Derivatives and stationary points · 导数与驻点
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| derivative/dɪˈrɪvətɪv/ | 导数 | dǎo shù |
What is the slope at one point?
- A curved road has different slopes at different positions. An average gradient cannot describe every point.
- This lesson studies derivative 导数: The instantaneous rate of change, also the gradient of a tangent.
某一点的斜率是什么？
- 弯曲道路在不同位置的斜率各不相同。平均梯度无法描述每一点的情况。
- 本课学习导数（derivative）：瞬时变化率，亦即切线的斜率。
Choose the mathematical structure
- For y=ax^n, dy/dx=anx^(n-1). A stationary point satisfies dy/dx=0. Check the sign change of the derivative, or the second derivative when it is nonzero, to classify 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
- 对于 y=ax^n，dy/dx=anx^(n-1)。驻点满足 dy/dx=0。通过检查导数的符号变化，或在二阶导数非零时利用二阶导数来分类该点。
- 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 derivative? · 以下哪项描述正确定义了导数？
The instantaneous rate of change, also the gradient of a tangent. · 瞬时变化率，亦即切线的斜率。
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=x³-3x, dy/dx=3x²-3. At x=2, the gradient is 9 and y=2. The tangent is y-2=9(x-2). The normal gradient is -1/9.
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。
对于 y=x³-3x，dy/dx=3x²-3。当 x=2 时，斜率为 9，y=2。切线方程为 y-2=9(x-2)。法线斜率为 -1/9。
Derivatives and stationary points · 导数与驻点
For y=ax^n, dy/dx=anx^(n-1) · 对于 y=ax^n，dy/dx=anx^(n-1)
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 y=x³-3x, find dy/dx at x=2. · 对于 y=x³-3x，求 x=2 处的 dy/dx。
dy/dx=3x²-3. At x=2 it is 12-3=9. · dy/dx=3x²-3。在 x=2 处，其值为 12-3=9。
Test a tempting shortcut
- A derivative gives a gradient, not the ordinate. Evaluate y and dy/dx separately at the supplied x-coordinate; the normal gradient is the negative reciprocal of a nonzero tangent gradient.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A tangent and normal at the same point always have the same gradient. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 导数给出的是斜率，而非纵坐标值。需在给定的 x 坐标处分别计算 y 和 dy/dx；法线斜率是非零切线斜率的负倒数。
- 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。
同一点的切线与法线具有相同的斜率。此说法错误。请说明它违反了哪一定义或假设。
For y=x³-3x, find y at x=2. · 对于 y=x³-3x，求 x=2 时的 y 值。
Substitute into the original curve: y(2)=8-6=2. · 代入原曲线：y(2)=8-6=2。
A tangent and normal at the same point always have the same gradient. · 同一点的切线和法线斜率并不相同。
A derivative gives a gradient, not the ordinate. Evaluate y and dy/dx separately at the supplied x-coordinate; the normal gradient is the negative reciprocal of a nonzero tangent gradient. · 导数给出的是斜率，而非纵坐标。需分别在给定的 x 坐标处计算 y 和 dy/dx；法线斜率为非零切线斜率的负倒数。
Interpret a new situation
- P1 covers polynomial derivatives, gradients, tangents and normals. Optimization and stationary-point classification are taught in P2; they are not introduced here.
- 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涵盖多项式导数、斜率、切线和法线。优化及驻点分类在P2中讲授；此处不作介绍。
- 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 y=4x³, find dy/dx at x=2. · 对于 y=4x³，求 x=2 处的 dy/dx。
The derivative is 12x²; at 2 it is 12×4=48. · 导数为 12x²；在 2 处，其值为 12×4=48。
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 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.
The instantaneous rate of change, also the gradient of a tangent. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- edexcel IAL纯数学；官方单元P1。其他单元的拓展内容已在范围审查中标识；这并非额外的加分要求。
- 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。
瞬时变化率，即切线的斜率。选择正确的关系式，展示解题方法，验证其适用条件并解释结果。