Chapter 8 Conics, Parametric Equations, and Polar Coordinates (圓錐曲線、參數方程式與極坐標)

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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

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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).

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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.

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Example 2 的示意圖

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Example 3 的示意圖

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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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Example 4 的示意圖

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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)

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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)

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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)

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直角坐標和極坐標的關係

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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

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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

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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

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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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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 .

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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

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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.

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示意圖 (承上例)

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Chapter 8 Conics, Parametric Equations, and Polar Coordinates (圓錐曲線、參數方程式與極坐標)

Full text

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

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

Section 8.1

Plane Curves and Parametric Equations

(平面曲線與參數方程式)

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).

Section 8.2

Parametric Equations and Calculus

(參數方程式及其微積分)

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.

Example 2 的示意圖

Example 3 的示意圖

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.

Example 4 的示意圖

Useful Formulas

Recall the following identities for the sine and cosine functions:

cos(α − β) = cos α cos β+ sin α sin β

cos(α+β) = cos α cos β − sin α sin β

sin(α − β) = sin α cos β − cos α sin β

sin(α+β) = sin α cos β+ cos α sin β

Section 8.3

Polar Coordinates and Polar Graphs (極坐標與極坐標圖形)

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.

極坐標的示意圖 (承上頁)

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 .

示意圖 (承上頁)

直角坐標和極坐標的關係

三瓣玫瑰線的動態示意圖

三瓣玫瑰線的動態示意圖

三瓣玫瑰線的動態示意圖

三瓣玫瑰線的動態示意圖

三瓣玫瑰線的動態示意圖

x = f(t) and y = g(t) ∀ t ∈ I, with dx/dt = f ^′ (t) ̸= 0 ∀ t ∈ I , then

dx/dt = g ^′ (t)

f ^′ (t) ≡ m(t) ∀ t ∈ I.

(2) the second derivative is given by d ² y

dx ² = d dx

dx/dt = m ^′ (t)

f ^′ (t) ∀ t ∈ I.

∫ _b

∫ _b

1 + [y ^′ (t) x ^′ (t)

] 2 [ x ^′ (t)dt

√ [x ^′ (t)] ² + [y ^′ (t)] ² dt =

√ [f ^′ (t)] ² + [g ^′ (t)] ² dt.

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.

(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 .

dx/dθ = f(θ) cos θ + f ^′ (θ) sin θ

−f(θ) sin θ + f ^′ (θ) cos θ .

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 .

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

2 [f(θ _i )] ² ∆θ _i ,

where ∆θ _i ≡ θ i − θ i −1 for i = 1, 2, . . . , n.

∫ _β

[f(θ)] ² dθ.

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

∫ _β

r ² + ( dr dθ

) ₂ dθ =

∫ _β

√ [f(θ)] ² + [f ^′ (θ)] ² dθ.

dθ = −f(θ) sin θ + f ^′ (θ) cos θ and dy

dθ = f(θ) cos θ + f ^′ (θ) sin θ.

[ − f(θ) sin θ + f ^′ (θ) cos θ ] 2

[ f(θ) cos θ + f ^′ (θ) sin θ ] 2

= [f(θ)] ² + [f ^′ (θ)] ² .

∫ _β √ ( dx

dθ ) ² + ( dy

dθ ) ² dθ =

∫ _β √

[f(θ)] ² + [f ^′ (θ)] ² dθ.