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Tang= ent and Arc Length in Polar Equations
------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0003.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Tang= ent and Arc Length in Polar Equations
Given= the polar function r =3D f(q), it may be realized as the parametric equations: =
x(q) =3D f(q)cos(q)    
y(q) =3D f(q)sin(q).
------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0004.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Tang= ent and Arc Length in Polar Equations
Given the parametric equations
x =3D x(q)        = ;            &n= bsp;  
y =3D y(q)        = ;           
the dervative        = ;            &n= bsp;    . 
dy
dx
=3D
dy/d<= /span>q<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
dx/d<= /span>q<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
Given= the polar function r =3D f(q), it may be realized as the parametric equations: =
x(q) =3D f(q)cos(q)    
y(q) =3D f(q)sin(q).
------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0004_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAf AAEAgAAAAAAAAAIFjI+pewUAOw== ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0004_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA+ AAEAgAAAAAAAAAIGjI+py+1bADt= ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0004_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcABMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBu AEoAgAAAAP8AAAL/hI+py+0Powsh0iuz3rzdynwi1ZXmiXyLqmDoCz8kmbghGOc6MK+499sJO7Og wegbKokVWs2CXEpvTub0KrNZsdxWddsN874csnhq3qTPSi2R3WWV1nAdjZ6to3HuDF5/8iXnFwW4 lTZSmCLUZ2JUhdFYI/lkN+j4I3KkufOncadIeBCZGDpnKrrZogbKxyeFClU5Gku7WduJCzQhGKVC eeWZZ5vSS6sJzCXsUZyp9cs5Zqh6uugcNDI9/Gbr9NyUrL1cvCg95nyrDTFObN56TK3Oe+gaHy5P XhZ/RH2Pv/+pT5FL/+YxQUKw4DpdsyYxLMiO0EN8/spMpHhxocKAd0sqbsy4jsdGPx2bjJQAcgK/ kxoZgWPZshNAmEkszaQ5KZc5nAZjRON5I1dKjEDZRCwK4yjSQEOXcnSqrClUC1OjVg0m9SozrbCy cs35dYjSsCHJMjJ7Fq1NtT69mnX7lm1budza0SVp9y5KEB71nuvr92/gBAUAADs= ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0005.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Tang= ent and Arc Length in Polar Equations
Given the parametric equations
x =3D x(q)        = ;            &n= bsp;  
y =3D y(q)        = ;           
the dervative        = ;            &n= bsp;    . 
Hence for <= /span>r =3D f(q), the slope at the point (r=3Df(q<= span style=3D'font-size:156%'>), q) is = ;
dy
dx
=3D
dy/d<= /span>q<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
dx/d<= /span>q<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
=3D
d(f(q)cos(q))/dq
d(f(q)sin(q))/dq
dy
dx
=3D
dy/d<= /span>q<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
dx/d<= /span>q<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
Given= the polar function r =3D f(q), it may be realized as the parametric equations: =
x(q) =3D f(q)cos(q)    
y(q) =3D f(q)sin(q).
------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0005_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAf AAEAgAAAAAAAAAIFjI+pewUAOw== ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0005_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh2QAFAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADW AAMAgAAAAAAAAAIijI+py+0Lopy02ouz3rz7D4biSJbmiabqyn7PC8fyTNdwAQA7 ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0005_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAf AAEAgAAAAAAAAAIFjI+pewUAOw== ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0005_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA+ AAEAgAAAAAAAAAIGjI+py+1bADt= ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0006.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Tang= ent and Arc Length in Polar Equations
Given the parametric equations
x =3D x(q)        = ;            &n= bsp;  
y =3D y(q)        = ;           
the dervative        = ;            &n= bsp;    . 
Given= the polar function r =3D f(q), it may be realized as the parametric equations: =
x(q) =3D f(q)cos(q)    
y(q) =3D f(q)sin(q).
Hence for <= /span>r =3D f(q), the slope at the point (r=3Df(q<= span style=3D'font-size:156%'>), q) is = ;
dy
dx
=3D
dy/d<= /span>q<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
dx/d<= /span>q<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
=3D
d(f(q)cos(q))/dq
d(f(q)sin(q))/dq
=3D
f '(= q<= span style=3D'font-size:133%'>)sin(q) + f(q<= span style=3D'font-size:133%'>)cos(q)=
f '(= q<= span style=3D'font-size:133%'>)cos(q) – f(<= /span>q<= span style=3D'font-size:133%'>)sin(q)=
dy
dx
=3D
dy/d<= /span>q<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
dx/d<= /span>q<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
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Slop= es in Polar Equations
Example: a. Find the slope at r =3D= cos(2q) at <= /span>q =3D p/6.
 
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Example: a. Find the slope at r =3D= cos(2q) at q =3D p/6.
 
(1/2,p/6)<= /div>
Slop= es in Polar Equations
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Example: a. Find the slope at r =3D= cos(2q) at q =3D p/6.
 
We have x(q) =3D cos(2q)cos(q) and y(q) =3D <= /span>cos(2q)sin(<= /span>q),
(1/2,p/6)<= /div>
Slop= es in Polar Equations
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Example: a. Find the slope at r =3D= cos(2q) at q =3D p/6.
 
We have x(q) =3D cos(2q)cos(q) and y(q) =3D <= /span>cos(2q)sin(<= /span>q), hence
dy
dx
=3D
dy/dq
dx/dq
=3D
(1/2,p/6)<= /div>
Slop= es in Polar Equations
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-2sin(2q)sin(q) + cos(2q)cos(q)
(1/2,p/6)<= /div>
Slop= es in Polar Equations
Example: a. Find the slope at r =3D= cos(2q) at q =3D p/6.
 
We have x(q) =3D cos(2q)cos(q) and y(q) =3D <= /span>cos(2q)sin(<= /span>q), hence
dy
dx
=3D
dy/dq
dx/dq
=3D
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AAIAgAAAAAAAAAIKjI+py+0Po5yoAAA7 ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0038.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
=3D
-2sin(2q)cos(q) – cos(2q)sin(<= /span>q)
(1/2,p/6)<= /div>
Slop= es in Polar Equations
-2sin(2q)sin(q) + cos(2q)cos(q)
Example: a. Find the slope at r =3D= cos(2q) at q =3D p/6.
 
We have x(q) =3D cos(2q)cos(q) and y(q) =3D <= /span>cos(2q)sin(<= /span>q), hence
dy
dx
=3D
dy/dq
dx/dq
=3D
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at q =3D p/6,
dy
dx
=3D
-2sin(p/3)sin(p/6) + cos(p/3)cos(p/6)
-2sin(p/3)cos(p/6) –= cos(p/3)sin(p/6)
(1/2,p/6)<= /div>
Slop= es in Polar Equations
=3D
-2sin(2q)cos(q) – cos(2q)sin(<= /span>q)
-2sin(2q)sin(q) + cos(2q)cos(q)
Example: a. Find the slope at r =3D= cos(2q) at q =3D p/6.
 
We have x(q) =3D cos(2q)cos(q) and y(q) =3D <= /span>cos(2q)sin(<= /span>q), hence
dy
dx
=3D
dy/dq
dx/dq
=3D
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=3D
-Ö3 * ½ + ½ * Ö3/2 
-Ö3* Ö3/2 – ½ * ½
(1/2,p/6)<= /div>
Slop= es in Polar Equations
at q =3D p/6,
dy
dx
=3D
-2sin(p/3)sin(p/6) + cos(p/3)cos(p/6)
-2sin(p/3)cos(p/6) –= cos(p/3)sin(p/6)
=3D
-2sin(2q)cos(q) – cos(2q)sin(<= /span>q)
-2sin(2q)sin(q) + cos(2q)cos(q)
Example: a. Find the slope at r =3D= cos(2q) at q =3D p/6.
 
We have x(q) =3D cos(2q)cos(q) and y(q) =3D <= /span>cos(2q)sin(<= /span>q), hence
dy
dx
=3D
dy/dq
dx/dq
=3D
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R0lGODlhIgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAf AAIAgAAAAAAAAAIGjI+py+1bADt= ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0012_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA+ AAIAgAAAAAAAAAIKjI+py+0Po5yoAAA7 ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0015.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
=3D
-Ö3 * ½ + ½ * Ö3/2 
-Ö3* Ö3/2 – ½ * ½
=3D
-2Ö3&= nbsp; +  Ö3&= nbsp;
-6 – 1
=3D
 Ö3&= nbsp;
7
(1/2,p/6)<= /div>
Slop= es in Polar Equations
at q =3D p/6,
dy
dx
=3D
-2sin(p/3)sin(p/6) + cos(p/3)cos(p/6)
-2sin(p/3)cos(p/6) –= cos(p/3)sin(p/6)
=3D
-2sin(2q)cos(q) – cos(2q)sin(<= /span>q)
-2sin(2q)sin(q) + cos(2q)cos(q)
Example: a. Find the slope at r =3D= cos(2q) at q =3D p/6.
 
We have x(q) =3D cos(2q)cos(q) and y(q) =3D <= /span>cos(2q)sin(<= /span>q), hence
dy
dx
=3D
dy/dq
dx/dq
=3D
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Example: b. Find where the slope is 0 for r =3D 1+ <= span style=3D'font-size:133%;color:red'>cos(q).
Slop= es in Polar Equations
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Example: b. Find where the slope is 0 for r =3D 1+<= span style=3D'font-size:133%;color:#339933'> cos(q).
Slop= es in Polar Equations
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Example: b. Find where the slope is 0 for r =3D 1+<= span style=3D'font-size:133%;color:#339933'> cos(q).
 We have x(q) =3D (1+cos(q))cos(q) and y(q) =3D <= /span>(1+cos(q))sin(= q).
Slop= es in Polar Equations
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dy
dx
=3D
dy/dq
dx/dq
We want
 =3D 0, or
dy
dq
 =3D 0.
Slop= es in Polar Equations
Example: b. Find where the slope is 0 for r =3D 1+<= span style=3D'font-size:133%;color:#339933'> cos(q).
 We have x(q) =3D (1+cos(q))cos(q) and y(q) =3D <= /span>(1+cos(q))sin(= q).
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file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0021.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
-sin(q)sin(q) +(1+ cos(= q))cos(q) =3D 0
dy
dq
=3D
Set
Slop= es in Polar Equations
dy
dx
=3D
dy/dq
dx/dq
We want
 =3D 0, or
dy
dq
 =3D 0.
Example: b. Find where the slope is 0 for r =3D 1+<= span style=3D'font-size:133%;color:#339933'> cos(q).
 We have x(q) =3D (1+cos(q))cos(q) and y(q) =3D <= /span>(1+cos(q))sin(= q).
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file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0021_image089.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAf AAIAgAAAAAAAAAIGjI+py+1bADt= ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0022.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
-S2 + C + C2 =3D 0
Slop= es in Polar Equations
-sin(q)sin(q) +(1+ cos(= q))cos(q) =3D 0
dy
dq
=3D
Set
dy
dx
=3D
dy/dq
dx/dq
We want
 =3D 0, or
dy
dq
 =3D 0.
Example: b. Find where the slope is 0 for r =3D 1+<= span style=3D'font-size:133%;color:#339933'> cos(q).
 We have x(q) =3D (1+cos(q))cos(q) and y(q) =3D <= /span>(1+cos(q))sin(= q).
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file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0022_image098.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAf AAIAgAAAAAAAAAIGjI+py+1bADt= ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0023.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
-S2 + C + C2 =3D 0
-(1 – C2) + C + C= 2 =3D 0
Slop= es in Polar Equations
-sin(q)sin(q) +(1+ cos(= q))cos(q) =3D 0
dy
dq
=3D
Set
dy
dx
=3D
dy/dq
dx/dq
We want
 =3D 0, or
dy
dq
 =3D 0.
Example: b. Find where the slope is 0 for r =3D 1+<= span style=3D'font-size:133%;color:#339933'> cos(q).
 We have x(q) =3D (1+cos(q))cos(q) and y(q) =3D <= /span>(1+cos(q))sin(= q).
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dq
-S2 + C + C2 =3D 0
-(1 – C2) + C + C= 2 =3D 0
2C2 + C – 1 =3D 0
(2C – 1)(C + 1) =3D 0<= /div>
Slop= es in Polar Equations
-sin(q)sin(q) +(1+ cos(= q))cos(q) =3D 0
dy
dq
=3D
Set
dy
dx
=3D
dy/dq
dx/dq
We want
 =3D 0, or
dy
dq
 =3D 0.
Example: b. Find where the slope is 0 for r =3D 1+<= span style=3D'font-size:133%;color:#339933'> cos(q).
 We have x(q) =3D (1+cos(q))cos(q) and y(q) =3D <= /span>(1+cos(q))sin(= q).
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file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0024_image116.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAf AAIAgAAAAAAAAAIGjI+py+1bADt= ------=_NextPart_01C97CE7.72CE92C0 Content-Location: file:///C:/D9125D13/42Tangentandarc-lengthinpolarcoordinates_files/slide0025.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
-S2 + C + C2 =3D 0
-(1 – C2) + C + C= 2 =3D 0
2C2 + C – 1 =3D 0
(2C – 1)(C + 1) =3D 0<= /div>
cos(q) =3D ½ à q =3D ±p/3
Slop= es in Polar Equations
-sin(q)sin(q) +(1+ cos(= q))cos(q) =3D 0
dy
dq
=3D
Set
dy
dx
=3D
dy/dq
dx/dq
We want
 =3D 0, or
dy
dq
 =3D 0.
Example: b. Find where the slope is 0 for r =3D 1+<= span style=3D'font-size:133%;color:#339933'> cos(q).
 We have x(q) =3D (1+cos(q))cos(q) and y(q