Perform one Newton step for x²−2=0 from x₀=1.
Answer and explanation
1.5
x₁=1−(1−2)/2=1.5.
Subjects
Numerical methods
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Sample exercises
Use one trapezoid over [0,2] to approximate the integral of x².
Answer and explanation
4
The interval width is 2 and the average endpoint height is (0+4)/2=2.
For f(x)=x² on [0,2], is the one-panel trapezoid estimate above or below the exact integral?
- Not a numerical value
- Above
- Always equal
- Below
Answer and explanation
Above
The exact value is 8/3; the chord lies above the convex graph.
Explain catastrophic cancellation using subtraction of nearly equal approximate values.
Answer and explanation
Leading digits cancel, leaving the small difference dependent on comparatively uncertain trailing digits. Relative error in the result can become large.
Credit a loss of relative accuracy and the role of nearly equal inputs.
An approximation is 2.01 and the reference value is 2. Find absolute error.
Answer and explanation
0.01
Absolute error is abs(2.01−2).
Which property describes sensitivity inherent in the problem rather than just the chosen algorithm?
- Source code indentation
- Conditioning
- Iteration count alone
- Font size
Answer and explanation
Conditioning
Conditioning measures how outputs respond to small changes in data.
Study notes
Numerical approximation and conditioning
Truncation error comes from approximation; round-off comes from finite-precision arithmetic.
UniversityNumerical approximation and conditioning · revision notes
Truncation error comes from approximation; round-off comes from finite-precision arithmetic.
UniversityOne Newton iteration
A single iteration is an approximation, not a guaranteed exact root. The derivative at x₀ must be nonzero.
UniversityOne Newton iteration · formula reference
A single iteration is an approximation, not a guaranteed exact root. The derivative at x₀ must be nonzero.
UniversityComposite trapezoid approximation
This is an approximation to the integral. For a convex quadratic it overestimates the exact area.
UniversityComposite trapezoid approximation · formula reference
This is an approximation to the integral. For a convex quadratic it overestimates the exact area.
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