AP 微积分预备 · 知识点 4
8:04
含参数、向量与矩阵的函数
看这个过山车。在任意时刻，车厢都有两个事实：它在哪里，以及它朝哪个方向运动。单一方程无法同时容纳两者。这颗星球运行的……
英语旁白 · 英中内嵌字幕
学习目标 4.1.A： 为以解析形式表示的参数函数构建图像或数值表。
来源：College Board AP 课程与考试说明
Unit 4 is part of the course but is not assessed on the AP Precalculus Exam (which covers only Units 1–3); it builds tools used in later courses.
A parametric function 参数函数 describes a curve by giving both coordinates as functions of a third variable, the parameter 参数 $t$: $\big(x(t),\,y(t)\big)$. As $t$ runs through its domain, the point traces a path – and, unlike $y=f(x)$, the path may loop or cross itself.
在参数形式中，x和y分别依赖于参数t。对每个陈述进行排序。
| English | 中文 | 拼音 |
|---|---|---|
| parametric function/ˌpærəˈmetrɪk ˈfʌŋkʃn/ | 参数函数 | cān shù hán shù |
| parameter/pəˈræmɪtə/ | 参数 | cān shù |
学习目标 4.2.A： 识别与位置相关的参数平面运动函数的关键特征。
来源：College Board AP 课程与考试说明
Reading $t$ as time, a parametric function models motion in the plane. $x(t)$ and $y(t)$ give the horizontal and vertical position; the point's direction of travel is the order in which the curve is drawn as $t$ increases (mark it with arrows).
Worked example. A particle moves with $x(t)=2t-1$ and $y(t)=t^2$ for $-1\le t\le 2$. Describe its path.
Make a table first:
| $t$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|
| $(x,y)$ | $(-3,1)$ | $(-1,0)$ | $(1,1)$ | $(3,4)$ |
To eliminate the parameter, solve $x=2t-1$ for $t=\tfrac{x+1}{2}$ and substitute: $y=\tfrac{(x+1)^2}{4}$. The path is the part of this parabola from $(-3,1)$ to $(3,4)$. Because $x$ increases with $t$, the particle moves left to right: down to the vertex $(-1,0)$ at $t=0$, then up.
Common mistake. Giving the whole parabola $y=\tfrac{(x+1)^2}{4}$ as the answer. The restriction $-1\le t\le 2$ becomes $-3\le x\le 3$, and the equation alone shows no direction.
随着参数的运行，点移动并描绘出一条平面曲线。
学习目标 4.3.A： 识别与方向及变化率相关的平面运动参数方程函数的关键特征。
来源：College Board AP 课程与考试说明
Over an interval of $t$, the average rate of change of $x$ is $\dfrac{\Delta x}{\Delta t}$ and of $y$ is $\dfrac{\Delta y}{\Delta t}$. Their signs tell you which way the point moves (right/left, up/down); together they describe the motion's speed and direction along the path.
Worked example. Find the average rates of change for the same particle on $0\le t\le 2$ and on $-1\le t\le 0$.
On the first interval it moves right and up on average; on the second it moves down while it moves right.
拖动两个向量,看它们的和。向量既有方向又有大小,不像一个普通的数。
学习目标 4.4.A： 用参数方程表示圆周运动或沿线段上的运动。
来源：College Board AP 课程与考试说明
A circle of radius $R$ centered at $(h,k)$ is $x=h+R\cos t,\ y=k+R\sin t$. A line through $(x_0,y_0)$ with direction $(a,b)$ is $x=x_0+at,\ y=y_0+bt$. Adjusting the coefficients changes the start point, speed, and direction of tracing.
Worked example. The circle with center $(3,-2)$ and radius $5$ is $x=3+5\cos t,\ y=-2+5\sin t$ for $0\le t<2\pi$. It starts at $(8,-2)$ and turns counterclockwise. Writing $y=-2-5\sin t$ instead traces the same circle clockwise.
Worked example. The segment from $(1,4)$ to $(7,1)$ has direction $(7-1,\,1-4)=(6,-3)$, so $x=1+6t,\ y=4-3t$ for $0\le t\le 1$. At $t=0$ the point is at $(1,4)$; at $t=1$ it reaches $(7,1)$.
学习目标 4.5.A： 绘制包含两个变量的方程的图像。
学习目标 4.5.B： 确定隐式定义函数中两个相关量如何共同变化。
来源：College Board AP 课程与考试说明
An implicit 隐式 equation relates $x$ and $y$ without solving for either, such as $x^2+y^2=25$. Its graph may fail the vertical-line test (not a function), so it is often split into pieces or described parametrically.
隐式方程在不解出y的情况下关联x和y。对每一项进行分类。
| English | 中文 | 拼音 |
|---|---|---|
| implicit/ɪmˈplɪsɪt/ | 隐式 | yǐn shì |
学习目标 4.6.A： 用解析法表示具有水平或垂直对称性的圆锥曲线。
来源：College Board AP 课程与考试说明
Conic sections 圆锥曲线 – circles, ellipses, parabolas, and hyperbolas – are the curves from slicing a cone, each given by a quadratic equation in $x$ and $y$. Their standard forms reveal centers, vertices, axes, and asymptotes.
Worked example. Identify $4x^2+9y^2-16x+18y-11=0$.
Complete the square in each variable:
Divide by $36$: $\dfrac{(x-2)^2}{9}+\dfrac{(y+1)^2}{4}=1$. This is an ellipse 椭圆 with center $(2,-1)$, reaching $3$ units left and right (to $(-1,-1)$ and $(5,-1)$) and $2$ units up and down (to $(2,1)$ and $(2,-3)$).
Worked example. The hyperbola 双曲线 $\dfrac{x^2}{16}-\dfrac{y^2}{9}=1$ has vertices $(\pm4,0)$ and asymptotes $y=\pm\tfrac34x$. The minus sign makes it a hyperbola, and the positive $x^2$ term means it opens left and right.
Common mistake. Forgetting to subtract the constants that completing the square adds ($-16$ and $-9$ above), or dividing only one term by $36$.
平方项告诉你这是哪种圆锥曲线。对每个方程进行分类。
| English | 中文 | 拼音 |
|---|---|---|
| Conic sections/ˈkɒnɪk ˈsekʃnz/ | 圆锥曲线 | yuán zhuī qū xiàn |
| ellipse/ɪˈlɪps/ | 椭圆 | tuǒ yuán |
| hyperbola/haɪˈpɜːbələ/ | 双曲线 | shuāng qū xiàn |
学习目标 4.7.A： 用参数方程表示平面上的曲线。
学习目标 4.7.B： 用参数方程表示圆锥曲线。
来源：College Board AP 课程与考试说明
Many implicit curves can be parametrized – rewritten as $\big(x(t),y(t)\big)$ – which makes them easier to graph and to treat as motion. The circle above is the basic example; ellipses use $x=h+A\cos t,\ y=k+B\sin t$.
For the ellipse above, $x=2+3\cos t,\ y=-1+2\sin t$, $0\le t<2\pi$.
当角扫过时,隐式圆 x 平方加 y 平方等于 4 变成参数形式 x = 2 cos t, y = 2 sin t。
学习目标 4.8.A： 识别向量的特征。
学习目标 4.8.B： 确定涉及向量的和与积。
学习目标 4.8.C： 确定给定向量的单位向量。
学习目标 4.8.D： 确定向量之间夹角的度量以及参与向量加法的向量的模。
来源：College Board AP 课程与考试说明
A vector 向量 has both magnitude 大小 (length) and direction. Write it by components $\langle a,b\rangle$. Add vectors component-by-component; scale by multiplying each component; the magnitude is $\sqrt{a^2+b^2}$. Vectors model displacements, velocities, and forces.
Worked example. Let $\vec u=\langle 3,-4\rangle$ and $\vec v=\langle -1,2\rangle$. Then $\vec u+2\vec v=\langle 3-2,\,-4+4\rangle=\langle 1,0\rangle$, and $\lvert\vec u\rvert=\sqrt{9+16}=5$. The unit vector in the direction of $\vec u$ divides by its magnitude: $\tfrac15\vec u=\left\langle \tfrac35,-\tfrac45\right\rangle$.
向量具有大小和方向。添加两个向量时，将它们头尾相接；合向量从第一个向量的尾部指向最后一个向量的头部。拖动每个向量以查看总和。
| English | 中文 | 拼音 |
|---|---|---|
| vector/ˈvektə/ | 向量 | xiàng liàng |
| magnitude/ˈmæɡnɪtjuːd/ | 大小 | dà xiǎo |
学习目标 4.9.A： 用向量值函数表示平面运动。
来源：College Board AP 课程与考试说明
A vector-valued function 向量值函数 outputs a vector for each input, e.g. $\vec{r}(t)=\langle x(t),y(t)\rangle$ – the same information as a parametric function, packaged as a moving position vector.
向量值函数输出向量；将分量向量相加即可得到合向量。
| English | 中文 | 拼音 |
|---|---|---|
| vector-valued function/ˈvektə ˈvæljuːd ˈfʌŋkʃn/ | 向量值函数 | xiàng liàng zhí hán shù |
学习目标 4.10.A： 确定两个矩阵的乘积。
来源：College Board AP 课程与考试说明
A matrix 矩阵 is a rectangular array of numbers. Add matrices of the same size entry-by-entry; multiply a matrix by a compatible one by combining rows with columns. Matrices store and transform data compactly.
Worked example.
In the other order the product is $\begin{bmatrix}3 & 6\\ 1 & 4\end{bmatrix}$, which is different. Matrix multiplication is not commutative: order matters.
2x2矩阵移动平面的每一点；这个矩阵使其发生侧向剪切。
| English | 中文 | 拼音 |
|---|---|---|
| matrix/ˈmeɪtrɪks/ | 矩阵 | jǔ zhèn |
学习目标 4.11.A： 确定 $2 \times 2$ 矩阵的逆。
学习目标 4.11.B： 应用行列式的值来判断可逆性及向量性质。
来源：College Board AP 课程与考试说明
The determinant 行列式 of a $2\times 2$ matrix $\begin{bmatrix} a & b \\ c & d\end{bmatrix}$ is $ad-bc$. A matrix has an inverse 逆矩阵 exactly when its determinant is nonzero; the inverse undoes the matrix, and it solves matrix equations (like a reciprocal for numbers).
Worked example. For $A=\begin{bmatrix}3 & 1\\ 2 & 4\end{bmatrix}$, the determinant is $ad-bc=(3)(4)-(1)(2)=10$. Since it is nonzero, $A$ is invertible, and $A^{-1}=\dfrac{1}{10}\begin{bmatrix}4 & -1\\ -2 & 3\end{bmatrix}$ (swap the diagonal, negate the off-diagonal, divide by the determinant).
Worked example. Solve $3x+y=7,\ 2x+4y=8$ with $A^{-1}$. In matrix form $A\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}7\\8\end{bmatrix}$, so
Check: $3(2)+1=7$ and $2(2)+4(1)=8$.
行列式告诉你矩阵缩放面积的程度；在这里它将面积放大了四倍。
| English | 中文 | 拼音 |
|---|---|---|
| determinant/dɪˈtɜːmɪnənt/ | 行列式 | háng liè shì |
| inverse/ɪnˈvɜːs/ | 逆矩阵 | nì jǔ zhèn |
学习目标 4.12.A： 利用 $2 \times 2$ 矩阵确定线性变换的输出向量。
来源：College Board AP 课程与考试说明
A $2\times 2$ matrix acts as a linear transformation 线性变换 of the plane – rotating, reflecting, stretching, or shearing points – by multiplying each position vector. The determinant measures how the transformation scales area (and its sign tells whether orientation flips).
Worked example. The matrix $\begin{bmatrix}2 & 0\\ 0 & 3\end{bmatrix}$ stretches the plane by $2$ horizontally and $3$ vertically, sending the vector $\begin{bmatrix}1\\1\end{bmatrix}$ to $\begin{bmatrix}2\\3\end{bmatrix}$. Its determinant $2\times3=6$ means every area is multiplied by $6$, so the unit square becomes a $2\times3$ rectangle.
Worked example. The rotation by angle $\theta$ counterclockwise about the origin is $\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}$. For $\theta=90^\circ$ it is $\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$, which sends $\begin{bmatrix}3\\1\end{bmatrix}$ to $\begin{bmatrix}-1\\3\end{bmatrix}$. Its determinant is $0\cdot0-(-1)(1)=1$: a rotation keeps every area the same.
一个$2\times2$矩阵将每个点映射到新位置，从而拉伸、旋转或剪切整个网格。更改条目并观察单位正方形的变换。
| English | 中文 | 拼音 |
|---|---|---|
| linear transformation/ˈlɪnɪə trænsfɔːˈmeɪʃn/ | 线性变换 | xiàn xìng biàn huàn |
学习目标 4.13.A： 确定线性变换与矩阵之间的关联。
学习目标 4.13.B： 确定两个线性变换的组合。
学习目标 4.13.C： 确定线性变换的逆。
来源：College Board AP 课程与考试说明
Because a matrix maps input vectors to output vectors, it is a function on the plane. Composing transformations corresponds to multiplying their matrices, and the inverse matrix reverses the mapping.
一个矩阵接收输入向量并返回输出向量——这是作用于向量的函数。
学习目标 4.14.A： 使用矩阵构建涉及两个状态之间转换的情境模型。
学习目标 4.14.B： 应用矩阵模型预测 $n$ 次转换步骤的未来和过去状态。
来源：College Board AP 课程与考试说明
Matrices model systems that step forward in stages – for example, populations moving between states each year. Repeated multiplication by a transition matrix advances the model one step at a time, so matrix powers predict long-run behavior.
Worked example. Each year $90\%$ of a town's residents who use bus A keep using it and $10\%$ switch to bus B; $20\%$ of bus B users switch to A and $80\%$ stay. Write the model with each column holding where one group goes:
After one year, $620$ use A and $380$ use B. The numbers stop changing when the flows balance, $0.1A=0.2B$, so $A=2B$. Out of $1000$ residents, that is about $667$ on A and $333$ on B in the long run.
矩阵用于模拟变换、系统和转换。对每项任务进行分类。
逐步完成，配合即时检查练习。
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