Root finding and numerical integration · 求根与数值积分。
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| 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? · 以下哪项描述正确定义了迭代法？
A repeated update in which each new approximation is calculated from the previous one. · 一种重复更新过程，其中每个新近似值均从前一个近似值计算得出。
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 · 求根与数值积分。
For equal strip width h, the trapezium estimate is h/2 times the sum of the two endpoint heights plus twice the internal heights · 对于等宽条带 h，梯形法则估计值为 h/2 乘以两个端点高度之和加上两倍的内部高度之和。
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find the strip width for two equal strips from 0 to 2. · 求从 0 到 2 的两条等宽条带的宽度。
Strip width=(2-0)/2=1. · 条带宽度 = (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. · 使用两个梯形条估算从 0 到 2 的 x² 积分值。
h=1 with heights 0,1,4 gives (1/2)(0+2×1+4)=3. · h=1，高度为 0,1,4，则 (1/2)(0+2×1+4)=3。
An increasing function always gives a trapezium overestimate. · 增函数总是导致梯形法则高估面积。
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. · 使用等宽条带并将内部高度计算两次。曲率决定估计值是偏大还是偏小；仅凭函数递增这一条件并不足以确定。
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. · 求从 0 到 2 的 x² 精确积分值。
The exact integral is [x³/3] from 0 to 2=8/3. · 精确积分为 [x³/3] 从 0 到 2 = 8/3。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
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.