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Chapter 10: Problem 24

Sketch the polar curve. $$r=e^{\theta}, \quad-\pi \leq \theta \leq \pi.$$

Short Answer

Expert verified
To sketch the polar curve \(r=e^\theta\) for \(-\pi\leq\theta\leq\pi\), we analyze the behavior of the curve in different sections of the domain. Recognizing that the exponential function is always positive, we sketch key points for \(-\pi\leq\theta<0\) and \(0\leq\theta\leq\pi\) separately. The curve will tend towards the origin for \(\theta\) approaching \(-\pi\), intersect the x-axis at (1,0) for \(\theta=0\), and undergo exponential growth as the angle increases from \(0\) to \(\pi\). By connecting these key points smoothly, the resulting sketch is a spiral-like curve that starts from the origin and grows exponentially in both directions.

Step by step solution

01

Analyze the polar curve

First, it is important to note that the equation is in the form \(r = e^\theta\), where r is the distance from the origin and \(\theta\) is the angle in radians. The range of values for \(\theta\) is -蟺 to 蟺, which covers the entire domain of the polar graph.
02

Determine the function's behavior

Keep in mind that the exponential function is always positive. With this polar function, we notice that for \(\theta\) in the range: - (-鈭�, -蟺], the radius should be decaying towards 0 as \(\theta\) approaches -蟺. - (-蟺, 0), the curve grows in the opposite direction as in the previous range. - (0, 蟺), the curve undergoes exponential growth as the angle increases.
03

Sketch key points for -蟺 to 0

Now we will focus on the range \( -\pi \leq \theta < 0 \). For this range of angles, key points to consider are: 1. At \(\theta = -\pi\), the curve should be tending towards the origin. 2. A point halfway between -蟺 and 0, say \(\theta = -\pi/2\), gives us \(r = e^{-\pi/2}\), which is greater than 0 and less than the value at \(\theta = 0\). 3. At \(\theta = 0\), \(r = e^0 = 1\), so the curve will intercept the x-axis at (1,0) in Cartesian coordinates.
04

Sketch key points for 0 to 蟺

Next, we will sketch the polar function for the range \(0 \leq \theta \leq \pi\). Key points to consider are: 1. At \(\theta = 0\), as mentioned before, r = 1. 2. A point halfway between 0 and 蟺, say \(\theta = \pi/2\), gives us \( r = e^{\pi/2}\), which is greater than \( e^0 = 1\). 3. At \(\theta = \pi\), \(r = e^\pi\), which is greater than \(e^{\pi/2}\) and \(r = 1\).
05

Combine key points

Using the key points we found, sketch the curve. 1. Start at the origin, where the curve tends towards 0 as \(\theta\) approaches -蟺. 2. At \(\theta = -\pi/2\), the curve is at some radius between 0 and 1. 3. At \(\theta = 0\), the curve intersects the x-axis at (1,0). 4. Between \(\theta = 0\) and \(\theta = \pi/2\), the radius increases exponentially. 5. At \(\theta = \pi/2\), the radius is greater than 1. 6. At \(\theta = \pi\), the radius is greater than it was at \(\theta = \pi/2\). Finish the sketch by connecting all the key points smoothly. The result is a spiral-like curve that starts from the origin, grows exponentially in the opposite direction from -蟺 to 0, then grows exponentially from 0 to 蟺.

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Key Concepts

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

Understanding the Exponential Function in Polar Coordinates
In the given exercise, the equation is represented as \( r = e^{\theta} \). This involves the exponential function, which is a key part of the equation. Exponential functions have a constant base, which in this case is the number \( e \) (approximately 2.718), raised to the power of the variable \( \theta \). This mathematical phenomenon signifies growth or decay, dependent on the sign and values of \( \theta \).

  • When \( \theta \) increases, the exponential function grows rapidly due to its base \( e \).
  • Conversely, when \( \theta \) decreases, especially negatively, \( e^{\theta} \) decays towards zero; this is notable in the behavior of the polar curve as \( \theta \) approaches -蟺.

Understanding the exponential growth and decay offers insight into how the radius \( r \) changes and helps in predicting the curve's pattern on the polar graph.
Importance of Radians in Polar Equations
In the realm of polar coordinates, radians play a crucial role as they measure angles around the pole (the origin). An angle expressed in radians is directly tied to the radius of a circle, which makes it more natural and convenient for calculations in mathematics, especially in trigonometry and calculus.

  • The problem specifies \( \theta \) ranges from -蟺 to 蟺, covering a complete circle and allowing us to visualize one complete cycle of the curve.
  • Using radians ensures that the transformation from polar to Cartesian coordinates is seamless and understandable.

Learning to interpret and utilize radians in problems like this enhances understanding of how functions behave across different segments of the circle, aiding in accurately sketching curves.
Steps to Sketch a Polar Curve
Polar curve sketching involves plotting points based on polar equations, from which the curve emerges as these points connect. For the equation \( r = e^{\theta} \), sketching involves understanding how the curve behaves as \( \theta \) varies.

  • Identify key points by calculating \( r \) for significant angles within the given range \(-\pi \leq \theta \leq \pi\).
  • Plot these calculated points since they serve as guides for sketching the overall shape of the curve.
  • Note the direction and speed the curve unfolds with increasing \( \theta \).

The aim is to construct a continuous curve with no sharp turns, ensuring smooth transitions between points. This methodical approach aids in creating an accurate visual representation of the exponential curve.
Analyzing the Radius Behavior in Polar Graphs
Understanding how the radius \( r \) behaves as \( \theta \) changes provides key insights into sketching the curve. In this context, \( r \) is defined through an exponential function, \( r = e^{\theta} \), meaning it is influenced directly by \( \theta \):

  • For \( \theta < 0 \), \( r \) decreases, approaching 0 toward \( \theta = -\pi \). This shows the curve tightening up to the origin from the left side of the polar graph.
  • At \( \theta = 0 \), \( r = 1 \), indicating the curve stretches outward, reflecting away from the origin.
  • As \( \theta \) grows toward \( \pi \), the radius \( r \) expands exponentially, suggesting a spiraling outward motion.

By observing these radius changes, we recognize patterns which not only help in sketching but also in predicting behaviors of similar polar curves. This understanding is foundational for moving confidently with more complex polar equation challenges.

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