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Determine whether the given function is periodic. If so, find its fundamental period. $$ x^{2} $$

Short Answer

Expert verified
If yes, what is its fundamental period? Answer: No, the function \(f(x) = x^2\) is not a periodic function, as there is no non-zero value of T for which the function repeats itself.

Step by step solution

01

Understand the function

We have the quadratic function \(f(x) = x^2\). Recall that a periodic function repeats its values in regular intervals or periods.
02

Search for a Period

We will search for a non-zero value of T such that \(f(x + T) = f(x)\) for all x. Let's analyze the expression using the properties of exponents: $$ f(x + T) = (x + T)^2 \\ (x + T)^2 \stackrel{?}{=} x^2 \quad (\text{testing if equal for all } x) $$ For any arbitrary real number x, there is no fixed non-zero value \(T\) that satisfies the equation above for all x. Therefore, we can conclude that the function is not periodic.
03

Conclude

The function \(f(x) = x^2\) is not a periodic function since there is no non-zero value of T for which the function repeats itself.

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Key Concepts

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

Understanding Periodic Functions
In mathematics, a periodic function is one that repeats its values at regular intervals, known as the function's period. One of the simplest examples of a periodic function is the sine function, which repeats every interval of \(2\pi\). To determine whether a function is periodic, we must find if there exists a non-zero number \(T\), known as the fundamental period, such that the equation \(f(x + T) = f(x)\) holds true for all values of \(x\).

When analyzing a function for periodicity, it is important to verify the condition over its entire domain. If no such \(T\) exists, the function is classified as non-periodic, and it does not have a fundamental period. Understanding this concept is crucial when studying functions in trigonometry, signal processing, and various other applications in science and engineering.
Quadratic Functions and Their Properties
Quadratic functions are polynomial functions of degree two, generally expressed in the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These functions are graphically represented by parabolas, which are symmetric U-shaped curves. The vertex of the parabola marks the maximum or minimum point of the function, depending on the direction the parabola opens.

A key characteristic of quadratic functions is that they do not exhibit periodic behavior. Their graphs extend indefinitely without repeating patterns or intervals, as evidenced in the function \(f(x) = x^2\). Despite their non-periodic nature, quadratic functions are widely used to model phenomena in physics, economics, and many other fields, making them an essential element in the study of algebra.
Properties of Exponents
Exponents represent the power to which a number, known as the base, is raised. They are part of the fundamental tools in algebra and appear in various mathematical contexts. The properties of exponents dictate how to manipulate expressions involving powers and are essential for simplifying and solving equations.

For instance, when multiplying numbers with the same base, the exponents are added: \(a^m \cdot a^n = a^{m+n}\). Conversely, when dividing, the exponents are subtracted: \(\frac{a^m}{a^n} = a^{m-n}\). There's also the power of a power property: \((a^m)^n = a^{m \cdot n}\). These rules are indispensable when dealing with periodic functions because they aid in assessing whether a function's period could exist based on the transformation of the function's input by some period \(T\). In exploring periodicity in quadratic functions, we see that there isn't a consistent way to apply these properties to find a repeating interval, reinforcing the conclusion that quadratic functions are not periodic.

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Most popular questions from this chapter

The total energy \(E(t)\) of the vibrating string is given as a function of time by $$ E(t)=\int_{0}^{L}\left[\frac{1}{2} \rho u_{t}^{2}(x, t)+\frac{1}{2} T u_{x}^{2}(x, t)\right] d x ; $$ the first term is the kinetic energy due to the motion of the string, and the second term is the potential energy created by the displacement of the string away from its equilibrium position. For the displacement \(u(x, t)\) given by Eq. \((20),\) that is, for the solution of the string problem with zero initial velocity, show that $$ E(t)=\frac{\pi^{2} T}{4 L} \sum_{n=1}^{\infty} n^{2} c_{n}^{2} $$ Note that the right side of Eq. (ii) does not depend on \(t .\) Thus the total energy \(E\) is a constant, and therefore is conserved during the motion of the string. Hint: Use Parseval's equation (Problem 37 of Section 10.4 and Problem 17 of Section \(10.3)\), and recall that \(a^{2}=T / \rho .\)

Consider the equation $$ a u_{x x}-b u_{t}+c u=0 $$ where \(a, b,\) and \(c\) are constants. (a) Let \(u(x, t)=e^{\delta t} w(x, t),\) where \(\delta\) is constant, and find the corresponding partial differential equation for \(w\). (b) If \(b \neq 0\), show that \(\delta\) can be chosen so that the partial differential equation found in part (a) has no term in \(w\). Thus, by a change of dependent variable, it is possible to reduce Eq. (i) to the heat conduction equation.

Find the solution of the heat conduction problem $$ \begin{aligned} 100 u_{x x} &=u_{t}, & 00 \\ u(0, t) &=0, & u(1, t)=0, & t>0 \\ u(x, 0) &=\sin 2 \pi x-\sin 5 \pi x, & 0 \leq x \leq 1 \end{aligned} $$

Carry out the following steps. Let \(L=10\) and \(a=1\) in parts (b) through (d). (a) Find the displacement \(u(x, t)\) for the given initial position \(f(x) .\) (b) Plot \(u(x, t)\) versus \(x\) for \(0 \leq x \leq 10\) and for several values of \(t\) between \(t=0\) and \(t=20\). (c) Plot \(u(x, t)\) versus \(t\) for \(0 \leq t \leq 20\) and for several values of \(x .\) (d) Construct an animation of the solution in time for at least one period. (e) Describe the motion of the string in a few sentences. \(f(x)=\left\\{\begin{array}{ll}{1,} & {L / 2-12)} \\ {0,} & {\text { otherwise }}\end{array}\right.\)

Determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations. $$ u_{x x}+u_{y y}+x u=0 $$

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