Root finding and numerical integration
導入済み| English | 意味 |
|---|---|
| iteration/ˌɪtəˈreɪʃn/ | 反復 |
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.
Choose the mathematical structure
- For equal strip width h, the trapezium estimate is h/2 times the sum of the two endpoint heights plus twice the internal heights.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
f(x)=1/x changes sign between -1 and 1. Why does this NOT prove a root in [-1,1]?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Strip width=(2-0)/2=1.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
An increasing function always gives a trapezium overestimate. This claim is false. Explain which definition or assumption it violates.
Use two strips for the trapezium estimate of integral x² from 0 to 2.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
h=1 with heights 0,1,4 gives (1/2)(0+2×1+4)=3.
An increasing function always gives a trapezium overestimate.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the exact integral of x² from 0 to 2.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
The exact integral is [x³/3] from 0 to 2=8/3.
Match each numerical method to its update or rule.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
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.
Use this in your course
- edexcel IAL pure mathematics; official unit P2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A repeated update in which each new approximation is calculated from the previous one. Choose the relationship, show the method, check its assumptions and interpret the result.