Root finding and numerical integration · Recherche de racines et intégration numérique
| English | Français |
|---|---|
| iteration/ˌɪtəˈreɪʃn/ | itération |
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? · Quelle description définit correctement l'itération ?
A repeated update in which each new approximation is calculated from the previous one. · Une mise à jour répétée dans laquelle chaque nouvelle approximation est calculée à partir de la précédente.
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 · Recherche de racines et intégration numérique
A continuous function with opposite signs at two endpoints has a root between them · Une fonction continue avec des signes opposés aux deux extrémités possède une racine entre elles
Compare the model with the worked case and explain one change. · Compare le modèle avec l'exemple résolu et explique un changement.
After one Newton step for x²-2 from x₀=1.5, find x₁. · Après une étape de Newton pour x²-2 à partir de x₀=1.5, trouvez 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. · Utilisez deux bandes pour l'estimation trapézoïdale de l'intégrale x² de 0 à 2.
h=1 with heights 0,1,4 gives (1/2)(0+2×1+4)=3. · h=1 avec des hauteurs 0,1,4 donne (1/2)(0+2×1+4)=3.
Every sign change proves a root, including one across a discontinuity. · Chaque changement de signe prouve l'existence d'une racine, y compris un changement à travers une discontinuité.
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. · Un changement de signe à travers une discontinuité ne prouve pas l'existence d'une racine. L'itération peut diverger ou former un cycle. La surestimation ou la sous-estimation de la règle trapézoïdale dépend de la courbure, et non seulement du fait que la fonction soit croissante.
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. · Trouvez l'intégrale exacte de x² de 0 à 2.
The exact integral is [x³/3] from 0 to 2=8/3. · L'intégrale exacte est [x³/3] de 0 à 2=8/3.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
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.