/*! 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 17 Consider \(|x|\) as a generalize... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 17

Consider \(|x|\) as a generalized function and find its derivative.

Short Answer

Expert verified
The derivative of \(|x|\) is a piecewise function, which is \(1\) for \(x > 0\), \(-1\) for \(x < 0\), and does not exist for \(x = 0\).

Step by step solution

01

Definition of Absolute Value Function

The absolute value function, \(|x|\), is a piecewise function defined as follows: \(|x| = x\) for \(x > 0\), \(|x| = -x\) for \(x < 0\), and \(|x| = 0\) for \(x = 0\). Based on this definition, we can describe \(|x|\) as a piecewise function.
02

Differentiation of the Piecewise Function

The derivative of \(|x|\) will also be a piecewise function. To find this, we take derivative of each piece separately. For \(x > 0\), the derivative of \(x\) is \(1\). For \(x < 0\), the derivative of \(-x\) is \(-1\).
03

Conclusion

In accordance with standard mathematical procedures, at \(x = 0\), the derivative of the function \(|x|\), denoted as \( \frac{d}{dx} |x| \), fails to exist because \(|x|\) has a corner at \(x = 0\), so it is not differentiable at that point. Thereby, the derivative of \(|x|\) can be described as a piecewise function: \( \frac{d}{dx} |x| = 1\) for \(x > 0\), \(\frac{d}{dx} |x| = -1\) for \(x < 0\), and \(\frac{d}{dx} |x|\) does not exist for \(x = 0\).

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.

Absolute Value Function
The absolute value function is a fundamental concept in mathematics, represented as \(|x|\). It's useful for expressing the magnitude of a number without reference to its sign. This function is crucial for handling real-world measurements where negative values don't make sense, like distance or speed.

The absolute value function is defined piecewise:
  • \(|x| = x\) for \(x > 0\)
  • \(|x| = -x\) for \(x < 0\)
  • \(|x| = 0\) for \(x = 0\)
This piecewise definition arises because the value of \(|x|\) changes depending on whether \(x \) is positive, negative, or zero.

Understanding absolute functions helps in managing different scenarios rooms where the output must remain positive, such as finding distances between points on graphs or maps.
Piecewise Function
A piecewise function is a function composed of multiple sub-functions, each defined on a particular interval of the main function's domain. These sub-functions allow handling different conditions or situations within a single framework.

### Characteristics of Piecewise Functions
  • Different Definitions: Each piece can have its own formula.
  • Specific Ranges: Each formula applies to a specific part of the domain.
  • Smoothness: May or may not be continuous or differentiable at the transition points.
For \(|x|\), it transitions at \(x = 0\) where it switches from \(x\) to \(-x\). Understanding how to break down functions into pieces allows students to analyze and solve complex mathematical problems more easily, focusing on smaller, simpler problems.
Derivative
The derivative is a measure of how a function changes as its input changes. In essence, it represents the slope of the tangent line at any point on the function鈥檚 graph.

### Basic Rules of Derivatives
  • For \(x > 0\): The derivative of \(x\) is \(1\).
  • For \(x < 0\): The derivative of \(-x\) is \(-1\).
  • At \(x = 0\): The derivative does not exist due to the corner formed by the absolute value function.
The non-existence of a derivative at certain points, like at \(x = 0\) for \(|x|\), is an important concept as it indicates places where the function is not differentiable. This often occurs in functions that have sharp turns or corners. Recognizing these points is key to understanding the behavior and characteristics of a function.

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

Write a density function for two point charges \(q_{1}\) and \(q_{2}\) located at \(\mathbf{r}=\mathbf{r}_{1}\) and \(\mathbf{r}=\mathbf{r}_{2}\), respectively.

Evaluate the following integrals: (a) \(\int_{-\infty}^{\infty} \delta\left(x^{2}-5 x+6\right)\left(3 x^{2}-7 x+2\right) d x\). (b) \(\int_{-\infty}^{\infty} \delta\left(x^{2}-\pi^{2}\right) \cos x d x\). (c) \(\int_{0.5}^{\infty} \delta(\sin \pi x)\left(\frac{2}{3}\right)^{x} d x\). (d) \(\int_{-\infty}^{\infty} \delta\left(e^{-x^{2}}\right) \ln x d x\).

Show that \(x \delta^{\prime}(x)=-\delta(x)\).

Prove the completeness of \(\mathbb{C}\), using the completeness of \(\mathbb{R}\).

Write a density function for four point charges \(q_{1}=q, q_{2}=-q, q_{3}=q\) and \(q_{4}=-q\), located at the corners of a square of side \(2 a\), lying in the \(x y\) plane, whose center is at the origin and whose first corner is at \((a, a)\).
See all solutions

Recommended explanations on Biology Textbooks

Biology Experiments

Read Explanation

Chemistry of Life

Read Explanation

Ecology

Read Explanation

Communicable Diseases

Read Explanation

Plant Biology

Read Explanation

Biological Processes

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.