- Topic
- Newton’s method
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Newton’s method practice question 1
Newton’s method improves an estimate of a root of by following the tangent line at to the point where it crosses the -axis. Which formula gives the next estimate ?
Each choice is a candidate for the -intercept of the tangent line at . What would you write down to find that intercept yourself?
Write the tangent line at , , then set and solve for .
Theory UBC Math 100 §10.1Key results
Worked solution
Strategy
The next estimate is the x-intercept of the tangent line at the current one. Write that line down and solve for where it is zero.
Working
The tangent line at xₙ matches the value and the slope of f there.
The next estimate is where this line meets the x-axis.
Solve for xₙ₊₁. This needs f′(xₙ) ≠ 0.
Answer
The answer is x sub n plus 1 equals x sub n minus f of x sub n over f prime of x sub n.
Check
On f(x) = x − 2, a line and so its own tangent, the formula takes x₀ = 5 to 5 − 3/1 = 2, the root, in one step.
Where this goes wrong
Adding the ratio instead of subtracting it. With a plus sign the estimate moves the right distance in the wrong direction, away from the root.
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.