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 using the false position method, follow these steps:
Step-by-Step Solution:
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Define the function:
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Choose initial guesses:
We need two points and such that and have opposite signs (i.e., the root lies between these values).For , let's choose:
Since and , there is a root between and .
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Apply the false position formula: The next approximation is given by the formula:
Substituting the values: Simplifying:
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Check the sign of : Since , we update to and continue the process with the interval .
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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:
- How does the false position method compare to the bisection method in terms of efficiency?
- How can we ensure convergence using the false position method?
- What are the conditions under which the false position method might fail to converge quickly?
- What are alternative methods for finding the roots of non-linear equations, such as Newton's method?
- 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)