CH 11

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)