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

2 學習內容 11.2 Parametric Curves11.2 Parametric Curves 11.3 Polar Coordinates11.3 Polar Coordinates 參數曲線 極座標

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 g ’ (t) = F ’ (f(t))f ’ (t) = F ’ (x)f ’ (t) if f ’ (t) ≠ 0

7 第二階導函數之參數表示法

8 A curve C is defined by the parametric equations x = t 2, y = t 3 – 3t. Show that C has two tangents at the point (3, 0) and find their equations. Example 1 (a) At the point (3, 0)  x = t 2 = 3, y = t 3 – 3t = 0 y = t 3 – 3t = t(t 2 – 3) = 0 when t = 0 or t = ±. 故曲線在 (3, 0) 這一點通過兩次

9 x = t 2, y = t 3 – 3t

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

11 vertical tangent t = 0  (0, 0).

12 A curve C is defined by x = t 2, y = t 3 – 3t. Determine where the curve is concave upward or downward. Example 1 (c) –The curve is concave upward when t > 0. –It is concave downward when t < 0.

13 A curve C is defined by x = t 2, y = t 3 – 3t. Sketch the curve. Example 1 (d)

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

15 θ = π/3 Tangent line

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

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

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)