Fitting and refining a function model
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| residual/rɪˈsɪdʒuːəl/ | sai số |
Which extra observation can help choose between them?
- Two different curve models fit the same three measurements.
- This lesson studies residual 残差: An observed output minus the output predicted by a model.
The method
- State inputs, outputs, units and the modelling domain.
- Fit parameters using the stated observations, then compare predictions with independent measurements.
- An exact fit to the calibration points does not prove the chosen function is correct everywhere.
- A residual records observed minus predicted output; model refinement needs evidence and a reasoned change.
The quadratic fits the arch points (0,0),(3,9),(6,0) exactly. Why is it not proven correct?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Both h=-x²+6x and the triangular model pass through the calibration points. The extra measurement at x=1 (5.2 m) favours the quadratic THERE - evidence, never a universal proof.
A worked case
For a proposed arch, x is horizontal distance and h its height, both in metres. Use the illustrative points (0,0),(3,9),(6,0), with 0≤x≤6. Let h=ax²+bx+c. Then c=0, 9a+3b=9 and 36a+6b=0 give a=−1,b=6. Thus h=−x²+6x=9−(x−3)², with maximum 9 m at x=3. A piecewise-linear alternative h=3x for 0≤x≤3 and h=18−3x for 3≤x≤6 fits the same three points. At x=1, an illustrative extra measurement 5.2 m gives residual 0.2 m for the quadratic but 2.2 m for the linear alternative. This favours the quadratic at that point, not a universal verdict.
Fitting and refining a function model
State inputs, outputs, units and the modelling domain
Check the evidence and conditions behind a transformed or fitted function.
Find a in the fitted quadratic h=ax²+bx+c.
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Subtract twice 9a+3b=9 from 36a+6b=0: 18a=−18, hence a=−1.
Where it goes wrong
- Do not use a fitted calibration point as independent validation.
- Outside 0≤x≤6, the quadratic can give negative height and no longer describes the proposed arch.
- A smaller residual at one point does not establish accuracy everywhere.
- The simpler model can be easier to interpret but still miss curvature.
Not true: A function that fits three calibration points is proven correct at every input. Fitting three points does not prove a model; it may fail at other inputs.
Find its maximum modelled height in metres.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Complete the square: h=9−(x−3)², with maximum 9.
A function that fits three calibration points is proven correct at every input.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Fitting three points does not prove a model; it may fail at other inputs.
Applying it
- For height at least 8 m, 9−(x−3)²≥8 gives 2≤x≤4: modelled horizontal width 2 m.
- Collect more intermediate measurements and estimate measurement uncertainty before relying on this clearance.
- Thickness and changes in shape can require a refined domain or formula; explain which observation motivates the refinement.
Find the modelled horizontal width where h≥8, in metres.
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The allowed interval is [2,4], with width 4−2=2 metres.
Match each model-fitting concept to its meaning.
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Fitting and validating are different stages; a calibration point cannot validate its own model.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Residual: An observed output minus the output predicted by a model.
- State inputs, outputs, units and the modelling domain.
- Watch for: do not use a fitted calibration point as independent validation.