive x-axis, which is known as the polar axis.

Any point in the plane can be represented by a pair (r, \theta), where r is the radial distance from the pole, and \theta is the angle measured in radians from the polar axis. Polar coordinates are particularly useful in situations where the relationship between two points is more straightforwardly described in terms of angles and distances, or where the systems exhibit radial symmetry.

Polar equations, such as r = \(\frac{2 e}{1-e \cos \theta}\), express the radial distance r as a function of the angle \theta and possibly other parameters, like the eccentricity e in the case of conic sections.

Transformation to Cartesian Coordinates

For better understanding or for different applications, polar coordinates can be converted to Cartesian coordinates using the equations:

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

This relation is key when transferring a problem from one coordinate system to another.

Graphing Utility

A graphing utility is a software or tool that helps plot equations onto a coordinate system. Graphing utilities can handle both Cartesian and polar coordinate systems, making them incredibly useful in visualizing complex equations, including conic sections.

When working with polar equations, graphing utilities allow students to quickly and accurately draw the curves associated with given equations. These tools often provide interactive features: by adjusting parameters, like the eccentricity e in our given exercise, users can observe real-time changes in the graph's shape, which offers a deeper conceptual understanding of the mathematical relationships involved.

Graphing utilities also aid in identifying curves by providing gridlines and points of reference on the graph. For any given equation, understanding the resulting graph's shape, symmetry, and intersections can give insights into solving related mathematical problems or applying these concepts in practical scenarios such as satellite trajectories or design of optical lenses.

Advantages in Education

For educators and students alike, graphing utilities are essential tools. They provide a visual aid that reinforces the connection between algebraic expressions and their geometric counterparts. By incorporating such tools in exercises like graphing r = \(\frac{2 e}{1-e \cos \theta}\), educators enhance learning experiences and support students in developing problem-solving skills.