Bisection method
Proposed Problems
Josept David Revuelta-Acosta, Ph.D
September 2024
The objective of the bisection method in solving equations is to find the root
(or zero) of a function within a specified interval. It is a numerical method to
solve nonlinear equations where the exact solution may be difficult or impossible
to obtain analytically. Specifically, it aims to find an approximation of the value
x such that f (x) = 0.
The following exercises help improve your understanding on this topic:
1. Let f (x) = 3(x + 1)(x − 12 )(x − 1). Use the bisection method on the
following intervals to find i10 .
a) [-2, 1.5] b) [-1.25, 2.5]
2. Use the bisection method to find solutions accurate to within 10−3 for
x3 − 7x2 + 14x − 6 = 0 on each interval.
a) [0, 1] b) [1, 3.2] c) [3.2, 4]
3. Use the bisection method to find solutions accurate to within 10−5 for the
following problems.
a) 3x − ex = 0 for 1 ≤ x ≤ 2
b) 2x + 3 cos x − ex = 0 for 0 ≤ x ≤ 1
c) x2 − 4x + 4 = ln x for 1 ≤ x ≤ 2 and 2≤x≤4
c) x + 1 = 2 sin(πx) for 0 ≤ x ≤ 0.5 and 0.5 ≤ x ≤ 1
4. Sketch the graphs of y = x and y = sinx. Use the bisection method to
find an approximation to within 10−5 to the first positive value of x with
x = 2 sin x.
5. Let f (x) = (x + 2)(x + 1)2 x(x − 1)3 (x − 2). To which zero of f does the
bisection method converge when applied to the following intervals?
a) [-1.5, 2.5] b) [-0.5, 2.4] c) [-0.5, 3] [-3, -0.5]
Instructions: Each exercise values 2.0 points and must be solved by hand
and scanned into a PDF file. Photos are NOT accepted.
