1 / 18

CH 11

CH 11. PARAMETRIC EQUATIONS AND POLAR COORDINATES 參數方程式與極座標. 學習內容. 參數曲線. 11.2 Parametric Curves 11.3 Polar Coordinates. 極座標. 11.2. Parametric Curves 參數曲線. 學習重點. 知道函數的參數式表示法 會求參數式曲線的切線斜率. 函數的表示法. 一般表示法 y = F ( x ) 參數 表示法 x = f ( t ) and y = g ( t ) 參數表示法代入一般式

ryan-mcdaniel
Télécharger la présentation

CH 11

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CH 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES 參數方程式與極座標

  2. 學習內容 參數曲線 • 11.2 Parametric Curves • 11.3 Polar Coordinates 極座標

  3. 11.2 Parametric Curves 參數曲線

  4. 學習重點 • 知道函數的參數式表示法 • 會求參數式曲線的切線斜率

  5. 函數的表示法 • 一般表示法 • y = F(x) • 參數表示法 • x = f(t) and y =g(t) • 參數表示法代入一般式 • g(t) = F(f(t))

  6. 導函數之參數表示法 If g, F, and f are differentiable and g(t) = F(f(t)), then the Chain Rule gives if f’(t) ≠ 0 • g’(t) = F’(f(t))f’(t) = F’(x)f’(t)

  7. 第二階導函數之參數表示法

  8. y = t3 – 3t = t(t2– 3) = 0 when t = 0 or t = ± . Example 1 (a) • A curve C is defined by the parametric equations x = t2, y = t3– 3t. Show that C has two tangents at the point (3, 0) and find their equations. x = t2 = 3, y = t3– 3t = 0 At the point (3, 0)  故曲線在(3, 0)這一點通過兩次

  9. x = t2, y = t3– 3t

  10. Example 1 (b) • A curve C is defined by x = t2, y = t3– 3t. Find the points on C where the tangent is horizontal or vertical. horizontal tangent • t2 = 1  t = ±1  (1, -2) and (1, 2)

  11. vertical tangent t = 0  (0, 0).

  12. Example 1 (c) • A curve C is defined by x = t2, y = t3– 3t. Determine where the curve is concave upward or downward. • The curve is concave upward when t > 0. • It is concave downward when t < 0.

  13. Example 1 (d) • A curve C is defined by x = t2, y = t3– 3t. Sketch the curve.

  14. Example 2 (a) Find the tangent to the cycloid x = r(θ– sin θ), y = r(1 – cos θ ) at the point where θ = π/3. θ = π/3

  15. θ = π/3 Tangent line

  16. Example 2 (b) At what points is the tangent horizontal? When is it vertical? x = r(θ– sin θ), y = r(1 – cos θ ) Horizontal tangent dy/dx = 0  sinθ = 0 and 1 – cos θ≠ 0 θ = (2n– 1)π, n an integer  ((2n – 1)πr, 2r).

  17. x = r(θ– sin θ), y = r(1 – cos θ ) Vertical tangent dy/dx = ∞ 1- cosθ = 0 θ = 2nπ, n an integer  (2nπ, 0).

  18. Q1 Find an equation of the tangent line to the curve x=t sin t, y=t cos t at t =11π. (a) y = 12π + x/(11π) (b) y = -11π + x/(11π) (c) y = 11π + x/(11π) (d) y = -12π + x/(11π)

More Related