14) Bisection Method for Approximating Zeros of a Function f We begin with two consecutive integers, a and a + 1, such that f(a) and f(a + 1) are of opposite sign. Evaluate f at the midpoint m, of a and a + 1. If f(m,) = 0, then mị is the zero of f, and we are finished. Otherwise, f (m,) is of opposite sign to either f(a) or f(a + 1). Suppose that it is f (a) and f(m1) that are of opposite sign. Now evaluate f at the midpoint m2 of a and m1. Repeat this process until the desired degree of accuracy is obtained. Note that each of these iterations places the zero in an interval whose length is half that of the previous interval. Use the bisection method to approximate the zero of f (x) = 8x* – 2x² + 5x – 1 in the interval [0,1] correct to three decimal places. Hint: The process ends when both endpoints agree to the desired number of decimal places.
14) Bisection Method for Approximating Zeros of a Function f We begin with two consecutive integers, a and a + 1, such that f(a) and f(a + 1) are of opposite sign. Evaluate f at the midpoint m, of a and a + 1. If f(m,) = 0, then mị is the zero of f, and we are finished. Otherwise, f (m,) is of opposite sign to either f(a) or f(a + 1). Suppose that it is f (a) and f(m1) that are of opposite sign. Now evaluate f at the midpoint m2 of a and m1. Repeat this process until the desired degree of accuracy is obtained. Note that each of these iterations places the zero in an interval whose length is half that of the previous interval. Use the bisection method to approximate the zero of f (x) = 8x* – 2x² + 5x – 1 in the interval [0,1] correct to three decimal places. Hint: The process ends when both endpoints agree to the desired number of decimal places.
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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![**Bisection Method for Approximating Zeros of a Function \( f \)**
We begin with two consecutive integers, \( a \) and \( a + 1 \), such that \( f(a) \) and \( f(a + 1) \) are of opposite sign. Evaluate \( f \) at the midpoint \( m_1 \) of \( a \) and \( a + 1 \). If \( f(m_1) = 0 \), then \( m_1 \) is the zero of \( f \), and we are finished. Otherwise, \( f(m_1) \) is of opposite sign to either \( f(a) \) or \( f(a + 1) \). Suppose that it is \( f(a) \) and \( f(m_1) \) that are of opposite sign. Now evaluate \( f \) at the midpoint \( m_2 \) of \( a \) and \( m_1 \). Repeat this process until the desired degree of accuracy is obtained. Note that each of these iterations places the zero in an interval whose length is half that of the previous interval.
Use the bisection method to approximate the zero of \( f(x) = 8x^4 - 2x^2 + 5x - 1 \) in the interval \([0, 1]\) correct to three decimal places.
**Hint:** The process ends when both endpoints agree to the desired number of decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb0e73cc8-5bc3-4bfd-b6e9-068f811e0db9%2F6ad46706-f30f-4daa-8ca0-392d482d5640%2Fu52lzir_processed.png&w=3840&q=75)
Transcribed Image Text:**Bisection Method for Approximating Zeros of a Function \( f \)**
We begin with two consecutive integers, \( a \) and \( a + 1 \), such that \( f(a) \) and \( f(a + 1) \) are of opposite sign. Evaluate \( f \) at the midpoint \( m_1 \) of \( a \) and \( a + 1 \). If \( f(m_1) = 0 \), then \( m_1 \) is the zero of \( f \), and we are finished. Otherwise, \( f(m_1) \) is of opposite sign to either \( f(a) \) or \( f(a + 1) \). Suppose that it is \( f(a) \) and \( f(m_1) \) that are of opposite sign. Now evaluate \( f \) at the midpoint \( m_2 \) of \( a \) and \( m_1 \). Repeat this process until the desired degree of accuracy is obtained. Note that each of these iterations places the zero in an interval whose length is half that of the previous interval.
Use the bisection method to approximate the zero of \( f(x) = 8x^4 - 2x^2 + 5x - 1 \) in the interval \([0, 1]\) correct to three decimal places.
**Hint:** The process ends when both endpoints agree to the desired number of decimal places.
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