Accuracy, bounds and compound measures
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| lower bound/ˈləʊə baʊnd/ | giới hạn dưới |
Is the printed measurement exact?
- A rectangular panel is labelled 8.0 cm by 5.0 cm, each to the nearest 0.1 cm. Its true area is not fixed at 40 cm².
- This lesson studies lower bound 下界: The smallest possible value consistent with a stated rounding rule.
The method
- A value rounded to the nearest unit u lies from stated value-u/2 up to, but usually not including, stated value+u/2.
- For positive quantities, combine extremes according to the operation.
For positive quantities, when is the upper bound of speed d/t achieved?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A quotient grows when the numerator rises and the denominator falls, so the largest speed pairs the upper distance bound with the lower time bound. Using the largest time underestimates it.
A worked case
The lengths satisfy 7.95≤L<8.05 and 4.95≤W<5.05. Since A=LW, 39.3525≤A<40.6525. For speed d/t, the largest speed uses the largest distance and smallest positive time.
Accuracy, bounds and compound measures
A value rounded to the nearest unit u lies from stated value-u/2 up to, but usually not including, stated value+u/2
Explain the different endpoint inclusions for a measurement rounded to 8.0 cm.
Find the lower bound of 3.4 rounded to the nearest 0.1.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Half the rounding unit is 0.05. Lower bound=3.4-0.05=3.35.
Where it goes wrong
- An upper bound is not automatically achieved.
- Dividing upper distance by upper time does not give the largest speed.
- Keep enough digits in intermediate calculations.
Not true: The greatest value of positive d/t uses the greatest d and greatest t. To make d/t as large as possible, use the greatest d and the least t; a larger t makes the quotient smaller.
Find the upper bound of 250 rounded to the nearest 10.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Half of 10 is 5. Upper bound=250+5=255; the upper endpoint is excluded.
To make d/t as large as possible, use the greatest d and the least t.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. To make d/t as large as possible, use the greatest d and the least t; a larger t makes the quotient smaller.
Applying it
- Distinguish measurement uncertainty from arithmetic rounding.
- A sensible reported precision cannot be finer than the measurements justify.
A distance is at most 105 m and time at least 20 s. Find the upper speed bound.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
For positive d and t, the largest quotient uses the greatest d and least t: 105/20=5.25.
Match each value rounded to the stated unit to its bounds.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A rounded value lies within half a unit of the stated figure; the upper bound itself is usually excluded.
In your course
- AQA A-level Mathematics (7357), specification section P.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Lower bound: The smallest possible value consistent with a stated rounding rule.
- A value rounded to the nearest unit u lies from stated value-u/2 up to, but usually not including, stated value+u/2.
- Watch for: an upper bound is not automatically achieved.