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Para= metric Equations
------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0003.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Para= metric 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)).  
------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0004.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Para= metric 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.  <= /span>
------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0005.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Para= metric 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.  <= /span>
Example: Plot the path of parametric equations
x(t) =3D Öt&= nbsp; , y(t) =3D t – 4 from t =3D 0 to t =3D 9.
------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0005_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAEAgAAAAABmAAIDjH8FADs= ------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0006.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Para= metric 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.  <= /span>
Example: Plot the path of parametric equations
x(t) =3D Öt&= nbsp; , y(t) =3D t – 4 from t =3D 0 to t =3D 9.
 5
3
9
 2
Ö6
6
 0
2
4
-3
1
1
-4
0
0
y
x
t
------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0006_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYQDDAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABf AMEAgQAAAAAAAABmAAECAwL/hI+py20CY3S02ouz3rz7D4biSJbmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kp1FMhqt9yu9wsOi8fksvmMTquzp0C7zTa55fB5yU7CY+P3t5yf 53cHiCWYRyiil1hnWPinGAIJIvmh1bj4eBnJSNc5SOkB2iHKYekZmHnq+Kk5ydmnmoi4GRs561o7 eVvZWrkb2hv6Oxo8OlxaXHq8Qcr8igr7uTZNXQ2W7JwaDX3YrOGdAY5hur3ajf39bL4uK37hbgFf Qc7NbitPge+g30Bvj8sqFy9t9TAFLGfwnEBgBP8NPFiQFkSHDCcmpGiMHwON/ws4KvB3MeQ9dOHU iQSoEKHElBFR7vGYACYCmQdArsSozJrOndNIjjN586QvmgaIAjBq06VQYUh9vluWbmFGp/GAKg06 UiqyhkuJccV6VVdTrcygliT7zexPtOHUvqM6z21VtuPkzoObz24+vPv07uPbz28/wBsFbyTc0XBH xB8Vf2Qc03FMyDMlz6Rck6fmzWQwF7X6sOVQz0dBVxTNlDRSyzVVs/5M9+lYlWJVm/ZKe3TsuRbB hn5p+2vY3+2C9x5+Gvjuu8KJ+069fG/z5F1zGmeJM9tx58inRv87HTdq77mp3+OMPv3r0t8Dzx5v vf3h8FvLk4evHXv1/Mrt16nX/5x4/eEX1XbmdReff/wVJ99i9C0YYIIEnmWggPul9V52GF43oIYU AogghLU1+NiDBU5YV4YXtqVihGW1GOKGJE5m4ocepqhejppxyKCCJ3a44loV/nejbDxm5aONPaL4 Vo1CMhnPejDqluSTQLr445JF8gYid15CV2WTQ4r45X1bMjdmlliyeOSIYXJ5ZYxKIgklml0eWKaE Z0qX5pxyptjmUDoOSk0BADs= ------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0006_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMwB/AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAy AH0AgQAAAAAAALvg4wECAwL/jI6py+0YzJu0imizhnL7iX1iE44md6ZXp4pl+73wJs9ZbVd4DrK8 7vv1hLcgkRRJKpfMpvPJBEin1Kr1is1qt9yu9wsOi8fksvmMTqvXZqj7DV8egXPKro7CO+56Pt5f BzgneERIZCiE+KPIw5jjaAM5IwkTZ3kZxaa5ydnp+QkaKjoaiml6qbdnlLrCykDZAqsim0J7YmuC O6LrsprKG+PbJ/xHHGg8eKrsRtrs/AwdLT1N3bV87eS6AOzBTYNcCH4onki+aN6I/qgeyT7pXgkf Kz9LX2t/i5+LzZ9U/Q8woMCBBKH1O6hNgTcNC4skTNDQQkQ6Dw9MtKNvV8ZeERUtbgzWsVXHi0NG fux2kF8BADs= ------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0012.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Para= metric 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.  <= /span>
Example: Plot the path of parametric equations
x(t) =3D Öt&= nbsp; , y(t) =3D t – 4 from t =3D 0 to t =3D 9.
 5
3
9
 2
Ö6
6
 0
2
4
-3
1
1
-4
0
0
y
x
t
------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0012_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYQDDAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABf AMEAgQAAAAAAAABmAAECAwL/hI+py20CY3S02ouz3rz7D4biSJbmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kp1FMhqt9yu9wsOi8fksvmMTquzp0C7zTa55fB5yU7CY+P3t5yf 53cHiCWYRyiil1hnWPinGAIJIvmh1bj4eBnJSNc5SOkB2iHKYekZmHnq+Kk5ydmnmoi4GRs561o7 eVvZWrkb2hv6Oxo8OlxaXHq8Qcr8igr7uTZNXQ2W7JwaDX3YrOGdAY5hur3ajf39bL4uK37hbgFf Qc7NbitPge+g30Bvj8sqFy9t9TAFLGfwnEBgBP8NPFiQFkSHDCcmpGiMHwON/ws4KvB3MeQ9dOHU iQSoEKHElBFR7vGYACYCmQdArsSozJrOndNIjjN586QvmgaIAjBq06VQYUh9vluWbmFGp/GAKg06 UiqyhkuJccV6VVdTrcygliT7zexPtOHUvqM6z21VtuPkzoObz24+vPv07uPbz28/wBsFbyTc0XBH xB8Vf2Qc03FMyDMlz6Rck6fmzWQwF7X6sOVQz0dBVxTNlDRSyzVVs/5M9+lYlWJVm/ZKe3TsuRbB hn5p+2vY3+2C9x5+Gvjuu8KJ+069fG/z5F1zGmeJM9tx58inRv87HTdq77mp3+OMPv3r0t8Dzx5v vf3h8FvLk4evHXv1/Mrt16nX/5x4/eEX1XbmdReff/wVJ99i9C0YYIIEnmWggPul9V52GF43oIYU AogghLU1+NiDBU5YV4YXtqVihGW1GOKGJE5m4ocepqhejppxyKCCJ3a44loV/nejbDxm5aONPaL4 Vo1CMhnPejDqluSTQLr445JF8gYid15CV2WTQ4r45X1bMjdmlliyeOSIYXJ5ZYxKIgklml0eWKaE Z0qX5pxyptjmUDoOSk0BADs= ------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0012_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMwBkAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAy AGMAgQAAAAAAALvg4wECAwLkjI+py60Co5y02hsN3rxD7YXiF4xmB55qla4u+cZCK590Pd54qO9o 6TP1gpgh0WI8UpJKCbNJckin1APgis1qt9yu9wsOi8fksvmMTqvX7Lb7DV9X5/QHlPOE5pt7Zf/4 RxQYNOhTuHOIk1izKNMY8/gS6TK5UndJF6e5ydnp+QkaKjoqimk6dYcHlHpRqeJqs8rKIjs7AStU a5uhuzvTu4ubA2wrLGLMQzyL7MH8cwqtQDpNXW19jZ2tLRbdjeC7pMzqrAp+K55KvqFehH7H3uqu J89H72cPiC+oT8hv6I/Im7cCADs= ------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0007.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Para= metric 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.  <= /span>
Example: Plot the path of parametric equations
x(t) =3D Öt&= nbsp; , y(t) =3D t – 4 from t =3D 0 to t =3D 9.
 5
3
9
 2
Ö6
6
 0
2
4
-3
1
1
-4
0
0
y
x
t
------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0007_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYQDDAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABf AMEAgQAAAAAAAABmAAECAwL/hI+py20CY3S02ouz3rz7D4biSJbmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kp1FMhqt9yu9wsOi8fksvmMTquzp0C7zTa55fB5yU7CY+P3t5yf 53cHiCWYRyiil1hnWPinGAIJIvmh1bj4eBnJSNc5SOkB2iHKYekZmHnq+Kk5ydmnmoi4GRs561o7 eVvZWrkb2hv6Oxo8OlxaXHq8Qcr8igr7uTZNXQ2W7JwaDX3YrOGdAY5hur3ajf39bL4uK37hbgFf Qc7NbitPge+g30Bvj8sqFy9t9TAFLGfwnEBgBP8NPFiQFkSHDCcmpGiMHwON/ws4KvB3MeQ9dOHU iQSoEKHElBFR7vGYACYCmQdArsSozJrOndNIjjN586QvmgaIAjBq06VQYUh9vluWbmFGp/GAKg06 UiqyhkuJccV6VVdTrcygliT7zexPtOHUvqM6z21VtuPkzoObz24+vPv07uPbz28/wBsFbyTc0XBH xB8Vf2Qc03FMyDMlz6Rck6fmzWQwF7X6sOVQz0dBVxTNlDRSyzVVs/5M9+lYlWJVm/ZKe3TsuRbB hn5p+2vY3+2C9x5+Gvjuu8KJ+069fG/z5F1zGmeJM9tx58inRv87HTdq77mp3+OMPv3r0t8Dzx5v vf3h8FvLk4evHXv1/Mrt16nX/5x4/eEX1XbmdReff/wVJ99i9C0YYIIEnmWggPul9V52GF43oIYU AogghLU1+NiDBU5YV4YXtqVihGW1GOKGJE5m4ocepqhejppxyKCCJ3a44loV/nejbDxm5aONPaL4 Vo1CMhnPejDqluSTQLr445JF8gYid15CV2WTQ4r45X1bMjdmlliyeOSIYXJ5ZYxKIgklml0eWKaE Z0qX5pxyptjmUDoOSk0BADs= ------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0007_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMwBJAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAy AEgAgQAAAAAAALvg4wECAwKxjI+py60Co5y02hsN3rxD7YXiF4xmB55qla4u+cZCK590Pd54qO9o 6TP1gpgh0WI8UpJKieMJjSYA1Kr1is1qt9yu9wsOi8fksvmMTqvX7DZaCo8/mj963b5h2vV0ftOv BHgkSEQYZOiDuKOIw1jjKAMZI/kiZxnnlqm5ydnp+QkaKhp6WQqFlweEiqS6utTq6gQbC0M7QemC u6KrwmszS+srBBwrnEPsaiyizGPqvFAAADs= ------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0008.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Para= metric 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.  <= /span>
Example: Plot the path of parametric equations
x(t) =3D Öt&= nbsp; , y(t) =3D t – 4 from t =3D 0 to t =3D 9.
 5
3
9
 2
Ö6
6
 0
2
4
-3
1
1
-4
0
0
y
x
t
------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0008_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYQDDAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABf AMEAgQAAAAAAAABmAAECAwL/hI+py20CY3S02ouz3rz7D4biSJbmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kp1FMhqt9yu9wsOi8fksvmMTquzp0C7zTa55fB5yU7CY+P3t5yf 53cHiCWYRyiil1hnWPinGAIJIvmh1bj4eBnJSNc5SOkB2iHKYekZmHnq+Kk5ydmnmoi4GRs561o7 eVvZWrkb2hv6Oxo8OlxaXHq8Qcr8igr7uTZNXQ2W7JwaDX3YrOGdAY5hur3ajf39bL4uK37hbgFf Qc7NbitPge+g30Bvj8sqFy9t9TAFLGfwnEBgBP8NPFiQFkSHDCcmpGiMHwON/ws4KvB3MeQ9dOHU iQSoEKHElBFR7vGYACYCmQdArsSozJrOndNIjjN586QvmgaIAjBq06VQYUh9vluWbmFGp/GAKg06 UiqyhkuJccV6VVdTrcygliT7zexPtOHUvqM6z21VtuPkzoObz24+vPv07uPbz28/wBsFbyTc0XBH xB8Vf2Qc03FMyDMlz6Rck6fmzWQwF7X6sOVQz0dBVxTNlDRSyzVVs/5M9+lYlWJVm/ZKe3TsuRbB hn5p+2vY3+2C9x5+Gvjuu8KJ+069fG/z5F1zGmeJM9tx58inRv87HTdq77mp3+OMPv3r0t8Dzx5v vf3h8FvLk4evHXv1/Mrt16nX/5x4/eEX1XbmdReff/wVJ99i9C0YYIIEnmWggPul9V52GF43oIYU AogghLU1+NiDBU5YV4YXtqVihGW1GOKGJE5m4ocepqhejppxyKCCJ3a44loV/nejbDxm5aONPaL4 Vo1CMhnPejDqluSTQLr445JF8gYid15CV2WTQ4r45X1bMjdmlliyeOSIYXJ5ZYxKIgklml0eWKaE Z0qX5pxyptjmUDoOSk0BADs= ------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0008_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMwAvAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAy AC0AgQAAAAAAALvg4wECAwJ6jI6py+0YzJu0imizhnL7iX1iE44md6ZXp4pl+73wJs9ZbVd4DrK8 7vv1hLcgkRRJKpfMpvPJBEin1Kr1is1qt9yu9wsOi8fksvmMTqvXZqj7DV8egXPKro7CO+56Pt5f BzgneERIZCiE+KPIw5jjaAM5IwkTZ3nJVAAAOw== ------=_NextPart_01C97CE7.48626F70 Content-Location: file:///C:/1E6C3F13/40ParametricEquations-Calculus_files/slide0009.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Para= metric 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.  <= /span>
Example: Plot the path of parametric equations
x(t) =3D Öt&= nbsp; , y(t) =3D t – 4 from t =3D 0 to t =3D 9.
 5
3
9
 2
Ö6
6
 0
2
4
-3
1
1
-4
0
0
y
x
t
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Para= metric 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.  <= /span>
Example: Plot the path of parametric equations
x(t) =3D Öt&= nbsp; , y(t) =3D t – 4 from t =3D 0 to t =3D 9.
 5
3
9
 2
Ö6
6
 0
2
4
-3
1
1
-4
0
0
y
x
t
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Para= metric 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.  <= /span>
Example: Plot the path of parametric equations
x(t) =3D Öt&= nbsp; , y(t) =3D t – 4 from t =3D 0 to t =3D 9.
 5
3
9
 2
Ö6
6
 0
2
4
-3
1
1
-4
0
0
y
x
t
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Para= metric 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.  <= /span>
Example: Plot the path of parametric equations
x(t) =3D Öt&= nbsp; , y(t) =3D t – 4 from t =3D 0 to t =3D 9.
 5
3
9
 2
Ö6
6
 0
2
4
-3
1
1
-4
0
0
y
x
t
t =3D = 0, (0,-4)
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Para= metric 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.  <= /span>
Example: Plot the path of parametric equations
x(t) =3D Öt&= nbsp; , y(t) =3D t – 4 from t =3D 0 to t =3D 9.
 5
3
9
 2
Ö6
6
 0
2
4
-3
1
1
-4
0
0
y
x
t
t =3D = 0, (0,-4)
t =3D = 1, (1,-3)
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