BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Solutions

Chapter
Section
BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

23-26 Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.

x 5 3 x 4 + x 3 x 2 x + 6 = 0

To determine

Find:

All the solutions of the equation correct to eight decimal places using Newton’s method.

Explanation

1) Concept:

Start with initial approximation x1  which is obtained from graph. Evaluate fx1and f'(x1) and apply the Newton’s formula. After the first iteration we get x2. Repeat the process with x1  replaced by x2. Calculate the approximations x1, x2, x3,  until the successive approximations xn, xn+1  agree to eight decimal places.

2) Formula:

i) Newton’s formula for nth approximation

xn+1=xn-fxnf'xn for n=1,2,3,

ii) Power rule of differentiation:

ddxxn=nxn-1

3) Calculation:

The equation is x5-3x4+x3-x2-x+6=0

Consider the function fx=x5-3x4+x3-x2-x+6

We find the x1  value using the graph.

From the graph we get initial roots are -1, 1.3 and 2.7

fx=x5-3x4+x3-x2-x+6

f'(x)=5x4-12x3+3x2-2x-1

xn+1=xn-fxnf'xn

xn+1=xn-x5-3x4+x3-x2-x+65x4-12x3+3x2-2x-1

i) First root

 x1= -1

x2=x1-x15-3x14+x13-x12-x1+65x14-12x13+3x12-2x1-1

 x2=-1-(-1)5-3-14+-13--12-(-1)+65(-1)4-12-13+3-12-2(-1)-1

x2 -1.04761905

fx2= (-1.04761905)5-3-1.047619054+-1.047619053--1.047619052-(-1.04761905)+6=-0.07508733

Evaluate f'x2

5-1.047619054-12-1.047619053+3-1.047619052- 2(-1.04761905)-1 =24.20756286

x3=-1.04761905-(-0.07508733) 24.20756286

x3 -1.04451724

Evaluate f(x3)

fx3= -1.044517245-3-1.044517244+-1.044517243--1.044517242-(-1.04451724)+6=-0.00033989

Evaluate f'x3

5-1.044517244-12-1.044517243+3-1.044517242- 2(-1.04451724)-1 =23.98868929

x4=-1.04451724-(-0.00033989) 23.98868929

x4-1.04450307

Evaluate f(x4)

fx4= -1.044503075-3-1.044503074+-1.044503073--1.044503072-(-1.04450307)+6=0.00000003

Evaluate f'x4

5-1.044503074-12-1.044503073+3-1.044503072- 2(-1.04450307)-1 =23.987692655

x4=-1.04450307-(0.00000003 ) 23.987692655

x5-1.04450307

x4-1.04450307 x5

ii) Second root:

x1= 1.3

x2=x1-x15-3x14+x13-x12-x1+65x14-12x13+3x12-2x1-1

 x2=1

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started
Sect-3.1 P-11ESect-3.1 P-12ESect-3.1 P-13ESect-3.1 P-14ESect-3.1 P-15ESect-3.1 P-16ESect-3.1 P-17ESect-3.1 P-18ESect-3.1 P-19ESect-3.1 P-20ESect-3.1 P-21ESect-3.1 P-22ESect-3.1 P-23ESect-3.1 P-24ESect-3.1 P-25ESect-3.1 P-26ESect-3.1 P-27ESect-3.1 P-28ESect-3.1 P-29ESect-3.1 P-30ESect-3.1 P-31ESect-3.1 P-32ESect-3.1 P-33ESect-3.1 P-34ESect-3.1 P-35ESect-3.1 P-36ESect-3.1 P-37ESect-3.1 P-38ESect-3.1 P-39ESect-3.1 P-40ESect-3.1 P-41ESect-3.1 P-42ESect-3.1 P-43ESect-3.1 P-44ESect-3.1 P-45ESect-3.1 P-46ESect-3.1 P-47ESect-3.1 P-48ESect-3.1 P-49ESect-3.1 P-50ESect-3.1 P-51ESect-3.1 P-52ESect-3.1 P-53ESect-3.1 P-54ESect-3.1 P-55ESect-3.1 P-56ESect-3.1 P-57ESect-3.1 P-58ESect-3.1 P-59ESect-3.1 P-60ESect-3.1 P-61ESect-3.1 P-62ESect-3.1 P-63ESect-3.1 P-64ESect-3.1 P-65ESect-3.1 P-66ESect-3.1 P-67ESect-3.1 P-68ESect-3.1 P-69ESect-3.1 P-70ESect-3.1 P-71ESect-3.1 P-72ESect-3.2 P-1ESect-3.2 P-2ESect-3.2 P-3ESect-3.2 P-4ESect-3.2 P-5ESect-3.2 P-6ESect-3.2 P-7ESect-3.2 P-8ESect-3.2 P-9ESect-3.2 P-10ESect-3.2 P-11ESect-3.2 P-12ESect-3.2 P-13ESect-3.2 P-14ESect-3.2 P-15ESect-3.2 P-16ESect-3.2 P-17ESect-3.2 P-18ESect-3.2 P-19ESect-3.2 P-20ESect-3.2 P-21ESect-3.2 P-22ESect-3.2 P-23ESect-3.2 P-24ESect-3.2 P-25ESect-3.2 P-26ESect-3.2 P-27ESect-3.2 P-28ESect-3.2 P-29ESect-3.2 P-30ESect-3.2 P-31ESect-3.2 P-32ESect-3.2 P-33ESect-3.2 P-34ESect-3.2 P-35ESect-3.2 P-36ESect-3.3 P-1ESect-3.3 P-2ESect-3.3 P-3ESect-3.3 P-4ESect-3.3 P-5ESect-3.3 P-6ESect-3.3 P-7ESect-3.3 P-8ESect-3.3 P-9ESect-3.3 P-10ESect-3.3 P-11ESect-3.3 P-12ESect-3.3 P-13ESect-3.3 P-14ESect-3.3 P-15ESect-3.3 P-16ESect-3.3 P-17ESect-3.3 P-18ESect-3.3 P-19ESect-3.3 P-20ESect-3.3 P-21ESect-3.3 P-22ESect-3.3 P-23ESect-3.3 P-24ESect-3.3 P-25ESect-3.3 P-26ESect-3.3 P-27ESect-3.3 P-28ESect-3.3 P-29ESect-3.3 P-30ESect-3.3 P-31ESect-3.3 P-32ESect-3.3 P-33ESect-3.3 P-34ESect-3.3 P-35ESect-3.3 P-36ESect-3.3 P-37ESect-3.3 P-38ESect-3.3 P-39ESect-3.3 P-40ESect-3.3 P-41ESect-3.3 P-42ESect-3.3 P-43ESect-3.3 P-44ESect-3.3 P-45ESect-3.3 P-46ESect-3.3 P-47ESect-3.3 P-48ESect-3.3 P-49ESect-3.3 P-50ESect-3.3 P-51ESect-3.3 P-52ESect-3.3 P-53ESect-3.3 P-54ESect-3.3 P-55ESect-3.3 P-56ESect-3.3 P-57ESect-3.3 P-58ESect-3.3 P-59ESect-3.3 P-60ESect-3.3 P-61ESect-3.3 P-62ESect-3.3 P-63ESect-3.3 P-64ESect-3.3 P-65ESect-3.3 P-66ESect-3.3 P-67ESect-3.3 P-68ESect-3.3 P-69ESect-3.3 P-70ESect-3.3 P-71ESect-3.3 P-72ESect-3.3 P-73ESect-3.3 P-74ESect-3.3 P-75ESect-3.3 P-76ESect-3.3 P-77ESect-3.4 P-1ESect-3.4 P-2ESect-3.4 P-3ESect-3.4 P-4ESect-3.4 P-5ESect-3.4 P-6ESect-3.4 P-7ESect-3.4 P-8ESect-3.4 P-9ESect-3.4 P-10ESect-3.4 P-11ESect-3.4 P-12ESect-3.4 P-13ESect-3.4 P-14ESect-3.4 P-15ESect-3.4 P-16ESect-3.4 P-17ESect-3.4 P-18ESect-3.4 P-19ESect-3.4 P-20ESect-3.4 P-21ESect-3.4 P-22ESect-3.4 P-23ESect-3.4 P-24ESect-3.4 P-25ESect-3.4 P-26ESect-3.4 P-27ESect-3.4 P-28ESect-3.4 P-29ESect-3.4 P-30ESect-3.4 P-31ESect-3.4 P-32ESect-3.4 P-33ESect-3.4 P-34ESect-3.4 P-35ESect-3.4 P-36ESect-3.4 P-37ESect-3.4 P-38ESect-3.4 P-39ESect-3.4 P-40ESect-3.4 P-41ESect-3.4 P-42ESect-3.4 P-43ESect-3.4 P-44ESect-3.4 P-45ESect-3.4 P-46ESect-3.4 P-47ESect-3.4 P-48ESect-3.4 P-49ESect-3.4 P-50ESect-3.4 P-51ESect-3.4 P-52ESect-3.4 P-53ESect-3.4 P-54ESect-3.4 P-55ESect-3.4 P-56ESect-3.4 P-57ESect-3.4 P-58ESect-3.4 P-59ESect-3.4 P-60ESect-3.4 P-61ESect-3.4 P-62ESect-3.4 P-63ESect-3.4 P-64ESect-3.4 P-65ESect-3.4 P-66ESect-3.4 P-67ESect-3.4 P-68ESect-3.4 P-69ESect-3.4 P-70ESect-3.4 P-71ESect-3.4 P-72ESect-3.4 P-73ESect-3.4 P-74ESect-3.5 P-1ESect-3.5 P-2ESect-3.5 P-3ESect-3.5 P-4ESect-3.5 P-5ESect-3.5 P-6ESect-3.5 P-7ESect-3.5 P-8ESect-3.5 P-9ESect-3.5 P-10ESect-3.5 P-11ESect-3.5 P-12ESect-3.5 P-13ESect-3.5 P-14ESect-3.5 P-15ESect-3.5 P-16ESect-3.5 P-17ESect-3.5 P-18ESect-3.5 P-19ESect-3.5 P-20ESect-3.5 P-21ESect-3.5 P-22ESect-3.5 P-23ESect-3.5 P-24ESect-3.5 P-25ESect-3.5 P-26ESect-3.5 P-27ESect-3.5 P-28ESect-3.5 P-29ESect-3.5 P-30ESect-3.5 P-31ESect-3.5 P-32ESect-3.5 P-33ESect-3.5 P-34ESect-3.5 P-35ESect-3.5 P-36ESect-3.5 P-37ESect-3.5 P-38ESect-3.5 P-39ESect-3.5 P-40ESect-3.5 P-41ESect-3.5 P-42ESect-3.5 P-43ESect-3.5 P-44ESect-3.5 P-45ESect-3.5 P-46ESect-3.5 P-47ESect-3.5 P-48ESect-3.5 P-49ESect-3.5 P-50ESect-3.5 P-51ESect-3.5 P-52ESect-3.5 P-53ESect-3.5 P-54ESect-3.5 P-55ESect-3.5 P-56ESect-3.5 P-57ESect-3.5 P-58ESect-3.5 P-59ESect-3.5 P-60ESect-3.6 P-1ESect-3.6 P-2ESect-3.6 P-3ESect-3.6 P-4ESect-3.6 P-5ESect-3.6 P-6ESect-3.6 P-7ESect-3.6 P-8ESect-3.6 P-9ESect-3.6 P-10ESect-3.6 P-11ESect-3.6 P-12ESect-3.6 P-13ESect-3.6 P-14ESect-3.6 P-15ESect-3.6 P-16ESect-3.6 P-17ESect-3.6 P-18ESect-3.6 P-19ESect-3.6 P-20ESect-3.6 P-21ESect-3.6 P-22ESect-3.6 P-23ESect-3.6 P-24ESect-3.6 P-25ESect-3.6 P-26ESect-3.6 P-27ESect-3.6 P-28ESect-3.7 P-1ESect-3.7 P-2ESect-3.7 P-3ESect-3.7 P-4ESect-3.7 P-5ESect-3.7 P-6ESect-3.7 P-7ESect-3.7 P-8ESect-3.7 P-9ESect-3.7 P-10ESect-3.7 P-11ESect-3.7 P-12ESect-3.7 P-13ESect-3.7 P-14ESect-3.7 P-15ESect-3.7 P-16ESect-3.7 P-17ESect-3.7 P-18ESect-3.7 P-19ESect-3.7 P-20ESect-3.7 P-21ESect-3.7 P-22ESect-3.7 P-23ESect-3.7 P-24ESect-3.7 P-25ESect-3.7 P-26ESect-3.7 P-27ESect-3.7 P-28ESect-3.7 P-29ESect-3.7 P-30ESect-3.7 P-31ESect-3.7 P-32ESect-3.7 P-33ESect-3.7 P-34ESect-3.7 P-35ESect-3.7 P-36ESect-3.7 P-37ESect-3.7 P-38ESect-3.7 P-39ESect-3.7 P-40ESect-3.7 P-41ESect-3.7 P-42ESect-3.7 P-43ESect-3.7 P-44ESect-3.7 P-45ESect-3.7 P-46ESect-3.7 P-47ESect-3.7 P-48ESect-3.7 P-49ESect-3.7 P-50ESect-3.7 P-51ESect-3.7 P-52ESect-3.7 P-53ESect-3.7 P-54ESect-3.7 P-55ESect-3.7 P-56ESect-3.7 P-57ESect-3.7 P-58ESect-3.7 P-59ESect-3.7 P-60ESect-3.7 P-61ESect-3.7 P-62ESect-3.7 P-63ESect-3.7 P-64ESect-3.7 P-65ESect-3.7 P-66ESect-3.7 P-67ESect-3.7 P-68ESect-3.7 P-69ESect-3.7 P-70ESect-3.7 P-71ESect-3.7 P-72ESect-3.7 P-73ESect-3.7 P-74ESect-3.7 P-75ESect-3.7 P-76ESect-3.7 P-77ESect-3.7 P-78ESect-3.7 P-79ESect-3.7 P-80ESect-3.8 P-1ESect-3.8 P-2ESect-3.8 P-3ESect-3.8 P-4ESect-3.8 P-5ESect-3.8 P-6ESect-3.8 P-7ESect-3.8 P-8ESect-3.8 P-9ESect-3.8 P-10ESect-3.8 P-11ESect-3.8 P-12ESect-3.8 P-13ESect-3.8 P-14ESect-3.8 P-15ESect-3.8 P-16ESect-3.8 P-17ESect-3.8 P-18ESect-3.8 P-19ESect-3.8 P-20ESect-3.8 P-21ESect-3.8 P-22ESect-3.8 P-23ESect-3.8 P-24ESect-3.8 P-25ESect-3.8 P-26ESect-3.8 P-27ESect-3.8 P-28ESect-3.8 P-29ESect-3.8 P-30ESect-3.8 P-31ESect-3.8 P-32ESect-3.8 P-33ESect-3.8 P-34ESect-3.8 P-35ESect-3.8 P-36ESect-3.8 P-37ESect-3.8 P-38ESect-3.8 P-39ESect-3.8 P-40ESect-3.9 P-1ESect-3.9 P-2ESect-3.9 P-3ESect-3.9 P-4ESect-3.9 P-5ESect-3.9 P-6ESect-3.9 P-7ESect-3.9 P-8ESect-3.9 P-9ESect-3.9 P-10ESect-3.9 P-11ESect-3.9 P-12ESect-3.9 P-13ESect-3.9 P-14ESect-3.9 P-15ESect-3.9 P-16ESect-3.9 P-17ESect-3.9 P-18ESect-3.9 P-19ESect-3.9 P-20ESect-3.9 P-21ESect-3.9 P-22ESect-3.9 P-23ESect-3.9 P-24ESect-3.9 P-25ESect-3.9 P-26ESect-3.9 P-27ESect-3.9 P-28ESect-3.9 P-29ESect-3.9 P-30ESect-3.9 P-31ESect-3.9 P-32ESect-3.9 P-33ESect-3.9 P-34ESect-3.9 P-35ESect-3.9 P-36ESect-3.9 P-37ESect-3.9 P-38ESect-3.9 P-39ESect-3.9 P-40ESect-3.9 P-41ESect-3.9 P-42ESect-3.9 P-43ESect-3.9 P-44ESect-3.9 P-45ESect-3.9 P-46ESect-3.9 P-47ESect-3.9 P-48ESect-3.9 P-49ESect-3.9 P-50ESect-3.9 P-51ESect-3.9 P-52ESect-3.9 P-53ESect-3.9 P-54ESect-3.9 P-55ESect-3.9 P-56ESect-3.9 P-57ESect-3.9 P-58ESect-3.9 P-59ESect-3.9 P-60ESect-3.9 P-61ESect-3.9 P-62ESect-3.9 P-63ESect-3.9 P-64ESect-3.9 P-65ESect-3.9 P-66ESect-3.9 P-67ESect-3.9 P-68ESect-3.9 P-69ESect-3.9 P-70ESect-3.9 P-71ESect-3.9 P-72ESect-3.9 P-73ESect-3.R P-1CCSect-3.R P-2CCSect-3.R P-3CCSect-3.R P-4CCSect-3.R P-5CCSect-3.R P-6CCSect-3.R P-7CCSect-3.R P-8CCSect-3.R P-9CCSect-3.R P-10CCSect-3.R P-11CCSect-3.R P-1TFQSect-3.R P-2TFQSect-3.R P-3TFQSect-3.R P-4TFQSect-3.R P-5TFQSect-3.R P-6TFQSect-3.R P-7TFQSect-3.R P-8TFQSect-3.R P-9TFQSect-3.R P-10TFQSect-3.R P-11TFQSect-3.R P-12TFQSect-3.R P-13TFQSect-3.R P-14TFQSect-3.R P-15TFQSect-3.R P-16TFQSect-3.R P-17TFQSect-3.R P-18TFQSect-3.R P-19TFQSect-3.R P-1ESect-3.R P-2ESect-3.R P-3ESect-3.R P-4ESect-3.R P-5ESect-3.R P-6ESect-3.R P-7ESect-3.R P-8ESect-3.R P-9ESect-3.R P-10ESect-3.R P-11ESect-3.R P-12ESect-3.R P-13ESect-3.R P-14ESect-3.R P-15ESect-3.R P-16ESect-3.R P-17ESect-3.R P-18ESect-3.R P-19ESect-3.R P-20ESect-3.R P-21ESect-3.R P-22ESect-3.R P-23ESect-3.R P-24ESect-3.R P-25ESect-3.R P-26ESect-3.R P-27ESect-3.R P-28ESect-3.R P-29ESect-3.R P-30ESect-3.R P-31ESect-3.R P-32ESect-3.R P-33ESect-3.R P-34ESect-3.R P-35ESect-3.R P-36ESect-3.R P-37ESect-3.R P-38ESect-3.R P-39ESect-3.R P-40ESect-3.R P-41ESect-3.R P-42ESect-3.R P-43ESect-3.R P-44ESect-3.R P-45ESect-3.R P-46ESect-3.R P-47ESect-3.R P-48ESect-3.R P-49ESect-3.R P-50ESect-3.R P-51ESect-3.R P-52ESect-3.R P-53ESect-3.R P-54ESect-3.R P-55ESect-3.R P-56ESect-3.R P-57ESect-3.R P-58ESect-3.R P-59ESect-3.R P-60ESect-3.R P-61ESect-3.R P-62ESect-3.R P-63ESect-3.R P-64ESect-3.R P-65ESect-3.R P-66ESect-3.R P-67ESect-3.R P-68ESect-3.P P-1PSect-3.P P-2PSect-3.P P-3PSect-3.P P-4PSect-3.P P-5PSect-3.P P-6PSect-3.P P-7PSect-3.P P-8PSect-3.P P-9PSect-3.P P-10PSect-3.P P-11PSect-3.P P-12PSect-3.P P-13PSect-3.P P-14PSect-3.P P-15PSect-3.P P-16PSect-3.P P-17PSect-3.P P-18PSect-3.P P-19PSect-3.P P-20PSect-3.P P-21PSect-3.P P-22PSect-3.P P-23PSect-3.P P-24P

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

Use Property 8 to estimate the value of the integral. 2(x2sinx)dx

Single Variable Calculus: Early Transcendentals, Volume I

Find an equation of the line that passes through the point (2, 2) and is parallel to the line 2x 4y 8 = 0.

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

15. use the trapezoidal Rule to approximate With n = 4 subintervals

Mathematical Applications for the Management, Life, and Social Sciences

Prove the statement using the , definition of a limit. limx2(x2+2x7)=1

Single Variable Calculus: Early Transcendentals

Proof Prove that for any real number y there exists x in (/2,/2) such that tanx=y.

Calculus: Early Transcendental Functions (MindTap Course List)

The fourth term of {an} defined by a1 = 3, and for n = 1, 2, 3, … is:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False: is a convergent series.

Study Guide for Stewart's Multivariable Calculus, 8th

Write each equation in exponential form. log381=4

College Algebra (MindTap Course List)

Describe the major elements of the APA ethical guidelines for nonhuman subjects in research.

Research Methods for the Behavioral Sciences (MindTap Course List)