Chapter 8 Conics, Parametric Equations, and Polar Coordinates - PowerPoint PPT Presentation

. . . . . . . . . . . . . . . Chapter 8 Conics, Parametric Equations, and Polar Coordinates Department of Mathematics, National Taiwan Normal University, Taiwan Spring 2019 Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . . . . 1/54 ( 圓錐曲線、參數方程式與極坐標 ) Hung-Yuan Fan ( 范洪源 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . 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 Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . . . . . 2/54 本章預定授課範圍 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . Section 8.1 Plane Curves and Parametric Equations Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . . 3/54 ( 平面曲線與參數方程式 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . and Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . 4/54 . . . . Def. ( 參數曲線的定義 ) Let I be an interval. A plane curve C is often defjned by the graph of the parametric equations ( 參數方程式 ) x = f ( t ) 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 . . . . . . . . . . . . . . . . . . . . . . 5/54 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . Section 8.2 Parametric Equations and Calculus Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . . 9/54 ( 參數方程式及其微積分 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . and dy (2) the second derivative is given by dx dx Chapter 8, Calculus B . . . . . . . . . . . . . 10/54 . . . . . . . . . . . Thm 8.1 ( 參數曲線的微分 ) If C is a smooth curve defjned by x = f ( t ) 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 dx = dy / dt dx / dt = g ′ ( t ) f ′ ( t ) ≡ m ( t ) ∀ t ∈ I . d 2 y = d ( dy / dx )/ dt = m ′ ( t ) ( dy ) dx 2 = d ∀ t ∈ I . dx / dt f ′ ( t ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . 13/54 Example 2 的示意圖 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . Thm 8.2 (Arc Length in Parametric Form) and given by a a Chapter 8, Calculus B . . . . . . . . . . . . . 19/54 . . . . . . . . . . . Let C be a smooth curve defjned by x = f ( t ) y = g ( t ) ∀ t ∈ I = [ a , b ] . If C does not intersect itself on I , then the arc length of C on I is ∫ b ∫ b [ x ′ ( t )] 2 + [ y ′ ( t )] 2 dt = [ f ′ ( t )] 2 + [ g ′ ( t )] 2 dt . √ √ s = Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . Useful Formulas Recall the following identities for the sine and cosine functions: Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . . 23/54 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 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . Section 8.3 Polar Coordinates and Polar Graphs Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . . 24/54 ( 極坐標與極坐標圖形 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . 25/54 Def. ( 極坐標的定義 ) The polar coordinates ( r , θ ) of a point P ( x , y ) ∈ R 2 is defjned 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 . Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . 26/54 極坐標的示意圖 ( 承上頁 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Notes Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . 27/54 . . . . (1) The polar coordinates of the pole is O = (0 , θ ) for any θ ∈ R . (2) The polar coordinates ( r , θ ) and ( r , θ + 2 n π ) represent the same point in R 2 , i.e., ( r , θ ) = ( r , θ + 2 n π ) ∀ n ∈ Z . (3) If r > 0 , then ( − r , θ ) = ( r , θ + (2 n + 1) π ) ∀ n ∈ Z . Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . 28/54 示意圖 ( 承上頁 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 8, Calculus B . . . . . . . . . . . . . . . . . . . . . . 30/54 直角坐標和極坐標的關係 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

Chapter 8 Conics, Parametric Equations, and Polar Coordinates - PowerPoint PPT Presentation

. . . . . . . . . . . . . . . Chapter 8 Conics, Parametric Equations, and Polar Coordinates Department of Mathematics, National Taiwan Normal University, Taiwan Spring 2019 Chapter 8, Calculus B . . . . . . . . . . .

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