MathematicsChapter 10: Numerical Methods
Sign Change and Root Bracketing
For continuous f, opposite signs at a and b guarantee at least one root in (a,b).
Learning objectives
- State the rule for sign change and root bracketing.
- Solve a sign change and root bracketing problem step by step.
- Check the result, restrictions, notation, and units.
Definition
For continuous f, opposite signs at a and b guarantee at least one root in (a,b).
Where / symbols
- Sign change and root bracketing
- For continuous f, opposite signs at a and b guarantee at least one root in (a,b).
CONCEPT AND RULE
For continuous f, opposite signs at a and b guarantee at least one root in (a,b).
METHOD
For sign change and root bracketing, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
f(x)=x^2-2 changes sign between 1 and 2.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve sign change and root bracketing
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Worked example
- Worked example:
- f(x)=x^2-2 changes sign between 1 and 2.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Sign change and root bracketing
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Sign Change and Root Bracketing
Sticky Notes · Formula and Worked Revision
Core Idea
- For continuous f, opposite signs at a and b guarantee at least one root in (a,b).
Rule / Formula
- For continuous f, opposite signs at a and b guarantee at least one root in (a,b).
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- f(x)=x^2-2 changes sign between 1 and 2.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- For continuous f, opposite signs at a and b guarantee at least one root in (a,b).
- Worked pattern: f(x)=x^2-2 changes sign between 1 and 2.
- Final check: domain, sign, notation, units, and reasonableness.
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