1 CH 11 CH 11 PARAMETRIC EQUATIONS PARAMETRIC EQUATIONS AND POLAR COORDINATES AND POLAR COORDINATES 參參參參參參參參參 參參參參參參參參參
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CH 11CH 11CH 11CH 11
PARAMETRIC EQUATIONS PARAMETRIC EQUATIONS AND POLAR COORDINATESAND POLAR COORDINATES參數方程式與極座標參數方程式與極座標
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學習內容• 11.2 11.2 Parametric Curves Parametric Curves • 11.3 Polar Coordinates11.3 Polar Coordinates
參數曲線極座標
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11.211.211.211.2
Parametric CurvesParametric Curves參數曲線
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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
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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))
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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)
'( )'( )
'( )
g tF x
f t
if f’(t) ≠ 0
if 0
dydy dxdt
dxdx dtdt
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第二階導函數之參數表示法
2
2
d dyd y d dy dt dx
dxdx dx dxdt
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•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.
Example 1 (a)
At the point (3, 0) x = t2 = 3, y = t3 – 3t = 0
y = t3 – 3t = t(t2 – 3) = 0 when t = 0 or t = ± .3
故曲線在 (3, 0)這一點通過兩次
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2/ 3 3 3 1
/ 2 2
dy dy dt tt
dx dx dt t t
33
1-3
2
3
dx
dy
3t
3-3
1--3-
2
3
dx
dy
3-t
3-x3y
3-x3-y
x = t2, y = t3 – 3t if 0
dydy dxdt
dxdx dtdt
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•A curve C is defined by x = t2, y = t3 – 3t. Find the points on C where the tangent is horizontal or vertical.
Example 1 (b)
2/ 3 3 3 1
/ 2 2
dy dy dt tt
dx dx dt t t
horizontal tangent
t
1-t
2
3 2
0dx
dy t2 = 1 t = ±1 (1, -2) and (1, 2)
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vertical tangent
t
1-t
2
3
dx
dy 2
t = 0 (0, 0).
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•A curve C is defined by x = t2, y = t3 – 3t. Determine where the curve is concave upward or downward.
Example 1 (c)
22 2
2 3
3 11 3 122 4
d dytd y dt dx t
dxdx t tdt
–The curve is concave upward when t > 0.–It is concave downward when t < 0.
2
2
d dyd y d dy dt dx
dxdx dx dxdt
t
1-t
2
3 2
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•A curve C is defined by x = t2, y = t3 – 3t. Sketch the curve.
Example 1 (d)
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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)
/ sin sin
/ 1 cos 1 cos
dy dy d r
dx dx d r
1
2
sin / 3 3 / 23
1 cos / 3 1
dy
dx
θ = π/3
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3sin
3 3 3 2
1 cos3 2
x r r
ry r
θ = π/3
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2 3 2
or
3 23
r r ry x
x y r
Tangent line
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At what points is the tangent horizontal? When is it vertical?
Example 2 (b)
Horizontal tangent
cos-1
sin
dx
dy
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 )
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dy/dx = ∞ 1- cosθ = 0 θ = 2nπ, n an integer (2nπ, 0).
Vertical tangent
cos-1
sin
dx
dy x = r(θ – sin θ), y = r(1 – cos θ )
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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)