Root finding and numerical integration
| English | 中文 | Pinyin |
|---|---|---|
| iteration/ˌɪtəˈreɪʃn/ | 迭代 | dié dài |
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.
Which description correctly defines iteration?
A repeated update in which each new approximation is calculated from the previous one.
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.
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.
h=1 with heights 0,1,4 gives (1/2)(0+2×1+4)=3.
An increasing function always gives a trapezium overestimate.
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.
The exact integral is [x³/3] from 0 to 2=8/3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL 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.