What is bisection method?
In mathematics, the bisection method is a root-finding method that applies to any continuous functions for which one knows two values with opposite signs
The bisection method is used to find the roots of a polynomial equation.
• It separates the interval and subdivides the interval in which the root of the equation lies.
• The principle behind this method is the intermediate theorem for continuous functions.
• It works by narrowing the gap between the positive and negative intervals until it closes in on the correct answer.
• This method narrows the gap by taking the average of the positive and negative intervals.
• It is a simple method and it is relatively slow. The bisection method is also known as interval halving method, root-finding method, binary search method or dichotomy method.
Multiple choice questions
1. If a function is real and continuous in the region from a to b and f(a) and f(b) have opposite signs then there is no real root between a and b.
a)True
b)False
2. A function is given by x – e^-x = 0. Find the root between a = 0 and b =1 by using Bisection method.
a) 0.655
b) 0.665
c) 0.565
d) 0.656
3. Use Bisection Method to find out the root of x – sinx – 0.5 = 0 between 1 and 2.
a) 1.497
b) 1.947
c) 1.479
d) 1.974
4. The Bisection method is also known as ___________________.
a) Binary Chopping
b) Quaternary Chopping
c) Tri region Chopping
d) Hex region Chopping
5. The function on which we use the bisection method is not necessary to be continuous on the interval we choose.
a)True
b)False
6.If it is provided that f(3) = 4 is one of the initial points. What can be the choice of second point for solving by Bisection Method?
a) -5 such that f(-5) = -26
b) 0 such that f(0) = 5
c) -3 such that f(-3) = -2
d) 13 such that f(13) = 2
7.What is the percentage decrease in an interval containing root after iteration is applied by Bisection Method?
a) 20%
b) 30%
c) 40%
d) 50%
8.The algorithm provided to find the roots of the function using Bisection Method is given by ____________
a) Bolzano’s theorem
b) Mean Value theorem
c) Bisection theorem
d) Secant theorem
9. The Bisection method has which of the following convergences?
a) Linear
b) Quadratic
c) Cubic
d) Quaternary
10.Which of the following statements applies to the bisection method used for finding roots of functions?
a.Converges within a few iterations
b.Guaranteed to work for all continuous functions
c.Is faster than the Newton-Raphson method
d.Requires that there be no error in determining the sign of the function
11.Using Bisection method, negative root of x^3 - 4x + 9 = 0 correct to three decimal places is
a. -2.506
b. -2.706
c. - 2.406
d. None of these
12.2 and 4 such that f(2) = 4 and f(4) = 16 are appropriate initial points for the bisection method.
a)True
b)False
13.Using Bisection method find the root of cos(x) – x * e^x = 0 with a = 0 and b = 1.
a) 0.617
b) 0.527
c) 0.517
d) 0.717
14. Find the root of x^4-x-10 = 0 approximately upto 5 iterations using Bisection Method. Let a = 1.5 and b = 2.
a) 1.68
b) 1.86
c) 1.88
d) 1.66
15. Use Bisection Method to find out the root of x – sinx – 0.5 = 0 between 1 and 2.
a) 1.497
b) 1.947
c) 1.479
d) 1.974
16.Find the approximated value of x till 4 iterations for e^(-x) = 3 log(x) using Bisection Method
a) 1.197
b) 1.187
c) 1.167
d) 1.176
17.Using Bisection method find the root of 3x^(2) = 5x+2 in the interval [0,3].
a) 0.617
b) 0.527
c) 0.517
d) 0.717
18. If a function is defined at 2 points 3 and 7 as f(3)=8 and f(7)=12, it is sufficient to find the roots through Bisection Method.
a)True
b)False
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