Newton’s Method
Iteratively find roots using tangent lines.
Key idea
Iteratively find roots using tangent lines.
In depth
To solve f(x) = 0, repeatedly replace x with x − f(x)/f′(x). Each step follows the tangent line to where it crosses the x-axis. Converges extremely fast near a simple root.
Worked example
To find √2, use f = x² − 2: x₁ = 1.5, x₂ = 1.4167, x₃ = 1.41422.
Formula
Real-world uses
- Solving equations such as financial break-even or engineering design equations numerically in software
Common mistake
Starting where f′ is near zero, which sends the iteration far away.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Newton’s Method?
Iteratively find roots using tangent lines. It is a topic at the expert level in Calculus.
What is the formula for Newton’s Method?
The key formula is xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ).
Where is Newton’s Method used in real life?
Real-world examples: Solving equations such as financial break-even or engineering design equations numerically in software.
What should I learn before Newton’s Method?
Make sure you are comfortable with Curve Sketching, the previous topic in Calculus.
What is a common mistake with Newton’s Method?
Starting where f′ is near zero, which sends the iteration far away.
How can I practise Newton’s Method?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Newton’s Method?
Move on to Antiderivatives next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.