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2. Show that the equation xe = 1 has a root between 0.5 and 0.6. starting with 0.55 as a first estimate, use one iteration of the Newton-Raphson method to obtain a better approximation of this root giving your answer correct to 3 decimal places. 3. Show that the equation f(x) = 0, where f(x)= x' + x² – 2x – 1, has a root in the interval 1

2. Show that the equation xe = 1 has a root between 0.5 and 0.6. starting with 0.55 as a first estimate, use one iteration of the Newton-Raphson method to obtain a better approximation of this root giving your answer correct to 3 decimal places. 3. Show that the equation f(x) = 0, where f(x)= x' + x² – 2x – 1, has a root in the interval 1

Advanced Engineering Mathematics
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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2. Show that the equation xe = 1 has a root between 0.5 and 0.6. starting with 0.55 as a first estimate, use one iteration of the Newton-Raphson method to obtain a better approximation of this root giving your answer correct to 3 decimal places. 3. Show that the equation f(x) = 0, where f(x) = x° + x² – 2x – 1, has a root in the interval 1<x<2. Use Newton Raphson pròcedure method to find an approximation to this root correct to 2 decimal places. %3D 4. Show that the equation: x5 - x³ - of the Newton- Raphson method to find an approximation to this root correct to 2 decimal places. 5. Show that the equation f(x) = 0, where f(x) = 12x + 16x – 3x – 4, has a root in each of the intervals: -2 <x <-1, -1<x <0. Use Newton Raphson method with initial value -1.5 to find two further approximation to the root in the interval: -2 < x <-1, giving your final answer to two decimal places. 6. Show that x = -1 is a root of the equation f(x)= 0. where f(x) = x* + x' + 3x² – 9x – 12. Show also that another root of this equation lies in the interval 1<x<2. Starting with 1.5 as a first approximation, use the Newton-Raphson procedure twice to find another approximation to this root, giving your answver correct to three decimal places. 7. Show that a root of the equation:Inx = 1 1 = 0, has a root in the interval 1<x<2 use two iterations - lies in the interval between i and 1.6. starting - - 2 with x = 1.4 as a first approximation for the root correct to 2 decimal places 8. Taking x =-as the first approximation to the real root of the equation x'– 6x +1 = 0, use one iteration of Newton-Raphson's method to obtain the second approximation giving your answer to 2 decimal places. 9. Show that a root of the equation:e + 4x - 5 = 0 lies within the range 0 <x<1. using x = 0.6 as a first approximation to this root and applying one iteration of the Newton- Raphson method, find a second approximation giving your answer correct to 2 decimal places. 10. Show that the equation: x³ + x - 6 = 0 has a root between land 2. Using the Newton- Raphson method and taking 1.6 as the first approximation, determine by means of two iterations, two other approximations for the root, giving your answers correct to 3 decimal places. 11. Given that f(x) = x – 5+ x² In x, show that the equation: f(x) = 0 has a root in the interval: 2 < x < 3. Taking x = one iteration of the Newton Raphson's procedure to find a second approximation of this root, giving the answer correct to two decimal places. 12. Use one iteration of the Newton-Raphson method to find, to 3 decimai places, a second approximation to the real root of the equation x + 3x -1 = 0, taking 0.2 as the first approximation to the root of the 1 3 2x 2 as a first approximation to the root of the equation, use .3 equation(
Transcribed Image Text:2. Show that the equation xe = 1 has a root between 0.5 and 0.6. starting with 0.55 as a first estimate, use one iteration of the Newton-Raphson method to obtain a better approximation of this root giving your answer correct to 3 decimal places. 3. Show that the equation f(x) = 0, where f(x) = x° + x² – 2x – 1, has a root in the interval 1<x<2. Use Newton Raphson pròcedure method to find an approximation to this root correct to 2 decimal places. %3D 4. Show that the equation: x5 - x³ - of the Newton- Raphson method to find an approximation to this root correct to 2 decimal places. 5. Show that the equation f(x) = 0, where f(x) = 12x + 16x – 3x – 4, has a root in each of the intervals: -2 <x <-1, -1<x <0. Use Newton Raphson method with initial value -1.5 to find two further approximation to the root in the interval: -2 < x <-1, giving your final answer to two decimal places. 6. Show that x = -1 is a root of the equation f(x)= 0. where f(x) = x* + x' + 3x² – 9x – 12. Show also that another root of this equation lies in the interval 1<x<2. Starting with 1.5 as a first approximation, use the Newton-Raphson procedure twice to find another approximation to this root, giving your answver correct to three decimal places. 7. Show that a root of the equation:Inx = 1 1 = 0, has a root in the interval 1<x<2 use two iterations - lies in the interval between i and 1.6. starting - - 2 with x = 1.4 as a first approximation for the root correct to 2 decimal places 8. Taking x =-as the first approximation to the real root of the equation x'– 6x +1 = 0, use one iteration of Newton-Raphson's method to obtain the second approximation giving your answer to 2 decimal places. 9. Show that a root of the equation:e + 4x - 5 = 0 lies within the range 0 <x<1. using x = 0.6 as a first approximation to this root and applying one iteration of the Newton- Raphson method, find a second approximation giving your answer correct to 2 decimal places. 10. Show that the equation: x³ + x - 6 = 0 has a root between land 2. Using the Newton- Raphson method and taking 1.6 as the first approximation, determine by means of two iterations, two other approximations for the root, giving your answers correct to 3 decimal places. 11. Given that f(x) = x – 5+ x² In x, show that the equation: f(x) = 0 has a root in the interval: 2 < x < 3. Taking x = one iteration of the Newton Raphson's procedure to find a second approximation of this root, giving the answer correct to two decimal places. 12. Use one iteration of the Newton-Raphson method to find, to 3 decimai places, a second approximation to the real root of the equation x + 3x -1 = 0, taking 0.2 as the first approximation to the root of the 1 3 2x 2 as a first approximation to the root of the equation, use .3 equation(
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