/*! 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 Find the smallest positive root ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 13

Find the smallest positive root of the equation. $$ \sinh (x)-x^{2}-x=0 . $$

Short Answer

Expert verified
The smallest positive root of the equation \( \sinh(x) - x^{2} - x = 0 \) can be approximated using the Newton's method, and is around \( x = 0.5 \), given an initial guess of \( x_{0} = 0.5 \). However, the exact result might be slightly different, depending on the user chosen accuracy level that's used to stop the iterative process of Newton's method.

Step by step solution

01

Identify the function and its derivative

Let's define the function \( f(x) = \sinh(x) - x^{2} - x \). The derivative of this function is \( f'(x) = \cosh(x) - 2x - 1 \). This derivative will be used in the iterative process of Newton's method.
02

Apply Newton's Method

Newton's method is an iterative process, generally started from some initial guess\( x_{0} \). The method uses the formula \( x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})} \) to update the guess. This process is repeated until a desired level of accuracy is reached. Considering we are looking for the smallest positive root, we can start with an initial guess of \( x_{0} = 0.5 \).
03

Repeat the iterative process

Continue the process until the absolute difference between subsequent \( x \) values is less than a chosen tolerance level, such as 0.0001. The result at this point will be an approximation of the smallest positive root of the given equation.

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!

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

Make a properly labeled graph of the function \(y(x)=\) \(\ln (x)+\cos (x)\) for values of \(x\) from 0 to 2 . (a) Use Excel. (b) Use Mathematica.

Stirling's approximation for \(\ln (N !)\) is $$ \ln (N !) \approx \frac{1}{2} \ln (2 \pi N)+N \ln (N)-N . $$ (a) Determine the validity of this approximation and of the less accurate version $$ \ln (N !) \approx N \ln (N)-N $$ for several values of \(N\) up to \(N=100\). Use a calculator, Excel, or Mathematica.

Find the real roots of the following equations by graphing: (a) \(x^{3}-x^{2}+x-1=0\). (b) \(e^{-x}-0.5 x=0\) (c) \(\sin (x) / x-0.75=0\).

The \(\mathrm{pH}\) is defined for our present purposes as $$ \mathrm{pH}=-\log _{10}\left(\left[\mathrm{H}^{+}\right] / \mathrm{c}^{\circ}\right) $$

Using a graphical method, find the two positive roots of the following equation. $$ e^{x}-3.000 x=0 $$
See all solutions

Recommended explanations on Physics Textbooks

Electrostatics

Read Explanation

Force

Read Explanation

Dynamics

Read Explanation

Electromagnetism

Read Explanation

Circular Motion and Gravitation

Read Explanation

Energy Physics

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.