Problem 30 - Calculus Early Transcendentals

Question

\(29-48\) Sketch the curve with the given polar equation. $$r^{2}-3 r+2=0$$

Step-by-Step Solution

Answer

The curve is two concentric circles with radii 1 and 2.

Step 1: Identify the Quadratic Equation

The given polar equation is \( r^2 - 3r + 2 = 0 \). This is a quadratic equation in terms of \( r \). Our goal in this step is to identify this equation so that we can find the solutions for \( r \).

Step 2: Solve the Quadratic Equation

To find the values of \( r \), we'll solve the quadratic equation \( r^2 - 3r + 2 = 0 \). We can use the quadratic formula: \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = -3 \), and \( c = 2 \). First, calculate the discriminant: \( b^2 - 4ac = (-3)^2 - 4 \times 1 \times 2 = 9 - 8 = 1 \). Next, substitute these into the quadratic formula to get: \[ r = \frac{3 \pm 1}{2} \] Thus, \( r = 2 \) or \( r = 1 \).

Step 3: Interpret the Solutions in Polar Coordinates

The solutions \( r = 2 \) and \( r = 1 \) represent circles in polar coordinates. Specifically, \( r = 2 \) corresponds to a circle with radius 2 centered at the origin, and \( r = 1 \) corresponds to a circle with radius 1 centered at the origin.

Step 4: Sketch the Polar Graph

Draw the polar coordinate system. First, draw a circle with radius 1 centered at the origin (this represents \( r = 1 \)). Then, draw another circle with radius 2, also centered at the origin (this represents \( r = 2 \)). You should now have two concentric circles, one inside the other, with the smaller circle representing \( r = 1 \), and the larger one \( r = 2 \).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that's characterized by the highest degree of the variable being squared, which means it includes terms like \(x^2\). The general form of a quadratic equation is:

• \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).

The equation provided in the exercise is \(r^2 - 3r + 2 = 0\). Here, \(r\) is the variable, and the numbers 1, -3, and 2 are the coefficients corresponding to \(a\), \(b\), and \(c\) respectively.
This equation represents a key concept when working with polar coordinates as it helps find specific distances (in this case, \(r\)) from the origin.
To solve a quadratic equation like this one, you can either factor the equation if it easily results in integer solutions, or use the quadratic formula for more complex scenarios. In this example, applying the quadratic formula \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) simplified the process, leading to solutions \(r = 2\) and \(r = 1\). These are crucial as they are directly used in other components of the exercise, like plotting on the polar graph.

Polar Graph

A polar graph is one of many ways to represent mathematical relationships and functions. It's used predominantly when dealing with polar coordinates, which offers an alternate way to plot points using a radius and angle. In a polar coordinate system:

• Each point on the graph is determined by a radius \(r\) and an angle \(\theta\).
• The position of any point is a combination of how far it is from the origin and what angle it forms with a reference direction (usually the positive x-axis).

When interpreting a polar graph, such as the one requested in the exercise, it's important to recognize how solutions to the quadratic equation affect the graph. Here, the solutions \(r = 2\) and \(r = 1\) indicate circles with these radii centered at the origin in polar coordinates.
To construct this polar graph, you'll need a coordinate system:

• First, draw the circle representing \(r=1\), where every point on this circle is at a radius 1 from the origin.
• Then, draw a larger circle for \(r=2\), marking points that maintain a radius of 2 from the origin.

It's a visually intuitive approach to see how quadratic and polar systems overlap to create such symmetric and recognizable shapes.

Concentric Circles

Concentric circles are a simple geometric concept but can be quite useful in understanding polar graphs. Two or more circles are said to be concentric when they share the same center point. In this exercise, we encountered concentric circles as the graph of the solutions \(r = 1\) and \(r = 2\). Here’s how they work:

• The inner circle, represented by \(r = 1\), is closer to the origin. It includes all points that are exactly 1 unit away from the center (the origin in polar coordinates).
• The outer circle, denoted by \(r = 2\), holds all points 2 units from the same center, encompassing the first circle without overlapping in any way beyond sharing the center.

Concentric circles help demonstrate the idea of multiple radii extending out from a common central point, vividly illustrating the solutions' spatial relationships when plotted.
Studying these can help you visualize distances and relations in a polar context, where each circle's radius symbolizes a solution from our quadratic equation. They're perfect for recognizing symmetries, distances, and arrangements akin to real-world objects, like ripples in a pond or tree rings.