/*! 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 Approximate the real zeros of \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 20

Approximate the real zeros of \(g(x):=x^{4}-x-3\)

Short Answer

Expert verified
As this involves iterative approximation, the exact answer may vary due to different starting guesses and more or less number of iterations performed. The answer below indicates that with above steps, the roots of \(g(x) = x^{4} - x - 3\) can be closely approximated.

Step by step solution

01

Graphical Analysis

Sketch the graph of \(g(x) = x^{4} - x - 3\). This process can be aided by software or graphing calculators. From the graph, make a visual estimation of the x-values at the points where the function crosses the x-axis (where \(g(x) = 0\)). These are the approximate solutions.
02

Preparation for Newton-Raphson Method

To apply the Newton-Raphson method for improving the estimate, find the first derivative of \(g(x)\). That is, compute \(g'(x) = 4x^{3} - 1\). The Newton-Raphson formula, \(x_{new} = x_{old} - \frac{g(x_{old})}{g'(x_{old})}\), will use this derivative.
03

First Iteration of Newton's Method

Substitute the approximate solution from the first step as \(x_{old}\) and compute a new approximation \(x_{new}\) using the Newton-Raphson formula. Make sure to observe and record the difference between the two approximations. If this difference is larger than a certain small number (say 0.0001), continue the process.
04

Next Iterations

Repeat Step 3 until the difference between successive approximations is less than your chosen small number. The \(x_{new}\) obtained from the last iteration is the final approximation of the real root.
05

Repeat for Other Roots

Repeat the whole process using different initial estimates from the graph in Step 1 to find other real roots of the function.

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

Evaluate the following limits: (a) \(\lim _{x \rightarrow 0+} x^{2 x}(0, \infty)\), (b) \(\lim _{x \rightarrow 0}(1+3 / x)^{x} \quad(0, \infty)\), (c) \(\lim _{x \rightarrow \infty}(1+3 / x)^{x} \quad(0, \infty)\), (d) \(\lim _{x \rightarrow 0+}\left(\frac{1}{x}-\frac{1}{\operatorname{Arctan} x}\right) \quad(0, \infty)\).

Show that the function \(f(x):=x^{3}-2 x-5\) has a zero \(r\) in the interval \(I:=[2,2.2]\). If \(x_{1}:=2\) and if we define the sequence \(\left(x_{n}\right)\) using the Newton procedure, show that \(\left|x_{n+1}-r\right| \leq(0.7)\left|x_{n}-r\right|^{2}\). Show that \(x_{4}\) is accurate to within six decimal places.

Let \(f(x):=\cos \boldsymbol{a} x\) for \(x \in \mathbb{R}\) where \(a \neq 0\). Find \(f^{(n)}(x)\) for \(n \in \mathbb{N}, x \in \mathbb{R}\).

Suppose that \(f:[0,2] \rightarrow \mathbb{R}\) is continuous on \([0,2]\) and differentiable on \((0,2)\), and that \(f(0)=0, f(1)=1, f(2)=1\) (a) Show that there exists \(c_{1} \in(0,1)\) such that \(f^{\prime}\left(c_{1}\right)=1\). (b) Show that there exists \(c_{2} \in(1,2)\) such that \(f^{\prime}\left(c_{2}\right)=0\). (c) Show that there exists \(c \in(0,2)\) such that \(f^{\prime}(c)=1 / 3\).

Let \(I\) be an interval and let \(f: I \rightarrow \mathbb{R}\) be differentiable on \(I\). Show that if the derivative \(f^{\prime}\) is never 0 on \(I\), then either \(f^{\prime}(x)>0\) for all \(x \in I\) or \(f^{\prime}(x)<0\) for all \(x \in I\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

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.