10.1 Curves defined by parametric equations goo.gl/Xc8nkW 1
Chapter 10
Parametric Equations and
Polar Coordinates
10.1
Curves defined by parametric equations (page
640)
4rkcCjg732g 將曲線用參數式的 方法表達是從歐拉 那 個 時 候 開 始 的, 起初是把 t 想成 時間, 而曲線視為 質點隨時間變化時 運動的軌跡。 到後 期, 數學上再做抽 象化的結果, t 不 見 得 要 和 時 間 對 應, 它就只是一個 參數 (實數軸上的 一個變數)而已。 各位首先要學習的 是把以前所學的基 本曲線, 像是直線 或是圓錐曲線等改 用參數式表達。Definition 1 (page 640). Suppose that x and y are both given as functions of a third variable t (called a parameter, 參數) by the equations
x = f (t), y = g(t),
(called parameter equations, 參數方程). Each value of t determines a point (x, y), which we can plot in a coordinate plane. As t varies, the point (x, y) = (f (t), g(t)) varies and traces out a curve C, which we call a parametric curve (參數曲線).
Sometimes t can be realized as “time” and we can interpret (x, y) = (f (t), g(t)) as the position of a particle at time t, but in many cases, t does not necessarily represent time, it is just a variable.
Example 2. How do we express the following curves by parametric equations? Curve Parametric Equation
Straight line passing through (x0, y0)
( x = y =
Circle with center (x0, y0) and radius r
( x = y = Ellipse (x − x0) 2 a2 + (y − y0)2 b2 = 1 ( x = y = Parabola (x − x0)2 = 4p(y − y0) ( x = y = Hyperbola (x − x0) 2 a2 − (y − y0)2 b2 = 1 ( x = y = FeBAvGJnKE0 這幾個例子雖然軌 跡都是圓, 但是不 同的參數表達會有 些微的差別, 試以 質點運動軌跡的想 法徹底理解參數式 的表示與圖形的關 係。
Example 3. Compare the following parametric equations: (a) (x, y) = (cos t, sin t), 0 ≤ t ≤ 2π.
(b) (x, y) = (cos t, sin t), 0 ≤ t ≤ 4π. (c) (x, y) = (cos 2t, sin 2t), 0 ≤ t ≤ π. (d) (x, y) = (sin t, cos t), 0 ≤ t ≤ 2π.
2 10.1 Curves defined by parametric equations goo.gl/Xc8nkW
Example 4 (page 642). Sketch the curve x = sin t, y = sin2t. Solution.
The Cycloid, page 643
kIs5L1BDdOU 學習參數式最標準 的模型是擺線, 各 位首先要會將擺線 的 參 數 式 確 實 表 達, 之後會從擺線 開始探討微積分相 關的理論。
Example 5 (page 643). The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line is called a cycloid (擺線). See Figure 1.
x y
O
r θ
Figure 1: The cycloid.
If the circle has radius r and rolls along the x-axis and if one position of P is the origin, find parametric equations for the cycloid.
Solution.
There are many interesting problems related to cycloids.
Brachistochrone Problem (
最速降線問題
), page 644vwA58B0iXZs 擺線在數學的發展 上具有很重要的意 義, 如何製造一個 溜滑梯軌道在磨擦 力忽略不計的情況 下質點從頂部沿軌 道 最 快 到 達 底 部, 稱 為 最 速 降 線 問 題, 可用微積分的 方法推出這個軌道 是擺線。
Find the curve along which a particle will slide in the shortest time (under the influence of gravity) from a point A to a lower point B not directly beneath A.
A
B Figure 2: Brachistochrone Problem.
The Swiss mathematician John Bernoulli, who posed this problem in 1696, showed that among all possible curves that join A to B, the particle will take the least time sliding from A to B if the curve is part of an inverted arch of a cycloid.
10.1 Curves defined by parametric equations goo.gl/Xc8nkW 3
Tautochrone Problem (
等時降線
), page 644The Dutch physicist Huygens had already shown that the cycloid is also the solution to the tautochrone problem: no matter where a particle P is placed on an inverted cycloid, it takes the same time to slide to the bottom.
擺線的另一個重要 意 義 是 等 時 降 線: 在擺線上任意一處 放置球, 而球沿擺 線的軌道滾動至底 部 所 花 的 時 間 相 同。 P P P P Figure 3: Tautochrone Problem.
Huygens proposed that pendulum clocks should swing in cycloidal arcs because then the pendulum would take the same time to make a complete oscillation whether it swings through a wide or a small arc.
Graphing Devices, page 644
We can use graphing devices to sketch complicated curves. The curves shown in Figure 4 are almost impossible to produce by hand.
各位小時候應該有 玩過繁花規, 就是 一枝筆插在一個圓 形的洞洞板中繞著 另一個較大的圓形 滾動就會畫出像是 左圖的曲線。 你會 看到左圖的三條曲 線參數式都是由正 弦函數與餘弦函數 組成, 但是振幅與 頻率的不同就會造 出曲線的多樣性。 -1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 -2 -1 1 2 -1 -0.5 0.5 1
Figure 4: (a) x = sin t + 12cos 5t + 14sin 13t, y = cos t + 12 sin 5t + 14cos 13t, t ∈ [0, 2π]. (b) x = sin t +12sin 5t +14cos 2.3t, y = cos t +12 cos 5t +14sin 2.3t, t ∈ [0, 20π]. (c) x = sin t − sin 2.3t, y = cos t, t ∈ [0, 20π].
4 10.2 Calculus with Parametric Curves goo.gl/XrNZiR
10.2
Calculus with Parametric Curves (page 649)
We now apply the methods of calculus to parametric curves. In particular, we solve problem involving tangents, area, arc length, surface area, and volume.
Tangents, page 649
htjjuSTc3p4 這裡要了解的是當 曲線以參數式表達 時, y′(x) 或切線 斜率的找法。 注意 到 y′′(x) 的結果 較為複雜。Suppose f and g are differentiable functions and we want to find the tangent line at a point on the curve x = f (t), y = g(t), where y is also a differentiable function of x. Then the Chain Rule gives dy dt = dy dx dx dt ⇒ dy dx = dy dt dx dt if dx dt 6= 0.
We can compute the second derivative dxd2y2 as follows:
d2y dx2 = d dx dy dx = d dt dy dx dx dt = d dt dy dt dx dt dx dt = dx dt d2 y dt2 − dy dtd 2x dt2 dx dt 3
特別注意: d 2y dx2 6= d2 y dt2 d2x dt2 。 以擺線為例, 計算 在θ= π 3 處的切 線方程式。 並觀察 擺線具有水平切線 及鉛直切線的所在 位置。 Example 1 (page 650).(a) Find the tangent line to the cycloid x = r(θ − sin θ), y = r(1 − cos θ) at the point where θ = π3.
(b) At what points its tangents horizontal? When is it vertical? Solution.
10.2 Calculus with Parametric Curves goo.gl/XrNZiR 5
Areas, page 651
c21vmKnvk7E 因為函數的積分對 應到的幾何意義是 有方向有符號的面 積, 所以當曲線用 參數式表達時, 若 要 呈 現 面 積 大 小 時, 必須和圖形對 應, 上限與下限的 對應遵照其幾何意 義決定。We know that the area under a curve y = F (x) from a to b is A =Rb
aF (x) dx,where F (x) ≥ 0.
If the curve is traced out once by the parameter equations x = f (t) and y = g(t), α ≤ t ≤ β, then we can calculate an area formula by using the Substitution Rule for Definite Integrals as follows: A = Z b a y dx = Z β α g(t)f′(t) dt or Z α β g(t)f′(t) dt.
Example 2 (page 651). Find the area under one arch of the cycloid x = r(θ − sin θ), y = r(1 − cos θ).
Solution.
Arc Length, page 652
un2lblg-zFM 至於曲線用參數式 表 達 若 要 算 曲 線 長, 因為積分的函 數恆正, 所以積分 下限與上限就是由 小到大, 得到的結 果也會是正的值。
We already know how to find the length L of a curve C given in the form y = F (x), a ≤ x ≤ b. If F′(x) is continuous, then L = Z b a s 1 + dy dx 2 dx.
Suppose that C can also be described by the parametric equations x = f (t) and y = g(t), α ≤ t ≤ β, where dxdt = f′(t) > 0. This means that C is traversed once, from left to right, as t increases from α to β and f (α) = a, f (β) = b. Then we obtain
L = Z b a s 1 + dy dx 2 dx = Z β α s 1 + dy/dt dx/dt 2 dx dt dt = Z β α s dx dt 2 + dy dt 2 dt.
The above formula is generally true even if C can’t expressed in the form y = F (x).
Theorem 3 (page 649). If a curve C is described by the parametric equations x = f (t), y = g(t), α ≤ t ≤ β, where f′ andg′ are continuous on[α, β] and C is traversed exactly once as t increases from α to β, then the length of C is
L = Z β α s dx dt 2 + dy dt 2 dt.
6 10.2 Calculus with Parametric Curves goo.gl/XrNZiR
Example 4 (page 653). Find the length of one arch of the cycloid x = r(θ − sin θ), y = r(1 − cos θ). 在看影片之前可先 猜一猜擺線一拱的 弧長, 然後再透過 影片的學習確定你 的直覺與結果是否 一致。 Solution.
Surface Area, page 654
VRDOsKPystk 若曲線用參數式表 達, 要計算旋轉體 體 積 時, 也 是 按 照第八章的概念列 式, 積分的下限與 上限也是由小到大 (遵照 曲 線 弧 長 的 概念而得)。
In the same way as for arc length, we can obtain a formula for surface area. If the curve given by the parametric equations x = f (t), y = g(t), α ≤ t ≤ β, is rotated about the x-axis, where f′, g′ are continuous and g(t) ≥ 0, then the area of the resulting surface is given by
S = Z β α 2πy s dx dt 2 + dy dt 2 dt.
The general symbolic formulas S =R 2πy ds and S = R 2πx ds are still valid, but for para-metric curves we use
ds = s dx dt 2 + dy dt 2 dt.
Example 5 (page 654). Show that the surface area of a sphere of radius r is 4πr2.
習學如何用微積分 計算球的表面積。
Solution.
Volume
10.3 Polar Coordinates goo.gl/aV44Z4 7
10.3
Polar Coordinates (page 658)
-PLImkGhcxE 認識極坐標。 平面 中的點之所在位置 有 多 種 方 法 呈 現, 平常較為熟悉的是 直角坐標, 而極坐 標是改用極點到點 的有向距離 r 以 及廣義角 θ 這兩 個數字表達點的位 置。 注意到這裡的 r 可正可負, 而極 坐標是一個多值的 對應關係。
A coordinate system represents a point in the plane by an ordered pair of numbers. We usually use Cartesian coordinates (笛卡爾坐標, 直角坐標), which are directed distances from two perpendicular axes. Here we describe another coordinate system introduced by Newton, called the polar coordinate system (極坐標).
x x polar axis polar axis r r P(r, θ) P(r, θ) Q(−r, θ) = (r, θ + π) θ θ θ+ π O O
Figure 1: Polar coordinate system. We choose a point in the plane that is called the pole and is labeled O. Then we draw a ray starting at O called polar axis, which is usually corresponds to the positive x-axis in Cartesian coordinates.
Here are some remarks about the polar coordinate system.
• If P is any other point in the plane, let r be the distance from O to P and let θ be the angle between the polar axis and the line OP . Then the point P is represented by the ordered pair (r, θ) are called polar coordinates of P .
• We use the convention that an angle is positive if measured in the counterclockwise direction from the polar axis and negative in the clockwise direction. If P = O, then r = 0 and we agree that (0, θ) represents the pole for any value of θ.
• The points (−r, θ) an (r, θ) lie on the same line through O and at the same direction |r| from O, but on opposite sides of O. If r > 0, the point (r, θ) lies in the same quadrant as θ; if r < 0, it lies in the quadrant on the opposite side of the pole.
• Notice that (−r, θ) represents the same point as (r, θ + π).
• The connection between polar and Cartesian coordinates: 直角坐標與極坐標
的轉換關係必須確 實了解。 (a) cos θ = x r, sin θ = y r. (角度和直角坐標與半徑的關係) (b) x = r cos θ, y = r sin θ. (直角坐標用極坐標表達) (c) r2= x2+ y2, tan θ = y x. (極坐標可以用直角坐標表達)
8 10.3 Polar Coordinates goo.gl/aV44Z4
Polar Curves, page 660
r6H4TSozfkM 熟悉一些極坐標方 程 式 對 應 的 圖 形。 有些極坐標方程式 表達看似簡單, 但 是圖形卻不是那麼 好想像, 可利用數 學繪圖軟體多多體 會。
The graph of a polar equation (極坐標方程式) r = f (θ), or more generally F (r, θ) = 0, consists of all points P that have at least one representation (r, θ) whose coordinates satisfy the equation.
Example 1 (page 660-662). Plot the following curves represented by the polar equation. (a) r = 2. (b) θ = π4. (c) r = 2 cos θ. (d) r = 1 + sin θ. (e) r = cos 2θ. Solution.
Symmetry, page 663
4vJ_5RFWMHM 善用圖形的對稱性 還有方程式的平移 縮放旋轉理論可以 幫助我們把一些曲 線做連結。 特別是 (d)的部份我覺得 很值得仔細思考。When we sketch polar curves it is sometimes helpful to take advantage of symmetry.
(a) If a polar equation is unchanged when θ is replaced by −θ, the curve is symmetric about .
(b) If the equation is unchanged when r is replaced by −r, or when θ is replaced by θ + π, the curve is symmetric about .
(c) If the equation is unchanged when θ is replaced by π − θ, the curve is symmetric about .
10.3 Polar Coordinates goo.gl/aV44Z4 9
Tangents to Polar Curves, page 663
Uj7fqwrHBFo 當曲線用極坐標方 程 式 表 達 的 時 候, 把 θ 想成是參數, 再用直角坐標與極 坐標 的 轉 換 關 係, 可 以 得 到 曲線 的 參數式表達, 這麼 一 來 就 可 以 用 參 數 式 的 概 念 計 算 y′(x)。
To find a tangent line to a polar curve r = f (θ), we regard θ as a parameter and write its parametric equations as
(
x = r cos θ = f (θ) cos θ y = r sin θ = f (θ) sin θ.
Using the method for finding slopes of parametric curves and the Product Rule, we have dy
dx = (1)
• Horizontal tangents: (provided that dxdθ 6= 0).
• Vertical tangents: (provided that dydθ 6= 0). • Tangent lines at the pole: we put r = 0 into formula (1) and get
dy dx= 9Ex-VwbwLRI 心臟線是研究極坐 標方程式的標準模 型, 藉此熟悉極坐 標並推得其切線方 程。 Example 2 (page 664).
(a) For the cardioid r = 1 + sin θ, find the slope of the tangent line when θ = π3. (b) Find the points on the cardioid where the tangent line is horizontal or vertical.
10 10.3 Polar Coordinates goo.gl/aV44Z4
Graphing Polar Curves with graphing Devices
我們可用數學繪圖 軟體幫助了解極坐 標 方 程 式 的 圖 形, 下方有幾個很有特 色的曲線, 各位不 妨將方程式輸入到 數學繪圖軟體看看 它的長相。
We can use graphing devices, Desmos Calculator for example, to sketch complicated curves. The curves shown in Figure 2 are almost impossible to produce by hand.
-1 -0.5 0.5 1 -1 -0.5 0.5 1 -2 -1 1 2 -1.5 -1 -0.5 0.5 1 1.5 -1 -0.5 0.5 1 -1 -0.5 0.5 1
Figure 2: (a) r = sin2(2.4θ)+cos4(2.4θ), θ ∈ [−2π 2.4 ,
22π
2.4]. (b) r = sin
2(1.2θ)+cos3(6θ), θ ∈
[0, 6π] (c) r = sin 85θ , θ ∈ [0, 10π].
Some interesting curves and their polar equations.
(a) r = a sin(bθ): rose or rhodonea curve (玫瑰線).
(b) r = a + bθ: Archimedean spiral (阿基米德螺線;等速螺線). (c) r = aebθ: logarithmic spiral (對數螺線).
(d) r2 = sin 2θ: lemniscate (雙紐線).
(e) r = esin θ− 2 cos(4θ): butterfly curve (蝶形線). (f) r = 1 + c sin θ: limacons de Pascal. (帕斯卡蝸線). (g) r = 1 + 2 sin(θ2): nephroid of Freeth.
(h) r =p1 − 0.8 sin2θ: hippopede. (i) r = | tan θ|| cot θ|: Valentine curve.
10.4 Areas and Length in Polar Coordinates goo.gl/j1z6kd 11
10.4
Areas and Length in Polar Coordinates (page
669)
In this section, we develop the formula for the area of the region whose boundary is given by a polar equation. xNsXx2N01hA 我們可以用極坐標 方程式計算區域面 積, 若 是 對 角 度 進行分割, 再取樣 本 點, 透過 扇 形 的面積加總後取極 限, 也可以得到面 積的積分公式。
Example 1 (page 669). The area of a sector of a circle with the radius r and the radian θ is A = .
Example 2 (page 669). Find the area of a region R bounded by the polar curve r = r(θ) and by rays θ = a and θ = b, where r(θ) is a positive continuous function and 0 < b − a ≤ 2π.
Solution.
Example 3 (page 670). Find the area enclosed by one loop of the four-leaved rose r = cos 2θ. Solution. 更一般地, 當區域 是由兩個極坐標方 程 式 圍 住 時, 其 區域面積也可以寫 出, 基本上就是大 扇形面積減掉小扇 形面積的概念。
Example 4(page 671). Find the area of the region R bounded by curves with polar equations r = f (θ), r = g(θ), θ = a, and θ = b, where f (θ) ≥ g(θ) ≥ 0 and 0 < b − a ≤ 2π.
12 10.4 Areas and Length in Polar Coordinates goo.gl/j1z6kd
To find all points of intersection of two polar curves, it is recommended that you draw the graphs of both curves.
Example 5 (page 671). Find all points of intersection of the curves r = cos 2θ and r = 12.
4XFn7sacQDY 兩個極坐標方程式 的交點實際上是一 個非 常 困 難 的 問 題, 這是因為極坐 標是多值函數。 現 在我們有數學繪圖 軟體的輔助, 可以 幫助我們更容易了 解 交 點 的 所 在 位 置。 Solution.
Arc Length, page 671
HZEzEdYxZm4
曲線弧長的計算方 式, 也是將問題轉 變成參數式之後再 進行計算。
To find the length of a polar curve r = r(θ), a ≤ θ ≤ b, we regard θ as a parameter and write the parameter equations of the curves as
(
x = r cos θ = r(θ) cos θ y = r sin θ = r(θ) sin θ.
Then using the Product Rule and differentiating with respect to θ, we obtain dx dθ = dy dθ = so dx dθ 2 + dy dθ 2
= (r′(θ))2cos2θ − 2r · (r′(θ))2cos θ sin θ + r2sin2θ + (r′(θ))2sin2θ + 2r · (r′(θ))2sin θ cos θ + r2cos2θ
=
Assuming that f′(θ) is continuous, we can write the arc length as
L = Z b a s dx dθ 2 + dy dθ 2 dθ =
10.4 Areas and Length in Polar Coordinates goo.gl/j1z6kd 13
Example 6 (page 672). Find the length of the cardioid r = 1 + sin θ. 看影片前不妨先猜 一 猜 心 臟 線 的 弧 長, 再透過影片的 解說對照你的直覺 與結果是否一致。
Solution.
Surface Area, page 674
9Ex-VwbwLRI 極坐標方程式的旋 轉 體 表 面 積 公 式, 也是將問題轉化為 參 數 式 之 後 再重 現。
The area of the surface generated by rotating the polar curve r = f (θ), a ≤ θ ≤ b (where f′(θ) is continuous and 0 ≤ a < b ≤ π) about the polar axis is
Surface area =
The area of the surface generated by rotating the polar curve r = f (θ), a ≤ θ ≤ b (where f′(θ) is continuous and 0 ≤ a < b ≤ π) about the line θ = π2 is
Surface area =
