/*! 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 13 Draw a sketch of the graph of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 13

Draw a sketch of the graph of the given equation.\(r=e^{\theta}\) (logarithmic spiral)

Short Answer

Expert verified
Plot points for different \(\theta\) values, connect them smoothly to form a spiral, and label the graph.

Step by step solution

01

Understand the Equation

The given equation is a polar equation in the form of a logarithmic spiral, given by \(r = e^{\theta}\). In this equation, \(r\) represents the radius and \(\theta\) represents the angle.
02

Calculate Key Points

To sketch the graph, calculate the values of \(r\) for different values of \(\theta\). For instance, when \(\theta = 0\), \(r = e^0 = 1\). When \(\theta = \frac{\pi}{4}\), \(r = e^{\frac{\pi}{4}}\), and so on. Make a table of \(\theta\) and corresponding values of \(r\).
03

Plot Key Points

Using the values computed in the previous step, plot the key points on the polar coordinate system. Start from \(\theta = 0\) and keep plotting points for increasing \(\theta\).
04

Draw the Spiral

Connect the plotted points smoothly to form the spiral. The curve starts at \(r = 1\) when \(\theta = 0\) and grows exponentially as \(\theta\) increases.
05

Label the Graph

Make sure to label the graph with the equation \(r = e^{\theta}\) and mark key points for clarity.

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 Coordinates
Polar coordinates are a way to represent points in the plane using a radius and an angle instead of the usual x and y coordinates (Cartesian coordinates). In polar coordinates, each point is determined by:
  • Radius (r): The distance from the origin to the point.
  • Angle (胃): The counterclockwise angle from the positive x-axis to the point.
Think of it like describing the position of a point using 'how far' and 'which direction' instead of 'over and up'. In the equation given in the exercise, r = 别镑胃, the radius r changes depending on the angle 胃.
As 胃 increases, you move around the origin in a counterclockwise direction, and as it changes, the distance from the origin changes too.
Exponential Function
The exponential function is an important mathematical function characterized by the constant e. It's written as e^x, where e is approximately 2.71828. In the context of the given exercise, the exponential function appears as 别镑胃.
The key feature of the exponential function is:
  • It grows rapidly as the input (胃) increases.
  • When the input is zero (胃 = 0), the exponential function equals 1 (i.e., e^0 = 1).
  • As 胃 increases, r = 别镑胃 starts at 1 and increases exponentially.
This means that for small increases in 胃, the radius r increases very quickly, creating the spiral effect in the polar coordinate system.
Graph Sketching
Graph sketching is the process of drawing a curve on a coordinate system based on a given function. For the equation r = 别镑胃 in polar coordinates, follow these steps:
  1. Calculate key points by substituting different values of 胃 into the equation to find the corresponding r. For example:
    - When 胃 = 0, r = 1.
    - When 胃 = 蟺/4, r = e^{蟺/4}.
    - When 胃 = 蟺/2, r = e^{蟺/2}.
  2. Plot the key points on a polar graph. Keep in mind each point is represented by a pair (r, 胃).

  3. Connect these points smoothly to visualize the graph. You will notice the logarithmic spiral shape forming, starting close to the origin and spiraling outwards exponentially.

  4. Label your graph clearly with the equation r = 别镑胃 and mark some key points to help visualize its growth.
This way, you can accurately sketch the logarithmic spiral described by the polar equation.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the area of the region enclosed by the graph of the given equation.\(r=4 \sin ^{2} \frac{1}{2} \theta\)

Find a polar equation of the graph having the given cartesian equation.\(x^{2}-y^{2}=16\)

Find a polar equation of the graph having the given cartesian equation.\(y=\frac{2 x}{x^{2}+1}\)

Find the area of the region which is inside the graph of the first equation and outside the graph of the second equation.\(\left\\{\begin{array}{l}r=2 \sin \theta \\ r=\sin \theta+\cos \theta\end{array}\right.\)

Draw a sketch of the graph of the given equation.\(r \cos \theta=-5\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.