Integrals, area and accumulation · 积分、面积与累积量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| antiderivative/ˌæntɪdɪˈrɪvətɪv/ | 原函数 | yuán hán shù |
How much change has accumulated?
- A velocity graph tells us motion at an instant. How can we recover the displacement accumulated over time?
- This lesson studies antiderivative 原函数: A function whose derivative equals the given integrand.
累积了多少变化量？
- 速度图告诉我们某一时刻的运动状态。我们如何恢复随时间累积的位移？
- 本课研究原函数：其导数等于给定被积函数的函数。
Choose the mathematical structure
- Integrate polynomial powers by increasing the index by one and dividing by the new index. Include an arbitrary constant and use a supplied point to determine it.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
选择数学结构
- 对幂函数进行积分时，指数加一后除以新指数。加上任意常数，并利用给定点确定该常数的具体数值。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
Which description correctly defines antiderivative? · 下列哪项描述正确定义了原函数？
A function whose derivative equals the given integrand. · 其导数等于给定被积函数的函数。
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.
An antiderivative of 3x² is x³+C because differentiating x³ gives 3x². If F(1)=5, then 1+C=5, so C=4 and F(x)=x³+4.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
3x² 的一个原函数是 x³+C，因为对 x³ 求导得 3x²。若 F(1)=5，则 1+C=5，因此 C=4，F(x)=x³+4。
Integrals, area and accumulation · 积分、面积与累积量
Integrate polynomial powers by increasing the index by one and dividing by the new index · 多项式幂函数的积分方法是将指数加一，再除以新的指数。
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。
For F prime(x)=3x², F(1)=5, find C in F(x)=x³+C. · 对于 F'(x)=3x²，已知 F(1)=5，求 F(x)=x³+C 中的常数 C。
F(1)=1+C=5, so C=4. · 由 F(1)=1+C=5，得 C=4。
Test a tempting shortcut
- An indefinite integral needs a constant. Integrating each term changes its power; copying the derivative rule gives the wrong result.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every indefinite integral has only one possible antiderivative. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 不定积分需要包含一个常数。逐项积分会改变其幂次；直接套用导数规则会导致错误结果。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
每个不定积分只有一个可能的原函数。此说法错误。请说明它违反了哪一定义或假设。
For that F, find F(2). · 对于该 F，求 F(2)。
Use F(x)=x³+4: F(2)=8+4=12. · 使用 F(x)=x³+4：F(2)=8+4=12。
Every indefinite integral has only one possible antiderivative. · 每一个不定积分只有一个可能的原函数。
An indefinite integral needs a constant. Integrating each term changes its power; copying the derivative rule gives the wrong result. · 不定积分必须包含常数项。对每一项分别积分会改变其指数；直接套用求导法则会导致错误结果。
Interpret a new situation
- P1 uses indefinite polynomial integration and a point to determine the constant. Definite integrals, areas and the trapezium rule are P2 content.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
解读新情境
- P1使用不定多项式积分及一个点来确定常数。定积分、面积及梯形法则属于P2的内容。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
For that F, find F(0). · 对于该 F，求 F(0)。
F(0)=0+4=4.
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 mathematics; official unit P1. 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 function whose derivative equals the given integrand. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- edexcel IAL数学；官方单元P1。其他单元的补充内容在范围审查中已注明；不属于额外加分要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
其导数等于给定被积函数的函数。请选择正确的关系式，展示解题方法，检查适用条件并解读结果。