9.1 (10.1) Parametric Curves ( 參數曲線 )

Transcription

1 9.1 (10.1) Parametric Curves ( 參數曲線 ) [Ex]Sketch and identify the curve defined by parametric equations x= 6 t, y = t, t 4 (a) Sketch the curve by using the parametric equations to plot points: (b) Eliminate the parameter to find a Cartesian equation ( 笛卡兒方程式 ) of the curve: Substituting into the expression for x, we have t = y x= 6 t = 6 ( y) = 6 4y x = 6 4 y So the curve represented by the given parametric equations is. The point (,-1) is called the initial point and (-10,) is called the terminal point. 起點終點

2 [Ex] What curve is represented by the parametric equations 0 t? x = cos t, y = sin t, (a) Sketch the curve by using the parametric equations to plot points: t x y 0 0 /3 1 / / 0 - (b) Eliminate the parameter to find a Cartesian equation of the curve: Observe that x y ( t ) ( t ) + = cos + sin = 4 So the curve represented by the given parametric equations is the unit circle 單位圓. x + y =4 4

3 [Ex]What curve is represented by the parametric equations 4 (a) x = t 1, y = t and (b) x = t 1, y = t. (a) (b) ( x+ 1) = y+ ( x+ 1) = y+, x 1

4 [Ex] Find parametric equations for the portion of the parabola y = x from (-1, 1) to (3, 9). Any equation of the form y = f (x) can be converted to parametric form by simply letting x equal t. Here, this gives us y = x = t. x= t y = t t,, for 1 3. Then is a parametric representation of the curve. [Ex5] [Ex5] Sketch the curve with parametric equations x = sin t, y = sin t. ( sin ) y = t = x So the curve represented by the given parametric equations is the parabola. y = x, 1 x 1

5 Exercise Sketch the curve with parametric Equations (a) x = cos t, y = 3sin t, (b) x = +4cos t, y = 3 + 4sin t and (c) x = 3cos t, y = 3sin t all for 0 t. (a) x =cost t, y =3sint t (b) x = +4cos t, y =3+4sint t (c) x =3cost t, y =3sint x y + = 1 3 ( x ) + ( y 3) = 16 ( x ) + ( y ) = 9

6 The Cycloid ( 擺線 ) [Ex7][Ex7] The curve traces out ( 描繪 ) by a point on the circumference ( 圓周 ) of a circle as the circle rolls ( 滾動 ) along a straight line is called a cycloid, as shown below. If the circle has radius r and rolls along the x-axis and if one position of P is the origin, find parametric equations for the cycloid. ( sin ) x = rθ rsinθ = r θ θ ( ) y = r rcosθ = r 1 cos θ, θ R

7 9. (10.) Calculus with Parametric Curves ( 參數曲線的微積分 ) Tangents ( 切線 ) dy dy d dy dt =, if 0 d y d dy dt =, if 0 dt = dt dt dt [Ex1][Ex1] 3 A curve C is defined by the parametric equations x = t y = t 3t. (a) Show that C has tangents at the point (3,0) and find their equations. (b) Find the points on C where the tangent is horizontal or vertical. (c) Determine where the curve is concave upward or downward. (d) Sketch the curve.,

8 (a) { x = t =3 dy dy dt 3t = 3 t = ± 3 and = = t y = t 3t = 0 dt t t dy ± 6 = = ± 3. So the equations of the tangents at (3,0) are t=± 3 3 y = 3 ( x 3) and y = 3 ( x 3. ) (b) Horizontal tangent: dy = 0 dy dt = 0 t = ± 1 So the corresponding points ( 相對應的點 ) on C are (1,-) and (1,). Vertical tangent: dt = t = 0 t = 0 So the corresponding points on C is (0,0). (c) d dy 3 1 (d) d y dt 1 + = t = t dt t < 0 t > 0 d y + the curve C C.D C.U +

9 [Ex][Ex] (a) Find the tangent to the cycloid x = r ( θ sinθ ) and y = r ( 1 cosθ ) at the point where θ = 3. (b) At what point is the tangent horizontal? When is it vertical? (a) dy dy dθ r sinθ sinθ dy = = = = dθ r (1 cos θ ) (1 cos θ ) θ= 3 When θ = 3, we have x = r( 3 3 ) and y = r. So the equation of the tangent is y r = 3( x r( 3 3 )) 3 (b) horizontal tangent: dy = 0 sinθ = 0 and 1 cosθ 0 θ = (n + 1) So the corresponding points are (( n + 1) r, ) r. vertical tangent: dθ = 0 θ = n dy sinθ cosθ dy lim = lim = lim = and lim = θ n + θ n + (1 cosθ ) θ n + sinθ θ n So the corresponding points are ( n r,0).

10 Exercise Find the slope of the tangent line to the path of the Scrambler x = cos t +sint t, y = sin t + cos t at (a) t = 0 ; (b) t = 4 and (c) the point (0, -3). Ans: (a) dy t= (b) dy 0 = 1 = t= 4 (c) θ lim ( 3 ) dy = This says that the slope of the tangent line at (0,-3) is undefined. The tangent line at the point (0, -3) is vertical.

11 Areas b The area under the curve y = F ( x ) from a to b is A =. F ( x) a If x = f(), t y = g() t, and the curve is traversed ( 橫越 ) once as t increases from α to β, then b A = y = g () t f () t dt a A = y = g () t f () t dt β α b α a ( ( ), ( )) β if ( f ( α ), g( α )) is the leftmost endpoint. if f β g β is the leftmost endpoint. [Ex3][Ex3] Find the area under one arch ( 拱形 ) of the cycloid x = r( θ sin θ) y = r( 1 cosθ) r A y = r(1 cos θ ) r(1 cosθ ) dθ = 0 0 ( ) = r (1 cos θ ) dθ = r 1 cosθ + cos θ dθ = r 1 cos θ + (1 + cos θ ) dθ = r θ sinθ + sinθ = r = 3r 4 0,.

12 Arc length Suppose that C is given by the parametric equations x = f(), t y = g() t, α t β and dt = f () t > 0, then the arc length of C L β dy = ds = dt α + dt dt, where a = f ( α ), b = f ( β ). [Ex5][Ex5] g y x = r ( θ sin θ ), y = r ( 1 cosθ ) Find the length of arch of the cycloid,. dy L = + d = (1- r cos + r sin d 0 θ θ θ θ dθ dθ ( )) ( ) 0 = r( 1 cosθ + cos θ + sin θ) dθ = r ( 1 cosθ) dθ 0 0 = r 4sin ( θ ) dθ = r sin( θ ) dθ 0 0 [ θ ] 0 = r cos( ) = r(+ ) = 8r

13 [Ex]Find the arc length of the plane curve x = cos 5t, y = sin 7t, for 0 t. Lissajous curve

14 9.3 (10.3) Polar Coordinates r ( ) P(, r θ ) = P x, y i y r = x + y x = rcosθ y = rsinθ O θ x tan θ = y x r = [Ex4][Ex4] What curve is represented by the polar equation 極方程? [Ex] Sketch the polar curve θ = 3.

15 [Ex6][Ex6] (a) Sketch the curve with polar equation r = cos. θ (b) Find a Cartesian equation for this curve. (a) Sketch the curve by using the parametric equations to plot points: (b) r = cosθ r = rcosθ x + y = x Completing the square ( 配方 ), we have ( x 1) + y = 1 which is an equation of a circle with center (1,0) and radius 1.

16 [Ex7][Ex7]Sketch the curve r = 1+ sinθ. [Ex8][Ex8]Sketch ] the curve r = cos θ.

17 [Ex] Sketch the curve r = θ. [Ex] Sketch the curve r = 3+ cosθ. [Ex] Sketch the curve r = 1 sin θ.

18 The graph of r = 1+ csinθ

19 Tangents to Polar Curves ( 極座標曲線的切線 ) dy dy dr sinθ + r cosθ = dθ = dθ, if 0 dr cosθ r sinθ d θ dθ dθ [Ex9][Ex9] (a) For the cardioid ( 心臟線 ) r = 1+ sinθ of Example 7, find the slope of the tangent line when θ = 3. (b) Find the points on the cardioid where the tangent is horizontal or vertical. Since r = 1+ sinθ, we have x= rcos θ = (1 + sin θ) cosθ and y = r sin θ = (1 + sin θ )sin θ,so dy d ((1 + sin θ)sin θ) dy dθ cos sin (1 sin )cos (1 sin )cos = dθ θ θ + + θ θ + θ θ = = = d ((1+ sin θ)cos θ) cosθ cos θ (1 + sin θ)sin θ (1 + sin θ)(1 sin θ) dθ dθ

20 (a) The slope of the tangent at the point where θ = 3 is dy cosθsin θ + (1 + sin θ) cosθ = = 1 θ= 3 cosθcos θ (1 + sin θ)sinθ θ= 3 (b) Observe that dy (1 sin θ)cosθ 0 θ,,, dθ = + = = (1 sin θ)(1 sin θ) 0 θ,, 6 6 dθθ = + = = ( ) ( ) ( ) ( ) So, there are horizontal tangents at,,1, 7 6,1,11 6 and vertical tangents at (3, 6) and 3,5 6. Besides, dy (1 + sin θ ) cosθ 1 sinθ lim = lim lim = lim = (1 sin θ ) (1+ sin θ) 3 cosθ θ (3 ) θ (3 ) θ (3 ) θ (3 ) Similarly, lim dy 1 sinθ = lim = 3 cosθ + + θ (3 ) θ (3 ) ( ) Thus, there is a vertical tangent at 0,3 which is the pole.

21 Exercise Find the slope of the tangent line to the three-leaf rose ( 三瓣玫瑰線 ) r at θ = 0and θ = 4. = sin 3θ Ans: (1) dy θ = 0 = 0 () dy θ = 4 1 =

22 9.4 (10.4) Areas and Lengths in Polar Coordinates ( 極座標曲線的面積與弧長 ) A 1 1 θ θ θ dθ n b * lim ( f( i )) Δ = ( f( )) n 1 a i=1 = = The area of the region bounded by the polar curve r = f ( θ ) and the rays ( θ ) θ θ, 0, = a and = b where f and 0< b a, is b 1 b 1 A = ( f ( θ )) dθ or A = rdθ a a

23 [Ex1][Ex1] Find the area enclosed by one loop of the four-leaved rose r = cos θ. Let r = 0 i.e. cosθ = 0 θ =, θ = 4 and θ = A = rd = ( cos ) d 4 4 θ θ θ = ( cos ) d = ( 1+ cos 4 ) d sin 4 = sin 4 θ + θ = θ θ θ θ Exercise Find the area enclosed by one loop of the three-leaved rose r = sin 3θ. Ans: 1

24 [Ex][Ex] Find the area of the region that lies inside the circle r = 3sinθ and outside the cardioid r = 1+ sinθ. r = 3sin θ 1 5 sinθ = θ = or θ = r = 1+ sinθ A= ( 3sin θ ) ( 1 + sin θ ) d θ = ( 8sin 1 sin ) d θ θ θ = ( ) 3 4cosθ sinθ dθ 6 1 = 3 θ sin θ + cos θ = [ ] Exercise Find the area of the region that lies inside the circle r = and outside the cardioid r = 3+ cosθ Ans: 3

25 [Ex3][Ex3] Find all points of intersection of the curves r = cos θ and r = 1. (1) r r = cos θ cos θ = θ =,,, = () So there are four points of intersection: ,,,,,,, r = cos θ 1 cosθ = θ = r = 1 4 5,,, So there are another four points of fintersection: ti ,,,,,,,

26 Arc length The length of a polar curve r = f( θ ), a θ b, is b b dy dr L = d r d a + θ = + θ dθ dθ a dθ f, if is continuous. [Ex4][Ex4] Find the length of the cardioid r = 1+ sinθ. dr L = () r + dθ 0 dθ = ( 1+ si nθ ) + ( cosθ ) dθ Exercise 0 = + sin θ d θ = = 8 0 Set up the integral for the arc length of the curve r = sin 3. θ