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# 39 parametric equations

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### 39 parametric equations

1. 1. Parametric Equations
2. 2. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)).
3. 3. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter.
4. 4. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9.
5. 5. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5
6. 6. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5
7. 7. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5
8. 8. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5
9. 9. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5
10. 10. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5
11. 11. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5
12. 12. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5 t = 0, (0,-4)
13. 13. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5 t = 0, (0,-4) t = 1, (1,-3)
14. 14. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5 t = 0, (0,-4) t = 1, (1,-3) t = 4, (2,0)
15. 15. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5 t = 1, (1,-3) t = 0, (0,-4) t = 4, (2,0) t = 6, ( 6,2) t = 9, (3,5)
16. 16. Parametric Equations We may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time t as (x(t), y(t)). The equations x(t), y(t) are called parametric equations and the variable t is called the parameter. Example: Plot the path of parametric equations x(t) = t , y(t) = t – 4 from t = 0 to t = 9. t x y 0 0 -4 1 1 -3 4 2 0 6 6 2 9 3 5 t = 1, (1,-3) t = 0, (0,-4) t = 4, (2,0) t = 6, ( 6,2) t = 9, (3,5)
17. 17. Parametric Equations Sometime its possible to find the x and y equation for the path given by the parametric equations as in this case. t = 1, (1,-3) t = 0, (0,-4) t = 4, (2,0) t = 6, ( 6,2) t = 9, (3,5)
18. 18. Parametric Equations Sometime its possible to find the x and y equation for the path given by the parametric equations as in this case. Example: Find the x&y equation given by the parametric equations x(t) = t , y(t) = t – 4 t = 1, (1,-3) t = 0, (0,-4) t = 4, (2,0) t = 6, ( 6,2) t = 9, (3,5)
19. 19. Parametric Equations Sometime its possible to find the x and y equation for the path given by the parametric equations as in this case. Example: Find the x&y equation given by the parametric equations x(t) = t , y(t) = t – 4 t = 1, (1,-3) t = 0, (0,-4) t = 4, (2,0) t = 6, ( 6,2) t = 9, (3,5) Since x(t) = t, we've x2 = t.
20. 20. Parametric Equations Sometime its possible to find the x and y equation for the path given by the parametric equations as in this case. Example: Find the x&y equation given by the parametric equations x(t) = t , y(t) = t – 4 t = 1, (1,-3) t = 0, (0,-4) t = 4, (2,0) t = 6, ( 6,2) t = 9, (3,5) Since x(t) = t, we've x2 = t. Hence y = x2 – 4 is the x&y equation of the curve.
21. 21. Parametric Equations Sometime its possible to find the x and y equation for the path given by the parametric equations as in this case. Example: Find the x&y equation given by the parametric equations x(t) = t , y(t) = t – 4 t = 1, (1,-3) t = 0, (0,-4) t = 4, (2,0) t = 6, ( 6,2) t = 9, (3,5) Since x(t) = t, we've x2 = t. Hence y = x2 – 4 is the x&y equation of the curve. In general, the parametric equations do not generate the entire x&y graph.
22. 22. Parametric Equations More generally, the parameter variable does not have to be time.
23. 23. Parametric Equations More generally, the parameter variable does not have to be time. Example: Graph the parametric equations x( ) = -3cos( ), y( ) = 3sin( ) for from 0 to 2 .
24. 24. Parametric Equations More generally, the parameter variable does not have to be time. Example: Graph the parametric equations x( ) = -3cos( ), y( ) = 3sin( ) for from 0 to 2 . Note that x2 + y2 = (-3cos( ))2+(3sin( ))2
25. 25. Parametric Equations More generally, the parameter variable does not have to be time. Example: Graph the parametric equations x( ) = -3cos( ), y( ) = 3sin( ) for from 0 to 2 . Note that x2 + y2 = (-3cos( ))2+(3sin( ))2 = 9cos2( ) + 9sin2( ) = 9
26. 26. Parametric Equations More generally, the parameter variable does not have to be time. Example: Graph the parametric equations x( ) = -3cos( ), y( ) = 3sin( ) for from 0 to 2 . Note that x2 + y2 = (-3cos( ))2+(3sin( ))2 = 9cos2( ) + 9sin2( ) = 9 or x2 + y2 = 9. Hence the path is the circle with r = 3.
27. 27. Parametric Equations More generally, the parameter variable does not have to be time. Example: Graph the parametric equations x( ) = -3cos( ), y( ) = 3sin( ) for from 0 to 2 . Note that x2 + y2 = (-3cos( ))2+(3sin( ))2 = 9cos2( ) + 9sin2( ) = 9 or x2 + y2 = 9. Hence the path is the circle with r = 3. x y 0 -3 0 /2 0 3 3 0
28. 28. Parametric Equations More generally, the parameter variable does not have to be time. Example: Graph the parametric equations x( ) = -3cos( ), y( ) = 3sin( ) for from 0 to 2 . Note that x2 + y2 = (-3cos( ))2+(3sin( ))2 = 9cos2( ) + 9sin2( ) = 9 or x2 + y2 = 9. Hence the path is the circle with r = 3. x y 0 -3 0 /2 0 3 3 0
29. 29. Parametric Equations More generally, the parameter variable does not have to be time. Example: Graph the parametric equations x( ) = -3cos( ), y( ) = 3sin( ) for from 0 to 2 . Note that x2 + y2 = (-3cos( ))2+(3sin( ))2 = 9cos2( ) + 9sin2( ) = 9 or x2 + y2 = 9. Hence the path is the circle with r = 3. x y 0 -3 0 /2 0 3 3 0
30. 30. Parametric Equations More generally, the parameter variable does not have to be time. Example: Graph the parametric equations x( ) = -3cos( ), y( ) = 3sin( ) for from 0 to 2 . Note that x2 + y2 = (-3cos( ))2+(3sin( ))2 = 9cos2( ) + 9sin2( ) = 9 or x2 + y2 = 9. Hence the path is the circle with r = 3. x y 0 -3 0 /2 0 3 3 0
31. 31. Parametric Equations In general, the parametric equations x( ) = r*cos( ), y( ) = r*sin( ) for from 0 to 2 is the circle of radius r. r
32. 32. Parametric Equations The parametric equations x( ) = r*cos( ), y( ) = r*sin( ) for from 0 to 2 is the circle of radius r. r The parametric equations x( ) = a*cos( ), y( ) = b*sin( ) for from 0 to 2 is an ellipse. a b
33. 33. Parametric Equations Parametrize x&y curves.
34. 34. Parametric Equations Parametrize x&y curves. Given y=f(x), we may put it into the "standard" parametric form as x = t, y = f(t).
35. 35. Parametric Equations Parametrize x&y curves. Given y=f(x), we may put it into the "standard" parametric form as x = t, y = f(t). Example: For the equation y = x2. The standard parametric equations for it is x(t) = t y(t) = t2
36. 36. Parametric Equations Parametrize x&y curves. Given y=f(x), we may put it into the "standard" parametric form as x = t, y = f(t). Example: For the equation y = x2. The standard parametric equations for it is x(t) = t y(t) = t2 Another set of parametric equations for it is x(t) = t3 y(t) = t6
37. 37. Parametric Equations Parametrize polar curves.
38. 38. Parametric Equations Parametrize polar curves. Given the polar function r = f( ), a point on the curve with polar coordinate (r=f( ), ) has the corresponding (x, y) coordinate with x = r*cos( ) y = r*sin( )
39. 39. Parametric Equations Parametrize polar curves. Given the polar function r = f( ), a point on the curve with polar coordinate (r=f( ), ) has the corresponding (x, y) coordinate with x = r*cos( ) y = r*sin( ) r=f( ) (r, ) (x=rcos( ), y=rsin( ))
40. 40. Parametric Equations Parametrize polar curves. Given the polar function r = f( ), a point on the curve with polar coordinate (r=f( ), ) has the corresponding (x, y) coordinate with x = r*cos( ) x( ) = f( )cos( ) y = r*sin( ) y( ) = f( )sin( ) or { r=f( ) (r, ) (x=rcos( ), y=rsin( ))
41. 41. Parametric Equations Parametrize polar curves. Given the polar function r = f( ), a point on the curve with polar coordinate (r=f( ), ) has the corresponding (x, y) coordinate with x = r*cos( ) x( ) = f( )cos( ) y = r*sin( ) y( ) = f( )sin( ) or { r=f( ) (r, ) (x=rcos( ), y=rsin( )) This is the standard parametrization of the polar function r=f( ) with as the parameter.
42. 42. Parametric Equations Parametrize polar curves. Example: Parametrize the polar function r = 1 – sin( )
43. 43. Parametric Equations Parametrize polar curves. Example: Parametrize the polar function r = 1 – sin( ) The standard parametrization is x = r*cos( ) y = r*sin( )
44. 44. Parametric Equations Parametrize polar curves. Example: Parametrize the polar function r = 1 – sin( ) The standard parametrization is x = r*cos( ) y = r*sin( ) x( ) = (1 – sin( ))cos( ) y( ) = (1 – sin( ))sin( ) or as:
45. 45. Parametric Equations Tangent Lines for Parametric Curves.
46. 46. Parametric Equations Tangent Lines for Parametric Curves. Given the parametric equations x = x(t) y = y(t) then the derivative dy dx = dy/dt dx/dt
47. 47. Parametric Equations Tangent Lines for Parametric Curves. Given the parametric equations x = x(t) y = y(t) then the derivative dy dx = dy/dt dx/dt Example: Given x(t) = t3, y(t) = t2. Find the slope of the tangent at the (-8, 4).
48. 48. Parametric Equations Tangent Lines for Parametric Curves. Given the parametric equations x = x(t) y = y(t) then the derivative dy dx = dy/dt dx/dt dy dx = dy/dt dx/dt = 2t 3t2 = 2 3t Example: Given x(t) = t3, y(t) = t2. Find the slope of the tangent at the (-8, 4).
49. 49. Parametric Equations Tangent Lines for Parametric Curves. Given the parametric equations x = x(t) y = y(t) then the derivative dy dx = dy/dt dx/dt dy dx = dy/dt dx/dt = 2t 3t2 = 2 3t The point (-8, 4) corresponds to t = -2. Example: Given x(t) = t3, y(t) = t2. Find the slope of the tangent at the (-8, 4).
50. 50. Parametric Equations Tangent Lines for Parametric Curves. Given the parametric equations x = x(t) y = y(t) then the derivative dy dx = dy/dt dx/dt dy dx = dy/dt dx/dt = 2t 3t2 = 2 3t The point (-8, 4) corresponds to t = -2. 2 3t -1 3 =Hence dy dx = Example: Given x(t) = t3, y(t) = t2. Find the slope of the tangent at the (-8, 4).
51. 51. Parametric Equations Tangent Lines for Parametric Curves. Given the parametric equations x = x(t) y = y(t) then the derivative dy dx = dy/dt dx/dt Example: Given x(t) = t3, y(t) = t2. Find the slope of the tangent at the (-8, 4). dy dx = dy/dt dx/dt = 2t 3t2 = 2 3t The point (-8, 4) corresponds to t = -2. 2 3t -1 3 =Hence dy dx = (-8, 4) -1 3 dy dx =
52. 52. Parametric Equations Arc Length for Parametric Curves.
53. 53. Parametric Equations Arc Length for Parametric Curves. Given the parametric equations x = x(t), y = y(t) from t = a to t = b where x'(t) and y'(t) are continuous, then the arc length is dttytx bt at 22 ))('())('(
54. 54. Parametric Equations Arc Length for Parametric Curves. Given the parametric equations x = x(t), y = y(t) from t = a to t = b where x'(t) and y'(t) are continuous, then the arc length is Example: Given x(t) = t3, y = t2 from t = 0 to t =1, find the arc length. 1 3 1 2 dttytx bt at 22 ))('())('(
55. 55. Parametric Equations Arc Length for Parametric Curves. Given the parametric equations x = x(t), y = y(t) from t = a to t = b where x'(t) and y'(t) are continuous, then the arc length is Example: Given x(t) = t3, y = t2 from t = 0 to t =1, find the arc length. We have x'(t) = t2, y'(t) = t, hence the arc length is 1 3 1 2 dttt t t 1 0 222 )()( dttytx bt at 22 ))('())('(
56. 56. Parametric Equations Arc Length for Parametric Curves. Given the parametric equations x = x(t), y = y(t) from t = a to t = b where x'(t) and y'(t) are continuous, then the arc length is Example: Given x(t) = t3, y = t2 from t = 0 to t =1, find the arc length. We have x'(t) = t2, y'(t) = t, hence the arc length is 1 3 1 2 dtttdtttdttt t t t t t t 1 0 2 1 0 24 1 0 222 1)()( dttytx bt at 22 ))('())('(
57. 57. Parametric Equations Arc Length for Parametric Curves. Given the parametric equations x = x(t), y = y(t) from t = a to t = b where x'(t) and y'(t) are continuous, then the arc length is Example: Given x(t) = t3, y = t2 from t = 0 to t =1, find the arc length. We have x'(t) = t2, y'(t) = t, hence the arc length is 1 3 1 2 dtttdtttdttt t t t t t t 1 0 2 1 0 24 1 0 222 1)()( Substitution: Set u= t2+1 dttytx bt at 22 ))('())('(
58. 58. Parametric Equations Arc Length for Parametric Curves. Given the parametric equations x = x(t), y = y(t) from t = a to t = b where x'(t) and y'(t) are continuous, then the arc length is Example: Given x(t) = t3, y = t2 from t = 0 to t =1, find the arc length. We have x'(t) = t2, y'(t) = t, hence the arc length is dttytx bt at 22 ))('())('( 1 3 1 2 3 1 3 2 |)1( 3 1 1)()( 2/31 0 2/32 1 0 2 1 0 24 1 0 222 t dtttdtttdttt t t t t t t Substitution: Set u= t2+1