学科

AP 微积分预备

AP 微积分预备涵盖多项式与有理函数、指数与对数函数、三角与极坐标函数,以及含参数、向量与矩阵的函数。它的目的是在微积分把"函数"当作既有知识之前,先把这一概念打牢。

这门课围绕一种习惯而非一份主题清单组织:对每一类函数,都要能从图像、数值、解析式和文字四个角度描述,并在这四种表示之间自如切换。考题正是据此设计——要求你根据表格为某个关于函数的结论辩护,或解释某个参数对图像的影响。

变化率贯穿全课,是通往微积分的直接桥梁,因此应当把它视为核心,而非众多主题之一。本站笔记按 College Board 的单元编排,每单元一页,每类函数的图像、表格与方程并列呈现。

AP 微积分预备历年真题

训练全部词汇

多项式函数与有理函数

单独打开本主题页面

讲义词汇表

1.1

协同变化

大纲

Learning Objective	Essential Knowledge
1.1.A Describe how the input and output values of a function vary together by comparing function values.	1.1.A.1 A function is a mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value. The set of input values is called the domain of the function, and the set of output values is called the range of the function. The variable representing input values is called the independent variable, and the variable representing output values is called the dependent variable. 1.1.A.2 The input and output values of a function vary in tandem according to the function rule, which can be expressed graphically, numerically, analytically, or verbally. 1.1.A.3 A function is increasing over an interval of its domain if, as the input values increase, the output values always increase. That is, for all $a$ and $b$ in the interval, if $a < b$, then $f(a) < f(b)$. 1.1.A.4 A function is decreasing over an interval of its domain if, as the input values increase, the output values always decrease. That is, for all $a$ and $b$ in the interval, if $a < b$, then $f(a) > f(b)$.
1.1.B Construct a graph representing two quantities that vary with respect to each other in a contextual scenario.	1.1.B.1 The graph of a function displays a set of input-output pairs and shows how the values of the function's input and output values vary. 1.1.B.2 A verbal description of the way aspects of phenomena change together can be the basis for constructing a graph. 1.1.B.3 The graph of a function is concave up on intervals in which the rate of change is increasing. 1.1.B.4 The graph of a function is concave down on intervals in which the rate of change is decreasing. 1.1.B.5 The graph intersects the $x$-axis when the output value is zero. The corresponding input values are said to be zeros of the function.

Learning Objective

Essential Knowledge

1.1.A
Describe how the input and output values of a function vary together by comparing function values.

1.1.A.1 A function is a mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value. The set of input values is called the domain of the function, and the set of output values is called the range of the function. The variable representing input values is called the independent variable, and the variable representing output values is called the dependent variable.
1.1.A.2 The input and output values of a function vary in tandem according to the function rule, which can be expressed graphically, numerically, analytically, or verbally.
1.1.A.3 A function is increasing over an interval of its domain if, as the input values increase, the output values always increase. That is, for all $a$ and $b$ in the interval, if $a < b$, then $f(a) < f(b)$.
1.1.A.4 A function is decreasing over an interval of its domain if, as the input values increase, the output values always decrease. That is, for all $a$ and $b$ in the interval, if $a < b$, then $f(a) > f(b)$.

1.1.B
Construct a graph representing two quantities that vary with respect to each other in a contextual scenario.

1.1.B.1 The graph of a function displays a set of input-output pairs and shows how the values of the function's input and output values vary.
1.1.B.2 A verbal description of the way aspects of phenomena change together can be the basis for constructing a graph.
1.1.B.3 The graph of a function is concave up on intervals in which the rate of change is increasing.
1.1.B.4 The graph of a function is concave down on intervals in which the rate of change is decreasing.
1.1.B.5 The graph intersects the $x$-axis when the output value is zero. The corresponding input values are said to be zeros of the function.

来源:美国大学理事会 AP 课程与考试说明

一个函数(function)是一个把每个输入映射到恰好一个输出的规则。允许的输入的集合是定义域(domain);输出的集合是值域(range)。输入变量是自变量(independent variable)而输出变量是因变量(dependent variable)。一个函数规则能用图象、数值、解析式(一个公式),或文字展示。

随着输入变化,输出"齐头并进"地——一起——变化。在一个区间上,一个函数是:

递增(increasing),若每当 $a 时,那么 $f(a)(更大的输入、更大的输出);
递减(decreasing),若每当 $a 时,那么 $f(a)>f(b)$(更大的输入、更小的输出)。

一个函数的图象显示它所有的输入-输出对,所以你能从图片直接读这个行为。

探索

Watch two quantities change together

y = ax² + bx + c

Precalculus studies how an output changes in tandem with its input. Read the curve left to right: where it is steep, $y$ changes a lot for a small change in $x$; where it is flat, $y$ barely moves.

词汇表训练

英文	中文	拼音
function	函数	hán shù
domain	定义域	dìng yì yù
range	值域	zhí yù
independent variable	自变量	zì biàn liàng
dependent variable	因变量	yīn biàn liàng
increasing	递增	dì zēng
decreasing	递减	dì jiǎn

练习卷

1.2

变化率

大纲

Learning Objective	Essential Knowledge
1.2.A Compare the rates of change at two points using average rates of change near the points.	1.2.A.1 The average rate of change of a function over an interval of the function's domain is the constant rate of change that yields the same change in the output values as the function yielded on that interval of the function's domain. It is the ratio of the change in the output values to the change in input values over that interval. 1.2.A.2 The rate of change of a function at a point quantifies the rate at which output values would change were the input values to change at that point. The rate of change at a point can be approximated by the average rates of change of the function over small intervals containing the point, if such values exist. 1.2.A.3 The rates of change at two points can be compared using average rate of change approximations over sufficiently small intervals containing each point, if such values exist.
1.2.B Describe how two quantities vary together at different points and over different intervals of a function.	1.2.B.1 Rates of change quantify how two quantities vary together. 1.2.B.2 A positive rate of change indicates that as one quantity increases or decreases, the other quantity does the same. 1.2.B.3 A negative rate of change indicates that as one quantity increases, the other decreases.

Learning Objective

Essential Knowledge

1.2.A
Compare the rates of change at two points using average rates of change near the points.

1.2.A.1 The average rate of change of a function over an interval of the function's domain is the constant rate of change that yields the same change in the output values as the function yielded on that interval of the function's domain. It is the ratio of the change in the output values to the change in input values over that interval.
1.2.A.2 The rate of change of a function at a point quantifies the rate at which output values would change were the input values to change at that point. The rate of change at a point can be approximated by the average rates of change of the function over small intervals containing the point, if such values exist.
1.2.A.3 The rates of change at two points can be compared using average rate of change approximations over sufficiently small intervals containing each point, if such values exist.

1.2.B
Describe how two quantities vary together at different points and over different intervals of a function.

1.2.B.1 Rates of change quantify how two quantities vary together.
1.2.B.2 A positive rate of change indicates that as one quantity increases or decreases, the other quantity does the same.
1.2.B.3 A negative rate of change indicates that as one quantity increases, the other decreases.

来源:美国大学理事会 AP 课程与考试说明

一个函数在一个区间上的平均变化率(average rate of change)是输出的变化除以输入的变化——会给出相同总变化的单个恒定的率。在 $[a,b]$ 上:

$$\text{avg rate} = \frac{f(b)-f(a)}{b-a}.$$

一个点处的变化率(rate of change at a point)测量输出恰好在那个输入处变化多快。你用点周围小区间上的平均率近似它。比较两个点,有更大平均率(在足够小的区间上)的那个变化更快。一个正的率意味着两个量朝相同方向移动;一个负的率意味着它们朝相反方向移动。

Worked example. 对于 $f(x)=x^2$,在 $[1,4]$ 上的平均变化率是 $\dfrac{f(4)-f(1)}{4-1}=\dfrac{16-1}{3}=5$。在 $[1,2]$ 上它是 $\dfrac{4-1}{1}=3$ ——率本身变化,这正是为什么 $f$ 不是线性的。

词汇表训练

英文	中文	拼音
average rate of change	平均变化率	píng jūn biàn huà lǜ

练习卷

1.3

线性函数与二次函数的变化率

大纲

Learning Objective	Essential Knowledge
1.3.A Determine the average rates of change for sequences and functions, including linear, quadratic, and other function types.	1.3.A.1 For a linear function, the average rate of change over any length input-value interval is constant. 1.3.A.2 For a quadratic function, the average rates of change over consecutive equal-length input-value intervals can be given by a linear function. 1.3.A.3 The average rate of change over the closed interval $[a, b]$ is the slope of the secant line from the point $(a, f(a))$ to $(b, f(b))$.
1.3.B Determine the change in the average rates of change for linear, quadratic, and other function types.	1.3.B.1 For a linear function, since the average rates of change over consecutive equal-length input-value intervals can be given by a constant function, these average rates of change for a linear function are changing at a rate of zero. 1.3.B.2 For a quadratic function, since the average rates of change over consecutive equal-length input-value intervals can be given by a linear function, these average rates of change for a quadratic function are changing at a constant rate. 1.3.B.3 When the average rate of change over equal-length input-value intervals is increasing for all small-length intervals, the graph of the function is concave up. When the average rate of change over equal-length input-value intervals is decreasing for all small-length intervals, the graph of the function is concave down.

Learning Objective

Essential Knowledge

1.3.A
Determine the average rates of change for sequences and functions, including linear, quadratic, and other function types.

1.3.A.1 For a linear function, the average rate of change over any length input-value interval is constant.
1.3.A.2 For a quadratic function, the average rates of change over consecutive equal-length input-value intervals can be given by a linear function.
1.3.A.3 The average rate of change over the closed interval $[a, b]$ is the slope of the secant line from the point $(a, f(a))$ to $(b, f(b))$.

1.3.B
Determine the change in the average rates of change for linear, quadratic, and other function types.

1.3.B.1 For a linear function, since the average rates of change over consecutive equal-length input-value intervals can be given by a constant function, these average rates of change for a linear function are changing at a rate of zero.
1.3.B.2 For a quadratic function, since the average rates of change over consecutive equal-length input-value intervals can be given by a linear function, these average rates of change for a quadratic function are changing at a constant rate.
1.3.B.3 When the average rate of change over equal-length input-value intervals is increasing for all small-length intervals, the graph of the function is concave up. When the average rate of change over equal-length input-value intervals is decreasing for all small-length intervals, the graph of the function is concave down.

来源:美国大学理事会 AP 课程与考试说明

在 $[a,b]$ 上的平均变化率是从 $(a,f(a))$ 到 $(b,f(b))$ 的割线(secant line)的斜率。

对于一个线性(linear)函数,在任何区间上的平均变化率是恒定的——所以率变化的率是零。
对于一个二次(quadratic)函数,在等长区间上的平均变化率本身形成一个线性模式——所以那些率以一个恒定的率变化。

这个"率的率"思想区别函数类型:一个恒定的二阶差标志一个二次。

探索

Explore a quadratic's changing rate

y = ax² + bx + c

Move the sliders and watch the parabola 抛物线 tilt. Its average rate of change is not constant — the slope is positive on one side of the vertex 顶点 and negative on the other. Notice how $a$ opens it up or down.

词汇表训练

英文	中文	拼音
secant line	割线	gē xiàn
linear	线性	xiàn xìng
quadratic	二次	èr cì

练习卷

1.4

多项式函数与变化率

大纲

Learning Objective	Essential Knowledge
1.4.A Identify key characteristics of polynomial functions related to rates of change.	1.4.A.1 A nonconstant polynomial function of $x$ is any function representation that is equivalent to the analytical form $p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_2 x^2 + a_1 x + a_0$, where $n$ is a positive integer, $a_i$ is a real number for each $i$ from $1$ to $n$, and $a_n$ is nonzero. The polynomial has degree $n$, the leading term is $a_n x^n$, and the leading coefficient is $a_n$. A constant is also a polynomial function of degree zero. 1.4.A.2 Where a polynomial function switches between increasing and decreasing, or at the included endpoint of a polynomial with a restricted domain, the polynomial function will have a local, or relative, maximum or minimum output value. Of all local maxima, the greatest is called the global, or absolute, maximum. Likewise, the least of all local minima is called the global, or absolute, minimum. 1.4.A.3 Between every two distinct real zeros of a nonconstant polynomial function, there must be at least one input value corresponding to a local maximum or local minimum. 1.4.A.4 Polynomial functions of an even degree will have either a global maximum or a global minimum. 1.4.A.5 Points of inflection of a polynomial function occur at input values where the rate of change of the function changes from increasing to decreasing or from decreasing to increasing. This occurs where the graph of a polynomial function changes from concave up to concave down or from concave down to concave up.

Learning Objective

Essential Knowledge

1.4.A
Identify key characteristics of polynomial functions related to rates of change.

1.4.A.1 A nonconstant polynomial function of $x$ is any function representation that is equivalent to the analytical form $p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_2 x^2 + a_1 x + a_0$, where $n$ is a positive integer, $a_i$ is a real number for each $i$ from $1$ to $n$, and $a_n$ is nonzero. The polynomial has degree $n$, the leading term is $a_n x^n$, and the leading coefficient is $a_n$. A constant is also a polynomial function of degree zero.
1.4.A.2 Where a polynomial function switches between increasing and decreasing, or at the included endpoint of a polynomial with a restricted domain, the polynomial function will have a local, or relative, maximum or minimum output value. Of all local maxima, the greatest is called the global, or absolute, maximum. Likewise, the least of all local minima is called the global, or absolute, minimum.
1.4.A.3 Between every two distinct real zeros of a nonconstant polynomial function, there must be at least one input value corresponding to a local maximum or local minimum.
1.4.A.4 Polynomial functions of an even degree will have either a global maximum or a global minimum.
1.4.A.5 Points of inflection of a polynomial function occur at input values where the rate of change of the function changes from increasing to decreasing or from decreasing to increasing. This occurs where the graph of a polynomial function changes from concave up to concave down or from concave down to concave up.

来源:美国大学理事会 AP 课程与考试说明

一个非常数多项式(polynomial)有形式

$$p(x)=a_n x^n + a_{n-1}x^{n-1}+\cdots+a_1 x + a_0,\quad a_n\neq 0.$$

它的次数(degree)是 $n$、它的首项(leading term)是 $a_n x^n$,而它的首项系数(leading coefficient)是 $a_n$。一个常数是一个 $0$ 次的多项式。

关键特征:

一个多项式在递增和递减之间切换的地方,它有一个局部(相对)极值(local (relative) extremum)。只有当多项式确实有全局极值时,最大的局部极大值才是全局(绝对)最大值(global (absolute) maximum)——但大多数多项式是无界的(每个奇次多项式都会跑向 $\pm\infty$),所以它们根本没有全局最大值或最小值。
在任何两个不同的实零点之间有至少一个局部极大值或极小值。
一个偶次(even-degree)多项式有一个全局最大值或一个全局最小值。
一个拐点(point of inflection)是图象改变凹凸性的地方——从上凹(concave up)到下凹(concave down)或反过来——即变化率在递增和递减之间切换的地方。
一个函数有对称性,当它是偶函数(even)时——$f(-x)=f(x)$,图象关于 $y$ 轴对称(像 $x^2$、$x^4$)——或是奇函数(odd)时——$f(-x)=-f(x)$,图象关于原点旋转对称(像 $x^3$、$x^5$)。用代入 $-x$ 并化简来检验;后面,$\cos\theta$ 是偶的而 $\sin\theta$ 是奇的。

词汇表训练

英文	中文	拼音
polynomial	多项式	duō xiàng shì
degree	次数	cì shù
leading term	首项	shǒu xiàng
leading coefficient	首项系数	shǒu xiàng xì shù
local (relative) extremum	局部极值	jú bù jí zhí
global (absolute) maximum	全局最大值	quán jú zuì dà zhí
point of inflection	拐点	guǎi diǎn
concave up	上凹	shàng āo
concave down	下凹	xià āo
even	偶函数	ǒu hán shù
odd	奇函数	jī hán shù

练习卷

1.5

多项式函数与复数零点

大纲

Learning Objective	Essential Knowledge
1.5.A Identify key characteristics of a polynomial function related to its zeros when suitable factorizations are available or with technology.	1.5.A.1 If $a$ is a complex number and $p(a) = 0$, then $a$ is called a zero of the polynomial function $p$, or a root of $p(x) = 0$. If $a$ is a real number, then $(x - a)$ is a linear factor of $p$ if and only if $a$ is a zero of $p$. 1.5.A.2 If a linear factor $(x - a)$ is repeated $n$ times, the corresponding zero of the polynomial function has a multiplicity $n$. A polynomial function of degree $n$ has exactly $n$ complex zeros when counting multiplicities. 1.5.A.3 If $a$ is a real zero of a polynomial function $p$, then the graph of $y = p(x)$ has an $x$-intercept at the point $(a, 0)$. Consequently, real zeros of a polynomial can be endpoints for intervals satisfying polynomial inequalities. 1.5.A.4 If $a + bi$ is a non-real zero of a polynomial function $p$, then its conjugate $a - bi$ is also a zero of $p$. 1.5.A.5 If the real zero, $a$, of a polynomial function has even multiplicity, then the signs of the output values are the same for input values near $x = a$. For these polynomial functions, the graph will be tangent to the $x$-axis at $x = a$. 1.5.A.6 The degree of a polynomial function can be found by examining the successive differences of the output values over equal-interval input values. The degree of the polynomial function is equal to the least value $n$ for which the successive $n$th differences are constant.
1.5.B Determine if a polynomial function is even or odd.	1.5.B.1 An even function is graphically symmetric over the line $x = 0$ and analytically has the property $f(-x) = f(x)$. If $n$ is even, then a polynomial of the form $p(x) = a_n x^n$, where $n \geq 1$ and $a_n \neq 0$, is an even function. 1.5.B.2 An odd function is graphically symmetric about the point $(0, 0)$ and analytically has the property $f(-x) = -f(x)$. If $n$ is odd, then a polynomial of the form $p(x) = a_n x^n$, where $n \geq 1$ and $a_n \neq 0$, is an odd function.

Learning Objective

Essential Knowledge

1.5.A
Identify key characteristics of a polynomial function related to its zeros when suitable factorizations are available or with technology.

1.5.A.1 If $a$ is a complex number and $p(a) = 0$, then $a$ is called a zero of the polynomial function $p$, or a root of $p(x) = 0$. If $a$ is a real number, then $(x - a)$ is a linear factor of $p$ if and only if $a$ is a zero of $p$.
1.5.A.2 If a linear factor $(x - a)$ is repeated $n$ times, the corresponding zero of the polynomial function has a multiplicity $n$. A polynomial function of degree $n$ has exactly $n$ complex zeros when counting multiplicities.
1.5.A.3 If $a$ is a real zero of a polynomial function $p$, then the graph of $y = p(x)$ has an $x$-intercept at the point $(a, 0)$. Consequently, real zeros of a polynomial can be endpoints for intervals satisfying polynomial inequalities.
1.5.A.4 If $a + bi$ is a non-real zero of a polynomial function $p$, then its conjugate $a - bi$ is also a zero of $p$.
1.5.A.5 If the real zero, $a$, of a polynomial function has even multiplicity, then the signs of the output values are the same for input values near $x = a$. For these polynomial functions, the graph will be tangent to the $x$-axis at $x = a$.
1.5.A.6 The degree of a polynomial function can be found by examining the successive differences of the output values over equal-interval input values. The degree of the polynomial function is equal to the least value $n$ for which the successive $n$th differences are constant.

1.5.B
Determine if a polynomial function is even or odd.

1.5.B.1 An even function is graphically symmetric over the line $x = 0$ and analytically has the property $f(-x) = f(x)$. If $n$ is even, then a polynomial of the form $p(x) = a_n x^n$, where $n \geq 1$ and $a_n \neq 0$, is an even function.
1.5.B.2 An odd function is graphically symmetric about the point $(0, 0)$ and analytically has the property $f(-x) = -f(x)$. If $n$ is odd, then a polynomial of the form $p(x) = a_n x^n$, where $n \geq 1$ and $a_n \neq 0$, is an odd function.

来源:美国大学理事会 AP 课程与考试说明

$p$ 的一个零点(zero)(或根(root))是一个 $p(a)=0$ 的数 $a$。对于一个实 $a$,$(x-a)$ 恰好当 $a$ 是一个零点时是 $p$ 的一个因式。若因式 $(x-a)$ 被重复 $n$ 次,那个零点有重数(multiplicity)$n$。计入重数,一个 $n$ 次多项式恰好有 $n$ 个复数(complex)零点。

一个实零点 $a$ 在 $(a,0)$ 给出一个 $x$ 截距;实零点是 $p(x)\ge 0$ 或 $\le 0$ 的区间的端点。
非实零点成共轭(conjugate)对:若 $a+bi$ 是一个零点,$a-bi$ 也是。
在一个偶重数的实零点处图象与 $x$ 轴相切(tangent)(它触及但不穿越);在奇重数它穿越。
次数等于等距输出的第 $n$ 个逐次差(successive differences)变得恒定的最小 $n$。

词汇表训练

英文	中文	拼音
zero	零点	líng diǎn
root	根	gēn
multiplicity	重数	chóng shù
complex	复数	fù shù
conjugate	共轭	gòng è

练习卷

1.6

多项式函数与末端行为

大纲

Learning Objective	Essential Knowledge
1.6.A Describe end behaviors of polynomial functions.	1.6.A.1 As input values of a nonconstant polynomial function increase without bound, the output values will either increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to \infty} p(x) = \infty$ or $\lim_{x \to \infty} p(x) = -\infty$. 1.6.A.2 As input values of a nonconstant polynomial function decrease without bound, the output values will either increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to -\infty} p(x) = \infty$ or $\lim_{x \to -\infty} p(x) = -\infty$. 1.6.A.3 The degree and sign of the leading term of a polynomial determines the end behavior of the polynomial function, because as the input values increase or decrease without bound, the values of the leading term dominate the values of all lower-degree terms.

Learning Objective

Essential Knowledge

1.6.A
Describe end behaviors of polynomial functions.

1.6.A.1 As input values of a nonconstant polynomial function increase without bound, the output values will either increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to \infty} p(x) = \infty$ or $\lim_{x \to \infty} p(x) = -\infty$.
1.6.A.2 As input values of a nonconstant polynomial function decrease without bound, the output values will either increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to -\infty} p(x) = \infty$ or $\lim_{x \to -\infty} p(x) = -\infty$.
1.6.A.3 The degree and sign of the leading term of a polynomial determines the end behavior of the polynomial function, because as the input values increase or decrease without bound, the values of the leading term dominate the values of all lower-degree terms.

来源:美国大学理事会 AP 课程与考试说明

末端行为(end behavior)描述随着输入无界地增长一个函数走向哪里。对于一个非常数多项式,随着 $x\to\pm\infty$ 输出走向 $+\infty$ 或 $-\infty$,例如写 $\lim_{x\to\infty}p(x)=\infty$。哪个方向完全取决于首项 $a_n x^n$,因为对于大的 $|x|$ 它主导所有低次的项:$a_n$ 的符号以及 $n$ 是偶还是奇固定两端。

探索

See how the leading term sets the end behavior

y = ax³ + bx² + cx + d

The leading term decides the ends. For a cubic with $a>0$ the graph falls on the left and rises on the right; make $a<0$ and the two ends swap. As $x\to\pm\infty$ the highest-power term dominates every other term.

词汇表训练

英文	中文	拼音
End behavior	末端行为	mò duān xíng wéi

练习卷

1.7

有理函数与末端行为

大纲

Learning Objective	Essential Knowledge
1.7.A Describe end behaviors of rational functions.	1.7.A.1 A rational function is analytically represented as a quotient of two polynomial functions and gives a measure of the relative size of the polynomial function in the numerator compared to the polynomial function in the denominator for each value in the rational function's domain. 1.7.A.2 The end behavior of a rational function will be affected most by the polynomial with the greater degree, as its values will dominate the values of the rational function for input values of large magnitude. For input values of large magnitude, a polynomial is dominated by its leading term. Therefore, the end behavior of a rational function can be understood by examining the corresponding quotient of the leading terms. 1.7.A.3 If the polynomial in the numerator dominates the polynomial in the denominator for input values of large magnitude, then the quotient of the leading terms is a nonconstant polynomial, and the original rational function has the end behavior of that polynomial. If that polynomial is linear, then the graph of the rational function has a slant asymptote parallel to the graph of the line. 1.7.A.4 If neither polynomial in a rational function dominates the other for input values of large magnitude, then the quotient of the leading terms is a constant, and that constant indicates the location of a horizontal asymptote of the graph of the original rational function. 1.7.A.5 If the polynomial in the denominator dominates the polynomial in the numerator for input values of large magnitude, then the quotient of the leading terms is a rational function with a constant in the numerator and nonconstant polynomial in the denominator, and the graph of the original rational function has a horizontal asymptote at $y = 0$. 1.7.A.6 When the graph of a rational function $r$ has a horizontal asymptote at $y = b$, where $b$ is a constant, the output values of the rational function get arbitrarily close to $b$ and stay arbitrarily close to $b$ as input values increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to \infty} r(x) = b$ or $\lim_{x \to -\infty} r(x) = b$.

Learning Objective

Essential Knowledge

1.7.A
Describe end behaviors of rational functions.

1.7.A.1 A rational function is analytically represented as a quotient of two polynomial functions and gives a measure of the relative size of the polynomial function in the numerator compared to the polynomial function in the denominator for each value in the rational function's domain.
1.7.A.2 The end behavior of a rational function will be affected most by the polynomial with the greater degree, as its values will dominate the values of the rational function for input values of large magnitude. For input values of large magnitude, a polynomial is dominated by its leading term. Therefore, the end behavior of a rational function can be understood by examining the corresponding quotient of the leading terms.
1.7.A.3 If the polynomial in the numerator dominates the polynomial in the denominator for input values of large magnitude, then the quotient of the leading terms is a nonconstant polynomial, and the original rational function has the end behavior of that polynomial. If that polynomial is linear, then the graph of the rational function has a slant asymptote parallel to the graph of the line.
1.7.A.4 If neither polynomial in a rational function dominates the other for input values of large magnitude, then the quotient of the leading terms is a constant, and that constant indicates the location of a horizontal asymptote of the graph of the original rational function.
1.7.A.5 If the polynomial in the denominator dominates the polynomial in the numerator for input values of large magnitude, then the quotient of the leading terms is a rational function with a constant in the numerator and nonconstant polynomial in the denominator, and the graph of the original rational function has a horizontal asymptote at $y = 0$.
1.7.A.6 When the graph of a rational function $r$ has a horizontal asymptote at $y = b$, where $b$ is a constant, the output values of the rational function get arbitrarily close to $b$ and stay arbitrarily close to $b$ as input values increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to \infty} r(x) = b$ or $\lim_{x \to -\infty} r(x) = b$.

来源:美国大学理事会 AP 课程与考试说明

一个有理函数(rational function)是两个多项式的一个商,$r(x)=\dfrac{\text{numerator}}{\text{denominator}}$。它的末端行为由首项的商支配:

分子次数 > 分母次数: 商是一个非常数多项式,而 $r$ 遵循那个多项式的末端行为。若那个商是线性的,图象有一条斜渐近线(slant asymptote)。
相等的次数: 商是一个常数,给出一条水平渐近线(horizontal asymptote)$y=$ 首项系数之比。
分子次数 < 分母次数: 商趋于 $0$,所以水平渐近线是 $y=0$。

在一条水平渐近线 $y=b$ 处,输出得到并保持任意地接近 $b$:$\lim_{x\to\pm\infty}r(x)=b$。

词汇表训练

英文	中文	拼音
rational function	有理函数	yǒu lǐ hán shù
slant asymptote	斜渐近线	xié jiàn jìn xiàn
horizontal asymptote	水平渐近线	shuǐ píng jiàn jìn xiàn

练习卷

1.8

有理函数与零点

大纲

Learning Objective	Essential Knowledge
1.8.A Determine the zeros of rational functions.	1.8.A.1 The real zeros of a rational function correspond to the real zeros of the numerator for such values in its domain. 1.8.A.2 The real zeros of both polynomial functions of a rational function $r$ are endpoints or asymptotes for intervals satisfying the rational function inequalities $r(x) \geq 0$ or $r(x) \leq 0$.

来源:美国大学理事会 AP 课程与考试说明

一个有理函数的实零点是它分子仍然在定义域里的实零点。这些零点,连同分母的零点,把数轴分成你在解不等式 $r(x)\ge 0$ 或 $r(x)\le 0$ 时测试的区间。

练习卷

1.9

有理函数与竖直渐近线

大纲

Learning Objective	Essential Knowledge
1.9.A Determine vertical asymptotes of graphs of rational functions.	1.9.A.1 If the value $a$ is a real zero of the polynomial function in the denominator of a rational function and is not also a real zero of the polynomial function in the numerator, then the graph of the rational function has a vertical asymptote at $x = a$. Furthermore, a vertical asymptote also occurs at $x = a$ if the multiplicity of $a$ as a real zero in the denominator is greater than its multiplicity as a real zero in the numerator. 1.9.A.2 Near a vertical asymptote, $x = a$, of a rational function, the values of the polynomial function in the denominator are arbitrarily close to zero, so the values of the rational function $r$ increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to a^+} r(x) = \infty$ or $\lim_{x \to a^+} r(x) = -\infty$ for input values near $a$ and greater than $a$, and $\lim_{x \to a^-} r(x) = \infty$ or $\lim_{x \to a^-} r(x) = -\infty$ for input values near $a$ and less than $a$.

Learning Objective

Essential Knowledge

1.9.A
Determine vertical asymptotes of graphs of rational functions.

1.9.A.1 If the value $a$ is a real zero of the polynomial function in the denominator of a rational function and is not also a real zero of the polynomial function in the numerator, then the graph of the rational function has a vertical asymptote at $x = a$. Furthermore, a vertical asymptote also occurs at $x = a$ if the multiplicity of $a$ as a real zero in the denominator is greater than its multiplicity as a real zero in the numerator.
1.9.A.2 Near a vertical asymptote, $x = a$, of a rational function, the values of the polynomial function in the denominator are arbitrarily close to zero, so the values of the rational function $r$ increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to a^+} r(x) = \infty$ or $\lim_{x \to a^+} r(x) = -\infty$ for input values near $a$ and greater than $a$, and $\lim_{x \to a^-} r(x) = \infty$ or $\lim_{x \to a^-} r(x) = -\infty$ for input values near $a$ and less than $a$.

来源:美国大学理事会 AP 课程与考试说明

一条垂直渐近线(vertical asymptote)在 $x=a$ 出现,当 $a$ 是分母的一个零点但不被分子约去时——更精确地,当它在分母里的重数超过它在分子里的重数时。它附近,分母近乎零,所以 $r$ 射向 $\pm\infty$:$\lim_{x\to a^{\pm}}r(x)=\pm\infty$。检查每一侧,因为两侧能走相反的方向。

Worked example. 描述 $r(x)=\dfrac{2x^2+3}{x^2-1}$。分子和分母有相等的次数,所以水平渐近线是 $y=\dfrac{2}{1}=2$。分母 $x^2-1$ 在 $x=\pm1$ 为零而两者都不约去,所以在 $x=1$ 和 $x=-1$ 有垂直渐近线。

探索

Explore a vertical asymptote

y = a/(x − b) + c

This is $y = \dfrac{a}{x-b} + c$. The graph shoots off toward $\pm\infty$ at the vertical asymptote 竖直渐近线 $x = b$ (where the bottom is zero) and levels off toward the horizontal asymptote 水平渐近线 $y = c$. Slide $b$ and $c$ to move each line.

词汇表训练

英文	中文	拼音
vertical asymptote	垂直渐近线	chuí zhí jiàn jìn xiàn

练习卷

1.10

有理函数与空洞

大纲

Learning Objective	Essential Knowledge
1.10.A Determine holes in graphs of rational functions.	1.10.A.1 If the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator, then the graph of the rational function has a hole at the corresponding input value. 1.10.A.2 If the graph of a rational function $r$ has a hole at $x = c$, then the location of the hole can be determined by examining the output values corresponding to input values sufficiently close to $c$. If input values sufficiently close to $c$ correspond to output values arbitrarily close to $L$, then the hole is located at the point with coordinates $(c, L)$. The corresponding mathematical notation is $\lim_{x \to c} r(x) = L$. It should be noted that $\lim_{x \to c^-} r(x) = \lim_{x \to c^+} r(x) = \lim_{x \to c} r(x) = L$.

Learning Objective

Essential Knowledge

1.10.A
Determine holes in graphs of rational functions.

1.10.A.1 If the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator, then the graph of the rational function has a hole at the corresponding input value.
1.10.A.2 If the graph of a rational function $r$ has a hole at $x = c$, then the location of the hole can be determined by examining the output values corresponding to input values sufficiently close to $c$. If input values sufficiently close to $c$ correspond to output values arbitrarily close to $L$, then the hole is located at the point with coordinates $(c, L)$. The corresponding mathematical notation is $\lim_{x \to c} r(x) = L$. It should be noted that $\lim_{x \to c^-} r(x) = \lim_{x \to c^+} r(x) = \lim_{x \to c} r(x) = L$.

来源:美国大学理事会 AP 课程与考试说明

一个空洞(hole)(可去点)在 $x=c$ 出现,当一个因式约去时——分子里零点 $c$ 的重数至少是它在分母里的重数。图象缺失一个单一的点。它的高度是化简后函数的极限:若 $c$ 附近的输入给出 $L$ 附近的输出,空洞在 $(c,L)$,而 $\lim_{x\to c}r(x)=L$。

词汇表训练

英文	中文	拼音
hole	空洞	kōng dòng

练习卷

1.11

多项式与有理表达式的等价表示

大纲

Learning Objective	Essential Knowledge
1.11.A Rewrite polynomial and rational expressions in equivalent forms.	1.11.A.1 Because the factored form of a polynomial or rational function readily provides information about real zeros, it can reveal information about $x$-intercepts, asymptotes, holes, domain, and range. 1.11.A.2 The standard form of a polynomial or rational function can reveal information about end behaviors of the function. 1.11.A.3 The information extracted from different analytic representations of the same polynomial or rational function can be used to answer questions in context.
1.11.B Determine the quotient of two polynomial functions using long division.	1.11.B.1 Polynomial long division is an algebraic process similar to numerical long division involving a quotient and remainder. If the polynomial $f$ is divided by the polynomial $g$, then $f$ can be rewritten as $f(x) = g(x)q(x) + r(x)$, where $q$ is the quotient, $r$ is the remainder, and the degree of $r$ is less than the degree of $g$. 1.11.B.2 The result of polynomial long division is helpful in finding equations of slant asymptotes for graphs of rational functions.
1.11.C Rewrite the repeated product of binomials using the binomial theorem.	1.11.C.1 The binomial theorem utilizes the entries in a single row of Pascal's Triangle to more easily expand expressions of the form $(a + b)^n$, including polynomial functions of the form $p(x) = (x + c)^n$, where $c$ is a constant.

来源:美国大学理事会 AP 课程与考试说明

同一个表达式,以不同的方式写,揭示不同的特征:

因式形式显示实零点,因而 $x$ 截距、空洞、垂直渐近线和定义域。
标准形式(展开)显示次数和首项,因而末端行为。

多项式长除法(polynomial long division)重写 $f(x)=g(x)\,q(x)+r(x)$,其中 $q$ 是商(quotient)而 $r$ 是余数(remainder)(次数小于 $g$)。当分子的次数比分母的大一时,商给出一条斜渐近线的方程。

词汇表训练

英文	中文	拼音
Polynomial long division	多项式长除法	duō xiàng shì zhǎng chú fǎ
quotient	商	shāng
remainder	余数	yú shù

练习卷

1.12

函数的变换

大纲

Learning Objective	Essential Knowledge
1.12.A Construct a function that is an additive and/or multiplicative transformation of another function.	1.12.A.1 The function $g(x) = f(x) + k$ is an additive transformation of the function $f$ that results in a vertical translation of the graph of $f$ by $k$ units. 1.12.A.2 The function $g(x) = f(x + h)$ is an additive transformation of the function $f$ that results in a horizontal translation of the graph of $f$ by $-h$ units. 1.12.A.3 The function $g(x) = a f(x)$, where $a \neq 0$, is a multiplicative transformation of the function $f$ that results in a vertical dilation of the graph of $f$ by a factor of $\|a\|$. If $a < 0$, the transformation involves a reflection over the $x$-axis. 1.12.A.4 The function $g(x) = f(bx)$, where $b \neq 0$, is a multiplicative transformation of the function $f$ that results in a horizontal dilation of the graph of $f$ by a factor of $\left\| \dfrac{1}{b} \right\|$. If $b < 0$, the transformation involves a reflection over the $y$-axis. 1.12.A.5 Additive and multiplicative transformations can be combined, resulting in combinations of horizontal and vertical translations and dilations. 1.12.A.6 The domain and range of a function that is a transformation of a parent function may be different from those of the parent function.

Learning Objective

Essential Knowledge

1.12.A
Construct a function that is an additive and/or multiplicative transformation of another function.

1.12.A.1 The function $g(x) = f(x) + k$ is an additive transformation of the function $f$ that results in a vertical translation of the graph of $f$ by $k$ units.
1.12.A.2 The function $g(x) = f(x + h)$ is an additive transformation of the function $f$ that results in a horizontal translation of the graph of $f$ by $-h$ units.
1.12.A.3 The function $g(x) = a f(x)$, where $a \neq 0$, is a multiplicative transformation of the function $f$ that results in a vertical dilation of the graph of $f$ by a factor of $|a|$. If $a < 0$, the transformation involves a reflection over the $x$-axis.
1.12.A.4 The function $g(x) = f(bx)$, where $b \neq 0$, is a multiplicative transformation of the function $f$ that results in a horizontal dilation of the graph of $f$ by a factor of $\left| \dfrac{1}{b} \right|$. If $b < 0$, the transformation involves a reflection over the $y$-axis.
1.12.A.5 Additive and multiplicative transformations can be combined, resulting in combinations of horizontal and vertical translations and dilations.
1.12.A.6 The domain and range of a function that is a transformation of a parent function may be different from those of the parent function.

来源:美国大学理事会 AP 课程与考试说明

函数图象的变换

一个变换(transformation)从一个旧函数 $f$ 构建一个新函数:

平移(translations)(移动):$f(x)+k$ 上/下移;$f(x-h)$ 右/左移。
伸缩(dilations)(拉伸):$a\,f(x)$ 竖直拉伸;$f(bx)$ 水平拉伸。
反射(reflections):$-f(x)$ 翻过 $x$ 轴;$f(-x)$ 翻过 $y$ 轴。

组合加性移动和乘性拉伸以为一个已知形状的移动、缩放版本建模。

探索

Explore shifts, stretches, and flips

Choose translate, reflect, rotate, or enlarge and change the amount. Watch which features stay the same — a transformation 变换 moves the whole graph without changing its underlying rule.

词汇表训练

英文	中文	拼音
transformation	变换	biàn huàn
Translations	平移	píng yí
Dilations	伸缩	shēn suō
Reflections	反射	fǎn shè

练习卷

1.13

函数模型选择与假设阐述

大纲

Learning Objective	Essential Knowledge
1.13.A Identify an appropriate function type to construct a function model for a given scenario.	1.13.A.1 Linear functions model data sets or aspects of contextual scenarios that demonstrate roughly constant rates of change. 1.13.A.2 Quadratic functions model data sets or aspects of contextual scenarios that demonstrate roughly linear rates of change, or data sets that are roughly symmetric with a unique maximum or minimum value. 1.13.A.3 Geometric contexts involving area or two dimensions can often be modeled by quadratic functions. Geometric contexts involving volume or three dimensions can often be modeled by cubic functions. 1.13.A.4 Polynomial functions model data sets or contextual scenarios with multiple real zeros or multiple maxima or minima. 1.13.A.5 A polynomial function of degree $n$ models data sets or contextual scenarios that demonstrate roughly constant nonzero $n$th differences. 1.13.A.6 A polynomial function of degree $n$ or less can be used to model a graph of $n + 1$ points with distinct input values. 1.13.A.7 A piecewise-defined function consists of a set of functions defined over nonoverlapping domain intervals and is useful for modeling a data set or contextual scenario that demonstrates different characteristics over different intervals.
1.13.B Describe assumptions and restrictions related to building a function model.	1.13.B.1 A model may have underlying assumptions about what is consistent in the model. 1.13.B.2 A model may have underlying assumptions about how quantities change together. 1.13.B.3 A model may require domain restrictions based on mathematical clues, contextual clues, or extreme values in the data set. 1.13.B.4 A model may require range restrictions, such as rounding values, based on mathematical clues, contextual clues, or extreme values in the data set.

Learning Objective

Essential Knowledge

1.13.A
Identify an appropriate function type to construct a function model for a given scenario.

1.13.A.1 Linear functions model data sets or aspects of contextual scenarios that demonstrate roughly constant rates of change.
1.13.A.2 Quadratic functions model data sets or aspects of contextual scenarios that demonstrate roughly linear rates of change, or data sets that are roughly symmetric with a unique maximum or minimum value.
1.13.A.3 Geometric contexts involving area or two dimensions can often be modeled by quadratic functions. Geometric contexts involving volume or three dimensions can often be modeled by cubic functions.
1.13.A.4 Polynomial functions model data sets or contextual scenarios with multiple real zeros or multiple maxima or minima.
1.13.A.5 A polynomial function of degree $n$ models data sets or contextual scenarios that demonstrate roughly constant nonzero $n$th differences.
1.13.A.6 A polynomial function of degree $n$ or less can be used to model a graph of $n + 1$ points with distinct input values.
1.13.A.7 A piecewise-defined function consists of a set of functions defined over nonoverlapping domain intervals and is useful for modeling a data set or contextual scenario that demonstrates different characteristics over different intervals.

1.13.B
Describe assumptions and restrictions related to building a function model.

1.13.B.1 A model may have underlying assumptions about what is consistent in the model.
1.13.B.2 A model may have underlying assumptions about how quantities change together.
1.13.B.3 A model may require domain restrictions based on mathematical clues, contextual clues, or extreme values in the data set.
1.13.B.4 A model may require range restrictions, such as rounding values, based on mathematical clues, contextual clues, or extreme values in the data set.

来源:美国大学理事会 AP 课程与考试说明

选择一个模型(model)意味着把一个函数类型匹配到一个量如何变化。一个线性模型拟合一个恒定的变化率;一个二次拟合一个恒定的二阶差;一个多项式拟合有几个转折的数据。在几何情形里,维数是一个强提示:关于面积(二维)的量通常是二次的,而关于体积(三维)的量通常是三次的——例如一个盒子的体积作为一个切割长度的函数。陈述你的模型依赖的假设(assumptions)(例如,模式继续),因为一个模型只在那些假设成立的地方有效。

词汇表训练

英文	中文	拼音
model	模型	mó xíng
assumptions	假设	jiǎ shè

练习卷

1.14

函数模型的构建与应用

大纲

Learning Objective	Essential Knowledge
1.14.A Construct a linear, quadratic, cubic, quartic, polynomial of degree $n$, or related piecewise-defined function model.	1.14.A.1 A model can be constructed based on restrictions identified in a mathematical or contextual scenario. 1.14.A.2 A model of a data set or a contextual scenario can be constructed using transformations of the parent function. 1.14.A.3 A model of a data set can be constructed using technology and regressions, including linear, quadratic, cubic, and quartic regressions. 1.14.A.4 A piecewise-defined function model can be constructed through a combination of modeling techniques.
1.14.B Construct a rational function model based on a context.	1.14.B.1 Data sets and aspects of contextual scenarios involving quantities that are inversely proportional can often be modeled by rational functions. For example, the magnitudes of both gravitational force and electromagnetic force between objects are inversely proportional to the objects' squared distance.
1.14.C Apply a function model to answer questions about a data set or contextual scenario.	1.14.C.1 A model can be used to draw conclusions about the modeled data set or contextual scenario, including answering key questions and predicting values, rates of change, average rates of change, and changing rates of change. Appropriate units of measure should be extracted or inferred from the given context.

来源:美国大学理事会 AP 课程与考试说明

要构建一个模型,用给定的特征——点、截距、带重数的零点、末端行为——写函数,然后用它在上下文里回答问题。总是对照对情况有意义的定义域检查答案,并用它们的现实世界单位解释输出。

一个分段函数(piecewise-defined function)在定义域的不重叠区间上用不同的规则。要求它的值,选区间包含输入的那个分支。例如,

$$f(x)=\begin{cases} x^2, & x<0\\ 2x, & x\ge 0\end{cases}$$

从第一个分支给出 $f(-3)=9$,但从第二个给出 $f(3)=6$——在行为在一个阈值处改变时有用(一个分级价格、一个改变的速度限制)。

词汇表训练

英文	中文	拼音
piecewise-defined function	分段函数	fēn duàn hán shù

练习卷

1.14

考试技巧

通过它的变化率读一个函数:平均率 = 割线斜率;一个恒定的率意味着线性,一个线性变化的率意味着二次。
因式分解一个多项式以求零点(x 截距)及其重数;偶重数触及、奇穿越坐标轴。
末端行为由首项设定;一个有理函数的渐近线来自顶部和底部的次数。
从关键特征(点、零点、末端行为)构建模型并陈述你的假设。
精确地描述变换:移动 $f(x-h)+k$、拉伸 $af(bx)$、反射 $-f(x)$ / $f(-x)$。

指数函数与对数函数

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讲义词汇表

2.1

等差数列与等比数列的变化

大纲

Learning Objective	Essential Knowledge
2.1.A Express arithmetic sequences found in mathematical and contextual scenarios as functions of the whole numbers.	2.1.A.1 A sequence is a function from the whole numbers to the real numbers. Consequently, the graph of a sequence consists of discrete points instead of a curve. 2.1.A.2 Successive terms in an arithmetic sequence have a common difference, or constant rate of change. 2.1.A.3 The general term of an arithmetic sequence with a common difference $d$ is denoted by $a_n$ and is given by $a_n = a_0 + dn$, where $a_0$ is the initial value, or by $a_n = a_k + d(n - k)$, where $a_k$ is the $k$th term of the sequence.
2.1.B Express geometric sequences found in mathematical and contextual scenarios as functions of the whole numbers.	2.1.B.1 Successive terms in a geometric sequence have a common ratio, or constant proportional change. 2.1.B.2 The general term of a geometric sequence with a common ratio $r$ is denoted by $g_n$ and is given by $g_n = g_0 r^n$, where $g_0$ is the initial value, or by $g_n = g_k r^{(n-k)}$, where $g_k$ is the $k$th term of the sequence. 2.1.B.3 Increasing arithmetic sequences increase equally with each step, whereas increasing geometric sequences increase by a larger amount with each successive step.

Learning Objective

Essential Knowledge

2.1.A
Express arithmetic sequences found in mathematical and contextual scenarios as functions of the whole numbers.

2.1.A.1 A sequence is a function from the whole numbers to the real numbers. Consequently, the graph of a sequence consists of discrete points instead of a curve.
2.1.A.2 Successive terms in an arithmetic sequence have a common difference, or constant rate of change.
2.1.A.3 The general term of an arithmetic sequence with a common difference $d$ is denoted by $a_n$ and is given by $a_n = a_0 + dn$, where $a_0$ is the initial value, or by $a_n = a_k + d(n - k)$, where $a_k$ is the $k$th term of the sequence.

2.1.B
Express geometric sequences found in mathematical and contextual scenarios as functions of the whole numbers.

2.1.B.1 Successive terms in a geometric sequence have a common ratio, or constant proportional change.
2.1.B.2 The general term of a geometric sequence with a common ratio $r$ is denoted by $g_n$ and is given by $g_n = g_0 r^n$, where $g_0$ is the initial value, or by $g_n = g_k r^{(n-k)}$, where $g_k$ is the $k$th term of the sequence.
2.1.B.3 Increasing arithmetic sequences increase equally with each step, whereas increasing geometric sequences increase by a larger amount with each successive step.

来源:美国大学理事会 AP 课程与考试说明

等比级数与收敛

一个数列(sequence)是一个从整数到实数的函数,所以它的图象是离散点,不是一条曲线。

一个等差数列(arithmetic sequence)有一个公差(common difference)$d$(一个恒定的变化率):$a_n = a_0 + dn$,或从一个已知项,$a_n = a_k + d(n-k)$。
一个等比数列(geometric sequence)有一个公比(common ratio)$r$(一个恒定的比例变化):$g_n = g_0\,r^{\,n}$,或 $g_n = g_k\,r^{\,(n-k)}$。

一个递增的等差数列每步增长相同的数额;一个递增的等比数列每步增长一个更大的数额。

Worked example. 一个 $a_0=3$ 和 $d=5$ 的等差数列有 $a_4=3+5(4)=23$。一个 $g_0=2$ 和 $r=3$ 的等比数列有 $g_4=2\cdot3^4=162$ ——加法与乘法使等比的遥遥领先。

探索

Compare arithmetic and geometric growth

An arithmetic sequence adds a fixed step each term (linear); a geometric sequence multiplies by a fixed ratio (exponential). Change the ratio and watch the terms explode or decay.

词汇表训练

英文	中文	拼音
sequence	数列	shù liè
arithmetic sequence	等差数列	děng chā shù liè
common difference	公差	gōng chāi
geometric sequence	等比数列	děng bǐ shù liè
common ratio	公比	gōng bǐ

练习卷

2.2

线性函数与指数函数的变化

大纲

Learning Objective	Essential Knowledge
2.2.A Construct functions of the real numbers that are comparable to arithmetic and geometric sequences.	2.2.A.1 Linear functions of the form $f(x) = b + mx$ are similar to arithmetic sequences of the form $a_n = a_0 + dn$, as both can be expressed as an initial value ($b$ or $a_0$) plus repeated addition of a constant rate of change, the slope ($m$ or $d$). 2.2.A.2 Similar to arithmetic sequences of the form $a_n = a_k + d(n - k)$, which are based on a known difference, $d$, and a $k$th term, linear functions can be expressed in the form $f(x) = y_i + m(x - x_i)$ based on a known slope, $m$, and a point, $(x_i, y_i)$. 2.2.A.3 Exponential functions of the form $f(x) = ab^x$ are similar to geometric sequences of the form $g_n = g_0 r^n$, as both can be expressed as an initial value ($a$ or $g_0$) times repeated multiplication by a constant proportion ($b$ or $r$). 2.2.A.4 Similar to geometric sequences of the form $g_n = g_k r^{(n-k)}$, which are based on a known ratio, $r$, and a $k$th term, exponential functions can be expressed in the form $f(x) = y_i r^{(x - x_i)}$ based on a known ratio, $r$, and a point, $(x_i, y_i)$. 2.2.A.5 Sequences and their corresponding functions may have different domains.

Learning Objective

Essential Knowledge

2.2.A
Construct functions of the real numbers that are comparable to arithmetic and geometric sequences.

2.2.A.1 Linear functions of the form $f(x) = b + mx$ are similar to arithmetic sequences of the form $a_n = a_0 + dn$, as both can be expressed as an initial value ($b$ or $a_0$) plus repeated addition of a constant rate of change, the slope ($m$ or $d$).
2.2.A.2 Similar to arithmetic sequences of the form $a_n = a_k + d(n - k)$, which are based on a known difference, $d$, and a $k$th term, linear functions can be expressed in the form $f(x) = y_i + m(x - x_i)$ based on a known slope, $m$, and a point, $(x_i, y_i)$.
2.2.A.3 Exponential functions of the form $f(x) = ab^x$ are similar to geometric sequences of the form $g_n = g_0 r^n$, as both can be expressed as an initial value ($a$ or $g_0$) times repeated multiplication by a constant proportion ($b$ or $r$).
2.2.A.4 Similar to geometric sequences of the form $g_n = g_k r^{(n-k)}$, which are based on a known ratio, $r$, and a $k$th term, exponential functions can be expressed in the form $f(x) = y_i r^{(x - x_i)}$ based on a known ratio, $r$, and a point, $(x_i, y_i)$.
2.2.A.5 Sequences and their corresponding functions may have different domains.

来源:美国大学理事会 AP 课程与考试说明

数列有连续的表亲:

线性函数(linear functions)$f(x)=b+mx$ 映照等差数列——一个初始值加斜率 $m$ 的重复相加。点形式:$f(x)=y_i+m(x-x_i)$。
指数函数(exponential functions)$f(x)=ab^x$ 映照等比数列——一个初始值乘以底数 $b$ 的重复相乘。点形式:$f(x)=y_i\,r^{\,(x-x_i)}$。

区别:线性 = 重复相加、指数 = 重复相乘。(一个数列和它的函数可能有不同的定义域。)

词汇表训练

英文	中文	拼音
exponential function	指数函数	zhǐ shù hán shù

练习卷

2.3

指数函数

大纲

Learning Objective	Essential Knowledge
2.3.A Identify key characteristics of exponential functions.	2.3.A.1 The general form of an exponential function is $f(x) = ab^x$, with the initial value $a$, where $a \neq 0$, and the base $b$, where $b > 0$, and $b \neq 1$. When $a > 0$ and $b > 1$, the exponential function is said to demonstrate exponential growth. When $a > 0$ and $0 < b < 1$, the exponential function is said to demonstrate exponential decay. 2.3.A.2 When the natural numbers are input values in an exponential function, the input value specifies the number of factors of the base to be applied to the function's initial value. The domain of an exponential function is all real numbers. 2.3.A.3 Because the output values of exponential functions in general form are proportional over equal-length input-value intervals, exponential functions are always increasing or always decreasing, and their graphs are always concave up or always concave down. Consequently, exponential functions do not have extrema except on a closed interval, and their graphs do not have points of inflection. 2.3.A.4 If the values of the additive transformation function $g(x) = f(x) + k$ of any function $f$ are proportional over equal-length input-value intervals, then $f$ is exponential.

Learning Objective

Essential Knowledge

2.3.A
Identify key characteristics of exponential functions.

2.3.A.1 The general form of an exponential function is $f(x) = ab^x$, with the initial value $a$, where $a \neq 0$, and the base $b$, where $b > 0$, and $b \neq 1$. When $a > 0$ and $b > 1$, the exponential function is said to demonstrate exponential growth. When $a > 0$ and $0 < b < 1$, the exponential function is said to demonstrate exponential decay.
2.3.A.2 When the natural numbers are input values in an exponential function, the input value specifies the number of factors of the base to be applied to the function's initial value. The domain of an exponential function is all real numbers.
2.3.A.3 Because the output values of exponential functions in general form are proportional over equal-length input-value intervals, exponential functions are always increasing or always decreasing, and their graphs are always concave up or always concave down. Consequently, exponential functions do not have extrema except on a closed interval, and their graphs do not have points of inflection.
2.3.A.4 If the values of the additive transformation function $g(x) = f(x) + k$ of any function $f$ are proportional over equal-length input-value intervals, then $f$ is exponential.

来源:美国大学理事会 AP 课程与考试说明

一般的指数函数(exponential function)是 $f(x)=ab^x$,初始值(initial value)$a\neq 0$ 而底数(base)$b>0,\ b\neq 1$。定义域:所有实数。

$a>0,\ b>1$ 给出指数增长(exponential growth);$a>0,\ 0 给出指数衰减(exponential decay)。
输出在等长输入区间上成比例。所以一个指数总是递增或总是递减,而且总是上凹或总是下凹——它没有极值(除了在一个闭区间上)而且没有拐点。

Bacteria under a microscope: exponential models describe populations that grow by a constant factor each time step

探索

Explore an exponential curve

y = a·e^(bx) + c

An exponential function changes by a constant factor over equal steps, so it rises (or decays) ever faster. The constant $c$ sets the horizontal asymptote it hugs.

词汇表训练

英文	中文	拼音
initial value	初始值	chū shǐ zhí
base	底数	dǐ shù
exponential growth	指数增长	zhǐ shù zēng zhǎng
exponential decay	指数衰减	zhǐ shù shuāi jiǎn

练习卷

2.4

指数函数的运算

大纲

Learning Objective	Essential Knowledge
2.4.A Rewrite exponential expressions in equivalent forms.	2.4.A.1 The product property for exponents states that $b^m b^n = b^{(m+n)}$. Graphically, this property implies that every horizontal translation of an exponential function, $f(x) = b^{(x+k)}$, is equivalent to a vertical dilation, $f(x) = b^{(x+k)} = b^x b^k = ab^x$, where $a = b^k$. 2.4.A.2 The power property for exponents states that $\left(b^m\right)^n = b^{(mn)}$. Graphically, this property implies that every horizontal dilation of an exponential function, $f(x) = b^{(cx)}$, is equivalent to a change of the base of an exponential function, $f(x) = \left(b^c\right)^x$, where $b^c$ is a constant and $c \neq 0$. 2.4.A.3 The negative exponent property states that $b^{-n} = \dfrac{1}{b^n}$. 2.4.A.4 The value of an exponential expression involving an exponential unit fraction, such as $b^{(1/k)}$ where $k$ is a natural number, is the $k$th root of $b$, when it exists.

Learning Objective

Essential Knowledge

2.4.A
Rewrite exponential expressions in equivalent forms.

2.4.A.1 The product property for exponents states that $b^m b^n = b^{(m+n)}$. Graphically, this property implies that every horizontal translation of an exponential function, $f(x) = b^{(x+k)}$, is equivalent to a vertical dilation, $f(x) = b^{(x+k)} = b^x b^k = ab^x$, where $a = b^k$.
2.4.A.2 The power property for exponents states that $\left(b^m\right)^n = b^{(mn)}$. Graphically, this property implies that every horizontal dilation of an exponential function, $f(x) = b^{(cx)}$, is equivalent to a change of the base of an exponential function, $f(x) = \left(b^c\right)^x$, where $b^c$ is a constant and $c \neq 0$.
2.4.A.3 The negative exponent property states that $b^{-n} = \dfrac{1}{b^n}$.
2.4.A.4 The value of an exponential expression involving an exponential unit fraction, such as $b^{(1/k)}$ where $k$ is a natural number, is the $k$th root of $b$, when it exists.

来源:美国大学理事会 AP 课程与考试说明

指数规则重塑指数表达式并连接到图象变换:

积: $b^m b^n = b^{m+n}$。一个水平移动 $b^{x+k}$ 等于一个竖直拉伸 $ab^x$,$a=b^k$。
幂: $(b^m)^n = b^{mn}$。一个水平拉伸 $b^{cx}$ 等于一个底数改变 $(b^c)^x$。
负指数: $b^{-n}=\dfrac{1}{b^n}$。
单位分数指数: $b^{1/k}=\sqrt[k]{b}$(第 $k$ 个方根(root))。

词汇表训练

英文	中文	拼音
root	方根	fāng gēn

练习卷

2.5

指数函数的情境与数据建模

大纲

Learning Objective	Essential Knowledge
2.5.A Construct a model for situations involving proportional output values over equal-length input-value intervals.	2.5.A.1 Exponential functions model growth patterns where successive output values over equal-length input-value intervals are proportional. When the input values are whole numbers, exponential functions model situations of repeated multiplication of a constant to an initial value. 2.5.A.2 A constant may need to be added to the dependent variable values of a data set to reveal a proportional growth pattern. 2.5.A.3 An exponential function model can be constructed from an appropriate ratio and initial value or from two input-output pairs. The initial value and the base can be found by solving a system of equations resulting from the two input-output pairs. 2.5.A.4 Exponential function models can be constructed by applying transformations to $f(x) = ab^x$ based on characteristics of a contextual scenario or data set. 2.5.A.5 Exponential function models can be constructed for a data set with technology using exponential regressions. 2.5.A.6 The natural base $e$, which is approximately $2.718$, is often used as the base in exponential functions that model contextual scenarios.

Learning Objective

Essential Knowledge

2.5.A
Construct a model for situations involving proportional output values over equal-length input-value intervals.

2.5.A.1 Exponential functions model growth patterns where successive output values over equal-length input-value intervals are proportional. When the input values are whole numbers, exponential functions model situations of repeated multiplication of a constant to an initial value.
2.5.A.2 A constant may need to be added to the dependent variable values of a data set to reveal a proportional growth pattern.
2.5.A.3 An exponential function model can be constructed from an appropriate ratio and initial value or from two input-output pairs. The initial value and the base can be found by solving a system of equations resulting from the two input-output pairs.
2.5.A.4 Exponential function models can be constructed by applying transformations to $f(x) = ab^x$ based on characteristics of a contextual scenario or data set.
2.5.A.5 Exponential function models can be constructed for a data set with technology using exponential regressions.
2.5.A.6 The natural base $e$, which is approximately $2.718$, is often used as the base in exponential functions that model contextual scenarios.

来源:美国大学理事会 AP 课程与考试说明

指数为在等间隔上以一个恒定比例增长(重复相乘)的量建模。

从一个比和一个初始值,或从两个点(为 $a$ 和 $b$ 解这个系统)构建一个模型。
有时一个常数必须被加到数据以揭示比例模式。
在技术上用指数回归(exponential regression)以拟合一个数据集。
自然底数(natural base)$e\approx 2.718$ 是现实世界模型的标准底数。

词汇表训练

英文	中文	拼音
exponential regression	指数回归	zhǐ shù huí guī
natural base	自然底数	zì rán dǐ shù

练习卷

2.6

竞争函数模型的验证

大纲

Learning Objective	Essential Knowledge
2.6.A Construct linear, quadratic, and exponential models based on a data set.	2.6.A.1 Two variables in a data set that demonstrate a slightly changing rate of change can be modeled by linear, quadratic, and exponential function models. 2.6.A.2 Models can be compared based on contextual clues and applicability to determine which model is most appropriate.
2.6.B Validate a model constructed from a data set.	2.6.B.1 A model is justified as appropriate for a data set if the graph of the residuals of a regression, the residual plot, appear without pattern. 2.6.B.2 The difference between the predicted and actual values is the error in the model. Depending on the data set and context, it may be more appropriate to have an underestimate or overestimate for any given interval.

来源:美国大学理事会 AP 课程与考试说明

当一个变化率只略微转变时,线性、二次和指数模型可能都看似拟合。用上下文和拟合质量选择:

一个模型是合适的,若它的残差图(residual plot)显示没有模式(一个残差(residual)是实际减预测)。
误差是预测和实际之间的差距;上下文决定一个高估还是低估更安全。

词汇表训练

英文	中文	拼音
residual plot	残差图	cán chà tú
residual	残差	cán chà

练习卷

2.7

复合函数

大纲

Learning Objective	Essential Knowledge
2.7.A Evaluate the composition of two or more functions for given values.	2.7.A.1 If $f$ and $g$ are functions, the composite function $f \circ g$ maps a set of input values to a set of output values such that the output values of $g$ are used as input values of $f$. For this reason, the domain of the composite function is restricted to those input values of $g$ for which the corresponding output value is in the domain of $f$. $(f \circ g)(x)$ can also be represented as $f(g(x))$. 2.7.A.2 Values for the composite function $f \circ g$ can be calculated or estimated from the graphical, numerical, analytical, or verbal representations of $f$ and $g$ by using output values from $g$ as input values for $f$. 2.7.A.3 The composition of functions is not commutative; that is, $f \circ g$ and $g \circ f$ are typically different functions; therefore, $f(g(x))$ and $g(f(x))$ are typically different values. 2.7.A.4 If the function $f(x) = x$ is composed with any function $g$, the resulting composite function is the same as $g$; that is, $g(f(x)) = f(g(x)) = g(x)$. The function $f(x) = x$ is called the identity function. When composing two functions, the identify function acts in the same way as $0$, the additive identity, when adding two numbers and $1$, the multiplicative identity, when multiplying two numbers.
2.7.B Construct a representation of the composition of two or more functions.	2.7.B.1 Function composition is useful for relating two quantities that are not directly related by an existing formula. 2.7.B.2 When analytic representations of the functions $f$ and $g$ are available, an analytic representation of $f(g(x))$ can be constructed by substituting $g(x)$ for every instance of $x$ in $f$. 2.7.B.3 A numerical or graphical representation of $f \circ g$ can often be constructed by calculating or estimating values for $(x, f(g(x)))$.
2.7.C Rewrite a given function as a composition of two or more functions.	2.7.C.1 Functions given analytically can often be decomposed into less complicated functions. When properly decomposed, the variable in one function should replace each instance of the function with which it was composed. 2.7.C.2 An additive transformation of a function, $f$, that results in vertical and horizontal translations can be understood as the composition of $g(x) = x + k$ with $f$. 2.7.C.3 A multiplicative transformation of a function, $f$, that results in vertical and horizontal dilations can be understood as the composition of $g(x) = kx$ with $f$.

来源:美国大学理事会 AP 课程与考试说明

复合函数(composite function)$(f\circ g)(x)=f(g(x))$ 把 $g$ 的输出喂进 $f$。它的定义域是 $g$ 的、输出位于 $f$ 定义域里的那些输入。

复合不可交换:$f(g(x))$ 和 $g(f(x))$ 通常不同。
要解析地构建 $f(g(x))$,把 $f$ 里的每个 $x$ 代入 $g(x)$。
恒等函数(identity function)$f(x)=x$ 在复合下使任何函数不变。
一个函数也能被分解成更简单的片段——对把一个加性移动看作与 $x+k$ 复合、或一个伸缩看作与 $kx$ 复合有用。

Matryoshka dolls: function composition nests layers — evaluate the inside first

词汇表训练

英文	中文	拼音
composite function	复合函数	fù hé hán shù
identity function	恒等函数	héng děng hán shù

练习卷

2.8

反函数

大纲

Learning Objective	Essential Knowledge
2.8.A Determine the input-output pairs of the inverse of a function.	2.8.A.1 On a specified domain, a function, $f$, has an inverse function, or is invertible, if each output value of $f$ is mapped from a unique input value. The domain of a function may be restricted in many ways to make the function invertible. 2.8.A.2 An inverse function can be thought of as a reverse mapping of the function. An inverse function, $f^{-1}$, maps the output values of a function, $f$, on its invertible domain to their corresponding input values; that is, if $f(a) = b$, then $f^{-1}(b) = a$. Alternately, on its invertible domain, if a function consists of input-output pairs $(a, b)$, then the inverse function consists of input-output pairs $(b, a)$.
2.8.B Determine the inverse of a function on an invertible domain.	2.8.B.1 The composition of a function, $f$, and its inverse function, $f^{-1}$, is the identity function; that is, $f\left(f^{-1}(x)\right) = f^{-1}(f(x)) = x$. 2.8.B.2 On a function's invertible domain, the function's range and domain are the inverse function's domain and range, respectively. The inverse of the table of values of $y = f(x)$ can be found by reversing the input-output pairs; that is, $(a, b)$ corresponds to $(b, a)$. 2.8.B.3 The inverse of the graph of the function $y = f(x)$ can be found by reversing the roles of the $x$- and $y$-axes; that is, by reflecting the graph of the function over the graph of the identity function $h(x) = x$. 2.8.B.4 The inverse of the function can be found by determining the inverse operations to reverse the mapping. One method for finding the inverse of the function $f$ is reversing the roles of $x$ and $y$ in the equation $y = f(x)$, then solving for $y = f^{-1}(x)$. 2.8.B.5 In addition to limiting the domain of a function to obtain an inverse function, contextual restrictions may also limit the applicability of an inverse function.

Learning Objective

Essential Knowledge

2.8.A
Determine the input-output pairs of the inverse of a function.

2.8.A.1 On a specified domain, a function, $f$, has an inverse function, or is invertible, if each output value of $f$ is mapped from a unique input value. The domain of a function may be restricted in many ways to make the function invertible.
2.8.A.2 An inverse function can be thought of as a reverse mapping of the function. An inverse function, $f^{-1}$, maps the output values of a function, $f$, on its invertible domain to their corresponding input values; that is, if $f(a) = b$, then $f^{-1}(b) = a$. Alternately, on its invertible domain, if a function consists of input-output pairs $(a, b)$, then the inverse function consists of input-output pairs $(b, a)$.

2.8.B
Determine the inverse of a function on an invertible domain.

2.8.B.1 The composition of a function, $f$, and its inverse function, $f^{-1}$, is the identity function; that is, $f\left(f^{-1}(x)\right) = f^{-1}(f(x)) = x$.
2.8.B.2 On a function's invertible domain, the function's range and domain are the inverse function's domain and range, respectively. The inverse of the table of values of $y = f(x)$ can be found by reversing the input-output pairs; that is, $(a, b)$ corresponds to $(b, a)$.
2.8.B.3 The inverse of the graph of the function $y = f(x)$ can be found by reversing the roles of the $x$- and $y$-axes; that is, by reflecting the graph of the function over the graph of the identity function $h(x) = x$.
2.8.B.4 The inverse of the function can be found by determining the inverse operations to reverse the mapping. One method for finding the inverse of the function $f$ is reversing the roles of $x$ and $y$ in the equation $y = f(x)$, then solving for $y = f^{-1}(x)$.
2.8.B.5 In addition to limiting the domain of a function to obtain an inverse function, contextual restrictions may also limit the applicability of an inverse function.

来源:美国大学理事会 AP 课程与考试说明

反函数是关于 y=x 的反射

指数与对数互为反函数

一个函数在每个输出来自一个唯一输入的定义域上是可逆(invertible)的(你可以限制定义域来强制这个)。反函数(inverse function)$f^{-1}$ 反转这个映射:若 $f(a)=b$ 那么 $f^{-1}(b)=a$。

$f\big(f^{-1}(x)\big)=f^{-1}\big(f(x)\big)=x$(复合给出恒等)。
定义域和值域交换:在一张表里把每个 $(a,b)$ 反转成 $(b,a)$;把图象在线 $y=x$ 上反射。
要求一个公式:在 $y=f(x)$ 里交换 $x$ 和 $y$,然后解 $y$。上下文可能进一步限制反函数在哪里适用。

词汇表训练

英文	中文	拼音
invertible	可逆	kě nì
inverse function	反函数	fǎn hán shù

练习卷

2.9

对数表达式

大纲

Learning Objective	Essential Knowledge
2.9.A Evaluate logarithmic expressions.	2.9.A.1 The logarithmic expression $\log_b c$ is equal to, or represents, the value that the base $b$ must be exponentially raised to in order to obtain the value $c$. That is, $\log_b c = a$ if and only if $b^a = c$, where $a$ and $c$ are constants, $b > 0$, and $b \neq 1$. (when the base of a logarithmic expression is not specified, it is understood as the common logarithm with a base of $10$) 2.9.A.2 The values of some logarithmic expressions are readily accessible through basic arithmetic while other values can be estimated through the use of technology. 2.9.A.3 On a logarithmic scale, each unit represents a multiplicative change of the base of the logarithm. For example, on a standard scale, the units might be $0, \; 1, \; 2, \; \ldots$, while on a logarithmic scale, using logarithm base $10$, the units might be $10^0, \; 10^1, \; 10^2, \; \ldots$.

Learning Objective

Essential Knowledge

2.9.A
Evaluate logarithmic expressions.

2.9.A.1 The logarithmic expression $\log_b c$ is equal to, or represents, the value that the base $b$ must be exponentially raised to in order to obtain the value $c$. That is, $\log_b c = a$ if and only if $b^a = c$, where $a$ and $c$ are constants, $b > 0$, and $b \neq 1$. (when the base of a logarithmic expression is not specified, it is understood as the common logarithm with a base of $10$)
2.9.A.2 The values of some logarithmic expressions are readily accessible through basic arithmetic while other values can be estimated through the use of technology.
2.9.A.3 On a logarithmic scale, each unit represents a multiplicative change of the base of the logarithm. For example, on a standard scale, the units might be $0, \; 1, \; 2, \; \ldots$, while on a logarithmic scale, using logarithm base $10$, the units might be $10^0, \; 10^1, \; 10^2, \; \ldots$.

来源:美国大学理事会 AP 课程与考试说明

对数(logarithm)回答"什么指数?":$\log_b c = a$ 恰好意味着 $b^a = c$(以 $b>0,\ b\neq 1$)。一个未写的底数意味着常用对数(common logarithm)(底数 $10$)。在一个对数尺度上,每个单位是底数的一个乘性步(……,$10^0,10^1,10^2,$ ……)。

词汇表训练

英文	中文	拼音
logarithm	对数	duì shù
common logarithm	常用对数	cháng yòng duì shù

练习卷

2.10

指数函数的反函数

大纲

Learning Objective	Essential Knowledge
2.10.A Construct representations of the inverse of an exponential function with an initial value of 1.	2.10.A.1 The general form of a logarithmic function is $f(x) = a\log_b x$, with base $b$, where $b > 0$, $b \neq 1$, and $a \neq 0$. 2.10.A.2 The way in which input and output values vary together have an inverse relationship in exponential and logarithmic functions. Output values of general-form exponential functions change proportionately as input values increase in equal-length intervals. However, input values of general-form logarithmic functions change proportionately as output values increase in equal-length intervals. Alternately, exponential growth is characterized by output values changing multiplicatively as input values change additively, whereas logarithmic growth is characterized by output values changing additively as input values change multiplicatively. 2.10.A.3 $f(x) = \log_b x$ and $g(x) = b^x$, where $b > 0$ and $b \neq 1$, are inverse functions. That is, $g(f(x)) = f(g(x)) = x$. 2.10.A.4 The graph of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$, is a reflection of the graph of the exponential function $g(x) = b^x$, where $b > 0$ and $b \neq 1$, over the graph of the identity function $h(x) = x$. 2.10.A.5 If $(s, \; t)$ is an ordered pair of the exponential function $g(x) = b^x$, where $b > 0$ and $b \neq 1$, then $(t, \; s)$ is an ordered pair of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$.

Learning Objective

Essential Knowledge

2.10.A
Construct representations of the inverse of an exponential function with an initial value of 1.

2.10.A.1 The general form of a logarithmic function is $f(x) = a\log_b x$, with base $b$, where $b > 0$, $b \neq 1$, and $a \neq 0$.
2.10.A.2 The way in which input and output values vary together have an inverse relationship in exponential and logarithmic functions. Output values of general-form exponential functions change proportionately as input values increase in equal-length intervals. However, input values of general-form logarithmic functions change proportionately as output values increase in equal-length intervals. Alternately, exponential growth is characterized by output values changing multiplicatively as input values change additively, whereas logarithmic growth is characterized by output values changing additively as input values change multiplicatively.
2.10.A.3 $f(x) = \log_b x$ and $g(x) = b^x$, where $b > 0$ and $b \neq 1$, are inverse functions. That is, $g(f(x)) = f(g(x)) = x$.
2.10.A.4 The graph of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$, is a reflection of the graph of the exponential function $g(x) = b^x$, where $b > 0$ and $b \neq 1$, over the graph of the identity function $h(x) = x$.
2.10.A.5 If $(s, \; t)$ is an ordered pair of the exponential function $g(x) = b^x$, where $b > 0$ and $b \neq 1$, then $(t, \; s)$ is an ordered pair of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$.

来源:美国大学理事会 AP 课程与考试说明

对数是指数的反函数:$y=b^x$ 和 $y=\log_b x$ 互相撤销,所以它们的图象是在 $y=x$ 上的反射。因此 $\log_b(b^x)=x$ 而 $b^{\log_b x}=x$。自然对数(natural logarithm)$\ln x = \log_e x$ 是 $e^x$ 的反函数。

词汇表训练

英文	中文	拼音
natural logarithm	自然对数	zì rán duì shù

练习卷

2.11

对数函数

大纲

Learning Objective	Essential Knowledge
2.11.A Identify key characteristics of logarithmic functions.	2.11.A.1 The domain of a logarithmic function in general form is any real number greater than zero, and its range is all real numbers. 2.11.A.2 Because logarithmic functions are inverses of exponential functions, logarithmic functions are also always increasing or always decreasing, and their graphs are either always concave up or always concave down. Consequently, logarithmic functions do not have extrema except on a closed interval, and their graphs do not have points of inflection. 2.11.A.3 The additive transformation function $g(x) = f(x + k)$, where $k \neq 0$, of a logarithmic function $f$ in general form does not have input values that are proportional over equal-length output-value intervals. However, if the input values of the additive transformation function, $g(x) = f(x + k)$, of any function $f$ are proportional over equal-length output value intervals, then $f$ is logarithmic. 2.11.A.4 With their limited domain, logarithmic functions in general form are vertically asymptotic to $x = 0$, with an end behavior that is unbounded. That is, for a logarithmic function in general form, $\lim\limits_{x \to 0^+} a\log_b x = \pm\infty$ and $\lim\limits_{x \to \infty} a\log_b x = \pm\infty$.

Learning Objective

Essential Knowledge

2.11.A
Identify key characteristics of logarithmic functions.

2.11.A.1 The domain of a logarithmic function in general form is any real number greater than zero, and its range is all real numbers.
2.11.A.2 Because logarithmic functions are inverses of exponential functions, logarithmic functions are also always increasing or always decreasing, and their graphs are either always concave up or always concave down. Consequently, logarithmic functions do not have extrema except on a closed interval, and their graphs do not have points of inflection.
2.11.A.3 The additive transformation function $g(x) = f(x + k)$, where $k \neq 0$, of a logarithmic function $f$ in general form does not have input values that are proportional over equal-length output-value intervals. However, if the input values of the additive transformation function, $g(x) = f(x + k)$, of any function $f$ are proportional over equal-length output value intervals, then $f$ is logarithmic.
2.11.A.4 With their limited domain, logarithmic functions in general form are vertically asymptotic to $x = 0$, with an end behavior that is unbounded. That is, for a logarithmic function in general form, $\lim\limits_{x \to 0^+} a\log_b x = \pm\infty$ and $\lim\limits_{x \to \infty} a\log_b x = \pm\infty$.

来源:美国大学理事会 AP 课程与考试说明

对数函数(logarithmic function)$f(x)=\log_b x$ 有定义域 $x>0$ 而值域是所有实数。它总是递增(对 $b>1$)或总是递减(对 $0)、总是朝一个方向凹,在 $x=0$ 有一条垂直渐近线——指数的水平渐近线的镜像。它对大的 $x$ 增长得很慢。

探索

Explore a logarithmic curve

y = a·ln(x − b) + c

A logarithm is the inverse of an exponential: it grows without bound but ever slower, with a vertical asymptote where its input hits zero.

词汇表训练

英文	中文	拼音
logarithmic function	对数函数	duì shù hán shù

练习卷

2.12

对数函数的运算

大纲

Learning Objective	Essential Knowledge
2.12.A Rewrite logarithmic expressions in equivalent forms.	2.12.A.1 The product property for logarithms states that $\log_b(xy) = \log_b x + \log_b y$. Graphically, this property implies that every horizontal dilation of a logarithmic function, $f(x) = \log_b(kx)$, is equivalent to a vertical translation, $f(x) = \log_b(kx) = \log_b k + \log_b x = a + \log_b x$, where $a = \log_b k$. 2.12.A.2 The power property for logarithms states that $\log_b x^n = n\log_b x$. Graphically, this property implies that raising the input of a logarithmic function to a power, $f(x) = \log_b x^k$, results in a vertical dilation, $f(x) = \log_b x^k = k\log_b x$. 2.12.A.3 The change of base property for logarithms states that $\log_b x = \dfrac{\log_a x}{\log_a b}$, where $a > 0$ and $a \neq 1$. This implies that all logarithmic functions are vertical dilations of each other. 2.12.A.4 The function $f(x) = \ln x$ is a logarithmic function with the natural base $e$; that is, $\ln x = \log_e x$.

Learning Objective

Essential Knowledge

2.12.A
Rewrite logarithmic expressions in equivalent forms.

2.12.A.1 The product property for logarithms states that $\log_b(xy) = \log_b x + \log_b y$. Graphically, this property implies that every horizontal dilation of a logarithmic function, $f(x) = \log_b(kx)$, is equivalent to a vertical translation, $f(x) = \log_b(kx) = \log_b k + \log_b x = a + \log_b x$, where $a = \log_b k$.
2.12.A.2 The power property for logarithms states that $\log_b x^n = n\log_b x$. Graphically, this property implies that raising the input of a logarithmic function to a power, $f(x) = \log_b x^k$, results in a vertical dilation, $f(x) = \log_b x^k = k\log_b x$.
2.12.A.3 The change of base property for logarithms states that $\log_b x = \dfrac{\log_a x}{\log_a b}$, where $a > 0$ and $a \neq 1$. This implies that all logarithmic functions are vertical dilations of each other.
2.12.A.4 The function $f(x) = \ln x$ is a logarithmic function with the natural base $e$; that is, $\ln x = \log_e x$.

来源:美国大学理事会 AP 课程与考试说明

对数性质反转指数规则:

$$\log_b(xy)=\log_b x+\log_b y,\quad \log_b\!\frac{x}{y}=\log_b x-\log_b y,\quad \log_b(x^n)=n\log_b x.$$

换底(change-of-base)公式让你能用技术计算任何对数:$\log_b x = \dfrac{\log x}{\log b}=\dfrac{\ln x}{\ln b}$。

词汇表训练

英文	中文	拼音
change-of-base	换底	huàn dǐ

练习卷

2.13

指数与对数方程和不等式

大纲

Learning Objective

Essential Knowledge

2.13.A
Solve exponential and logarithmic equations and inequalities.

2.13.A.1 Properties of exponents, properties of logarithms, and the inverse relationship between exponential and logarithmic functions can be used to solve equations and inequalities involving exponents and logarithms.
2.13.A.2 When solving exponential and logarithmic equations found through analytical or graphical methods, the results should be examined for extraneous solutions precluded by the mathematical or contextual limitations.
2.13.A.3 Logarithms can be used to rewrite expressions involving exponential functions in different ways that may reveal helpful information. Specifically, $b^x = c^{(\log_c b)(x)}$.

2.13.B
Construct the inverse function for exponential and logarithmic functions.

2.13.B.1 The function $f(x) = ab^{(x-h)} + k$ is a combination of additive transformations of an exponential function in general form. The inverse of $y = f(x)$ can be found by determining the inverse operations to reverse the mapping.
2.13.B.2 The function $f(x) = a\log_b(x - h) + k$ is a combination of additive transformations of a logarithmic function in general form. The inverse of $y = f(x)$ can be found by determining the inverse operations to reverse the mapping.

来源:美国大学理事会 AP 课程与考试说明

要解,用指数和对数互为反函数的事实:

隔离指数,然后对两侧取一个对数(用幂性质把指数拿下来)。
隔离对数,然后对两侧取指数。
总是检查增根(extraneous solutions)——一个对数的自变量必须保持正。

Worked example. 解 $2\cdot3^x=54$。除以 $2$:$3^x=27=3^3$,所以 $x=3$。当两侧不是整齐的幂时,改取对数:$5^x=20$ 给出 $x=\dfrac{\ln 20}{\ln 5}\approx1.86$。

练习卷

2.14

对数函数的情境与数据建模

大纲

Learning Objective

Essential Knowledge

2.14.A
Construct a logarithmic function model.

2.14.A.1 Logarithmic functions are inverses of exponential functions and can be used to model situations involving proportional growth, or repeated multiplication, where the input values change proportionally over equal-length output-value intervals. Alternately, if the output value is a whole number, it indicates how many times the initial value has been multiplied by the proportion.
2.14.A.2 A logarithmic function model can be constructed from an appropriate proportion and a real zero or from two input-output pairs.
2.14.A.3 Logarithmic function models can be constructed by applying transformations to $f(x) = a\log_b x$ based on characteristics of a context or data set.
2.14.A.4 Logarithmic function models can be constructed for a data set with technology using logarithmic regressions.
2.14.A.5 The natural logarithm function is often useful in modeling real-world phenomena.
2.14.A.6 Logarithmic function models can be used to predict values for the dependent variable.

来源:美国大学理事会 AP 课程与考试说明

对数为在巨大的乘性范围上变化的量(声音、酸强度、地震)建模。从数据构建一个对数模型,并对一个数据集用对数回归(logarithmic regression)。因为一个对数压缩大值,它把比例增长变成一个直线模式——半对数图背后的思想。

练习卷

2.15

半对数图

大纲

Learning Objective	Essential Knowledge
2.15.A Determine if an exponential model is appropriate by examining a semi-log plot of a data set.	2.15.A.1 In a semi-log plot, one of the axes is logarithmically scaled. When the $y$-axis of a semi-log plot is logarithmically scaled, data or functions that demonstrate exponential characteristics will appear linear. 2.15.A.2 An advantage of semi-log plots is that a constant never needs to be added to the dependent variable values to reveal that an exponential model is appropriate.
2.15.B Construct the linearization of exponential data.	2.15.B.1 Techniques used to model linear functions can be applied to a semi-log graph. 2.15.B.2 For an exponential model of the form $y = ab^x$, the corresponding linear model for the semi-log plot is $y = (\log_n b)x + \log_n a$, where $n > 0$ and $n \neq 1$. Specifically, the linear rate of change is $\log_n b$, and the initial linear value is $\log_n a$.

来源:美国大学理事会 AP 课程与考试说明

一个半对数图(semi-log plot)把输出放在一个对数轴上而输入放在一个正常轴上。在这些轴上,一个指数函数 $y=ab^x$ 变成一条直线,因为 $\log y = \log a + (\log b)\,x$ 在 $x$ 里是线性的。所以:若数据在一个半对数图上看起来线性,一个指数模型拟合;线的斜率给出 $\log b$ 而它的截距给出 $\log a$。

词汇表训练

英文	中文	拼音
semi-log plot	半对数图	bàn duì shù tú

练习卷

2.15

考试技巧

把等差(加一个公差)与等比(乘一个公比)数列及它们的线性/指数函数表亲区分开。
指数 = 重复相乘,所以它最终超过任何线性或多项式模型。
对数是指数的反函数($\log_b c=a\Leftrightarrow b^a=c$);用它来解 $b^x=k$。
应用对数律(积、商、幂)和换底公式;一个对数的自变量必须是正的。
在一个半对数图上一个指数模型变成一条直线。

三角函数与极坐标函数

单独打开本主题页面

讲义词汇表

3.1

周期现象

大纲

Learning Objective

Essential Knowledge

3.1.A
Construct graphs of periodic relationships based on verbal representations.

3.1.A.1 A periodic relationship can be identified between two aspects of a context if, as the input values increase, the output values demonstrate a repeating pattern over successive equal-length intervals.
3.1.A.2 The graph of a periodic relationship can be constructed from the graph of a single cycle of the relationship.

3.1.B
Describe key characteristics of a periodic function based on a verbal representation.

3.1.B.1 The period of the function is the smallest positive value $k$ such that $f(x + k) = f(x)$ for all $x$ in the domain. Consequently, the behavior of a periodic function is completely determined by any interval of width $k$.
3.1.B.2 The period can be estimated by investigating successive equal-length output values and finding where the pattern begins to repeat.
3.1.B.3 Periodic functions take on characteristics of other functions, such as intervals of increase and decrease, different concavities, and various rates of change. However, with periodic functions, all characteristics found in one period of the function will be in every period of the function.

来源:美国大学理事会 AP 课程与考试说明

一个关系是周期性(periodic)的,若它的输出模式在规则的输入步重复。周期(period)是使所有 $x$ 满足 $f(x+k)=f(x)$ 的最小正 $k$。你能通过复制一个单一的周期段(cycle)构建整个图象,并通过找到模式重复的间隔多远来估计周期。在每个周期段内一个周期函数仍有递增/递减和极大值/极小值的区间。

sin 和 cos 在 -1 和 1 之间起伏;tan 在 90 和 270 度断裂

词汇表训练

英文	中文	拼音
periodic	周期性	zhōu qī xìng
period	周期	zhōu qī
cycle	周期段	zhōu qī duàn

练习卷

3.2

正弦、余弦与正切

大纲

Learning Objective

Essential Knowledge

3.2.A
Determine the sine, cosine, and tangent of an angle using the unit circle.

3.2.A.1 In the coordinate plane, an angle is in standard position when the vertex coincides with the origin and one ray coincides with the positive $x$-axis. The other ray is called the terminal ray. Positive and negative angle measures indicate rotations from the positive $x$-axis in the counterclockwise and clockwise direction, respectively. Angles in standard position that share a terminal ray differ by an integer number of revolutions.
3.2.A.2 The radian measure of an angle in standard position is the ratio of the length of the arc of a circle centered at the origin subtended by the angle to the radius of that same circle. For a unit circle, which has radius $1$, the radian measure is the same as the length of the subtended arc.
3.2.A.3 Given an angle in standard position and a circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The sine of the angle is the ratio of the vertical displacement of $P$ from the $x$-axis to the distance between the origin and point $P$. Therefore, for a unit circle, the sine of the angle is the $y$-coordinate of point $P$.
3.2.A.4 Given an angle in standard position and a circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The cosine of the angle is the ratio of the horizontal displacement of $P$ from the $y$-axis to the distance between the origin and point $P$. Therefore, for a unit circle, the cosine of the angle is the $x$-coordinate of point $P$.
3.2.A.5 Given an angle in standard position, the tangent of the angle is the slope, if it exists, of the terminal ray. Because the slope of the terminal ray is the ratio of the vertical displacement to the horizontal displacement over any interval, the tangent of the angle is the ratio of the $y$-coordinate to the $x$-coordinate of the point at which the terminal ray intersects the unit circle; alternately, it is the ratio of the angle's sine to its cosine.

来源:美国大学理事会 AP 课程与考试说明

一个角处于标准位置(standard position),当它的顶点在原点而它的始边位于正 $x$ 轴上时。它的弧度(radian)度量是一个单位圆上的弧长。对于终边遇到一个半径 $r$ 的圆的点 $P$:

$$\cos\theta = \frac{x}{r},\qquad \sin\theta = \frac{y}{r},\qquad \tan\theta = \frac{y}{x}\ (\text{the slope of the terminal ray}).$$

探索

Read sine and cosine off the unit circle

On the unit circle a point at angle $\theta$ has coordinates $(\cos\theta,\ \sin\theta)$. Spin the angle to watch sine and cosine trace out as the height and width.

词汇表训练

英文	中文	拼音
standard position	标准位置	biāo zhǔn wèi zhì
radian	弧度	hú dù

练习卷

3.3

正弦与余弦函数值

大纲

Learning Objective

Essential Knowledge

3.3.A
Determine coordinates of points on a circle centered at the origin.

3.3.A.1 Given an angle of measure $\theta$ in standard position and a circle with radius $r$ centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The coordinates of point $P$ are $(r\cos\theta, r\sin\theta)$.
3.3.A.2 The geometry of isosceles right and equilateral triangles, while attending to the signs of the values based on the quadrant of the angle, can be used to find exact values for the cosine and sine of angles that are multiples of $\dfrac{\pi}{4}$ and $\dfrac{\pi}{6}$ radians and whose terminal rays do not lie on an axis.

来源:美国大学理事会 AP 课程与考试说明

在一个半径 $r$ 的圆上,$x=r\cos\theta$ 而 $y=r\sin\theta$。特殊角处($0,\tfrac{\pi}{6},\tfrac{\pi}{4},\tfrac{\pi}{3},\tfrac{\pi}{2},\dots$)的精确值来自等腰直角和等边三角形(equilateral triangle)几何,符号由角的象限(quadrant)设定。

词汇表训练

英文	中文	拼音
equilateral triangle	等边三角形	děng biān sān jiǎo xíng
quadrant	象限	xiàng xiàn

练习卷

3.4

正弦与余弦函数图像

大纲

Learning Objective

Essential Knowledge

3.4.A
Construct representations of the sine and cosine functions using the unit circle.

3.4.A.1 Given an angle of measure $\theta$ in standard position and a unit circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The sine function, $f(\theta) = \sin\theta$, gives the $y$-coordinate, or vertical displacement from the $x$-axis, of point $P$. The domain of the sine function is all real numbers.
3.4.A.2 As the input values, or angle measures, of the sine function increase, the output values oscillate between $-1$ and $1$, taking every value in between and tracking the vertical distance of points on the unit circle from the $x$-axis.
3.4.A.3 Given an angle of measure $\theta$ in standard position and a unit circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The cosine function, $f(\theta) = \cos\theta$, gives the $x$-coordinate, or horizontal displacement from the $y$-axis, of point $P$. The domain of the cosine function is all real numbers.
3.4.A.4 As the input values, or angle measures, of the cosine function increase, the output values oscillate between $-1$ and $1$, taking every value in between and tracking the horizontal distance of points on the unit circle from the $y$-axis.

来源:美国大学理事会 AP 课程与考试说明

单位圆画出正弦曲线

在单位圆(unit circle)($r=1$)上,$\sin\theta$ 是 $y$ 坐标而 $\cos\theta$ 是 $x$ 坐标。随着 $\theta$ 增加,两者以周期 $2\pi$ 在 $-1$ 和 $1$ 之间平滑地振荡(oscillate)。正弦从 $0$ 开始(上升);余弦从 $1$ 开始。它们是移动了 $\tfrac{\pi}{2}$ 的同一个波。

探索

Graph a sine wave

Unrolling the circle gives the wave $y=\sin x$: it repeats every $2\pi$ (its period) and swings between $-1$ and $1$ (its amplitude).

词汇表训练

英文	中文	拼音
unit circle	单位圆	dān wèi yuán
oscillate	振荡	zhèn dàng

练习卷

3.5

正弦型函数

大纲

Learning Objective

Essential Knowledge

3.5.A
Identify key characteristics of the sine and cosine functions.

3.5.A.1 A sinusoidal function is any function that involves additive and multiplicative transformations of $f(\theta) = \sin\theta$. The sine and cosine functions are both sinusoidal functions, with $\cos\theta = \sin\left(\theta + \dfrac{\pi}{2}\right)$.
3.5.A.2 The period and frequency of a sinusoidal function are reciprocals. The period of $f(\theta) = \sin\theta$ and $g(\theta) = \cos\theta$ is $2\pi$, and the frequency is $\dfrac{1}{2\pi}$.
3.5.A.3 The amplitude of a sinusoidal function is half the difference between its maximum and minimum values. The amplitude of $f(\theta) = \sin\theta$ and $g(\theta) = \cos\theta$ is $1$.
3.5.A.4 The midline of the graph of a sinusoidal function is determined by the average, or arithmetic mean, of the maximum and minimum values of the function. The midline of the graphs of $y = \sin\theta$ and $y = \cos\theta$ is $y = 0$.
3.5.A.5 As input values increase, the graphs of sinusoidal functions oscillate between concave down and concave up.
3.5.A.6 The graph of $y = \sin\theta$ has rotational symmetry about the origin and is therefore an odd function. The graph of $y = \cos\theta$ has reflective symmetry over the $y$-axis and is therefore an even function.

来源:美国大学理事会 AP 课程与考试说明

一个正弦型函数(sinusoidal function)是正弦(或余弦)的任何变换。它的关键特征:

周期和频率(frequency)互为倒数;$\sin\theta$ 有周期 $2\pi$。
振幅(amplitude)= 最大和最小输出之间距离的一半。
中线(midline)$y=d$ = 最大和最小的平均(水平的中心线)。

图象每半个周期段交替上凹和下凹。

Worked example. 对于 $f(\theta)=3\sin(2\theta)+1$:振幅是 $3$、周期是 $\dfrac{2\pi}{2}=\pi$,而中线是 $y=1$。所以最大输出是 $1+3=4$ 而最小是 $1-3=-2$。

词汇表训练

英文	中文	拼音
sinusoidal function	正弦型函数	zhèng xián xíng hán shù
frequency	频率	pín lǜ
Amplitude	振幅	zhèn fú
Midline	中线	zhōng xiàn

练习卷

3.6

正弦型函数的变换

大纲

Learning Objective

Essential Knowledge

3.6.A
Identify the amplitude, vertical shift, period, and phase shift of a sinusoidal function.

3.6.A.1 Functions that can be written in the form $f(\theta) = a\sin(b(\theta + c)) + d$ or $g(\theta) = a\cos(b(\theta + c)) + d$, where $a, b, c,$ and $d$ are real numbers and $a \neq 0$, are sinusoidal functions and are transformations of the sine and cosine functions. Additive and multiplicative transformations are the same for both sine and cosine because the cosine function is a phase shift of the sine function by $-\dfrac{\pi}{2}$ units.
3.6.A.2 The graph of the additive transformation $g(\theta) = \sin\theta + d$ of the sine function $f(\theta) = \sin\theta$ is a vertical translation of the graph of $f$, including its midline, by $d$ units. The same transformation of the cosine function yields the same result.
3.6.A.3 The graph of the additive transformation $g(\theta) = \sin(\theta + c)$ of the sine function $f(\theta) = \sin\theta$ is a horizontal translation, or phase shift, of the graph of $f$ by $-c$ units. The same transformation of the cosine function yields the same result.
3.6.A.4 The graph of the multiplicative transformation $g(\theta) = a\sin\theta$ of the sine function $f(\theta) = \sin\theta$ is a vertical dilation of the graph of $f$ and differs in amplitude by a factor of $|a|$. The same transformation of the cosine function yields the same result.
3.6.A.5 The graph of the multiplicative transformation $g(\theta) = \sin(b\theta)$ of the sine function $f(\theta) = \sin\theta$ is a horizontal dilation of the graph of $f$ and differs in period by a factor of $\left|\dfrac{1}{b}\right|$. The same transformation of the cosine function yields the same result.
3.6.A.6 The graph of $y = f(\theta) = a\sin(b(\theta + c)) + d$ has an amplitude of $|a|$ units, a period of $\left|\dfrac{1}{b}\right|2\pi$ units, a midline vertical shift of $d$ units from $y = 0$, and a phase shift of $-c$ units. The same transformations of the cosine function yield the same results.

来源:美国大学理事会 AP 课程与考试说明

正弦函数的振幅、周期与相位

一般的正弦型函数是

$$f(\theta)=a\sin\big(b(\theta+c)\big)+d \quad\text{or}\quad a\cos\big(b(\theta+c)\big)+d,$$

其中 $|a|$ 是振幅、$\dfrac{2\pi}{|b|}$ 是周期、$-c$ 是水平相移(phase shift)(括号里的 $+c$ 把图像向左移 $c$),而 $d$ 是竖直移动(vertical shift)(中线)。一个负的 $a$ 反射这个波。

探索

Stretch and shift a sinusoid

In $y=a\sin(bx+c)+d$: $a$ sets the amplitude, $b$ the period, $c$ a horizontal phase shift, and $d$ a vertical shift. Change each and watch the wave respond.

词汇表训练

英文	中文	拼音
phase shift	相移	xiāng yí

练习卷

3.7

正弦型函数的情境与数据建模

大纲

Learning Objective

Essential Knowledge

3.7.A
Construct sinusoidal function models of periodic phenomena.

3.7.A.1 The smallest interval of input values over which the maximum or minimum output values start to repeat, that is, the input-value interval between consecutive maxima or consecutive minima, can be used to determine or estimate the period and frequency for a sinusoidal function model.
3.7.A.2 The maximum and minimum output values can be used to determine or estimate the amplitude and vertical shift for a sinusoidal function model.
3.7.A.3 An actual pair of input-output values can be compared to pairs of input-output values produced by a sinusoidal function model to determine or estimate a phase shift for the model.
3.7.A.4 Sinusoidal function models can be constructed for a data set with technology by estimating key values or using sinusoidal regressions.
3.7.A.5 Sinusoidal functions that model a data set are frequently only useful over their contextual domain and can be used to predict values of the dependent variable from values of the independent variable.

来源:美国大学理事会 AP 课程与考试说明

要为周期数据(潮汐、日照、温度)建模:从模式重复的地方读周期、从最大和最小得到振幅和中线,并设定相移使一个峰落在正确的输入处。技术拟合一个正弦回归。这样一个模型只在它的上下文定义域上可信。

练习卷

3.8

正切函数

大纲

Learning Objective

Essential Knowledge

3.8.A
Construct representations of the tangent function using the unit circle.

3.8.A.1 Given an angle of measure $\theta$ in standard position and a unit circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The tangent function, $f(\theta) = \tan\theta$, gives the slope of the terminal ray.
3.8.A.2 Because the slope of the terminal ray is the ratio of the change in the $y$-values to the change in the $x$-values between any two points on the ray, the tangent function is also the ratio of the sine function to the cosine function. Therefore, $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, where $\cos\theta \neq 0$.

3.8.B
Describe key characteristics of the tangent function.

3.8.B.1 Because the slope values of the terminal ray repeat every one-half revolution of the circle, the tangent function has a period of $\pi$.
3.8.B.2 The tangent function demonstrates periodic asymptotic behavior at input values $\theta = \dfrac{\pi}{2} + k\pi$, for integer values of $k$, because $\cos\theta = 0$ at those values.
3.8.B.3 The tangent function increases and its graph changes from concave down to concave up between consecutive asymptotes.

来源:美国大学理事会 AP 课程与考试说明

$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ 是终边的斜率。它的斜率值每半圈重复,所以它的周期是 $\pi$。它在 $\cos\theta=0$ 的地方(在 $\theta=\tfrac{\pi}{2}+k\pi$)有垂直渐近线,而它在连续的渐近线之间总是递增,在每个零点切换凹凸性。

练习卷

3.9

反三角函数

大纲

Learning Objective

Essential Knowledge

3.9.A
Construct analytical and graphical representations of the inverse of the sine, cosine, and tangent functions over a restricted domain.

3.9.A.1 For inverse trigonometric functions, the input and output values are switched from their corresponding trigonometric functions, so the output value of an inverse trigonometric function is often interpreted as an angle measure and the input is a value in the range of the corresponding trigonometric function.
3.9.A.2 The inverse trigonometric functions are called arcsine, arccosine, and arctangent (also represented as $\sin^{-1}x, \cos^{-1}x,$ and $\tan^{-1}x$). Because the corresponding trigonometric functions are periodic, they are only invertible if they have restricted domains.
3.9.A.3 In order to define their respective inverse functions, the domain of the sine function is restricted to $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$, the cosine function to $[0, \pi]$, and the tangent function to $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$.

来源:美国大学理事会 AP 课程与考试说明

反三角函数(inverse trigonometric functions)——反正弦(arcsine)、反余弦(arccosine)、反正切(arctangent)——反转三角函数,所以它们取一个比并返回一个角。因为正弦、余弦和正切重复,它们的定义域在一个反函数能存在之前必须被限制(e.g. 正弦到 $[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$)。

词汇表训练

英文	中文	拼音
inverse trigonometric functions	反三角函数	fǎn sān jiǎo hán shù
arcsine	反正弦	fǎn zhèng xián

练习卷

3.10

三角方程与不等式

大纲

Learning Objective

Essential Knowledge

3.10.A
Solve equations and inequalities involving trigonometric functions.

3.10.A.1 Inverse trigonometric functions are useful in solving equations and inequalities involving trigonometric functions, but solutions may need to be modified due to domain restrictions.
3.10.A.2 Because trigonometric functions are periodic, there are often infinitely many solutions to trigonometric equations.
3.10.A.3 In trigonometric equations and inequalities arising from a contextual scenario, there is often a domain restriction that can be implied from the context, which limits the number of solutions.

来源:美国大学理事会 AP 课程与考试说明

用反三角函数来解三角方程。因为这些函数是周期性的,通常有无穷多个解——通过加周期的倍数写通解,然后保留在要求的(常常是上下文的)定义域里的那些。

Worked example. 在 $[0,2\pi)$ 上解 $2\sin\theta=1$。那么 $\sin\theta=\tfrac12$,而在单位圆上正弦在第一和第二象限为正,所以 $\theta=\dfrac{\pi}{6}$ 或 $\dfrac{5\pi}{6}$。在所有实数上你会给每个加 $2\pi k$。

练习卷

3.11

正割、余割与余切函数

大纲

Learning Objective

Essential Knowledge

3.11.A
Identify key characteristics of functions that involve quotients of the sine and cosine functions.

3.11.A.1 The secant function, $f(\theta) = \sec\theta$, is the reciprocal of the cosine function, where $\cos\theta \neq 0$.
3.11.A.2 The cosecant function, $f(\theta) = \csc\theta$, is the reciprocal of the sine function, where $\sin\theta \neq 0$.
3.11.A.3 The graphs of the secant and cosecant functions have vertical asymptotes where cosine and sine are zero, respectively, and have a range of $(-\infty, -1] \cup [1, \infty)$.
3.11.A.4 The cotangent function, $f(\theta) = \cot\theta$, is the reciprocal of the tangent function, where $\tan\theta \neq 0$. Equivalently, $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$, where $\sin\theta \neq 0$.
3.11.A.5 The graph of the cotangent function has vertical asymptotes for domain values where $\tan\theta = 0$ and is decreasing between consecutive asymptotes.

来源:美国大学理事会 AP 课程与考试说明

这些是主要三个的倒数(reciprocals):

$$\sec\theta=\frac{1}{\cos\theta},\qquad \csc\theta=\frac{1}{\sin\theta},\qquad \cot\theta=\frac{1}{\tan\theta}.$$

正割和余割在余弦和正弦为零的地方有垂直渐近线;余切在正切为零的地方(正弦为零的地方)有它们。

词汇表训练

英文	中文	拼音
reciprocals	倒数	dào shǔ

练习卷

3.12

三角函数的等价表示

大纲

Learning Objective	Essential Knowledge
3.12.A Rewrite trigonometric expressions in equivalent forms with the Pythagorean identity.	3.12.A.1 The Pythagorean Theorem can be applied to right triangles with points on the unit circle at coordinates $(\cos\theta, \sin\theta)$, resulting in the Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$. 3.12.A.2 The Pythagorean identity can be algebraically manipulated into other forms involving trigonometric functions, such as $\tan^2\theta = \sec^2\theta - 1$, and can be used to establish other trigonometric relationships, such as $\arcsin x = \arccos\left(\sqrt{1 - x^2}\right)$, with appropriate domain restrictions.
3.12.B Rewrite trigonometric expressions in equivalent forms with sine and cosine sum identities.	3.12.B.1 The sum identity for sine is $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$. 3.12.B.2 The sum identity for cosine is $\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$. 3.12.B.3 The sum identities for sine and cosine can also be used as difference and double-angle identities. 3.12.B.4 Properties of trigonometric functions, known trigonometric identities, and other algebraic properties can be used to verify additional trigonometric identities.
3.12.C Solve equations using equivalent analytic representations of trigonometric functions.	3.12.C.1 A specific equivalent form involving trigonometric expressions can make information more accessible. 3.12.C.2 Equivalent trigonometric forms may be useful in solving trigonometric equations and inequalities.

来源:美国大学理事会 AP 课程与考试说明

毕达哥拉斯恒等式(Pythagorean identity)来自单位圆上的 $x^2+y^2=1$:

$$\sin^2\theta+\cos^2\theta=1,$$

它重新排列成带 $\sec$ 和 $\tan$ 的形式。和角公式(sum identities):

$$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta,\qquad \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta,$$

也给出差角和二倍角(double-angle)恒等式(设 $\beta=\alpha$)。

词汇表训练

英文	中文	拼音
Pythagorean identity	毕达哥拉斯恒等式	bì dá gē lā sī héng děng shì
sum identities	和角公式	hé jiǎo gōng shì
double-angle	二倍角	èr bèi jiǎo

练习卷

3.13

三角学与极坐标

大纲

Learning Objective

Essential Knowledge

3.13.A
Determine the location of a point in the plane using both rectangular and polar coordinates.

3.13.A.1 The polar coordinate system is based on a grid of circles centered at the origin and on lines through the origin. Polar coordinates are defined as an ordered pair, $(r, \theta)$, such that $|r|$ represents the radius of the circle on which the point lies, and $\theta$ represents the measure of an angle in standard position whose terminal ray includes the point. In the polar coordinate system, the same point can be represented many ways.
3.13.A.2 The coordinates of a point in the polar coordinate system, $(r, \theta)$, can be converted to coordinates in the rectangular coordinate system, $(x, y)$, using $x = r\cos\theta$ and $y = r\sin\theta$.
3.13.A.3 The coordinates of a point in the rectangular coordinate system, $(x, y)$, can be converted to coordinates in the polar coordinate system, $(r, \theta)$, using $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan\left(\dfrac{y}{x}\right)$ for $x > 0$ or $\theta = \arctan\left(\dfrac{y}{x}\right) + \pi$ for $x < 0$.
3.13.A.4 A complex number can be understood as a point in the complex plane and can be determined by its corresponding rectangular or polar coordinates. When the complex number has the rectangular coordinates $(a, b)$, it can be expressed as $a + bi$. When the complex number has polar coordinates $(r, \theta)$, it can be expressed as $(r\cos\theta) + i(r\sin\theta)$.

来源:美国大学理事会 AP 课程与考试说明

极坐标(polar coordinates)由 $(r,\theta)$ 定位一个点——在角 $\theta$ 距原点的距离 $r$。用以下转换

$$x=r\cos\theta,\quad y=r\sin\theta,\qquad r=\sqrt{x^2+y^2},\quad \tan\theta=\frac{y}{x}.$$

一个复数(complex number)是点 $(x,y)=x+yi$,同样能用 $r$ 和 $\theta$ 写。

词汇表训练

英文	中文	拼音
Polar coordinates	极坐标	jí zuò biāo
complex number	复数	fù shù

练习卷

3.14

极坐标函数图像

大纲

Learning Objective

Essential Knowledge

3.14.A
Construct graphs of polar functions.

3.14.A.1 The graph of the function $r = f(\theta)$ in polar coordinates consists of input-output pairs of values where the input values are angle measures and the output values are radii.
3.14.A.2 The domain of the polar function $r = f(\theta)$, given graphically, can be restricted to a desired portion of the function by selecting endpoints corresponding to the desired angle and radius.
3.14.A.3 When graphing polar functions in the form of $r = f(\theta)$, changes in input values correspond to changes in angle measure from the positive $x$-axis, and changes in output values correspond to changes in distance from the origin.

来源:美国大学理事会 AP 课程与考试说明

极坐标曲线的描绘

$r=f(\theta)$ 的图象是随着 $\theta$ 扫过所有的点 $(r,\theta)$。限制 $\theta$ 的定义域只画曲线的一部分。随着 $\theta$ 增加,$r$ 增长或缩小,描出螺线、圆、玫瑰线和蚶线。

探索

Plot a curve in polar coordinates

In polar form a point is a distance $r$ at angle $\theta$. Letting $r$ depend on $\theta$ traces shapes a rule in $x,y$ can't — like this cardioid.

练习卷

3.15

极坐标函数的变化率

大纲

Learning Objective

Essential Knowledge

3.15.A
Describe characteristics of the graph of a polar function.

3.15.A.1 If a polar function, $r = f(\theta)$, is positive and increasing or negative and decreasing, then the distance between $f(\theta)$ and the origin is increasing.
3.15.A.2 If a polar function, $r = f(\theta)$, is positive and decreasing or negative and increasing, then the distance between $f(\theta)$ and the origin is decreasing.
3.15.A.3 For a polar function, $r = f(\theta)$, if the function changes from increasing to decreasing or decreasing to increasing on an interval, then the function has a relative extremum on the interval corresponding to a point relatively closest to or farthest from the origin.
3.15.A.4 The average rate of change of $r$ with respect to $\theta$ over an interval of $\theta$ is the ratio of the change in the radius values to the change in $\theta$ over an interval of $\theta$. Graphically, the average rate of change indicates the rate at which the radius is changing per radian.
3.15.A.5 The average rate of change of $r$ with respect to $\theta$ over an interval of $\theta$ can be used to estimate values of the function within the interval.

来源:美国大学理事会 AP 课程与考试说明

随着 $\theta$ 增加,离原点的距离 $r$ 变化:

$r$ 为正且增加(或为负且减少)$\Rightarrow$ 远离原点移动;
$r$ 为正且减少(或为负且增加)$\Rightarrow$ 朝向原点移动。

$r$ 在递增和递减之间切换的地方,距离达到一个相对极值。$r$ 相对于 $\theta$ 的平均变化率,$\dfrac{\Delta r}{\Delta \theta}$,估计曲线在一个区间上向内或向外移动多快。

练习卷

3.15

考试技巧

用弧度和单位圆:$\cos\theta$ 和 $\sin\theta$ 是坐标,总是在 $-1$ 和 $1$ 之间。
对于一个正弦型函数 $a\sin(b(\theta+c))+d$:$|a|$ 是振幅、$\tfrac{2\pi}{|b|}$ 是周期、$d$ 是中线、相移是 $-c$($+c$ 向左移)。
通过从最大和最小读周期、振幅和中线为周期数据建模。
反三角函数需要一个限制的定义域并返回一个角;三角方程有无穷多个解(加周期)。
用 $x=r\cos\theta$、$y=r\sin\theta$ 转换极↔直角。

涉及参数、向量与矩阵的函数

单独打开本主题页面

讲义词汇表

4.1

参数方程函数

大纲

Learning Objective

Essential 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 Objective

Essential 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 Objective

Essential 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 Objective

Essential 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 Objective

Essential 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 Objective

Essential 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

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 Objective

Essential 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 Objective	Essential 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。

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 Objective

Essential 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。

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 Objective

Essential 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 Objective

Essential 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 Objective

Essential 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。

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 Objective	Essential 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 Objective

Essential 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。