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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
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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 • g(t) = F(f(t))
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)
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
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)
vertical tangent t = 0 (0, 0).
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.
Example 1 (d) • A curve C is defined by x = t2, y = t3– 3t. Sketch the curve.
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
θ = π/3 Tangent line
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).
x = r(θ– sin θ), y = r(1 – cos θ ) Vertical tangent dy/dx = ∞ 1- cosθ = 0 θ = 2nπ, n an integer (2nπ, 0).
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)