MathematicsChapter 10: Numerical Methods
Newton-Raphson Method
Use x_(n+1)=x_n-f(x_n)/f'(x_n); good starting values often give rapid convergence.
Learning objectives
- State the rule for newton-raphson method.
- Solve a newton-raphson method problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Use x(n+1)=xn-f(xn)/f'(xn); good starting values often give rapid convergence.
Formula
x0=1.5 gives x1
Where / symbols
- Newton-Raphson method
- Use x_(n+1)=x_n-f(x_n)/f'(x_n); good starting values often give rapid convergence.
CONCEPT AND RULE
Use x(n+1)=xn-f(xn)/f'(xn); good starting values often give rapid convergence.
METHOD
For newton-raphson method, 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
For x2-2, x0=1.5 gives x1=1.5+21.52=1.4167.
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 newton-raphson method
- 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:
- For x2-2, x0=1.5 gives x1=1.5+21.52=1.4167.
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
- Newton-Raphson method
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Newton-Raphson Method
Sticky Notes · Formula and Worked Revision
Core Idea
- Use x(n+1)=xn-f(xn)/f'(xn); good starting values often give rapid convergence.
Rule / Formula
- Use x(n+1)=xn-f(xn)/f'(xn); good starting values often give rapid convergence.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- For x2-2, x0=1.5 gives x1=1.5+21.52=1.4167.
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
- Use x(n+1)=xn-f(xn)/f'(xn); good starting values often give rapid convergence.
- Worked pattern: For x2-2, x0=1.5 gives x1=1.5+21.52=1.4167.
- Final check: domain, sign, notation, units, and reasonableness.
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