参数方程、极坐标与向量值函数

AP 微积分 BC · 第 9 主题

训练

讲义词汇表

9.1

定义参数方程与其求导

大纲

Enduring Understanding	Learning Objective	Essential Knowledge
CHA-3 Derivatives allow us to solve real-world problems involving rates of change.	CHA-3.G Calculate derivatives of parametric functions. BC ONLY	CHA-3.G.1 Methods for calculating derivatives of real-valued functions can be extended to parametric functions. BC ONLY CHA-3.G.2 For a curve defined parametrically, the value of $\dfrac{dy}{dx}$ at a point on the curve is the slope of the line tangent to the curve at that point. $\dfrac{dy}{dx}$, the slope of the line tangent to a curve defined using parametric equations, can be determined by dividing $\dfrac{dy}{dt}$ by $\dfrac{dx}{dt}$, provided $\dfrac{dx}{dt}$ does not equal zero. BC ONLY

来源:美国大学理事会 AP 课程与考试说明

参数方程(parametric equations)把 $x$ 和 $y$ 各作为一个参数(parameter)$t$(常常是时间)的函数给出:$x=x(t)$、$y=y(t)$。它们描出一条不必是 $x$ 的一个函数的曲线。曲线的斜率用链式法则求出:

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\qquad(dx/dt\neq0).$$

Worked example. 对于 $x=t^2$、$y=t^3-t$,求 $t=2$ 处的斜率。这里 $\dfrac{dx}{dt}=2t$ 而 $\dfrac{dy}{dt}=3t^2-1$,所以 $\dfrac{dy}{dx}=\dfrac{3t^2-1}{2t}$;在 $t=2$ 这是 $\dfrac{11}{4}$。

词汇表训练

英文	中文	拼音
Parametric equations	参数方程	cān shù fāng chéng

练习卷

9.2

参数方程的二阶导数

大纲

Enduring Understanding	Learning Objective	Essential Knowledge
CHA-3 Derivatives allow us to solve real-world problems involving rates of change.	CHA-3.G Calculate derivatives of parametric functions. BC ONLY	CHA-3.G.3 $\dfrac{d^2 y}{dx^2}$ can be calculated by dividing $\dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)$ by $\dfrac{dx}{dt}$. BC ONLY

来源:美国大学理事会 AP 课程与考试说明

二阶导数不是 $\dfrac{d^2y/dt^2}{d^2x/dt^2}$。相反,对 $t$ 求导一阶导数,然后再次除以 $dx/dt$:

$$\frac{d^2y}{dx^2}=\frac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{dx/dt}.$$

用它来测试一条参数曲线的凹凸性。

练习卷

9.3

求参数方程给定曲线的弧长

大纲

Enduring Understanding	Learning Objective	Essential Knowledge
CHA-6 Definite integrals allow us to solve problems involving the accumulation of change in length over an interval.	CHA-6.B Determine the length of a curve in the plane defined by parametric functions, using a definite integral. BC ONLY	CHA-6.B.1 The length of a parametrically defined curve can be calculated using a definite integral. BC ONLY

来源:美国大学理事会 AP 课程与考试说明

一条参数曲线在 $t$ 从 $a$ 到 $b$ 的长度是

$$L=\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt.$$

这是弧长的参数版本——在参数上把微小的斜边 $\sqrt{dx^2+dy^2}$ 相加。

练习卷

9.4

定义向量值函数与其求导

大纲

Enduring Understanding	Learning Objective	Essential Knowledge
CHA-3 Derivatives allow us to solve real-world problems involving rates of change.	CHA-3.H Calculate derivatives of vector-valued functions. BC ONLY	CHA-3.H.1 Methods for calculating derivatives of real-valued functions can be extended to vector-valued functions. BC ONLY

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)$\vec{r}(t)=\langle x(t),\,y(t)\rangle$ 为每个 $t$ 给出一个位置向量。按分量求导:速度是 $\vec{v}(t)=\langle x'(t),\,y'(t)\rangle$ 而加速度是 $\vec{a}(t)=\langle x''(t),\,y''(t)\rangle$。速率是模 $|\vec{v}|=\sqrt{x'^2+y'^2}$。

一个向量值直线:从 a 开始,然后滑动方向 b 的 t 倍

探索

Add vector-valued components

A vector-valued function packs an $x(t)$ and $y(t)$ into one vector. Differentiating each component gives the velocity vector, tangent to the path.

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

9.5

向量值函数的积分

大纲

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-8 Solving an initial value problem allows us to determine an expression for the position of a particle moving in the plane.	FUN-8.A Determine a particular solution given a rate vector and initial conditions. BC ONLY	FUN-8.A.1 Methods for calculating integrals of real-valued functions can be extended to parametric or vector-valued functions. BC ONLY

来源:美国大学理事会 AP 课程与考试说明

逐分量积分一个向量函数。给定加速度或速度加一个初始条件,积分每个分量并用条件求常数——从加速度恢复速度,或从速度恢复位置。

练习卷

9.6

用参数方程与向量值函数求解运动问题

大纲

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-8 Solving an initial value problem allows us to determine an expression for the position of a particle moving in the plane.	FUN-8.B Determine values for positions and rates of change in problems involving planar motion. BC ONLY	FUN-8.B.1 Derivatives can be used to determine velocity, speed, and acceleration for a particle moving along a curve in the plane defined using parametric or vector-valued functions. BC ONLY FUN-8.B.2 For a particle in planar motion over an interval of time, the definite integral of the velocity vector represents the particle's displacement (net change in position) over the interval of time, from which we might determine its position. The definite integral of speed represents the particle's total distance traveled over the interval of time. BC ONLY

来源:美国大学理事会 AP 课程与考试说明

对于一个在一个平面里移动的质点:位置是 $\langle x(t),y(t)\rangle$、速度和加速度是它的导数、速率是 $|\vec v|$,而在 $[a,b]$ 上的行进距离(distance traveled)是

$$\int_a^b\sqrt{x'(t)^2+y'(t)^2}\,dt.$$

考试技能: 这些平面运动问题几乎每年在 BC 自由回答上出现——熟练地求速率、一个较后时刻的位置(初始点加速度的积分),和总距离。

Worked example. 一个质点有位置 $\langle t^2,\ t^3-t\rangle$。它的速度是 $\langle 2t,\ 3t^2-1\rangle$,所以在 $t=1$ 速度是 $\langle 2,\ 2\rangle$ 而速率是 $\sqrt{2^2+2^2}=2\sqrt{2}$。它在 $t=2$ 的位置是 $\langle 4,\ 6\rangle$。

练习卷

9.7

定义极坐标与在极坐标形式下求导

大纲

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-3 Recognizing opportunities to apply derivative rules can simplify differentiation.	FUN-3.G Calculate derivatives of functions written in polar coordinates. BC ONLY	FUN-3.G.1 Methods for calculating derivatives of real-valued functions can be extended to functions in polar coordinates. BC ONLY FUN-3.G.2 For a curve given by a polar equation $r = f(\theta)$, derivatives of $r$, $x$, and $y$ with respect to $\theta$, and first and second derivatives of $y$ with respect to $x$ can provide information about the curve. BC ONLY

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

极坐标(polar coordinates)由一个点离原点的距离 $r$ 和角 $\theta$ 定位它:用 $x=r\cos\theta$、$y=r\sin\theta$ 转换。一条极曲线 $r=f(\theta)$ 是一条 $\theta$ 的参数曲线,所以它的斜率是

$$\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta},\quad\text{with } x=r\cos\theta,\ y=r\sin\theta.$$

极坐标由一个点的距离 r 和角 theta 给出它

探索

Plot a polar curve

In polar coordinates a point is a distance $r$ at angle $\theta$. Letting $r$ depend on $\theta$ traces curves like this cardioid.

词汇表训练

英文	中文	拼音
Polar coordinates	极坐标	jí zuò biāo

练习卷

9.8

求极坐标区域的面积或单条极坐标曲线所围面积

大纲

Enduring Understanding	Learning Objective	Essential Knowledge
CHA-5 Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.	CHA-5.D Calculate areas of regions defined by polar curves using definite integrals. BC ONLY	CHA-5.D.1 The concept of calculating areas in rectangular coordinates can be extended to polar coordinates. BC ONLY

来源:美国大学理事会 AP 课程与考试说明

一条极曲线 $r=f(\theta)$ 从 $\alpha$ 到 $\beta$ 扫出的面积是

$$A=\frac12\int_\alpha^\beta \big(f(\theta)\big)^2\,d\theta.$$

区域是薄的三角扇形的一个"扇子";选择正确的 $\theta$ 限(曲线在哪里开始和结束描出区域)是主要的挑战。

Worked example. 求玫瑰线 $r=2\sin(2\theta)$ 的一个花瓣的面积($\theta$ 从 $0$ 到 $\tfrac{\pi}{2}$ 描出)。用 $\sin^2 u=\tfrac12(1-\cos 2u)$,

$$A=\frac12\int_0^{\pi/2}(2\sin 2\theta)^2\,d\theta=\int_0^{\pi/2}(1-\cos 4\theta)\,d\theta=\left[\theta-\tfrac{\sin 4\theta}{4}\right]_0^{\pi/2}=\frac{\pi}{2}.$$

一个极区域的面积作为 r 平方 d theta 积分的一半扫出 — 阴影区域是从 $\alpha$ 到 $\beta$ 扫出的薄扇形的一个扇子;每个有面积 $\tfrac12 r^2\,d\theta$,所以总量是 $\tfrac12\int_\alpha^\beta r^2\,d\theta$。

练习卷

9.9

求两条极坐标曲线所围区域的面积

大纲

Enduring Understanding	Learning Objective	Essential Knowledge
CHA-5 Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.	CHA-5.D Calculate areas of regions defined by polar curves using definite integrals. BC ONLY	CHA-5.D.2 Areas of regions bounded by polar curves can be calculated with definite integrals. BC ONLY

来源:美国大学理事会 AP 课程与考试说明

对于一条外曲线 $r_1$ 和一条内曲线 $r_2$ 之间的面积,减去扇形:

$$A=\frac12\int_\alpha^\beta\big(r_1^2-r_2^2\big)\,d\theta.$$

先找到交角(令 $r_1=r_2$),并小心哪条曲线在每个区间上是外面的——它们能交换。

练习卷

9.9

考试技巧

对于参数曲线,$\tfrac{dy}{dx}=\tfrac{dy/dt}{dx/dt}$;速率是 $\sqrt{(dx/dt)^2+(dy/dt)^2}$。
一个向量值函数携带相同的信息——逐分量求导/积分它。
在极里,用 $x=r\cos\theta$、$y=r\sin\theta$ 转换;扫出的面积是 $\tfrac12\int r^2\,d\theta$。
注意描出的方向($dx/dt$ 的符号)并为极面积设定正确的 $\theta$ 限。
当有帮助时消去参数以恢复普通的 $y$-对-$x$ 形状。

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