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
- A continuous function with opposite signs at two endpoints has a root between them. Newton's method uses x next=x-f(x)/f prime(x), with a nonzero derivative. The trapezium rule approximates a definite integral using endpoint 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 f(x)=x²-2 and x₀=1.5, Newton gives x₁=1.5-(2.25-2)/3=1.4166667. With y=x² on [0,2] and two equal strips, h=1 and trapezium area=(1/2)(0+2×1+4)=3; the exact area is 8/3.
Root finding and numerical integration · Busca de raízes e integração numérica
A continuous function with opposite signs at two endpoints has a root between them · Uma função contínua com sinais opostos em dois extremos tem uma raiz entre eles
Compare the model with the worked case and explain one change. · Compare o modelo com o caso resolvido e explique uma mudança.
After one Newton step for x²-2 from x₀=1.5, find x₁. · Após um passo de Newton para x²-2 a partir de x₀=1.5, encontre x₁.
x₁=1.5-(1.5²-2)/(2×1.5)=17/12≈1.416667.
Test a tempting shortcut
- A sign change across a discontinuity does not prove a root. Iteration can diverge or cycle. The trapezium rule's overestimate or underestimate depends on curvature, not just whether the function increases.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every sign change proves a root, including one across a discontinuity. 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.
Every sign change proves a root, including one across a discontinuity. · Toda mudança de sinal comprova uma raiz, incluindo uma através de uma descontinuidade.
A sign change across a discontinuity does not prove a root. Iteration can diverge or cycle. The trapezium rule's overestimate or underestimate depends on curvature, not just whether the function increases. · Uma mudança de sinal através de uma descontinuidade não comprova uma raiz. A iteração pode divergir ou ciclar. A superestimação ou subestimação da regra do trapézio depende da curvatura, não apenas se a função aumenta.
Interpret a new situation
- Give a stopping criterion and report sensible accuracy. Confirm the approximated root by a sign bracket around the stated rounded answer; explain any failure of the chosen iteration.
- 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
- Current first-assessment-2021 Applications and Interpretation HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.