Math Problem Statement
find a positive root of the equation 2x=3+cosx by Bisection method.
Solution
To find the positive root of the equation using the Bisection Method, follow the steps below:
Steps:
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Rewrite the equation in the form : The root of the equation corresponds to the value of where .
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Choose an initial interval such that . This ensures that there is a root in the interval.
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Apply the Bisection Method formula to find the midpoint : Calculate and check the sign of .
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Update the interval:
- If , then the root lies in the interval , so update .
- If , then the root lies in the interval , so update .
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Repeat the process until the interval is sufficiently small (i.e., until is smaller than the desired tolerance).
Step-by-Step Calculation:
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Initial Interval: Let's find an interval where changes sign.
- For :
- For :
Since and , the function changes sign between and . Therefore, the interval is a suitable starting point.
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First Iteration:
- ,
- Midpoint
- : Since , the root lies in the interval . Update .
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Second Iteration:
- ,
- Midpoint
- : Since , the root lies in the interval . Update .
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Third Iteration:
- ,
- Midpoint
- : Since , the root lies in the interval . Update .
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Repeat the process until the interval is sufficiently small. Continue bisecting until the root is accurate to the desired number of decimal places.
Final Answer:
The Bisection Method will give an increasingly accurate approximation of the root within the interval . After several iterations, you'll converge on the root.
Would you like me to complete more iterations, or explain anything further?
5 Related Questions:
- What are the advantages of the Bisection Method over other numerical methods?
- How does the choice of initial interval affect the convergence of the Bisection Method?
- Can the Bisection Method be applied to non-continuous functions?
- How do you determine the tolerance level for the Bisection Method?
- What is the convergence rate of the Bisection Method compared to Newton's Method?
Tip: Always ensure that the function changes sign within the interval before applying the Bisection Method.
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Math Problem Analysis
Mathematical Concepts
Equations
Trigonometry
Numerical Methods
Formulas
Bisection Method formula
Theorems
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Suitable Grade Level
Higher Education
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