Calculus
International Baccalaureate · IB Diploma · Mathematics: Analysis and Approaches · SL Topic 5 2:11 English narration · English + 中文 subtitles burned in
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These are selected topics, not the whole syllabus, and every example is fictional.
这些是精选主题，不是完整大纲，所有例子都是虚构的。
The derivative is the gradient at a point; stationary points have gradient zero.
导数是某点处的斜率；驻点处斜率为零。
A definite integral is signed accumulation: area below the axis counts negative.
定积分是带符号的累积：轴下方的面积为负。
Take y equals x cubed minus 3 x.
取 y = x³ − 3x。
Its gradient is 3 x squared minus 3, which is zero at x equals 1 and minus 1.
斜率 3x² − 3 在 x = 1 和 −1 处为零。
At 1, y is minus 2 and the second derivative 6 x is positive: a local minimum.
x = 1 时 y = −2，二阶导数 6x 为正：极小值。
At minus 1, y is 2 and the second derivative is negative: a local maximum.
x = −1 时 y = 2，二阶导数为负：极大值。
The tangents there are flat.
那里的切线是水平的。
A zero derivative does not guarantee a maximum or minimum.
导数为零不保证是极大或极小值。
y equals x cubed has gradient 3 x squared, zero at the origin, but the gradient is positive on both sides: the curve flattens and keeps rising.
y = x³ 的斜率 3x² 在原点为零，但两侧斜率都为正：曲线变平后继续上升。
That is a point of inflection.
这是拐点。
On a restricted domain, also check the endpoints.
在限定的定义域上，还要检查端点。
A velocity is v equals 3 t squared minus 3, for t from 0 to 2.
速度 v = 3t² − 3，t 从 0 到 2。
It is negative until t equals 1, then positive.
t = 1 之前为负，之后为正。
The integral, t cubed minus 3 t from 0 to 2, gives a displacement of 2.
积分 ₀² 给出位移 2。
But the area below the axis is 2 and the area above is 4: the object moved back 2 and forward 4, a distance of 6.
但轴下方面积是 2，上方是 4：物体先后退 2 再前进 4，路程是 6。
Integrate term by term: raise each power by one and divide by the new power.
逐项积分：每个幂次加一，再除以新的幂次。
3 t squared becomes t cubed and minus 3 becomes minus 3 t.
3t² 变成 t³，−3 变成 −3t。
Evaluate from 0 to 2: 8 minus 6, then minus 0, which is 2 metres.
从 0 到 2 代入：8 − 6 − 0 = 2 米。
Differentiate back to check.
再求导检验。
The power rule fails for 1 over x, whose integral is ln of x.
幂法则对 1/x 不适用，它的积分是 ln x。
Check yourself.
自我检测。
The derivative of x cubed minus 3 x is 3 x squared minus 3.
x³ − 3x 的导数是 3x² − 3。
The integral of 3 t squared minus 3 from 0 to 2 is 8 minus 6: 2.
∫₀²(3t² − 3)dt = 8 − 6 = 2。