### Lecture 3.2: Estimating √3 Using the Bisection Method (Up to 3 Iterations)---#### Objective:Estimate \(\sqrt{3}\) (root 3) using the Bisection Method, up to 3 iterations.Given:- \(\sqrt{3} \approx x_3\)- Using \( a = 1 \) and \( b = 2 \)#### Theory (Intermediate Value Theorem):**Intermediate Value Theorem (IVT):**Let \( f(x) \) be continuous on the interval \([a, b]\), and assume that \( f(a) \cdot f(b) < 0 \). Then, there exists a number \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \).In mathematical notation:\[ \exists c \in (a, b) \ \text{such that} \ f(c) = 0 \]where \( c \) is a root for \( f(x) \).---#### Steps:1. **Choose Initial Interval:** - Initial interval is chosen as \([1, 2]\). 2. **Apply the Bisection Method:** - Use the properties of the function’s continuity and the Intermediate Value Theorem to iteratively narrow down the interval to identify the root.3. **Iterations:** - Perform the bisection method for up to 3 iterations to estimate \(\sqrt{3}\).---By leveraging the above principles and steps, one can estimate \(\sqrt{3}\) effectively using the bisection method. Each iteration will successively narrow the interval where the root lies, thus improving the accuracy of the estimate.

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Description: ### Lecture 3.2: Estimating √3 Using the Bisection Method (Up to 3 Iterations)---#### Objective:Estimate \(\sqrt{3}\) (root 3) using the Bisection Method, up to 3 iterations.Given:- \(\sqrt{3} \approx x_3\)- Using \( a = 1 \) and \( b = 2 \)#### The

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Advanced Math

Lec 3.2. estimate V3 applymg Bisection Method upto '3'teratinos ( S3? xg) n usng a-1, b-2 Thm ( Intewedrate Jalve Thauszet on Ea,b], and fla) flb)<0. Then, 7celab] svsh that f(a)=0 a is a root for fex) f6> be continvous 1

Lec 3.2. estimate V3 applymg Bisection Method upto '3'teratinos ( S3? xg) n usng a-1, b-2 Thm ( Intewedrate Jalve Thauszet on Ea,b], and fla) flb)<0. Then, 7celab] svsh that f(a)=0 a is a root for fex) f6> be continvous 1

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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