Root finding and numerical integration · Busca de raízes e integração numérica
| English | Português |
|---|---|
| iteration/ˌɪtəˈreɪʃn/ | iteração |
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? · Qual descrição define corretamente iteração?
A repeated update in which each new approximation is calculated from the previous one. · Uma atualização repetida em que cada nova aproximação é calculada a partir da anterior.
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 · Busca de raízes e integração numérica
For equal strip width h, the trapezium estimate is h/2 times the sum of the two endpoint heights plus twice the internal heights · Para largura de faixa igual h, a estimativa trapézio é h/2 vezes a soma das duas alturas nas extremidades mais duas vezes as alturas internas
Compare the model with the worked case and explain one change. · Compare o modelo com o caso resolvido e explique uma mudança.
Find the strip width for two equal strips from 0 to 2. · Encontre a largura da faixa para duas faixas iguais de 0 a 2.
Strip width=(2-0)/2=1. · Largura da faixa=(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. · Use duas faixas para a estimativa do trapézio da integral x² de 0 a 2.
h=1 with heights 0,1,4 gives (1/2)(0+2×1+4)=3. · h=1 com alturas 0,1,4 resulta em (1/2)(0+2×1+4)=3.
An increasing function always gives a trapezium overestimate. · Uma função crescente sempre gera uma superestimação trapézio.
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. · Use larguras de faixa iguais e inclua as alturas internas duas vezes. A curvatura determina se a estimativa está acima ou abaixo da área exata; ser crescente sozinho não garante isso.
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. · Encontre a integral exata de x² de 0 a 2.
The exact integral is [x³/3] from 0 to 2=8/3. · A integral exata é [x³/3] de 0 a 2=8/3.
Match each part of a complete solution to its purpose. · Associe cada parte de uma solução completa ao seu propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Uma suposição justifica o modelo; uma verificação testa o resultado; a interpretação conecta-o à pergunta.
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.