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Chapter 10: Problem 83

Sketch a graph of the polar equation. $$ r=4(1+\cos \theta) $$

Short Answer

Expert verified
The graph of the polar equation \(r = 4(1+\cos\theta)\) is a cardioid, with the cusp located at the origin pointing in the direction of the positive x-axis. It touches the origin when \(\theta= \pi\) and reaches its maximum 'r' = 8 when \(\theta = 0\) or \(2\pi\).

Step by step solution

01

Identify Symmetry

Check if the polar equation exhibits symmetry. It can be about the x-axis, y-axis or origin. For this equation, trying \(\theta\) and \(-\theta\) both give the same 'r'. Therefore, the graph has symmetry about the x-axis.
02

Locate Maximum and Minimum Values of \('r'\)

The minimum value of the function occurs when \(\cos \theta\) is at its minimum of -1. So, the minimum 'r' is \(4(1 - 1) = 0\), at \(\theta = \pi\). The maximum value of \(\cos \theta\) is 1, hence, our 'r' is maximum when \(\theta = 0\) or \(2\pi\), which is \(4(1 + 1) = 8\). Thus, the graph of the equation hits these extremities at \((r, \theta) = (0, \pi)\) and at \((r, \theta) = (8, 0), (8, 2\pi)\).
03

Sketch the Polar Graph

To plot these points, start by plotting the maximum 'r' at \(\theta = 0\) and \(2\pi\). This gives a dot 8 units along the x-axis in both directions as there's symmetry about the x-axis. Now, plot another point where 'r' = 0 at \(\theta = \pi\). This indicates a dip at the origin. So, the graph starts from the point (8,0), dips down to the origin at \((0, \pi)\), then goes back to the point (8,2\pi). The graph forms a cardioid shape (heart-like) facing in the direction of the positive x-axis.

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Key Concepts

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

Polar Coordinates
Polar coordinates form a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This reference point is known as the pole, typically corresponding to the origin in a Cartesian coordinate system, and the reference direction is along the positive x-axis.

In mathematics, polar coordinates are represented as \( r, \theta \) where \( r \) is the radius or the distance from the pole, and \( \theta \) is the angle measured in radians from the positive x-axis. Polar coordinates are especially useful in situations where the relationship between two points is better expressed by the angle and distance rather than x and y coordinates, such as in the case of circular or spiral patterns.
Cardioid Shape
The cardioid is a heart-shaped curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is also the graphic representation of many polar equations, specifically when the equation has the form \( r = a(1 + \cos \theta) \) or \( r = a(1 - \cos \theta) \).

In our exercise, the polar equation \( r=4(1+\cos \theta) \) produces a cardioid. As the angle \( \theta \) varies from 0 to \( 2\pi \) radians, the cardioid shape emerges. This shape is symmetrical about the x-axis and has a single cusp at the pole where the curve comes to a point.
Symmetry in Polar Graphs
In polar graphs, symmetry simplifies the drawing and understanding of curves. There are three types of symmetry to consider in polar graphs: symmetry about the x-axis, the y-axis, and the origin. A graph is symmetrical about the x-axis if replacing \( \theta \) with \( -\theta \) in the equation yields the same value of \( r \).

The polar equation \( r=4(1+\cos \theta) \) we are examining is symmetric about the x-axis, as indicated in Step 1 of the solution. When plotting, this symmetry allows us to reflect points across the x-axis, thus reducing the range of angles we need to evaluate to complete the graph.
Maximum and Minimum Values in Polar Equations
Identifying the maximum and minimum values of \( r \) in polar equations is crucial as it assists in understanding the extent of the graph. These values determine the farthest and nearest points of the graph to the pole. In the case of the equation \( r=4(1+\cos \theta) \) given in the exercise, the maximum value of \( r \) occurs at \( \theta = 0 \) and \( \theta = 2\pi \) where \( r \) equals 8, and the minimum value is 0 when \( \theta = \pi \) as the \( \cos \theta \) term reduces \( r \) to zero.

Finding these extremities is crucial for sketching a proper graph, as these points guide the shape and boundaries of the curve, ultimately helping to visualize the range and the overall appearance of the graph.

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Most popular questions from this chapter

(a) Use a graphing utility to graph each set of parametric equations. \(x=t-\sin t \quad x=2 t-\sin (2 t)\) \(y=1-\cos t \quad y=1-\cos (2 t)\) \(0 \leq t \leq 2 \pi \quad 0 \leq t \leq \pi\) (b) Compare the graphs of the two sets of parametric equations in part (a). If the curve represents the motion of a particle and \(t\) is time, what can you infer about the average speeds of the particle on the paths represented by the two sets of parametric equations? (c) Without graphing the curve, determine the time required for a particle to traverse the same path as in parts (a) and (b) if the path is modeled by \(x=\frac{1}{2} t-\sin \left(\frac{1}{2} t\right) \quad\) and \(\quad y=1-\cos \left(\frac{1}{2} t\right)\)

Find the area of the surface generated by revolving the curve about each given axis. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Interval }} \\\ x=4 \cos \theta, y=4 \sin \theta, &\quad 0 \leq \theta \leq \frac{\pi}{2}, \quad y \text { -axis } \end{array} $$

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. Folium of Descartes: \(x=\frac{3 t}{1+t^{3}}, \quad y=\frac{3 t^{2}}{1+t^{3}}\)

Find the area of the surface generated by revolving the curve about each given axis. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Interval }} \\\ x=\frac{1}{3} t^{3}, y=t+1, &\quad 1 \leq t \leq 2, \quad y \text { -axis }\end{array} $$

Use the parametric equations \(x=a(\theta-\sin \theta) \quad\) and \(\quad y=a(1-\cos \theta), a>0\) to answer the following. (a) Find \(d y / d x\) and \(d^{2} y / d x^{2}\). (b) Find the equations of the tangent line at the point where \(\theta=\pi / 6\) (c) Find all points (if any) of horizontal tangency. (d) Determine where the curve is concave upward or concave downward. (e) Find the length of one arc of the curve.
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