Maximum and Minimum Bounds
If measured inputs lie in stated intervals, output bounds are obtained by evaluating the extrema permitted by the model; for monotonic positive formulas this means selecting endpoint combinations according to whether each variable increases or decreases the result.
Why this shows up in the exam
Tolerance analysis · Checking propagated-error approximations · Bounding ratios and products
Learn the idea
Interval bounds provide a direct way to test worst-case results without losing sign information. Replace each measured input by its allowed low and high values, then ask which combination makes the output smallest or largest.
🧠 Memory hook: For a maximum, push each input in the direction that raises the result.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- x in [x_m-Delta x, x_m+Delta x] — input uncertainty interval
- f_min <= f(x_i) <= f_max — result interval from allowed extrema
How to approach it
- 1Write every input interval
- 2Determine monotonic direction or inspect extrema
- 3Compute low and high outputs before summarising
Common slip-ups that cost marks
- •Using the same endpoint for numerator and denominator automatically
- •Forgetting sign changes or nonmonotonic behaviour
- •Reporting symmetric error when bounds are strongly asymmetric
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A quantity Q is defined as Q = force^1 / speed^2. Which dimensional formula represents Q?
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