Chapter 2.5, Problem 57E

BuyFind*arrow_forward*

8th Edition

James Stewart

ISBN: 9781305270343

Textbook Problem

(a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root.

cos *x = x*^{3}

(a)

To determine

**To prove:** The equation

Explanation

**Theorem used: The Intermediate value Theorem**

Suppose that if *f* is continuous on the closed interval [*a*, *b*] and let *N* be any number between *c* in (*a*, *b*) such that

**Proof:**

Rewrite the equation as follows,

Consider the function

The function

Without loss of generality, take the subinterval

In order to show that there is a root of the equation *c* between 0 and 1 for which

Take

(b)

To determine

**To find:** An interval of length 0.01 that contains a root by using the calculator.

Sect-2.1 P-1ESect-2.1 P-2ESect-2.1 P-3ESect-2.1 P-4ESect-2.1 P-5ESect-2.1 P-6ESect-2.1 P-7ESect-2.1 P-8ESect-2.1 P-9ESect-2.2 P-1E

Sect-2.2 P-2ESect-2.2 P-3ESect-2.2 P-4ESect-2.2 P-5ESect-2.2 P-6ESect-2.2 P-7ESect-2.2 P-8ESect-2.2 P-9ESect-2.2 P-10ESect-2.2 P-11ESect-2.2 P-12ESect-2.2 P-13ESect-2.2 P-14ESect-2.2 P-15ESect-2.2 P-16ESect-2.2 P-17ESect-2.2 P-18ESect-2.2 P-19ESect-2.2 P-20ESect-2.2 P-21ESect-2.2 P-22ESect-2.2 P-23ESect-2.2 P-24ESect-2.2 P-25ESect-2.2 P-26ESect-2.2 P-27ESect-2.2 P-28ESect-2.2 P-29ESect-2.2 P-30ESect-2.2 P-31ESect-2.2 P-32ESect-2.2 P-33ESect-2.2 P-34ESect-2.2 P-35ESect-2.2 P-36ESect-2.2 P-37ESect-2.2 P-38ESect-2.2 P-39ESect-2.2 P-40ESect-2.2 P-41ESect-2.2 P-42ESect-2.2 P-43ESect-2.2 P-44ESect-2.2 P-45ESect-2.2 P-46ESect-2.2 P-47ESect-2.2 P-49ESect-2.2 P-50ESect-2.2 P-51ESect-2.2 P-52ESect-2.2 P-53ESect-2.2 P-54ESect-2.2 P-55ESect-2.3 P-1ESect-2.3 P-2ESect-2.3 P-3ESect-2.3 P-4ESect-2.3 P-5ESect-2.3 P-6ESect-2.3 P-7ESect-2.3 P-8ESect-2.3 P-9ESect-2.3 P-10ESect-2.3 P-11ESect-2.3 P-12ESect-2.3 P-13ESect-2.3 P-14ESect-2.3 P-15ESect-2.3 P-16ESect-2.3 P-17ESect-2.3 P-18ESect-2.3 P-19ESect-2.3 P-20ESect-2.3 P-21ESect-2.3 P-22ESect-2.3 P-23ESect-2.3 P-24ESect-2.3 P-25ESect-2.3 P-26ESect-2.3 P-27ESect-2.3 P-28ESect-2.3 P-29ESect-2.3 P-30ESect-2.3 P-31ESect-2.3 P-32ESect-2.3 P-33ESect-2.3 P-34ESect-2.3 P-35ESect-2.3 P-36ESect-2.3 P-37ESect-2.3 P-38ESect-2.3 P-39ESect-2.3 P-40ESect-2.3 P-41ESect-2.3 P-42ESect-2.3 P-43ESect-2.3 P-44ESect-2.3 P-45ESect-2.3 P-46ESect-2.3 P-47ESect-2.3 P-48ESect-2.3 P-49ESect-2.3 P-50ESect-2.3 P-51ESect-2.3 P-52ESect-2.3 P-53ESect-2.3 P-54ESect-2.3 P-55ESect-2.3 P-56ESect-2.3 P-57ESect-2.3 P-58ESect-2.3 P-59ESect-2.3 P-60ESect-2.3 P-61ESect-2.3 P-62ESect-2.3 P-63ESect-2.3 P-64ESect-2.3 P-65ESect-2.3 P-66ESect-2.4 P-1ESect-2.4 P-2ESect-2.4 P-3ESect-2.4 P-4ESect-2.4 P-5ESect-2.4 P-6ESect-2.4 P-7ESect-2.4 P-8ESect-2.4 P-9ESect-2.4 P-10ESect-2.4 P-11ESect-2.4 P-12ESect-2.4 P-13ESect-2.4 P-14ESect-2.4 P-15ESect-2.4 P-16ESect-2.4 P-17ESect-2.4 P-18ESect-2.4 P-19ESect-2.4 P-20ESect-2.4 P-21ESect-2.4 P-22ESect-2.4 P-23ESect-2.4 P-24ESect-2.4 P-25ESect-2.4 P-26ESect-2.4 P-27ESect-2.4 P-28ESect-2.4 P-29ESect-2.4 P-30ESect-2.4 P-31ESect-2.4 P-32ESect-2.4 P-33ESect-2.4 P-34ESect-2.4 P-36ESect-2.4 P-37ESect-2.4 P-38ESect-2.4 P-39ESect-2.4 P-40ESect-2.4 P-41ESect-2.4 P-42ESect-2.4 P-43ESect-2.4 P-44ESect-2.5 P-1ESect-2.5 P-2ESect-2.5 P-3ESect-2.5 P-4ESect-2.5 P-5ESect-2.5 P-6ESect-2.5 P-7ESect-2.5 P-8ESect-2.5 P-9ESect-2.5 P-10ESect-2.5 P-11ESect-2.5 P-12ESect-2.5 P-13ESect-2.5 P-14ESect-2.5 P-15ESect-2.5 P-16ESect-2.5 P-17ESect-2.5 P-18ESect-2.5 P-19ESect-2.5 P-20ESect-2.5 P-21ESect-2.5 P-22ESect-2.5 P-23ESect-2.5 P-24ESect-2.5 P-25ESect-2.5 P-26ESect-2.5 P-27ESect-2.5 P-28ESect-2.5 P-29ESect-2.5 P-30ESect-2.5 P-31ESect-2.5 P-32ESect-2.5 P-33ESect-2.5 P-34ESect-2.5 P-35ESect-2.5 P-36ESect-2.5 P-37ESect-2.5 P-38ESect-2.5 P-39ESect-2.5 P-40ESect-2.5 P-41ESect-2.5 P-42ESect-2.5 P-43ESect-2.5 P-44ESect-2.5 P-45ESect-2.5 P-46ESect-2.5 P-47ESect-2.5 P-48ESect-2.5 P-49ESect-2.5 P-50ESect-2.5 P-51ESect-2.5 P-52ESect-2.5 P-53ESect-2.5 P-54ESect-2.5 P-55ESect-2.5 P-56ESect-2.5 P-57ESect-2.5 P-58ESect-2.5 P-59ESect-2.5 P-60ESect-2.5 P-61ESect-2.5 P-62ESect-2.5 P-63ESect-2.5 P-64ESect-2.5 P-65ESect-2.5 P-66ESect-2.5 P-67ESect-2.5 P-68ESect-2.5 P-69ESect-2.5 P-70ESect-2.5 P-71ESect-2.5 P-72ESect-2.6 P-1ESect-2.6 P-2ESect-2.6 P-3ESect-2.6 P-4ESect-2.6 P-5ESect-2.6 P-6ESect-2.6 P-7ESect-2.6 P-8ESect-2.6 P-9ESect-2.6 P-10ESect-2.6 P-11ESect-2.6 P-12ESect-2.6 P-13ESect-2.6 P-14ESect-2.6 P-15ESect-2.6 P-16ESect-2.6 P-17ESect-2.6 P-18ESect-2.6 P-19ESect-2.6 P-20ESect-2.6 P-21ESect-2.6 P-22ESect-2.6 P-23ESect-2.6 P-24ESect-2.6 P-25ESect-2.6 P-26ESect-2.6 P-27ESect-2.6 P-28ESect-2.6 P-29ESect-2.6 P-30ESect-2.6 P-31ESect-2.6 P-32ESect-2.6 P-33ESect-2.6 P-34ESect-2.6 P-35ESect-2.6 P-36ESect-2.6 P-37ESect-2.6 P-38ESect-2.6 P-39ESect-2.6 P-40ESect-2.6 P-41ESect-2.6 P-42ESect-2.6 P-43ESect-2.6 P-44ESect-2.6 P-45ESect-2.6 P-46ESect-2.6 P-47ESect-2.6 P-48ESect-2.6 P-49ESect-2.6 P-50ESect-2.6 P-51ESect-2.6 P-52ESect-2.6 P-53ESect-2.6 P-54ESect-2.6 P-55ESect-2.6 P-56ESect-2.6 P-57ESect-2.6 P-58ESect-2.6 P-59ESect-2.6 P-60ESect-2.6 P-61ESect-2.6 P-62ESect-2.6 P-63ESect-2.6 P-64ESect-2.6 P-65ESect-2.6 P-66ESect-2.6 P-67ESect-2.6 P-68ESect-2.6 P-69ESect-2.6 P-70ESect-2.6 P-71ESect-2.6 P-72ESect-2.6 P-73ESect-2.6 P-74ESect-2.6 P-75ESect-2.6 P-76ESect-2.6 P-77ESect-2.6 P-78ESect-2.6 P-79ESect-2.6 P-80ESect-2.6 P-81ESect-2.7 P-1ESect-2.7 P-2ESect-2.7 P-3ESect-2.7 P-4ESect-2.7 P-5ESect-2.7 P-6ESect-2.7 P-7ESect-2.7 P-8ESect-2.7 P-9ESect-2.7 P-10ESect-2.7 P-11ESect-2.7 P-12ESect-2.7 P-13ESect-2.7 P-14ESect-2.7 P-15ESect-2.7 P-16ESect-2.7 P-17ESect-2.7 P-18ESect-2.7 P-19ESect-2.7 P-20ESect-2.7 P-21ESect-2.7 P-22ESect-2.7 P-23ESect-2.7 P-24ESect-2.7 P-25ESect-2.7 P-26ESect-2.7 P-27ESect-2.7 P-28ESect-2.7 P-29ESect-2.7 P-30ESect-2.7 P-31ESect-2.7 P-32ESect-2.7 P-33ESect-2.7 P-34ESect-2.7 P-35ESect-2.7 P-36ESect-2.7 P-37ESect-2.7 P-38ESect-2.7 P-39ESect-2.7 P-40ESect-2.7 P-41ESect-2.7 P-42ESect-2.7 P-43ESect-2.7 P-44ESect-2.7 P-45ESect-2.7 P-46ESect-2.7 P-47ESect-2.7 P-48ESect-2.7 P-49ESect-2.7 P-50ESect-2.7 P-51ESect-2.7 P-53ESect-2.7 P-54ESect-2.7 P-55ESect-2.7 P-56ESect-2.7 P-57ESect-2.7 P-58ESect-2.7 P-59ESect-2.7 P-60ESect-2.7 P-61ESect-2.8 P-1ESect-2.8 P-2ESect-2.8 P-3ESect-2.8 P-4ESect-2.8 P-5ESect-2.8 P-6ESect-2.8 P-7ESect-2.8 P-8ESect-2.8 P-9ESect-2.8 P-10ESect-2.8 P-11ESect-2.8 P-12ESect-2.8 P-13ESect-2.8 P-14ESect-2.8 P-15ESect-2.8 P-16ESect-2.8 P-17ESect-2.8 P-18ESect-2.8 P-19ESect-2.8 P-20ESect-2.8 P-21ESect-2.8 P-22ESect-2.8 P-23ESect-2.8 P-24ESect-2.8 P-25ESect-2.8 P-26ESect-2.8 P-27ESect-2.8 P-28ESect-2.8 P-29ESect-2.8 P-30ESect-2.8 P-31ESect-2.8 P-32ESect-2.8 P-33ESect-2.8 P-34ESect-2.8 P-35ESect-2.8 P-36ESect-2.8 P-37ESect-2.8 P-38ESect-2.8 P-39ESect-2.8 P-40ESect-2.8 P-41ESect-2.8 P-42ESect-2.8 P-43ESect-2.8 P-44ESect-2.8 P-45ESect-2.8 P-46ESect-2.8 P-47ESect-2.8 P-48ESect-2.8 P-49ESect-2.8 P-50ESect-2.8 P-51ESect-2.8 P-52ESect-2.8 P-53ESect-2.8 P-54ESect-2.8 P-55ESect-2.8 P-56ESect-2.8 P-57ESect-2.8 P-58ESect-2.8 P-59ESect-2.8 P-60ESect-2.8 P-61ESect-2.8 P-62ESect-2.8 P-63ESect-2.8 P-64ESect-2.8 P-65ESect-2.8 P-66ESect-2.8 P-67ECh-2 P-1RCCCh-2 P-2RCCCh-2 P-3RCCCh-2 P-4RCCCh-2 P-5RCCCh-2 P-6RCCCh-2 P-7RCCCh-2 P-8RCCCh-2 P-9RCCCh-2 P-10RCCCh-2 P-11RCCCh-2 P-12RCCCh-2 P-13RCCCh-2 P-14RCCCh-2 P-15RCCCh-2 P-16RCCCh-2 P-1RQCh-2 P-2RQCh-2 P-3RQCh-2 P-4RQCh-2 P-5RQCh-2 P-6RQCh-2 P-7RQCh-2 P-8RQCh-2 P-9RQCh-2 P-10RQCh-2 P-11RQCh-2 P-12RQCh-2 P-13RQCh-2 P-14RQCh-2 P-15RQCh-2 P-16RQCh-2 P-17RQCh-2 P-18RQCh-2 P-19RQCh-2 P-20RQCh-2 P-21RQCh-2 P-22RQCh-2 P-23RQCh-2 P-24RQCh-2 P-25RQCh-2 P-26RQCh-2 P-1RECh-2 P-2RECh-2 P-3RECh-2 P-4RECh-2 P-5RECh-2 P-6RECh-2 P-7RECh-2 P-8RECh-2 P-9RECh-2 P-10RECh-2 P-11RECh-2 P-12RECh-2 P-13RECh-2 P-14RECh-2 P-15RECh-2 P-16RECh-2 P-17RECh-2 P-18RECh-2 P-19RECh-2 P-20RECh-2 P-21RECh-2 P-22RECh-2 P-23RECh-2 P-24RECh-2 P-25RECh-2 P-26RECh-2 P-27RECh-2 P-28RECh-2 P-29RECh-2 P-30RECh-2 P-31RECh-2 P-32RECh-2 P-33RECh-2 P-34RECh-2 P-35RECh-2 P-36RECh-2 P-37RECh-2 P-38RECh-2 P-39RECh-2 P-40RECh-2 P-41RECh-2 P-42RECh-2 P-43RECh-2 P-44RECh-2 P-45RECh-2 P-46RECh-2 P-47RECh-2 P-48RECh-2 P-49RECh-2 P-50RECh-2 P-51RECh-2 P-52RECh-2 P-53RECh-2 P-54RECh-2 P-1PCh-2 P-2PCh-2 P-3PCh-2 P-4PCh-2 P-5PCh-2 P-6PCh-2 P-7PCh-2 P-8PCh-2 P-9PCh-2 P-10PCh-2 P-11PCh-2 P-12PCh-2 P-13PCh-2 P-14P

Applied Calculus

Finite Mathematics and Applied Calculus (MindTap Course List)

Precalculus: Mathematics for Calculus (Standalone Book)

Calculus: An Applied Approach (MindTap Course List)

Calculus: Early Transcendentals

Calculus: Early Transcendental Functions

Multivariable Calculus

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Essentials Of Statistics

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

Calculus (MindTap Course List)

Statistics for The Behavioral Sciences (MindTap Course List)

Understanding Basic Statistics

Single Variable Calculus

Probability and Statistics for Engineering and the Sciences

Mathematical Applications for the Management, Life, and Social Sciences

Intermediate Algebra

Mathematical Excursions (MindTap Course List)

Single Variable Calculus: Early Transcendentals

Elements Of Modern Algebra

Elementary Geometry For College Students, 7e

Trigonometry (MindTap Course List)

Contemporary Mathematics for Business & Consumers

Elementary Technical Mathematics

Calculus of a Single Variable

Calculus: Early Transcendental Functions (MindTap Course List)

Elementary Geometry for College Students

Finite Mathematics for the Managerial, Life, and Social Sciences

Study Guide for Stewart's Multivariable Calculus, 8th

Calculus (MindTap Course List)

Multivariable Calculus

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

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Statistics for Business & Economics, Revised (MindTap Course List)

Research Methods for the Behavioral Sciences (MindTap Course List)

Mathematics For Machine Technology

Introduction To Statistics And Data Analysis

Mathematics: A Practical Odyssey

College Algebra (MindTap Course List)

Research Methods for the Behavioral Sciences (MindTap Course List)