Cálculo
Diploma del Bachillerato Internacional · IB Diploma · Mathematics: Analysis and Approaches · HL Tema 5 5:24 Narración en inglés · Subtítulos en inglés + 中文 quemados en pantalla
Capítulos
Transcripción
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。
The chain rule multiplies by the inside derivative; products and quotients have their own rules.
链式法则要乘内层导数；乘积和商各有法则。
Integrate products by parts or by substitution, never factor by factor.
乘积用分部积分或换元，不能逐个因子积分。
A rate equation gives a family of curves; the starting value picks one.
变化率方程给出一族曲线；初始值选出其中一条。
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。
Differentiate 3 x plus 1 to the 4.
对 (3x + 1)⁴ 求导。
The outer power gives 4 times 3 x plus 1 cubed, and the chain rule multiplies by the inside derivative, 3: 12 times 3 x plus 1 cubed, which is 12 at x equals 0.
外层给出 4(3x + 1)³，链式法则再乘内层导数 3：12(3x + 1)³，x = 0 时为 12。
For a product, u prime v plus u v prime: x times x gives 2 x, not 1 times 1.
乘积用 u′v + uv′：x · x 的导数是 2x，不是 1 × 1。
The circle x squared plus y squared equals 25 is not y as a function of x.
圆 x² + y² = 25 并不是 y 关于 x 的函数。
Differentiate both sides anyway: 2 x plus 2 y times y prime equals 0, because each y term brings a d y by d x.
照样两边求导：2x + 2y·y′ = 0，因为每个 y 项都会带出 dy/dx。
So y prime is minus x over y.
所以 y′ = −x/y。
At 3, 4 the tangent has gradient minus 3 quarters, at right angles to the radius.
在 (3, 4) 处切线斜率 −¾，与半径垂直。
x e to the x cannot be integrated factor by factor.
x·eˣ 不能逐个因子积分。
By parts, take u equals x, so it simplifies, and v prime equals e to the x.
分部积分：取 u = x，求导后变简单；v′ = eˣ。
The integral is x e to the x minus the integral of e to the x: x e to the x minus e to the x plus C.
积分等于 xeˣ 减去 ∫eˣ：xeˣ − eˣ + C。
For 2 x over x squared plus 1, the inside derivative 2 x is already there: substitute w and get ln of x squared plus 1, plus C.
对 2x/(x² + 1)，内层导数 2x 已经在那里：令 w = x² + 1，得 ln(x² + 1) + C。
Put limits on it: the area under x e to the x from 0 to 1.
加上积分限：x·eˣ 在 0 到 1 下方的面积。
Evaluate x e to the x minus e to the x at 1, which is 0, and at 0, which is minus 1.
xeˣ − eˣ 在 1 处为 0，在 0 处为 −1。
Subtract: the area is exactly 1.
相减：面积恰好是 1。
In a definite substitution, either change the limits or return to x first.
定积分换元时，要么换积分限，要么先换回 x。
A population grows at a rate proportional to its size: d y by d x equals 0.5 y.
种群的增长率与其大小成正比：dy/dx = 0.5y。
Separate: d y over y equals 0.5 d x.
分离变量：dy/y = 0.5dx。
Integrate: ln y is 0.5 x plus a constant, so y is A e to the 0.5 x, a whole family.
积分：ln y = 0.5x + 常数，所以 y = Ae^(0.5x)，这是一族曲线。
The starting value y of 0 equal to 2 picks one: A is 2.
初始值 y(0) = 2 选出一条：A = 2。
Euler's method walks along the slope field in steps of h.
欧拉法沿斜率场以步长 h 前进。
Start at 0, 2: the slope 0.5 y is 1, so step to 2 plus 0.2 times 1, which is 2.2.
从 (0, 2) 出发：斜率 0.5y = 1，所以走到 2 + 0.2 × 1 = 2.2。
There the slope is 1.1, so the next value is 2.42.
那里斜率 1.1，下一个值 2.42。
The exact curve gives about 2.4428: each straight step falls a little below this curve.
精确曲线约为 2.4428：每个直步都略低于曲线。
Smaller steps track it more closely.
步长越小越贴近。
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。
The derivative of x times x is 2 x, not 1 times 1.
x · x 的导数是 2x，不是 1 × 1。
The integral of 2 x over x squared plus 1 is ln of x squared plus 1, plus C.
∫2x/(x² + 1)dx = ln(x² + 1) + C。
d y by d x equals 0.5 y with y of 0 equal to 2 gives y equals 2 e to the 0.5 x.
dy/dx = 0.5y，y(0) = 2，得 y = 2e^(0.5x)。