Math Problem Statement

Approximate root of the function f(x)=cosx-x using method of false position

Solution

The method of false position (also called the regula falsi method) is an iterative technique used to approximate the roots of a function. It works similarly to the bisection method but uses a secant line (linear interpolation) to approximate the root more efficiently.

To find the approximate root of f(x)=cos(x)xf(x) = \cos(x) - x using the false position method, follow these steps:

Step-by-Step Solution:

  1. Define the function:
    f(x)=cos(x)xf(x) = \cos(x) - x

  2. Choose initial guesses:
    We need two points x0x_0 and x1x_1 such that f(x0)f(x_0) and f(x1)f(x_1) have opposite signs (i.e., the root lies between these values).

    For f(x)=cos(x)xf(x) = \cos(x) - x, let's choose: x0=0andx1=π21.57x_0 = 0 \quad \text{and} \quad x_1 = \frac{\pi}{2} \approx 1.57

    • f(0)=cos(0)0=1f(0) = \cos(0) - 0 = 1
    • f(1.57)=cos(1.57)1.571.57f(1.57) = \cos(1.57) - 1.57 \approx -1.57

    Since f(0)>0f(0) > 0 and f(1.57)<0f(1.57) < 0, there is a root between x0=0x_0 = 0 and x1=1.57x_1 = 1.57.

  3. Apply the false position formula: The next approximation x2x_2 is given by the formula: x2=x1f(x1)(x1x0)f(x1)f(x0)x_2 = x_1 - \frac{f(x_1)(x_1 - x_0)}{f(x_1) - f(x_0)}

    Substituting the values: x2=1.57(1.57)(1.570)1.571x_2 = 1.57 - \frac{(-1.57)(1.57 - 0)}{-1.57 - 1} Simplifying: x2=1.571.57×1.572.57x_2 = 1.57 - \frac{-1.57 \times 1.57}{-2.57} x21.570.958x_2 \approx 1.57 - 0.958 x20.612x_2 \approx 0.612

  4. Check the sign of f(x2)f(x_2): f(0.612)=cos(0.612)0.6120.8190.612=0.207f(0.612) = \cos(0.612) - 0.612 \approx 0.819 - 0.612 = 0.207 Since f(x2)>0f(x_2) > 0, we update x0x_0 to x2x_2 and continue the process with the interval [0.612,1.57][0.612, 1.57].

  5. Iterate the process: Continue applying the false position formula and refining the interval until the desired level of accuracy is achieved.

Would you like more detailed steps for further iterations, or would you like to know how to implement this numerically?

Related Questions:

  1. How does the false position method compare to the bisection method in terms of efficiency?
  2. How can we ensure convergence using the false position method?
  3. What are the conditions under which the false position method might fail to converge quickly?
  4. What are alternative methods for finding the roots of non-linear equations, such as Newton's method?
  5. How does the slope of the secant line in false position differ from Newton's method?

Tip: The false position method works best when the function is monotonic (strictly increasing or decreasing) between the chosen points. If the function is highly non-linear, convergence might be slow.

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Math Problem Analysis

Mathematical Concepts

Numerical Methods
Root Finding
False Position Method
Trigonometric Functions

Formulas

False position formula: x2 = x1 - [f(x1)(x1 - x0)] / [f(x1) - f(x0)]
f(x) = cos(x) - x

Theorems

Intermediate Value Theorem (used for root existence between two points)

Suitable Grade Level

Undergraduate (Calculus I or Numerical Methods)