Math Problem Statement

Write down step by step maths solution to Compute the real root of f(x) = x − 2sin(x) = 0 correct upto 6 decimal places using Regula Falsi Method where output will be:1.8954942. solve x2 = x0 - [(x1 - x0)/(f(x1) - f(x0))] × f(x0).Where f(1) = 1 - 2sin(1) ≈ 1 - 1.6829 ≈ -0.6829 f(2) = 2 - 2sin(2) ≈ 2 - 1.8186 ≈ 0.1814

Solution

We aim to solve f(x) = x − 2sin(x) = 0 using the Regula Falsi Method (False Position Method). Here's the step-by-step procedure:


Step 1: Define the Function and Initial Guess

The function is:

f(x)=x2sin(x)f(x) = x - 2\sin(x)

From the problem:

  • Initial guesses are x0=1x_0 = 1 and x1=2x_1 = 2.
  • Evaluate f(x0)f(x_0) and f(x1)f(x_1):
    • f(1)=12sin(1)11.6829=0.6829f(1) = 1 - 2\sin(1) \approx 1 - 1.6829 = -0.6829,
    • f(2)=22sin(2)21.8186=0.1814f(2) = 2 - 2\sin(2) \approx 2 - 1.8186 = 0.1814.

f(x0)\therefore f(x_0) and f(x1)f(x_1) have opposite signs, confirming a root exists in the interval [1,2][1, 2].


Step

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Numerical Methods
Root-Finding Algorithms
Regula Falsi Method

Formulas

x2 = x0 - [(x1 - x0) / (f(x1) - f(x0))] × f(x0)
f(x) = x − 2sin(x)

Theorems

Intermediate Value Theorem

Suitable Grade Level

Undergraduate (Numerical Methods or Calculus Course)