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.
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.
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=1, the gradient is 0 and y=-2. The second derivative is 6x, positive at x=1, so this is a local minimum. At x=-1, y=2 and the second derivative is negative, giving a local maximum.
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 zero derivative does not always mean a maximum or minimum: y=x³ is stationary at 0 but continues increasing. An endpoint can also produce an extreme value on a restricted domain.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every point with zero derivative is a local maximum or minimum. This claim is false. Explain which definition or assumption it violates.
Find the y-coordinate of the stationary point at x=1. · 求 x=1 处驻点的 y 坐标。
Substitute into the original curve: y(1)=1-3=-2. · 代入原曲线方程：y(1)=1-3=-2。
Every point with zero derivative is a local maximum or minimum. · 导数为零的点不一定是局部最大值或最小值。
A zero derivative does not always mean a maximum or minimum: y=x³ is stationary at 0 but continues increasing. An endpoint can also produce an extreme value on a restricted domain. · 导数为零并不总是意味着最大值或最小值：y=x³ 在 0 处是驻点，但函数仍持续递增。端点在受限定义域上也可能产生极值。
Interpret a new situation
- At GCSE/IGCSE use only the polynomial scope allowed by the tier; do not add chain, product or quotient rules there. At advanced level, connect the derivative to rates and optimization with a valid domain.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
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
- Current first-assessment-2021 Analysis and Approaches 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.
The instantaneous rate of change, also the gradient of a tangent. Choose the relationship, show the method, check its assumptions and interpret the result.