剑桥 IGCSE IGCSE, A-Level & AP 练习单预览

涉及参数、向量与矩阵的函数

AP 微积分预备 · 第 4 主题

训练
讲义 词汇表
4.1

参数方程函数
大纲
Learning ObjectiveEssential Knowledge

4.1.A
Construct a graph or table of values for a parametric function represented analytically.

  • 4.1.A.1 A parametric function in $\mathbb{R}^2$, the set of all ordered pairs of two real numbers, consists of a set of two parametric equations in which two dependent variables, $x$ and $y$, are dependent on a single independent variable, $t$, called the parameter.
  • 4.1.A.2 Because variables $x$ and $y$ are dependent on the independent variable, $t$, the coordinates $(x_i,\, y_i)$ at time $t_i$ can be written as functions of $t$ and can be expressed as the single parametric function $f(t) = (x(t),\, y(t))$, where in this case $x$ and $y$ are names of two functions.
  • 4.1.A.3 A numerical table of values can be generated for the parametric function $f(t) = (x(t),\, y(t))$ by evaluating $x_i$ and $y_i$ at several values of $t_i$ within the domain.
  • 4.1.A.4 A graph of a parametric function can be sketched by connecting several points from the numerical table of values in order of increasing value of $t$.
  • 4.1.A.5 The domain of the parametric function $f$ is often restricted, which results in start and end points on the graph of $f$.

来源:美国大学理事会 AP 课程与考试说明

第 4 单元是课程的一部分但不在 AP 微积分预备考试中考查(考试只覆盖第 1–3 单元);它构建后面课程里使用的工具。

一个参数函数(parametric function)通过把两个坐标都作为第三个变量、参数(parameter)$t$ 的函数给出来描述一条曲线:$\big(x(t),\,y(t)\big)$。随着 $t$ 走过它的定义域,点描出一条路径——而,不像 $y=f(x)$,这条路径可能环绕或与自己交叉。

词汇表 训练
英文 中文 拼音
parametric function 参数函数 cān shù hán shù
parameter 参数 cān shù
练习卷
4.2

用参数方程函数为平面运动建模
大纲
Learning ObjectiveEssential Knowledge

4.2.A
Identify key characteristics of a parametric planar motion function that are related to position.

  • 4.2.A.1 A parametric function given by $f(t) = (x(t),\, y(t))$ can be used to model particle motion in the plane. The graph of this function indicates the position of a particle at time $t$.
  • 4.2.A.2 The horizontal and vertical extrema of a particle's motion can be determined by identifying the maximum and minimum values of the functions $x(t)$ and $y(t)$, respectively.
  • 4.2.A.3 The real zeros of the function $x(t)$ correspond to $y$-intercepts, and the real zeros of $y(t)$ correspond to $x$-intercepts.

来源:美国大学理事会 AP 课程与考试说明

把 $t$ 读作时间,一个参数函数为平面里的运动建模。$x(t)$ 和 $y(t)$ 给出水平和竖直位置;点行进的方向是随着 $t$ 增加曲线被画出的顺序(用箭头标记它)。

练习卷
4.3

参数方程函数与变化率
大纲
Learning ObjectiveEssential Knowledge

4.3.A
Identify key characteristics of a parametric planar motion function that are related to direction and rate of change.

  • 4.3.A.1 As the parameter increases, the direction of planar motion of a particle can be analyzed in terms of $x$ and $y$ independently. If $x(t)$ is increasing or decreasing, the direction of motion is to the right or left, respectively. If $y(t)$ is increasing or decreasing, the direction of motion is up or down, respectively.
  • 4.3.A.2 At any given point in the plane, the direction of planar motion may be different for different values of $t$.
  • 4.3.A.3 The same curve in the plane can be parametrized in different ways and can be traversed in different directions with different parametric functions.
  • 4.3.A.4 Over a given interval $[t_1,\, t_2]$ within the domain, the average rate of change can be computed for $x(t)$ and $y(t)$ independently. The ratio of the average rate of change of $y$ to the average rate of change of $x$ gives the slope of the graph between the points on the curve corresponding to $t_1$ and $t_2$, so long as the average rate of change of $x(t) \neq 0$.

来源:美国大学理事会 AP 课程与考试说明

在 $t$ 的一个区间上,$x$ 的平均变化率是 $\dfrac{\Delta x}{\Delta t}$ 而 $y$ 的是 $\dfrac{\Delta y}{\Delta t}$。它们的符号告诉你点向哪个方向移动(右/左、上/下);它们一起描述运动沿路径的速率和方向。

练习卷
4.4

参数定义的圆与直线
大纲
Learning ObjectiveEssential Knowledge

4.4.A
Express motion around a circle or along a line segment parametrically.

  • 4.4.A.1 A complete counterclockwise revolution around the unit circle that starts and ends at $(1,\, 0)$ and is centered at the origin can be modeled by $(x(t),\, y(t)) = (\cos t,\, \sin t)$ with domain $0 \leq t \leq 2\pi$.
  • 4.4.A.2 Transformations of the parametric function $(x(t),\, y(t)) = (\cos t,\, \sin t)$ can model any circular path traversed in the plane.
  • 4.4.A.3 A linear path along the line segment from the point $(x_1,\, y_1)$ to the point $(x_2,\, y_2)$ can be parametrized many ways, including using an initial position $(x_1,\, y_1)$ and rates of change for $x$ with respect to $t$ and $y$ with respect to $t$.

来源:美国大学理事会 AP 课程与考试说明

一个中心在 $(h,k)$ 的半径 $R$ 的圆是 $x=h+R\cos t,\ y=k+R\sin t$。一条通过 $(x_0,y_0)$、方向 $(a,b)$ 的线是 $x=x_0+at,\ y=y_0+bt$。调整系数改变起点、速率和描出的方向。

练习卷
4.5

隐函数
大纲
Learning ObjectiveEssential Knowledge

4.5.A
Construct a graph of an equation involving two variables.

  • 4.5.A.1 An equation involving two variables can implicitly describe one or more functions.
  • 4.5.A.2 An equation involving two variables can be graphed by finding solutions to the equation.
  • 4.5.A.3 Solving for one of the variables in an equation involving two variables can define a function whose graph is part or all of the graph of the equation.

4.5.B
Determine how the two quantities related in an implicitly defined function vary together.

  • 4.5.B.1 For ordered pairs on the graph of an implicitly defined function that are close together, if the ratio of the change in the two variables is positive, then the two variables simultaneously increase or both decrease; conversely, if the ratio is negative, then as one variable increases, the other decreases.
  • 4.5.B.2 The rate of change of $x$ with respect to $y$ or of $y$ with respect to $x$ can be zero, indicating vertical or horizontal intervals, respectively.

来源:美国大学理事会 AP 课程与考试说明

一个隐式(implicit)方程把 $x$ 和 $y$ 关联而不解出任一个,例如 $x^2+y^2=25$。它的图象可能不通过竖直线测试(不是一个函数),所以它常常被分成片段或参数地描述。

词汇表 训练
英文 中文 拼音
implicit 隐式 yǐn shì
练习卷
4.6

圆锥曲线
大纲
Learning ObjectiveEssential Knowledge

4.6.A
Represent conic sections with horizontal or vertical symmetry analytically.

  • 4.6.A.1 A parabola with vertex $(h,\, k)$ can, if $a \neq 0$, be represented analytically as $x - h = a(y - k)^2$ if it opens left or right, or as $y - k = a(x - h)^2$ if it opens up or down.
  • 4.6.A.2 An ellipse centered at $(h,\, k)$ with horizontal radius $a$ and vertical radius $b$ can be represented analytically as $\dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1$. A circle is a special case of an ellipse where $a = b$.
  • 4.6.A.3 A hyperbola centered at $(h,\, k)$ with vertical and horizonal lines of symmetry can be represented algebraically as $\dfrac{(x - h)^2}{a^2} - \dfrac{(y - k)^2}{b^2} = 1$ for a hyperbola opening left and right, or as $\dfrac{(y - k)^2}{b^2} - \dfrac{(x - h)^2}{a^2} = 1$ for a hyperbola opening up and down. The asymptotes are $y - k = \pm \dfrac{b}{a}(x - h)$.

来源:美国大学理事会 AP 课程与考试说明

圆锥曲线：一个圆锥，四条曲线

圆锥曲线(conic sections)——圆、椭圆、抛物线和双曲线——是切割一个圆锥得到的曲线,每个由一个 $x$ 和 $y$ 的二次方程给出。它们的标准形式揭示中心、顶点、轴和渐近线。

Elliptical orbits: conic sections describe closed paths with two foci — a key parametric model
Elliptical orbits: conic sections describe closed paths with two foci — a key parametric model
词汇表 训练
英文 中文 拼音
Conic sections 圆锥曲线 yuán zhuī qū xiàn
练习卷
4.7

隐函数的参数化
大纲
Learning ObjectiveEssential Knowledge

4.7.A
Represent a curve in the plane parametrically.

  • 4.7.A.1 A parametrization $(x(t),\, y(t))$ for an implicitly defined function will, when $x(t)$ and $y(t)$ are substituted for $x$ and $y$, respectively, satisfy the corresponding equation for every value of $t$ in the domain.
  • 4.7.A.2 If $f$ is a function of $x$, then $y = f(x)$ can be parametrized as $(x(t),\, y(t)) = (t, f(t))$. If $f$ is invertible, its inverse can be parametrized as $(x(t),\, y(t)) = (f(t),\, t)$ for an appropriate interval of $t$.

4.7.B
Represent conic sections parametrically.

  • 4.7.B.1 A parabola can be parametrized in the same way that any equation that can be solved for $x$ or $y$ can be parametrized. Equations that can be solved for $x$ can be parametrized as $(x(t),\, y(t)) = (f(t),\, t)$ by solving for $x$ and replacing $y$ with $t$. Equations that can be solved for $y$ can be parametrized as $(x(t),\, y(t)) = (t, f(t))$ by solving for $y$ and replacing $x$ with $t$.
  • 4.7.B.2 An ellipse can be parametrized using the trigonometric functions $x(t) = h + a\cos t$ and $y(t) = k + b\sin t$ for $0 \leq t \leq 2\pi$.
  • 4.7.B.3 A hyperbola can be parametrized using trigonometric functions. For a hyperbola that opens left and right, the functions are $x(t) = h + a\sec t$ and $y(t) = k + b\tan t$ for $0 \leq t \leq 2\pi$. For a hyperbola that opens up and down, the functions are $x(t) = h + a\tan t$ and $y(t) = k + b\sec t$ for $0 \leq t \leq 2\pi$.

来源:美国大学理事会 AP 课程与考试说明

许多隐式曲线能被参数化(parametrized)——重写为 $\big(x(t),y(t)\big)$ ——这使它们更容易画图并作为运动处理。上面的圆是基本例子;椭圆用 $x=h+A\cos t,\ y=k+B\sin t$。

练习卷
4.8

向量
大纲
Learning ObjectiveEssential Knowledge

4.8.A
Identify characteristics of a vector.

  • 4.8.A.1 A vector is a directed line segment. When a vector is placed in the plane, the point at the beginning of the line segment is called the tail, and the point at the end of the line segment is called the head. The length of the line segment is the magnitude of the vector.
  • 4.8.A.2 A vector $\overrightarrow{P_1 P_2}$ with two components can be plotted in the $xy$-plane from $P_1 = (x_1, y_1)$ to $P_2 = (x_2, y_2)$. The vector is identified by $a$ and $b$, where $a = x_2 - x_1$ and $b = y_2 - y_1$. The vector can be expressed as $\langle a,\, b \rangle$. A zero vector $\langle 0,\, 0 \rangle$ is the trivial case when $P_1 = P_2$.
  • 4.8.A.3 The direction of the vector is parallel to the line segment from the origin to the point with coordinates $(a,\, b)$. The magnitude of the vector is the square root of the sum of the squares of the components.
  • 4.8.A.4 For a vector represented geometrically in the plane, the components of the vector can be found using trigonometry.

4.8.B
Determine sums and products involving vectors.

  • 4.8.B.1 The multiplication of a constant and a vector results in a new vector whose components are found by multiplying the constant by each of the components of the original vector. The new vector is parallel to the original vector.
  • 4.8.B.2 The sum of two vectors in $\mathbb{R}^2$ is a new vector whose components are found by adding the corresponding components of the original vectors. The new vector can be represented graphically as a vector whose tail corresponds to the tail of the first vector and whose head corresponds to the head of the second vector when the second vector's tail is located at the first vector's head.
  • 4.8.B.3 The dot product of two vectors is the sum of the products of their corresponding components. That is, $\langle a_1,\, b_1 \rangle \cdot \langle a_2,\, b_2 \rangle = a_1 a_2 + b_1 b_2$.

4.8.C
Determine a unit vector for a given vector.

  • 4.8.C.1 A unit vector is a vector of magnitude $1$. A unit vector in the same direction as a given nonzero vector can be found by scalar multiplying the vector by the reciprocal of its magnitude.
  • 4.8.C.2 The vector $\langle a,\, b \rangle$ can be expressed as $a\vec{i} + b\vec{j}$ in $\mathbb{R}^2$, where $\vec{i}$ and $\vec{j}$ are unit vectors in the $x$ and $y$ directions, respectively. That is, $\vec{i} = \langle 1,\, 0 \rangle$ and $\vec{j} = \langle 0,\, 1 \rangle$.

4.8.D
Determine angle measures between vectors and magnitudes of vectors involved in vector addition.

  • 4.8.D.1 The dot product is geometrically equivalent to the product of the magnitudes of the two vectors and the cosine of the angle between them. Therefore, if the dot product of two nonzero vectors is zero, then the vectors are perpendicular.
  • 4.8.D.2 The Law of Sines and Law of Cosines can be used to determine side lengths and angle measures of triangles formed by vector addition.

来源:美国大学理事会 AP 课程与考试说明

矢量的分解

一个向量(vector)既有大小(magnitude)(长度)又有方向。用分量 $\langle a,b\rangle$ 写它。逐分量地相加向量;通过乘每个分量来缩放;大小是 $\sqrt{a^2+b^2}$。向量为位移、速度和力建模。

按三角形法则相加向量
按三角形法则相加向量
探索

Add two vectors tip to tail

A vector has magnitude and direction. Adding two vectors places them tip to tail; the resultant runs from the first tail to the last tip. Drag each to see the sum.

词汇表 训练
英文 中文 拼音
vector 向量 xiàng liàng
magnitude 大小 dà xiǎo
练习卷
4.9

向量值函数
大纲
Learning ObjectiveEssential Knowledge

4.9.A
Represent planar motion in terms of vector-valued functions.

  • 4.9.A.1 The position of a particle moving in a plane that is given by the parametric function $f(t) = (x(t),\, y(t))$ may be expressed as a vector-valued function, $p(t) = x(t)\vec{i} + y(t)\vec{j}$ or $p(t) = \langle x(t),\, y(t) \rangle$. The magnitude of the position vector at time $t$ gives the distance of the particle from the origin.
  • 4.9.A.2 The vector-valued function $v(t) = \langle x(t),\, y(t) \rangle$ can be used to express the velocity of a particle moving in a plane at different times, $t$. At time $t$, the sign of $x(t)$ indicates if the particle is moving right or left, and the sign of $y(t)$ indicates if the particle is moving up or down. The magnitude of the velocity vector at time $t$ gives the speed of the particle.

来源:美国大学理事会 AP 课程与考试说明

一个向量值函数(vector-valued function)为每个输入输出一个向量,例如 $\vec{r}(t)=\langle x(t),y(t)\rangle$ ——与一个参数函数相同的信息,打包为一个移动的位置向量。

一个向量值直线:从 a 开始,然后滑动方向 b 的 t 倍
一个向量值直线:从 a 开始,然后滑动方向 b 的 t 倍
词汇表 训练
英文 中文 拼音
vector-valued function 向量值函数 xiàng liàng zhí hán shù
练习卷
4.10

矩阵
大纲
Learning ObjectiveEssential Knowledge

4.10.A
Determine the product of two matrices.

  • 4.10.A.1 An $n \times m$ matrix is an array consisting of $n$ rows and $m$ columns.
  • 4.10.A.2 Two matrices can be multiplied if the number of columns in the first matrix equals the number of rows in the second matrix. The product of the matrices is a new matrix in which the component in the $i$th row and $j$th column is the dot product of the $i$th row of the first matrix and the $j$th column of the second matrix.

来源:美国大学理事会 AP 课程与考试说明

一个矩阵(matrix)是数字的一个矩形阵列。逐项相加相同大小的矩阵;通过把行与列组合把一个矩阵乘以一个兼容的矩阵。矩阵紧凑地存储和变换数据。

词汇表 训练
英文 中文 拼音
matrix 矩阵 jǔ zhèn
练习卷
4.11

矩阵的逆与行列式
大纲
Learning ObjectiveEssential Knowledge

4.11.A
Determine the inverse of a $2 \times 2$ matrix.

  • 4.11.A.1 The identity matrix, $I$, is a square matrix consisting of $1$s on the diagonal from the top left to bottom right and $0$s everywhere else.
  • 4.11.A.2 Multiplying a square matrix by its corresponding identity matrix results in the original square matrix.
  • 4.11.A.3 The product of a square matrix and its inverse, when it exists, is the identity matrix of the same size.
  • 4.11.A.4 The inverse of a $2 \times 2$ matrix, when it exists, can be calculated with or without technology.

4.11.B
Apply the value of the determinant to invertibility and vectors.

  • 4.11.B.1 The determinant of the matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $ad - bc$. The determinant can be calculated with or without technology and is denoted $\det(A)$.
  • 4.11.B.2 If a $2 \times 2$ matrix consists of two column or row vectors from $\mathbb{R}^2$, then the nonzero absolute value of the determinant of the matrix is the area of the parallelogram spanned by the vectors represented in the columns or rows of the matrix. If the determinant equals $0$, then the vectors are parallel.
  • 4.11.B.3 The square matrix $A$ has an inverse if and only if $\det(A) \neq 0$.

来源:美国大学理事会 AP 课程与考试说明

一个 $2\times 2$ 矩阵 $\begin{bmatrix} a & b \\ c & d\end{bmatrix}$ 的行列式(determinant)是 $ad-bc$。一个矩阵恰好当它的行列式非零时有一个逆矩阵(inverse);逆矩阵撤销这个矩阵,而它解矩阵方程(像数字的一个倒数)。

Worked example. 对于 $A=\begin{bmatrix}3 & 1\\ 2 & 4\end{bmatrix}$,行列式是 $ad-bc=(3)(4)-(1)(2)=10$。因为它非零,$A$ 可逆,而 $A^{-1}=\dfrac{1}{10}\begin{bmatrix}4 & -1\\ -2 & 3\end{bmatrix}$(交换对角线、取反副对角线、除以行列式)。

词汇表 训练
英文 中文 拼音
determinant 行列式 háng liè shì
inverse 逆矩阵 nì jǔ zhèn
练习卷
4.12

线性变换与矩阵
大纲
Learning ObjectiveEssential Knowledge

4.12.A
Determine the output vectors of a linear transformation using a $2 \times 2$ matrix.

  • 4.12.A.1 A linear transformation is a function that maps an input vector to an output vector such that each component of the output vector is the sum of constant multiples of the input vector components.
  • 4.12.A.2 A linear transformation will map the zero vector to the zero vector.
  • 4.12.A.3 A single vector in $\mathbb{R}^2$ can be expressed as a $2 \times 1$ matrix. A set of $n$ vectors in $\mathbb{R}^2$ can be expressed as a $2 \times n$ matrix.
  • 4.12.A.4 For a linear transformation, $L$, from $\mathbb{R}^2$ to $\mathbb{R}^2$, there is a unique $2 \times 2$ matrix, $A$, such that $L(\vec{v}) = A\vec{v}$ for vectors in $\mathbb{R}^2$. Conversely, for a given $2 \times 2$ matrix, $A$, the function $L(\vec{v}) = A\vec{v}$ is a linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$.
  • 4.12.A.5 Multiplication of a $2 \times 2$ transformation matrix, $A$, and a $2 \times n$ matrix of $n$ input vectors gives a $2 \times n$ matrix of the $n$ output vectors for the linear transformation $L(\vec{v}) = A\vec{v}$.

来源:美国大学理事会 AP 课程与考试说明

矩阵变换平面

一个 $2\times 2$ 矩阵作为平面的一个线性变换(linear transformation)——旋转、反射、拉伸,或错切点——通过乘每个位置向量。行列式测量变换如何缩放面积(而它的符号告诉是否取向翻转)。

Worked example. 矩阵 $\begin{bmatrix}2 & 0\\ 0 & 3\end{bmatrix}$ 把平面水平拉伸 $2$、竖直 $3$,把向量 $\begin{bmatrix}1\\1\end{bmatrix}$ 送到 $\begin{bmatrix}2\\3\end{bmatrix}$。它的行列式 $2\times3=6$ 意味着每个面积被乘以 $6$,所以单位正方形变成一个 $2\times3$ 矩形。

一个 2x2 矩阵把单位正方形映射到一个平行四边形;行列式是面积缩放
一个 2x2 矩阵把单位正方形映射到一个平行四边形;行列式是面积缩放
探索

Transform the plane with a matrix

A $2\times2$ matrix maps every point to a new one, so it stretches, rotates or shears the whole grid. Change the entries and watch the unit square transform.

词汇表 训练
英文 中文 拼音
linear transformation 线性变换 xiàn xìng biàn huàn
练习卷
4.13

作为函数的矩阵
大纲
Learning ObjectiveEssential Knowledge

4.13.A
Determine the association between a linear transformation and a matrix.

  • 4.13.A.1 The linear transformation mapping $\langle x,\, y \rangle$ to $\langle a_{11} x + a_{12} y,\, a_{21} x + a_{22} y \rangle$ is associated with the matrix $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$.
  • 4.13.A.2 The mapping of the unit vectors in a linear transformation provides valuable information for determining the associated matrix.
  • 4.13.A.3 The matrix $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ is associated with a linear transformation of vectors that rotates every vector an angle $\theta$ counterclockwise about the origin.
  • 4.13.A.4 The absolute value of the determinant of a $2 \times 2$ transformation matrix gives the magnitude of the dilation of regions in $\mathbb{R}^2$ under the transformation.

4.13.B
Determine the composition of two linear transformations.

  • 4.13.B.1 The composition of two linear transformations is a linear transformation.
  • 4.13.B.2 The matrix associated with the composition of two linear transformations is the product of the matrices associated with each linear transformation.

4.13.C
Determine the inverse of a linear transformation.

  • 4.13.C.1 Two linear transformations are inverses if their composition maps any vector to itself.
  • 4.13.C.2 If a linear transformation, $L$, is given by $L(\vec{v}) = A\vec{v}$, then its inverse transformation is given by $L^{-1}(\vec{v}) = A^{-1}\vec{v}$, where $A^{-1}$ is the inverse of the matrix $A$.

来源:美国大学理事会 AP 课程与考试说明

因为一个矩阵把输入向量映射到输出向量,它是平面上的一个函数。复合变换对应于乘它们的矩阵,而逆矩阵反转这个映射。

练习卷
4.14

用矩阵为情境建模
大纲
Learning ObjectiveEssential Knowledge

4.14.A
Construct a model of a scenario involving transitions between two states using matrices.

  • 4.14.A.1 A contextual scenario can indicate the rate of transitions between states as percent changes. A matrix can be constructed based on these rates to model how states change over discrete intervals.

4.14.B
Apply matrix models to predict future and past states for $n$ transition steps.

  • 4.14.B.1 The product of a matrix that models transitions between states and a corresponding state vector can predict future states.
  • 4.14.B.2 Repeated multiplication of a matrix that models the transitions between states and corresponding resultant state vectors can predict the steady state, a distribution between states that does not change from one step to the next.
  • 4.14.B.3 The product of the inverse of a matrix that models transitions between states and a corresponding state vector can predict past states.

来源:美国大学理事会 AP 课程与考试说明

矩阵为分阶段向前步进的系统建模——例如,每年在状态之间移动的种群。用一个转移矩阵(transition matrix)的重复相乘一次一步地推进模型,所以矩阵幂预测长期行为。

练习卷
4.14

考试技巧
  • 一个参数函数分别给出 $x(t)$ 和 $y(t)$ ——随着 $t$ 增加追踪运动的方向。
  • 一个向量有大小和方向;逐分量地相加并用 $\sqrt{a^2+b^2}$ 求大小。
  • 一个 $2\times2$ 矩阵的行列式是 $ad-bc$(注意负号);它在变换下缩放面积。
  • 一个 $2\times2$ 矩阵变换平面(旋转、反射、拉伸);只有当行列式非零时逆矩阵存在。
  • 注意这个单元不在 AP 微积分预备考试中考查(只考第 1–3 单元),但它支撑后面的课程。

本主题的互动课程

逐步学习,并即时检测练习。

AP 微积分预备历年真题

AP 微积分预备的更多主题

登录或创建账号

IGCSE, A-Level & AP