- Topic
- Newton’s method
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Newton’s method practice question 3
Newton’s method is used to find a root of , starting from . Which statement is correct?
Test the choices by trying to take the first step from yourself. What does the update formula need at 3?
Compute and and put them into . What goes wrong?
Theory UBC Math 100 §10.1Key results
Worked solution
Strategy
Try to take the first step. The formula divides by the slope at the starting point, so look at that slope first.
Working
Value and slope at the starting point: 3 is the vertex of the parabola.
The formula divides by zero, so x₁ is not defined.
Geometrically, the tangent at the vertex is horizontal and never crosses the x-axis.
Nothing else is wrong with the function: from x₀ = 2 or x₀ = 4, where f is −1, the method converges to 3 − √2 or 3 + √2, so a negative f(x₀) is no obstacle.
Answer
The answer is The method fails at the first step, because f prime of 3 is 0 and the tangent line is horizontal..
Check
Completing the square, f(x) = (x − 3)² − 2, so x = 3 is the vertex of the parabola, where the tangent is flat.
Where this goes wrong
Reading a flat tangent as a step of zero, as if the method could sit at x = 3. The step is f(3)/f′(3), and division by f′(3) = 0 is impossible: a horizontal tangent never reaches the x-axis, so there is no next estimate.
Leans on
Slope of a tangent line, from The definition of the derivative
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