/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 49 Graphing a Polar Equation In Exe... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 49

Graphing a Polar Equation In Exercises \(45-54\) , use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. $$r=\frac{2}{1+\cos \theta}$$

Short Answer

Expert verified
The graph of the polar equation \(r=\frac{2}{1+\cos \theta}\) is a circle. The graph is traced only once over the interval \(0 \leq \theta \leq 2\pi\).

Step by step solution

01

Familiarize yourself with the given equation

The given polar equation is \(r=\frac{2}{1+\cos \theta}\). This equation represents a shape in a polar coordinate system, which can be plotted using a graphing utility.
02

Graph the equation

Input the given polar equation into a graphing utility/tool. The utility will generate a visual representation of the equation on a polar grid that shows all possible points \(r , \theta\) where \(r\) is the radius or the distance from the pole (origin), and \(\theta\) is the angle from the polar axis.
03

Analyze the graph

After inputting the equation into the graphing tool, a plot is obtained. By analyzing this plot, observe the pattern and nature of the graph. Note how many times the graph makes a complete rotation.
04

Determine the interval

By inspection, it's seen that the graph completes one full rotation over the interval \(0 \leq \theta \leq 2\pi\). That is, it traces the same path once as \(\theta\) varies from 0 to \(2\pi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Equation
A polar equation is a mathematical expression that defines a relationship between the distance from a fixed point (called the pole or origin) and the angle from a fixed direction (usually the positive x-axis). Unlike a Cartesian coordinate system which uses x and y coordinates to define a point in a plane, polar coordinates use r (the radial distance) and 胃 (the angular displacement) to specify positions.
The polar equation given in this exercise is \[r = \frac{2}{1+\cos \theta}\]which is a type of conic section known as a lima莽on. This specific form involves a variable radius r that changes based on the angle 胃, forming a unique shape when plotted on a polar grid.
Graphing Utility
To visualize polar equations, it's essential to use a graphing utility or a calculator that offers polar mode plotting. These tools transform the equation into a visual graph that shows how points r (胃) are located in the polar coordinate system.
When you enter the polar equation \[r = \frac{2}{1+\cos \theta}\]into the utility, it calculates all possible values of r for each 胃 within a specified range.
The utility surfaces a graph that visually represents this relationship where you can observe line symmetries and curves unique to polar systems. Most graphing calculators or software like Desmos or GeoGebra allow users to switch views to polar plots and provide helpful tools to zoom and explore particular areas of the plot.
Interval for Theta
Identifying the correct interval for 胃 is crucial for analyzing how the graph is traced. This interval determines how much of the curve is drawn within the polar grid.
The interval's purpose is to confine the tracing of the plot to just a single complete rotation, ensuring the image doesn't overlap unnecessarily. For many polar equations, especially those producing symmetrical curves, the angular 胃 is often contained within \[0 \leq \theta \leq 2\pi\]This range spans a complete rotation around the circle, starting and ending at the same point, without retracing any part of the curve, allowing you to see the entire pattern once.
Complete Rotation
A complete rotation in polar coordinates refers to the graph returning to its initial position after tracing out its path according to the defined equation. For many equations such as \[r = \frac{2}{1+\cos \theta}\]this means the graph covers 360 degrees or 2蟺 radians, which is the full circle sweep.
When you input the interval 胃 from 0 to 2蟺 into the graphing utility, the graph completes one cycle.
This complete tracing ensures you observe the entire plot pattern without overlaps or partial paths being missing. Knowing the correct interval and identifying a complete rotation helps in understanding the full structure of the graph and checking for symmetries or specific features like loops.

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Most popular questions from this chapter

True or False? In Exercises \(113-116,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) represent the same point on the polar coordinate system, then \(\left|r_{1}\right|=\left|r_{2}\right|\)

True or False? In Exercises \(113-116,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(x>0,\) then the point \((x, y)\) on the rectangular coordinate system can be represented by \((r, \theta)\) on the polar coordinate system, where \(r=\sqrt{x^{2}+y^{2}}\) and \(\theta=\arctan (y / x)\)

Eccentricity In Exercises 67 and 68 , let \(r_{0}\) represent the distance from a focus to the nearest vertex, and let \(r_{1}\) represent the distance from the focus to the farthest vertex. Show that the eccentricity of an ellipse can be written as \(e=\frac{r_{1}-r_{0}}{r_{1}+r_{0}}\) Then show that \(\frac{r_{1}}{r_{0}}=\frac{1+e}{1-e}\)

Area and Arc Length in Polar Coordinates Spiral of Archimedes The curve represented by the equation \(r=a \theta,\) where \(a\) is a constant, is called the spiral of Archimedes. (a) Use a graphing utility to graph \(r=\theta,\) where \(\theta \geq 0 .\) What happens to the graph of \(r=a \theta\) as \(a\) increases? What happens if \(\theta \leq 0 ?\) (b) Determine the points on the spiral \(r=a \theta(a>0, \theta \geq 0),\) where the curve crosses the polar axis. (c) Find the length of \(r=\theta\) over the interval \(0 \leq \theta \leq 2 \pi\) (d) Find the area under the curve \(r=\theta\) for \(0 \leq \theta \leq 2 \pi\)

Finding the Area of a Polar Region Between Two Curves In Exercises \(37-44\) , use a graphing utility to graph the polar equations. Find the area of the given region analytically. Inside \(r=2 \cos \theta\) and outside \(r=1\)
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