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Parametric Equations
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)).
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter.
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(t) = t , y(t) = t – 4 from t = 0 to t = 9.
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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)
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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)
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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)
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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)
Parametric EquationsWe may describe the motion of a particle in the xy-plane by giving it's position (x, y) at time tas (x(t), y(t)). The equations x(t), y(t) are calledparametric equations and the variable t is called the parameter. Example: Plot the path of parametric equationsx(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)
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)
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)
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.
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.
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.
Parametric Equations
More generally, the parameter variable does not have to be time.
Parametric Equations
More generally, the parameter variable does not have to be time. Example: Graph the parametric equationsx() = -3cos(), y() = 3sin() for from 0 to 2.
Parametric Equations
More generally, the parameter variable does not have to be time. Example: Graph the parametric equationsx() = -3cos(), y() = 3sin() for from 0 to 2.
Note that x2 + y2 = (-3cos())2+(3sin())2
Parametric Equations
More generally, the parameter variable does not have to be time. Example: Graph the parametric equationsx() = -3cos(), y() = 3sin() for from 0 to 2.
Note that x2 + y2 = (-3cos())2+(3sin())2
= 9cos2() + 9sin2() = 9
Parametric Equations
More generally, the parameter variable does not have to be time. Example: Graph the parametric equationsx() = -3cos(), y() = 3sin() for from 0 to 2.
Note that x2 + y2 = (-3cos())2+(3sin())2
= 9cos2() + 9sin2() = 9or x2 + y2 = 9. Hence thepath is the circle with r = 3.
Parametric Equations
More generally, the parameter variable does not have to be time. Example: Graph the parametric equationsx() = -3cos(), y() = 3sin() for from 0 to 2.
Note that x2 + y2 = (-3cos())2+(3sin())2
= 9cos2() + 9sin2() = 9or x2 + y2 = 9. Hence thepath is the circle with r = 3.
x y
0 -3 0
/2 0 3 3 0
Parametric Equations
More generally, the parameter variable does not have to be time. Example: Graph the parametric equationsx() = -3cos(), y() = 3sin() for from 0 to 2.
Note that x2 + y2 = (-3cos())2+(3sin())2
= 9cos2() + 9sin2() = 9or x2 + y2 = 9. Hence thepath is the circle with r = 3.
x y
0 -3 0
/2 0 3 3 0
Parametric Equations
More generally, the parameter variable does not have to be time. Example: Graph the parametric equationsx() = -3cos(), y() = 3sin() for from 0 to 2.
Note that x2 + y2 = (-3cos())2+(3sin())2
= 9cos2() + 9sin2() = 9or x2 + y2 = 9. Hence thepath is the circle with r = 3.
x y
0 -3 0
/2 0 3 3 0
Parametric Equations
More generally, the parameter variable does not have to be time. Example: Graph the parametric equationsx() = -3cos(), y() = 3sin() for from 0 to 2.
Note that x2 + y2 = (-3cos())2+(3sin())2
= 9cos2() + 9sin2() = 9or x2 + y2 = 9. Hence thepath is the circle with r = 3.
x y
0 -3 0
/2 0 3 3 0
Parametric Equations
In general, the parametric equationsx() = ±r*cos(), y() = ±r*sin() for from 0 to 2 is the circle of radius r.
r
Parametric Equations
The parametric equationsx() = ±r*cos(), y() = ±r*sin() for from 0 to 2 is the circle of radius r.
r
The parametric equationsx() = ±a*cos(), y() = ±b*sin() for from 0 to 2 is an ellipse.
a
b
Parametric EquationsParametrize x&y curves.
Parametric EquationsParametrize x&y curves.Given y=f(x), we may put it into the "standard"parametric form as x = t, y = f(t).
Parametric EquationsParametrize 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) = ty(t) = t2
Parametric EquationsParametrize 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) = ty(t) = t2 Another set of parametric equations for it isx(t) = t3
y(t) = t6
Parametric EquationsParametrize polar curves.
Parametric EquationsParametrize polar curves.Given the polar function r = f(), a point on the curve with polar coordinate (r=f(), ) has the corresponding (x, y) coordinate withx = r*cos() y = r*sin()
Parametric EquationsParametrize polar curves.Given the polar function r = f(), a point on the curve with polar coordinate (r=f(), ) has the corresponding (x, y) coordinate withx = r*cos() y = r*sin()
r=f()
(r, )
(x=rcos(), y=rsin())
Parametric EquationsParametrize polar curves.Given the polar function r = f(), a point on the curve with polar coordinate (r=f(), ) has the corresponding (x, y) coordinate withx = r*cos() x() = f()cos() y = r*sin() y() = f()sin()
or {
r=f()
(r, )
(x=rcos(), y=rsin())
Parametric EquationsParametrize polar curves.Given the polar function r = f(), a point on the curve with polar coordinate (r=f(), ) has the corresponding (x, y) coordinate withx = 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.
Parametric EquationsParametrize polar curves.Example: Parametrize the polar function r = 1 – sin()
Parametric EquationsParametrize polar curves.Example: Parametrize the polar function r = 1 – sin() The standard parametrization isx = r*cos() y = r*sin()
Parametric EquationsParametrize polar curves.Example: Parametrize the polar function r = 1 – sin() The standard parametrization isx = r*cos() y = r*sin()
x() = (1 – sin())cos()y() = (1 – sin())sin()
or as:
Parametric EquationsTangent Lines for Parametric Curves.
Parametric EquationsTangent Lines for Parametric Curves.Given the parametric equationsx = x(t) y = y(t) then the derivative
dydx
=dy/dt
dx/dt
Parametric EquationsTangent Lines for Parametric Curves.Given the parametric equationsx = x(t) y = y(t) then the derivative
dydx
=dy/dt
dx/dtExample: Given x(t) = t3, y(t) = t2. Find the slope of the tangent at the (-8, 4).
Parametric EquationsTangent Lines for Parametric Curves.Given the parametric equationsx = x(t) y = y(t) then the derivative
dydx
=dy/dt
dx/dt
dydx =
dy/dtdx/dt =
2t3t2 =
23t
Example: Given x(t) = t3, y(t) = t2. Find the slope of the tangent at the (-8, 4).
Parametric EquationsTangent Lines for Parametric Curves.Given the parametric equationsx = x(t) y = y(t) then the derivative
dydx
=dy/dt
dx/dt
dydx =
dy/dtdx/dt =
2t3t2 =
23t
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).
Parametric EquationsTangent Lines for Parametric Curves.Given the parametric equationsx = x(t) y = y(t) then the derivative
dydx
=dy/dt
dx/dt
dydx =
dy/dtdx/dt =
2t3t2 =
23t
The point (-8, 4) corresponds to t = -2.
23t
-13=Hence dy
dx=
Example: Given x(t) = t3, y(t) = t2. Find the slope of the tangent at the (-8, 4).
Parametric EquationsTangent Lines for Parametric Curves.Given the parametric equationsx = x(t) y = y(t) then the derivative
dydx
=dy/dt
dx/dtExample: Given x(t) = t3, y(t) = t2. Find the slope of the tangent at the (-8, 4). dydx =
dy/dtdx/dt =
2t3t2 =
23t
The point (-8, 4) corresponds to t = -2.
23t
-13=Hence dy
dx=
(-8, 4)
-1
3
dy
dx=
Parametric EquationsArc Length for Parametric Curves.
Parametric EquationsArc 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
dttytxbt
at
22 ))('())('(
Parametric EquationsArc 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.
13
12
dttytxbt
at
22 ))('())('(
Parametric EquationsArc 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
13
12
dtttt
t
1
0
222 )()(
dttytxbt
at
22 ))('())('(
Parametric EquationsArc 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
13
12
dtttdtttdtttt
t
t
t
t
t
1
0
21
0
241
0
222 1)()(
dttytxbt
at
22 ))('())('(
Parametric EquationsArc 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
13
12
dtttdtttdtttt
t
t
t
t
t
1
0
21
0
241
0
222 1)()(
Substitution:Set u= t2+1
dttytxbt
at
22 ))('())('(
Parametric EquationsArc 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
dttytxbt
at
22 ))('())('(
13
12
3
1
3
2|)1(
3
1
1)()(
2/31
0
2/32
1
0
21
0
241
0
222
t
dtttdtttdtttt
t
t
t
t
t
Substitution:Set u= t2+1
