/*! 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 30 Use Gnuplot to graph the functio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 30

Use Gnuplot to graph the function \(y=\frac{\sin x}{x}\) from \(x=-4 \pi\) to \(x=4 \pi\). What happens at \(x=0\) ?

Short Answer

Expert verified
At \(x=0\), \(y=1\) because the function \(y=\frac{\sin x}{x}\) approaches 1 as \(x\) approaches 0.

Step by step solution

01

Open Gnuplot

Start by opening the Gnuplot application on your computer.
02

Set the plotting range

Define the range of the x-axis to be from \(-4\pi\) to \(4\pi\). You can do this by typing the following command:```plaintextset xrange [-4*pi:4*pi]```
03

Define the Function

Define the function \(\frac{\sin x}{x}\) by entering the following command:```plaintextf(x) = sin(x)/x```
04

Handle the singularity at x=0

Gnuplot cannot handle the division by zero directly. Define the function such that it's 1 at x=0, this way:```plaintextf(x) = (x==0) ? 1 : sin(x)/x```
05

Plot the Function

Now, plot the function using the command:```plaintextplot f(x) title 'sin(x)/x'```
06

Analyze the Graph

Observe the behavior of the function at \(-4\pi\) to \(4\pi\) and particularly at \(x=0\). The function should smoothly approach 1 as \(x\) approaches 0.

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.

gnuplot plotting
Gnuplot is a versatile tool that allows us to visualize mathematical functions. It's particularly powerful for creating plots of trigonometric functions and handling various plotting challenges efficiently. To plot the function \(y=\frac{\sin x}{x}\) from \(x=-4\pi\) to \(x=4\pi\), follow these steps:
  • Open Gnuplot on your computer.
  • Set the range for the x-axis by typing: set xrange [-4*pi:4*pi]
  • Define the function as: f(x) = sin(x)/x
  • Handle singularities by defining the function at \(x=0\) as: f(x) = (x==0) ? 1 : sin(x)/x
  • Finally, plot the function with: plot f(x) title 'sin(x)/x'

You should see a smooth curve representing the function from \(x=-4\pi\) to \(x=4\pi\). This gives immediate visual insight into the behavior of the function across this range.
handling singularities
Singularities are points where a function becomes undefined, often resulting in division by zero. In the function \(y=\frac{\sin x}{x}\), a singularity occurs at \(x=0\) because the denominator becomes zero.
To handle this in Gnuplot, we define the function such that it smoothly handles \(x=0\). By using the conditional definition f(x) = (x==0) ? 1 : sin(x)/x, we set the function to equal 1 at \(x=0\), avoiding the undefined value.
This approach ensures the plot remains accurate without abrupt breaks or errors, providing a better visualization of the function鈥檚 behavior at that critical point.
trigonometric function behavior
Trigonometric functions often exhibit periodic and oscillatory behavior. The function \(y=\frac{\sin x}{x}\) is an example that showcases key properties of trigonometric functions.
  • As \(x\) approaches zero, the function smoothly transitions to 1, reflecting the limit of \(\frac{\sin x}{x}\) as \(x\) approaches zero.
  • The function oscillates and gradually decays in amplitude as \(x\) moves away from zero, creating a wave-like pattern.
  • This behavior is typical of sinc functions, which are often used in signal processing and other applications.

By visualizing \(y=\frac{\sin x}{x}\) in Gnuplot, students can better grasp these behaviors and understand the value of techniques like handling singularities to improve their plots.

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 amplitude, period, and phase shift of the given function. Then graph one cycle of the function, either by hand or by using Gnuplot (see Appendix B). \(y=2+\cos (5 x+\pi)\)

Find the exact value of the given expression in radians. \(\cos ^{-1} 1\)

Find the amplitude, period, and phase shift of the given function. Then graph one cycle of the function, either by hand or by using Gnuplot (see Appendix B). \(y=\sin (2 \pi x-\pi)\)

Draw the graph of the given function for \(0 \leq x \leq 2 \pi\). \(y=2-\sin x\)

For any point \((x, y)\) on the unit circle and any angle \(\alpha\), show that the point \(R_{\alpha}(x, y)\) defined by \(R_{\alpha}(x, y)=(x \cos \alpha-y \sin \alpha, x \sin \alpha+y \cos \alpha)\) is also on the unit circle. What is the geometric interpretation of \(R_{\alpha}(x, y)\) ? Also, show that \(R_{-\alpha}\left(R_{\alpha}(x, y)\right)=(x, y)\) and \(R_{\beta}\left(R_{\alpha}(x, y)\right)=R_{\alpha+\beta}(x, y)\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

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.