A-level · Beta
Newton–Raphson
Move the initial guess and follow tangent steps towards a root.
Drag X along the horizontal track to move the initial guess. Add one tangent step at a time.
Tab to a ring or outcome, then use the arrow keys. Sliders also work with touch and keyboard.
Iteration limit reached. More steps may help; this is not proof of divergence.
- Completed iterations
- 3
- Latest residual
- 0.00205836
| j | xⱼ | f(xⱼ) |
|---|---|---|
| 0 | 1 | -1 |
| 1 | 1.5 | 0.875 |
| 2 | 1.34783 | 0.100682 |
| 3 | 1.3252 | 0.00205836 |
Blue is f(x) = x³ − x − 1. Tangents may leave the visible axes. A small residual is numerical evidence, not an exact algebraic proof.
Predict, test and explain
Step 1 of 3
Predict
For f(x) = x³ − x − 1 and x₀ = 1, what is x₁?
Explore and compare
Choose the investigation above. Predict what will change before you move a control.
Try it in the simulation. Change one thing at a time.
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Hints, self-check and connections
Try a self-check
Fixed values, separate from the diagram controls.
For f(x) = x³ − x − 1 and x₀ = 1, what is x₁?
Hint
A tangent meets the x-axis at the next Newton estimate.
Reveal answer
f(1) = −1 and f′(1) = 2, so x₁ = 1 − (−1)/2 = 1.5.
Build an explanation
Hint 1
A tangent meets the x-axis at the next Newton estimate.
Hint 2
Use x − f(x)/f′(x) before rounding.
Hint 3
A zero derivative makes that step undefined; not every initial guess converges.
Worked example
This example uses fixed values, separate from the diagram controls.
Take one Newton step from x₀ = 1.
- Evaluate f(1) = 1 − 1 − 1 = −1.
- Evaluate f′(1) = 3 − 1 = 2.
- Calculate 1 − (−1)/2.
Answer: x₁ = 1.5.
Connect this idea
Watch out: Subtracting a negative correction increases this first estimate.
Specification and learning route
AQA I2 · Edexcel Pure 9.3
A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.
Useful starting knowledge: Functions and roots; Graph interpretation; Iteration; Differentiation for Newton-Raphson; Definite integrals for trapezium rule.