Mathematics · University
One Newton iteration
A single iteration is an approximation, not a guaranteed exact root. The derivative at x₀ must be nonzero.
Method and conditions
A single iteration is an approximation, not a guaranteed exact root. The derivative at x₀ must be nonzero.
x₁ = x₀ − f(x₀)/f′(x₀)
Worked examples
Apply one Newton iteration to f(x) = x² − 30, starting at x₀ = 3. Enter x₁. Round the final result to 4 decimal places if needed.
f′(x₀) = 6; x₁ = 3 − (-21)/(6).
6.5
Apply one Newton iteration to f(x) = x² − 17, starting at x₀ = 20. Enter x₁. Round the final result to 4 decimal places if needed.
f′(x₀) = 40; x₁ = 20 − (383)/(40).
10.425
Try it
Apply one Newton iteration to f(x) = x² − 10, starting at x₀ = 3. Enter x₁. Round the final result to 4 decimal places if needed.
Apply one Newton iteration to f(x) = x² − 24, starting at x₀ = 18. Enter x₁. Round the final result to 4 decimal places if needed.
Check your work
Answers and marking notes
- 3.1667
f′(x₀) = 6; x₁ = 3 − (-1)/(6).
- 9.6667
f′(x₀) = 36; x₁ = 18 − (300)/(36).
Further reading and review references
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