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In which of the following categories can we put Bisection method?
- a)Bracket Solutions
- b)Graphical Solution
- c)Empirical Solutions
- d)Trial Solutions
Correct answer is option 'A'. Can you explain this answer?
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In which of the following categories can we put Bisection method?a)Bra...
Bracketing Methods:
- All bracketing methods always converge, whereas open methods (may sometimes diverge).
- We must start with an initial interval [a,b], where f(a) and f(b) have opposite signs.
- Since the graph y = f(x) of a continuous function is unbroken, it will cross the abscissa at a zero x = 'a' that lies somewhere within the interval [a,b].
- One of the ways to test a numerical method for solving the equation f(x) = 0 is to check its performance on a polynomial whose roots are known.
Bisection method:
Used to find the root for a function. Root of a function f(x) = a such that f(a)= 0
Property: if a function f(x) is continuous on the interval [a…b] and sign of f(a) ≠ sign of f(b). There is a value c belongs to [a…b] such that f(c) = 0, means c is a root in between [a….b]
Note:
Bisection method cut the interval into 2 halves and check which half contains a root of the equation.
1) Suppose interval [a, b] .
2) Cut interval in the middle to find m : m = (a + b)/2
3) sign of f(m) not matches with f(a), proceed the search in new interval.
Used to find the root for a function. Root of a function f(x) = a such that f(a)= 0
Property: if a function f(x) is continuous on the interval [a…b] and sign of f(a) ≠ sign of f(b). There is a value c belongs to [a…b] such that f(c) = 0, means c is a root in between [a….b]
Note:
Bisection method cut the interval into 2 halves and check which half contains a root of the equation.
1) Suppose interval [a, b] .
2) Cut interval in the middle to find m : m = (a + b)/2
3) sign of f(m) not matches with f(a), proceed the search in new interval.
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In which of the following categories can we put Bisection method?a)Bra...
Bisection method is a numerical method used to find the roots of a given equation. It is commonly used in various fields of engineering and science to solve nonlinear equations. In the context of the given options, the Bisection method can be categorized under "Bracket Solutions" (option A).
Explanation:
Bracket Solutions:
The bracketing method is a technique used to find the root of an equation within a specified interval. It involves narrowing down the interval containing the root by iteratively selecting smaller intervals that still bracket the root. The Bisection method is one such bracketing method.
The Bisection method works by initially selecting an interval [a, b], where the function changes sign or brackets the root. It then finds the midpoint of the interval and checks if the function value at the midpoint is close to zero. If it is, the midpoint is considered as the root. Otherwise, the interval is bisected by selecting a new interval [a, c] or [c, b], depending on the sign of the function value at the midpoint. This process is repeated iteratively until the desired level of accuracy is achieved.
Other Categories:
- Graphical Solution (option B): Graphical solution involves plotting the equation on a graph and visually identifying the intersection points with the x-axis. The Bisection method is not a graphical solution method as it does not involve plotting the equation or visually determining the root.
- Empirical Solutions (option C): Empirical solutions refer to approaches based on observations, experiments, or practical experience. The Bisection method is a numerical method based on mathematical principles and calculations, rather than empirical observations.
- Trial Solutions (option D): Trial solutions involve testing different values or guesses to find the root of an equation. The Bisection method is not a trial and error method as it systematically narrows down the interval containing the root.
Therefore, the correct category for the Bisection method is "Bracket Solutions" (option A) as it is a bracketing method used to find the roots of an equation within a specified interval.
Explanation:
Bracket Solutions:
The bracketing method is a technique used to find the root of an equation within a specified interval. It involves narrowing down the interval containing the root by iteratively selecting smaller intervals that still bracket the root. The Bisection method is one such bracketing method.
The Bisection method works by initially selecting an interval [a, b], where the function changes sign or brackets the root. It then finds the midpoint of the interval and checks if the function value at the midpoint is close to zero. If it is, the midpoint is considered as the root. Otherwise, the interval is bisected by selecting a new interval [a, c] or [c, b], depending on the sign of the function value at the midpoint. This process is repeated iteratively until the desired level of accuracy is achieved.
Other Categories:
- Graphical Solution (option B): Graphical solution involves plotting the equation on a graph and visually identifying the intersection points with the x-axis. The Bisection method is not a graphical solution method as it does not involve plotting the equation or visually determining the root.
- Empirical Solutions (option C): Empirical solutions refer to approaches based on observations, experiments, or practical experience. The Bisection method is a numerical method based on mathematical principles and calculations, rather than empirical observations.
- Trial Solutions (option D): Trial solutions involve testing different values or guesses to find the root of an equation. The Bisection method is not a trial and error method as it systematically narrows down the interval containing the root.
Therefore, the correct category for the Bisection method is "Bracket Solutions" (option A) as it is a bracketing method used to find the roots of an equation within a specified interval.
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In which of the following categories can we put Bisection method?a)Bracket Solutionsb)Graphical Solutionc)Empirical Solutionsd)Trial SolutionsCorrect answer is option 'A'. Can you explain this answer?
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