Section 4.13
Newton's Method
How does your calculator compute ? It doesn't guess randomly. It slides down tangent lines, getting roughly double the correct digits with every single step.
1
The Iterative Formula
The Recursive Step
Logic: We want to find where . Instead of solving the curve (hard), we solve the Tangent Line (easy, it's just ) to find a new intercept. Repeat until happy.
2
Worked Example
Solving a Cubic
Find the real root of starting at .
Newton's Pinball
Step 0: Refined Guess
Follow the red tangent line down satisfy . That intercept becomes the next guess.
Look how quickly the x-values converge to the intersection point!
Step 1: Setup
Step 2: Iteration 1
Plug in :
Step 3: Iteration 2
Plug in :
Step 4: Iteration 3
Plug in :
Step 5: Iteration 4
Plug in :
Converged to 5 decimals!
3
Level Up Examples
Example A: How to Calculate Roots
Derive the algorithm for by solving .
1. Formula:
2. Simplify:
.This is the ancient "Babylonian Method!"
3. Compute (Start with ):
Three steps and we have 4 correct decimals of
Example B: Failure Mode
What happens if ?
Zero Slope:
is Undefined.Geometry: The tangent line is horizontal and never hits the x-axis.
5
Practice Quiz
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