/*! 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 20 You are given \(f(x)\) on an int... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 20

You are given \(f(x)\) on an interval, say \(0< x< b\). Sketch several periods of the even function \(f_{c}\) of period \(2 b,\) the odd function \(f_{s}\) of period \(2 b,\) and the function \(f_{p}\) of period \(b\), each of which equals \(f(x)\) on \(0< x< b\). Expand each of the three functions in an appropriate Fourier series. $$f(x)=x^{2}, \quad 0< x< 1$$

Short Answer

Expert verified
Sketch even, odd, and periodic extensions, then use Fourier series formulas with calculated coefficients.

Step by step solution

01

- Define the given function

We are given the function: \( f(x) = x^2 \) on the interval \( 0 < x < 1 \).
02

- Sketch the even function \( f_c \)

The even extension of \( f(x) \) is done by mirroring \( f(x) = x^2 \) over the y-axis and then periodically repeating it with period \( 2b = 2 \). Therefore, on \( -1 < x < 0 \), \( f_c(x) = ( - x )^2 = x^2 \). The even extension over one period can be written as: \( f_c(x) = x^2, \text{ for } -1 < x < 1 \). Sketch the period from \( -1 \) to \( 1 \) and repeat.
03

- Sketch the odd function \( f_s \)

The odd extension of \( f(x) \) is done by negating \( f(x) \) over the y-axis and then periodically repeating it with period \( 2b = 2 \). Therefore, on \( -1 < x < 0 \), \( f_s(x) = -( - x )^2 = - x^2 \). The odd extension over one period can be written as: \( f_s(x) = \begin{cases} x^2, & 0 < x < 1 \-x^2, & -1 < x < 0 \end{cases} \). Sketch the period from \( -1 \) to \( 1 \) and repeat.
04

- Sketch the periodic function \( f_p \)

The periodic extension of \( f(x) = x^2 \) is by periodically repeating it with period \( b = 1 \). For \( f_p(x) \), the function is simply \( x^2 \) repeated every 1 unit. Thus, \( f_p(x) = x^2 \) for \( 0 < x < 1 \), then repeat.
05

- Find the Fourier series for \( f_c \)

The Fourier series for an even function of period \( 2 \) is as follows: \[ f_c(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos\left( \frac{n\pi x}{b} \right) \]. Calculate the coefficients: \[ a_0 = \frac{1}{b} \int_{0}^{b} f(x)\, dx = \frac{1}{1} \int_{0}^{1} x^2\, dx = \frac{1}{3} \]. \[ a_n = \frac{2}{b} \int_{0}^{b} f(x) \cos\left( \frac{n\pi x}{b} \right) dx = 2 \int_{0}^{1} x^2 \cos(n \pi x) dx \]. Evaluate the integral using integration by parts.
06

- Find the Fourier series for \( f_s \)

The Fourier series for an odd function of period \( 2 \) is as follows: \[ f_s(x) = \sum_{n=1}^{\infty} b_n \sin\left( \frac{n\pi x}{b} \right) \]. Calculate the coefficients: \[ b_n = \frac{2}{b} \int_{0}^{b} f(x) \sin \left( \frac{n\pi x}{b} \right) dx = 2 \int_{0}^{1} x^2 \sin(n \pi x) dx \]. Evaluate the integral using integration by parts.
07

- Find the Fourier series for \( f_p \)

The Fourier series for \( f(x) \) with period \( b = 1 \) is: \[ f_p(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos(2n\pi x) + b_n \sin(2n\pi x) \]. Calculate the coefficients: \[ a_0 = \int_{0}^{1} f(x) dx = \int_{0}^{1} x^2 dx = \frac{1}{3} \]. \[ a_n = \int_{0}^{1} f(x) \cos(2n\pi x) dx \]. \[ b_n = \int_{0}^{1} f(x) \sin(2n\pi x) dx \]. Evaluate the integrals using integration by parts.

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.

Even Functions
A function is called even if it satisfies the condition: \( f(x) = f(-x) \) for all \( x \) in its domain. This means that the graph of an even function is symmetric with respect to the y-axis. In our example, we extend the given function \( f(x) = x^2 \). Since \( ( - x )^2 = x^2 \), the extension remains the same on both sides of the y-axis. The periodic extension therefore mirrors around \( x = 0 \) and repeats every 2.
Odd Functions
A function is considered odd if it satisfies the condition \( f(-x) = - f(x) \) for all \( x \) in its domain. This implies the graph of an odd function is symmetric with respect to the origin. In the case of \( f(x) = x^2 \), the odd extension is created by reflecting and then negating the function on the negative side, resulting in \( -x^2 \). This function then repeats with a period of 2.
Periodic Functions
A periodic function repeats its values in regular intervals or periods. For a function \( f(x) \), if there exists a period \( T > 0 \) such that \( f(x + T) = f(x) \) for all \( x \), then \( f(x) \) is periodic with period \( T \). In this exercise, the given function \( f(x) = x^2 \) is periodically extended to have a period of 1. This means the function \( f(x) \) repeats itself every unit interval.
Fourier Coefficients
Fourier series approximates a periodic function using sines and cosines. To do this accurately, coefficients \( a_n \) and \( b_n \) are calculated. These are integrals that represent the weights of the cosine and sine terms. For even functions, only cosine terms are non-zero, and for odd functions, only sine terms are non-zero. To find these coefficients, integrals like: \[ a_n = \frac{2}{b} \int_{0}^{b} f(x) \text{cos}\bigg(\frac{n \pi x}{b}\bigg) \, dx \]. are used.
Integration by Parts
Integration by parts is a method used to solve integrals involving products of functions. It's based on the formula: \[ \int u \, dv = uv - \int v \, du \]. In the Fourier series, we often come across integrals like \( \int x^2 \cos(n \pi x) dx \) which can be evaluated using integration by parts. Here, we set \( u = x^2 \) and \( dv = \cos(n \pi x) dx \) to simplify the integral step-by-step.

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, frequency, and velocity amplitude for the motion of a particle whose distance \(s\) from the origin is the given function. $$s=2 \sin 3 t \cos 3 t$$

You are given a complex function \(z=f(t) .\) In each case, show that a particle whose coordinate is (a) \(x=\operatorname{Re} z,\) (b) \(y=\operatorname{Im} z\) is undergoing simple harmonic motion, and find the amplitude, period, frequency, and velocity amplitude of the motion. $$\quad z=2 e^{-i t / 2}$$

You are given \(f(x)\) on an interval, say \(0< x< b\). Sketch several periods of the even function \(f_{c}\) of period \(2 b,\) the odd function \(f_{s}\) of period \(2 b,\) and the function \(f_{p}\) of period \(b\), each of which equals \(f(x)\) on \(0< x< b\). Expand each of the three functions in an appropriate Fourier series. $$f(x)=\left\\{\begin{array}{ll} 1, & 0< x< 1 \\ 0, & 1< x< 3 \end{array}\right.$$

Find the average value of the function on the given interval. Use equation (4.8) if it applies. If an average value is zero, you may be able to decide this from a quick sketch which shows you that the areas above and below the \(x\) axis are the same. $$\cos x \text { on }(0,3 \pi)$$

Write an equation for a sinusoidal radio wave of amplitude 10 and frequency 600 kilohertz. Hint: The velocity of a radio wave is the velocity of light, \(c=3 \cdot 10^{8} \mathrm{m} / \mathrm{sec}.\)
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

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.