Root finding and numerical integration
Diperkenalkan| English | Makna |
|---|---|
| iteration/ˌɪtəˈreɪʃn/ | iterasi |
How reliable is an approximation?
- A nonlinear equation has no convenient exact solution. An approximation still needs evidence that it is reliable.
- This lesson studies iteration 迭代: A repeated update in which each new approximation is calculated from the previous one.
The method
- For equal strip width h, the trapezium estimate is h/2 times the sum of the two endpoint heights plus twice the internal heights.
f(x)=1/x changes sign between -1 and 1. Why does this NOT prove a root in [-1,1]?
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The intermediate value reasoning requires continuity on the whole interval. At the asymptote x=0 the function jumps from negative to positive with no root existing.
A worked case
For y=x² on [0,2] with two equal strips, h=1 and the heights are 0,1,4. The estimate is (1/2)(0+2×1+4)=3. The exact area is 8/3, so this convex curve gives an overestimate.
Root finding and numerical integration
For equal strip width h, the trapezium estimate is h/2 times the sum of the two endpoint heights plus twice the internal heights
Compare the model with the worked case and explain one change.
Find the strip width for two equal strips from 0 to 2.
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Strip width=(2-0)/2=1.
Where it goes wrong
- Use equal strip widths and include internal heights twice.
- Curvature decides whether the estimate is above or below the exact area; being increasing alone does not.
Not true: An increasing function gives a trapezium overestimate. A sign change shows a root only if the function is continuous there: 1/x changes sign across x=0 but has no root.
Use two strips for the trapezium estimate of integral x² from 0 to 2.
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h=1 with heights 0,1,4 gives (1/2)(0+2×1+4)=3.
An increasing function gives a trapezium overestimate.
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A sign change shows a root only if the function is continuous there: 1/x changes sign across x=0 but has no root.
Applying it
- P2 introduces the trapezium rule.
- Newton–Raphson and other root-finding methods belong to other named units; they are not part of this P2 lesson.
Find the exact integral of x² from 0 to 2. Give it to 3 significant figures.
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The exact integral is [x³/3] from 0 to 2=8/3.
Match each numerical method to its update or rule.
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Each method has its own conditions: bisection needs a continuous sign change, Newton needs a nonzero derivative, and the trapezium rule needs equal strip widths.
In your course
- Pearson Edexcel International A Level Mathematics, unit P2.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Iteration: A repeated update in which each new approximation is calculated from the previous one.
- For equal strip width h, the trapezium estimate is h/2 times the sum of the two endpoint heights plus twice the internal heights.
- Watch for: use equal strip widths and include internal heights twice.
Write from an exact integral of 12
- Integrate 6x from 0 to 2 exactly.
- An estimate can sit near this value and still differ.
What is the exact value of the integral of 6x from 0 to 2?
3 times 4 is 12.
What is the exact integral?
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3 x squared at 2 is 12. Other options — 3 (the lesson trapezium estimate); 8/3 (the lesson exact integral of x squared).