- Topic
- Newton’s method
- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 4
Newton’s method practice question 9
Newton’s method is applied to , whose only real root is near , starting from . What happens to the estimates?
Rather than predict, follow the estimates from yourself. What does each step need at the current estimate?
Compute from , then from . Keep going until you see a pattern.
, so and .
Theory UBC Math 100 §10.1Key results
Worked solution
Strategy
Take steps exactly until something repeats. Each step depends only on the current estimate, so once a value comes back, the same steps repeat forever.
Working
Value and slope at x₀ = 1. The slope is negative, so the tangent heads right as it falls.
The first step.
Value and slope at x₁ = 2.
Back where it started.
The pair repeats forever. Neither 1 nor 2 is a root (f is 2 and 1 there), and the root near −0.77 is never approached.
Answer
The answer is The estimates cycle 1, 2, 1, 2 and never approach the root..
Check
The tangent at 1, y = 4 − 2x, meets the axis at 2, and the tangent at 2, y = x − 1, meets it back at 1.
Where this goes wrong
Assuming the estimates must converge somewhere and stopping after one step. Newton’s method can cycle: here the tangent at 1 points to 2, and the tangent at 2 points straight back to 1.
Leans on
The power rule, from Differentiation rules
Saved in this browser only.