Exercise · ZW-BISECTION-METHOD-004 · ★☆☆ Basic
Which theorem guarantees that a root exists in an interval with a sign change?
Choose one option.
Show answer
- A. The intermediate value theorem
Explanation
The intermediate value theorem ensures a continuous function takes every value between f(a) and f(b). Taylor's and Rolle's theorems do not guarantee this existence here.
Review the concept: Bisection method
The bisection method is an iterative algorithm that approximates a root of a continuous function \( f(x) \) on an interval \( [a, b] \) with a sign change, halving the interval at each step.
\[c_n = \frac{a_n + b_n}{2},\quad |c_n - r| \le \frac{b-a}{2^{n+1}}\]
More practice problems: bisection method
- ZW-BISECTION-METHOD-001 What sign condition at the endpoints of [a, b] is needed to apply the bisection method?
- ZW-BISECTION-METHOD-002 What is the midpoint of the interval [2, 5]?
- ZW-BISECTION-METHOD-003 If the initial interval has length 4, what is its length after one iteration?
- ZW-BISECTION-METHOD-005 What must be evaluated at each iteration of the bisection method?
- ZW-BISECTION-METHOD-006 For f(x) = x³ − x − 2 on [1, 2], what is the first midpoint c?
- ZW-BISECTION-METHOD-007 For f(x) = x^3 - x - 2 on [1, 2] , f(1) = -2 and f(1.5) = -0.125 . In which subinterval is the root?
- ZW-BISECTION-METHOD-008 If the error after n iterations is at most (b - a)/2^n+1 , what is the smallest number of iteration…
- ZW-BISECTION-METHOD-009 For f(x) = cos(x) − x on [0, 1], what signs do f(0) and f(1) have?