- Topic
- Newton’s method
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Newton’s method practice question 2
Let . Starting from , use one step of Newton’s method to find .
What does one Newton step need to know about at the starting point ?
Compute and , then .
. Keep the sign of when you subtract the ratio.
A fraction, or a decimal to two decimal places.
Theory UBC Math 100 §10.1Key results
Worked solution
Strategy
One Newton step needs two numbers at the starting point, the value and the slope, and then the update formula.
Working
The value at x₀. It is not zero, so 1 is not the root.
The slope at x₀.
f is negative and rising at 1, so the root is to the right, and the step moves right.
Answer
The answer is five quarters, or 1.25.
Check
f(1.25) ≈ 0.20 is far closer to 0 than f(1) = −1, and the sign change puts the root between 1 and 1.25.
Where this goes wrong
Sending the step the wrong way, by adding f(1)/f′(1) or by losing the minus sign of f(1), which gives 3/4. Since f is negative and increasing at 1, x₁ has to be bigger than 1.
Leans on
The power rule, from Differentiation rules
Try a similar question
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