Chapter 8
Conics, Parametric Equations, and Polar Coordinates
(圓錐曲線、參數方程式與極坐標)
Hung-Yuan Fan (范洪源)
Department of Mathematics, National Taiwan Normal University, Taiwan
Spring 2019
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本章預定授課範圍
8.1 Plane Curves and Parametric Equations 8.2 Parametric Equations and Calculus 8.3 Polar Coordinates and Polar Graphs 8.4 Area and Arc Length in Polar Coordinates
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 2/54
Section 8.1
Plane Curves and Parametric Equations
(平面曲線與參數方程式)
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Def. (參數曲線的定義)
Let I be an interval. A plane curve C is often defined by the graph of the parametric equations (參數方程式)
x = f(t) and y = g(t) ∀ t ∈ I,
where f and g are conti. functions of t, and t is a parameter (參數).
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 4/54
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 6/54
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 8/54
Section 8.2
Parametric Equations and Calculus
(參數方程式及其微積分)
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Thm 8.1 (參數曲線的微分) If C is a smooth curve defined by
x = f(t) and y = g(t) ∀ t ∈ I, with dx/dt = f ′ (t) ̸= 0 ∀ t ∈ I , then
(1) the slope of C at the point (x, y) is given by dy
dx = dy/dt
dx/dt = g ′ (t)
f ′ (t) ≡ m(t) ∀ t ∈ I.
(2) the second derivative is given by d 2 y
dx 2 = d dx
(dy dx )
= d(dy/dx)/dt
dx/dt = m ′ (t)
f ′ (t) ∀ t ∈ I.
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 10/54
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 12/54
Example 2 的示意圖
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 14/54
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Example 3 的示意圖
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 16/54
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 18/54
Thm 8.2 (Arc Length in Parametric Form) Let C be a smooth curve defined by
x = f(t) and y = g(t) ∀ t ∈ I = [a, b].
If C does not intersect itself on I , then the arc length of C on I is given by
s =
∫ b
a
√
1 + (dy dx
) 2
dx =
∫ b
a
√
1 + [y ′ (t) x ′ (t)
] 2 [ x ′ (t)dt
]
=
∫ b a
√ [x ′ (t)] 2 + [y ′ (t)] 2 dt =
∫ b a
√ [f ′ (t)] 2 + [g ′ (t)] 2 dt.
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 20/54
Example 4 的示意圖
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 22/54
Useful Formulas
Recall the following identities for the sine and cosine functions:
1
cos(α − β) = cos α cos β+ sin α sin β
2
cos(α+β) = cos α cos β − sin α sin β
3
sin(α − β) = sin α cos β − cos α sin β
4
sin(α+β) = sin α cos β+ cos α sin β
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Section 8.3
Polar Coordinates and Polar Graphs (極坐標與極坐標圖形)
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 24/54
Def. (極坐標的定義)
The polar coordinates (r, θ) of a point P(x, y) ∈ R 2 is defined by r =directed distance from the ple (極點) O to P.
θ =directed angle, conterclockwise from the polar axis (極軸)
to the line OP.
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極坐標的示意圖 (承上頁)
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 26/54
Notes
(1) The polar coordinates of the pole is O = (0, θ) for any θ ∈ R.
(2) The polar coordinates (r, θ) and (r, θ + 2nπ) represent the same point in R 2 , i.e., (r, θ) = (r, θ + 2nπ) ∀ n ∈ Z .
(3) If r > 0, then ( −r, θ) = (r, θ + (2n + 1)π) ∀ n ∈ Z .
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示意圖 (承上頁)
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 28/54
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直角坐標和極坐標的關係
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 30/54
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 32/54
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 34/54
三瓣玫瑰線的動態示意圖
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三瓣玫瑰線的動態示意圖
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 35/54
三瓣玫瑰線的動態示意圖
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三瓣玫瑰線的動態示意圖
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 35/54
三瓣玫瑰線的動態示意圖
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三瓣玫瑰線的動態示意圖
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 35/54
Slope and Tangent Lines
Thm 8.5 (Slope in Polar Form)
If f is a diff. function of θ, then the slope of the tangent line to the graph of r = f(θ) at the point (r, θ) is
dy
dx = dy/dθ
dx/dθ = f(θ) cos θ + f ′ (θ) sin θ
−f(θ) sin θ + f ′ (θ) cos θ .
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 37/54
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 39/54
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函數 r = sin θ 的示意圖 (承上例)
The rectangular equation for r = f(θ) = sin θ is given by r = sin θ ⇒ r 2 = r sin θ ⇒ x 2 + y 2 = y
⇒ x 2 + (y − 1
2 ) 2 = ( 1 2 ) 2 .
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 41/54
Section 8.4
Area and Arc Length in Polar Coordinates
(極坐標上的圖形面積與弧長)
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示意圖
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 43/54
Consider a polar region given by
R = {(r, θ) | α ≤ θ ≤ β, 0 ≤ r ≤ f(θ)}
with 0 < β − α ≤ 2π. If the region R is partitioned into n polar sectors (極坐標扇形) by the rays r = θ i (i = 1, 2, . . . , n) with
α ≡ θ 0 < θ 1 < θ 2 < · · · < θ n −1 < θ n ≡ β, then the area of R should be
A = lim
n →∞
∑ n i=1
1
2 [f(θ i )] 2 ∆θ i ,
where ∆θ i ≡ θ i − θ i −1 for i = 1, 2, . . . , n.
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Thm 8.7 (Area in Polar Coordinates)
If f(θ) is conti. on [α, β] with 0 < β − α ≤ 2π , then the area of the polar region
R = {(r, θ) | α ≤ θ ≤ β, 0 ≤ r ≤ f(θ)}
is given by
A = 1 2
∫ β
α
[f(θ)] 2 dθ.
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 45/54
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 47/54
示意圖 (承上例)
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Arc Length in Polar Form
Thm 8.8 (Arc Length of a Polar Curve)
If f(θ) and f ′ (θ) are conti. on [α, β] with 0 < β − α ≤ 2π , then the arc length of a polar curve r = f(θ) from θ = α to θ = β is given by
s =
∫ β
α
√
r 2 + ( dr dθ
) 2 dθ =
∫ β
α
√ [f(θ)] 2 + [f ′ (θ)] 2 dθ.
Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 49/54
pf: From the proof of Thm 8.5, we know that
dx
dθ = −f(θ) sin θ + f ′ (θ) cos θ and dy
dθ = f(θ) cos θ + f ′ (θ) sin θ.
Then we immediately get (dx
dθ ) 2
+ (dy dθ
) 2
=
[ − f(θ) sin θ + f ′ (θ) cos θ ] 2
+
[ f(θ) cos θ + f ′ (θ) sin θ ] 2
= [f(θ)] 2 + [f ′ (θ)] 2 .
So, the arc length of the polar curve r = f(θ) is given by
s =
∫ β √ ( dx
dθ ) 2 + ( dy
dθ ) 2 dθ =
∫ β √
[f(θ)] 2 + [f ′ (θ)] 2 dθ.
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 51/54
心臟線的示意圖 (承上例)
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Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 8, Calculus B 53/54