Applications of Derivatives · Applied Problems
Newton's Method
Newton's method iteratively approximates roots of f(x) = 0. Each step uses the tangent line to get a better approximation.
Conditions. f'(xₙ) ≠ 0 at each step. Convergence depends on the initial guess.
- 3 variables
- 2 worked examples
- 15 in Applications of Derivatives
Variables
| Symbol | Name | Unit |
|---|---|---|
| xn | Current approximation Current x value | - |
| fxn | f(xₙ) Function value at xₙ | - |
| fpxn | f'(xₙ) Derivative at xₙ | - |
Worked examples
Use Newton's method with x₀ = 2 to approximate √5 (root of x²-5=0).
- f(x) = x²-5, f'(x) = 2x
- x₁ = 2 - (4-5)/(4) = 2 + 1/4 = 2.25
- x₂ = 2.25 - (5.0625-5)/(4.5) = 2.25 - 0.01389 ≈ 2.2361
Answer: x₂ ≈ 2.2361 (actual √5 ≈ 2.23607)
1 more worked examplePremium
Use Newton's method with x₀ = 1 to approximate ∛2 (root of x³ - 2 = 0).
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Practice
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