- Topic
- Newton’s method
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Newton’s method practice question 4
Use Newton’s method on with to approximate . Find .
Two steps means using the update formula twice. At which point is the tangent line drawn for each step?
The first step is , with .
The second step starts again from : compute and at the new point.
An exact fraction.
Theory UBC Math 100 §10.1Key results
Worked solution
Strategy
Take two steps, each with the value and the slope at the latest estimate. Keep the fractions exact; the denominators stay small.
Working
The first step, from x₀ = 2.
Value and slope at the new estimate.
Dividing by 9/2 is multiplying by 2/9: (1/16)(2/9) = 1/72.
Two steps already agree with √5 to four decimal places.
Answer
The answer is 161 over 72, about 2.23611.
Check
(161/72)² ≈ 5.0002, so x₂ squares to within 0.0002 of 5, as an estimate of √5 should.
Where this goes wrong
Reusing the slope f′(2) = 4 in the second step, which gives 143/64. Each step uses the tangent line at the latest estimate, so the slope must be recomputed at 9/4.
Leans on
Linear approximation, from Linear approximation
Try a similar question
Let . Starting from , use one step of Newton’s method to find .Saved in this browser only.