Derivatives and stationary points
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| derivative/dɪˈrɪvətɪv/ | đạo hàm |
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.
The method
- 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.
At x=0 the curve y=x³ has dy/dx=0. Why is it not a local maximum or minimum?
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The derivative 3x² is positive on both sides of x=0, so the point is stationary but the function continues to rise - a point of inflection. Classification needs the sign change or the second derivative, not merely dy/dx=0.
A worked case
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
For y=ax^n, dy/dx=anx^(n-1)
Compare the model with the worked case and explain one change.
For y=x³-3x, find dy/dx at x=2.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
dy/dx=3x²-3. At x=2 it is 12-3=9.
Where it goes wrong
- 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.
Not true: Every point with zero derivative is a local maximum or minimum. A zero derivative can also be a point of inflection: y=x³ at x=0 is neither a maximum nor a minimum.
Find the y-coordinate of the stationary point at x=1.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Substitute into the original curve: y(1)=1-3=-2.
A zero derivative can also be a point of inflection.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. A zero derivative can also be a point of inflection: y=x³ at x=0 is neither a maximum nor a minimum.
Applying it
- 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.
For y=4x³, find dy/dx at x=2.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The derivative is 12x²; at 2 it is 12×4=48.
Match each derivative fact to its statement.
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The power rule lowers the index by one; a zero first derivative flags a candidate, and the second derivative or a sign check classifies it.
In your course
- AQA A-level Mathematics (7357), specification section G.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Derivative: The instantaneous rate of change, also the gradient of a tangent.
- For y=ax^n, dy/dx=anx^(n-1).
- Watch for: A zero derivative does not always mean a maximum or minimum: y=x³ is stationary at 0 but continues increasing.