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دبلومة البكالوريا الدولية · IB Diploma · Mathematics: Applications and Interpretation · SL الموضوع 5 2:11 سرد باللغة الإنجليزية · ترجمة مدمجة بالإنجليزية + الصينية
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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。
Newton's method slides down tangents to a root; the trapezium rule adds strips for an area.
牛顿法沿切线逼近根；梯形法则把小条面积相加。
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。
To solve x squared minus 2 equals zero, start at 1.5.
解 x² − 2 = 0，从 1.5 出发。
There f is 0.25 and the gradient is 3.
那里 f = 0.25，斜率 3。
Follow the tangent down to the x-axis: x one is 1.5 minus 0.25 over 3, which is 1.41667.
沿切线下到 x 轴：x₁ = 1.5 − 0.25/3 = 1.41667。
One step already lands near root 2.
一步就已接近 √2。
Confirm a rounded root with a sign change either side.
用两侧变号来确认四舍五入后的根。
Estimate the area under y equals x squared from 0 to 2 with two strips of width 1.
用两条宽 1 的小条估计 y = x² 在 0 到 2 下方的面积。
The heights are 0, 1 and 4, so the trapezium rule gives a half times 0 plus 2 times 1 plus 4: 3.
高度是 0、1、4，梯形法则给出 ½ × (0 + 2 × 1 + 4) = 3。
The exact area is 8 thirds.
精确面积是 8/3。
The straight tops sit above this convex curve, so the rule overestimates.
梯形的直顶在这条凸曲线上方，所以高估。
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。
A sign change across a discontinuity, like 1 over x, proves nothing.
跨越间断点的变号（如 1/x）不能说明有根。