Upper & Lower Bounds Flashcards

All 6 cards in this deck

How do you write the error interval for a value that has been rounded?

e.g. a length is $90$ m correct to the nearest metre

Go half the rounding unit either side: $89.5 \text{ m} \le \text{length} < 90.5 \text{ m}$.

How do you write the error interval for a value that has been truncated?

e.g. $N$ truncated to 1 digit gives $7$

From the truncated value up to the next value: $7 \le N < 8$.

To find the lower bound of $a-b$, which bounds of $a$ and $b$ do you use?

$\text{LB}(a)-\text{UB}(b)$ — smallest $a$ take away largest $b$.

Which bounds do you use to test whether an average speed could have been greater than a stated value, when the distance and time are both rounded?

The upper bound of the speed: $\text{UB(distance)} \div \text{LB(time)}$, then compare it with the stated value.

What are the steps to give a calculated value to a suitable degree of accuracy by considering bounds?

e.g. $d=\dfrac{c^3}{8}$ where $c=10.9$ correct to 3 s.f.

1. Use $\text{LB}(c)=10.85$ and $\text{UB}(c)=10.95$ in the formula: $159.66\ldots$ and $164.11\ldots$
2. Round both bounds until they agree: both give $160$
3. State it with the reason: $d=160$ since both bounds round to $160$

True or false? If a distance is measured to the nearest 5 km instead of the nearest km, the maximum possible average speed for a fixed time interval increases.

True. The bounds are wider ($\pm 2.5$ km instead of $\pm 0.5$ km), so the upper bound of the distance is larger.