Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 6
Graph the polar equations. (a) \(r=e^{\theta / 2 \pi},\) for \(0 \leq \theta \leq 2 \pi\) (b) \(r=e^{-\theta / 2 \pi},\) for \(0 \leq \theta \leq 2 \pi\)
Short Answer
Expert verified
(a) The graph is an outward spiral; (b) the graph is an inward spiral.
Step by step solution
01
Introduction to Polar Equations
To graph polar equations, we need to understand how the polar coordinate system works. In polar coordinates, each point in the plane is determined by a distance from the origin (r) and an angle (胃) from the positive x-axis. We will graph the given polar equations over the specified interval for 胃.
02
Understanding the Equation (a)
The first equation is given as \(r = e^{\theta / 2 \pi}\). Here, r represents the radius, and 胃 varies from 0 to \(2\pi\). This creates a spiral pattern since the radius increases exponentially as 胃 increases.
03
Plotting Equation (a)
Begin at 胃 = 0, where \(r = e^{0} = 1\). As 胃 increases to \(2\pi\), calculate r at various points to see the pattern. For example, at 胃 = \(\pi\), \(r = e^{\pi / 2\pi} = e^{1/2}\), and at 胃 = \(2\pi\), \(r = e^{1} = e\). Plot these values to create a spiral that expands outward.
04
Understanding the Equation (b)
The second equation is \(r = e^{-\theta / 2 \pi}\). Like equation (a), 胃 varies from 0 to \(2\pi\). However, with a negative exponent, the radius decreases as 胃 increases, forming a spiral inward towards the origin.
05
Plotting Equation (b)
At 胃 = 0, \(r = e^{0} = 1\). As 胃 approaches \(2\pi\), calculate intermediate values, such as at 胃 = \(\pi\), \(r = e^{-1/2}\), and at 胃 = \(2\pi\), \(r = e^{-1}\). The graph creates a spiral that contracts inward towards the origin, decreasing in size.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Equations
Polar equations offer a unique way to represent curves using the polar coordinate system. This system defines each point in a 2-D plane with a radius \( r \) (distance from the origin) and an angle \( \theta \) (angular direction from the positive x-axis). Understanding this representation is crucial for graphing polar equations.For instance, a spiral pattern can be recognized through equations like \( r = e^{\theta / 2 \pi} \) or \( r = e^{-\theta / 2 \pi} \). These equations specify how the radius changes as \( \theta \) changes:
- In \( r = e^{\theta / 2 \pi} \), the radius grows as \( \theta \) increases due to the positive exponent, forming an outward spiral.
- Conversely, in \( r = e^{-\theta / 2 \pi} \), the radius shrinks as \( \theta \) grows due to the negative exponent, creating an inward spiral.
Graphing Spirals
Spirals are fascinating patterns often found in nature and mathematics. Graphing spirals in polar coordinates reveals how they expand or contract. A spiral's behavior can be dramatically different based on the function type and the exponent's sign.For an outward spiral like \( r = e^{\theta / 2 \pi} \):
- Start at \( \theta = 0 \) with \( r = 1 \). The radius increases as \( \theta \) moves towards \( 2\pi \).
- As \( \theta \) reaches \( \pi \), you calculate \( r = e^{1/2} \), gradually increasing in size until \( r = e \) at \( \theta = 2\pi \). This forms a smooth expanding spiral.
- Begin similarly at \( \theta = 0 \) with \( r = 1 \), but this time the radius diminishes towards the origin as \( \theta \) approaches \( 2\pi \).
- Intermediate points like \( \theta = \pi \) give \( r = e^{-1/2} \), further reducing to \( r = e^{-1} \) at \( 2\pi \). This results in a contracting spiral inward.
Exponential Growth and Decay
Exponential functions are powerful tools in mathematics, describing rapid growth or decay. Their properties are observable in numerous real-world applications from population dynamics to finance.Exponential growth, as in \( r = e^{\theta / 2 \pi} \):
- Characterized by the constant acceleration of increase.
- The function multiplies rapidly as \( \theta \) grows, showcasing how small incremental changes in \( \theta \) lead to disproportionately large changes in \( r \).
- Demonstrates the principle of steady decrease, approaching zero.
- This allows examination of phenomena that diminish over time or distance, providing insight into consumption, depreciation, and natural decay processes.
