### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Class Note for MATH 1432 at UH

### View Full Document

## 21

## 0

## Popular in Course

## Popular in Department

This 71 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 21 views.

## Reviews for Class Note for MATH 1432 at UH

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/06/15

Polar Coordinates Polar Coordinate System ray 9 polar axis ray67r Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 2 1 9 Polar Coordinates Polar Coordinate System ray 6 polar axis ray67r Frame of Reference In the polar coordinate system the frame of reference is a point 0 that we call the pole and a ray that emanates from it that we call the polar axis H Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 2 1 9 Polar Coordinates Polar Coordinate System ray 6 6 The purpose of the polar A coordinates is to represent polar axis curves that have symmetry about a point or spiral about a point ray67r Frame of Reference In the polar coordinate system the frame of reference is a point 0 that we call the pole and a ray that emanates from it that we call the polar axis in Jiwen He University of Houston Math 1432 Setti on 26625 Lecture 12 February 21 2003 2 1 9 Polar Coordinates 1 Polar Coordinates polar axis polar axis 5 ray67r raygir Definition A point is given polar coordinates r7 6 iff it lies at a distance lrl from the pole a long the ray 6 if r 2 O and along the ray 6 7r if r lt O Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 19 Polar Coordinates l 39 39 Points in Polar Coordinates ray 6 1 6 In 6 27 I 6 47 219 219 polar axis polar ms 0 2 6 7 2 6 94 ray67r Points in Polar Coordinates Jiwen He University of Houston Math 1432 Section 26626 Lecture 3912 February 21 2008 Polar Coordinates l 39 39 Points in Polar Coordinates ray 6 1 6 In 6 27 I 6 47 219 219 polar axis polar ms 0 2 6 7 2 6 94 ray67r Points in Polar Coordinates o O 06 for all 6 Jiwen He University of Houston Math 1432 Section 26626 Lecture 3912 February 21 2008 Polar Coordinates w Points in Polar Coordinates ray 6 1 6 In 6 27 I 6 47 219 219 polar axis polar ms 0 2 6 7 2 6 94 ray67r Points in Polar Coordinates o O 06 for all 6 o r6 r6 2n7r for all integers n Jiwen He University of Houston Math 1432 Section 26626 Lecture 3912 February 21 2008 4 1 9 Polar Coordinates w Points in Polar Coordinates ray 6 1 6 In 6 27 I 6 47 219 219 polar axis polar ms 0 2 6 7 2 6 94 ray67r Points in Polar Coordinates o O 06 for all 6 o r6 r6 2n7r for all integers n o r 6 r6 7T Jiwen He University of Houston Math 1432 Section 26626 Lecture 3912 February 21 2008 4 1 9 Polar Coordinates Relation to Rectangular Coord l x y polar axis A Relation to Rectangular Coordinates Jiwen He University of Houston Math 1432 Section 26526 Lecture 12 February 21 2008 5 19 Polar Coordinates Relation to Rectangular Coord l x y polar axis A Relation to Rectangular Coordinates o x rcosQ y rsin6 Jiwen He University of Houston Math 1432 Section 26526 Lecture 12 February 21 2008 5 19 Polar Coordinates 7 Relation to Rectangular Coord l x y polar axis A Jiwen He University of Houston Math 1432 Section 26526 Lecture 12 February 21 2008 5 19 Polar Coordinates 7 Relation to Rectangular Coord l x y polar axis A oxrcos yrsin6 gt x2yr7 tang X o rX2y26tan1 H Jiwen He University of Houston Math 1432 Section 26526 Lecture 12 February 21 2008 5 19 Polar Coordinates Polar Coordinates Relation to Rectanr Circles in Polar Coordinates r22 r 2 r4sin6 1392 40030 Circles in Polar Coordinates In rectangular coordinates In polar coordinates Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 6 19 Polar Coordinates Polar Coordinates Relation to Rectanr Circles in Polar Coordinates r22 r 2 r4sin6 1392 40030 Circles in Polar Coordinates In rectangular coordinates In polar coordinates X2y2za2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 6 19 Polar Coordinates Polar Coordinates Relation to Rectanr Circles in Polar Coordinates r22 r 2 r4sin6 1392 40030 Circles in Polar Coordinates In rectangular coordinates In polar coordinates X2y2za2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 6 19 Polar Coordinates Polar Coordinates Relation to Rectanr Circles in Polar Coordinates r22 r 2 r4sin6 1392 40030 Circles in Polar Coordinates In rectangular coordinates In polar coordinates X2y2za2 r23 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 Polar Coordinates Polar Coordinates Relation to Rectanr Circles in Polar Coordinates r22 r 2 r4sin6 r 4 0056 Circles in Polar Coordinates In rectangular coordinates In polar coordinates X2y2za2 r23 x2 l y a2a2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 Polar Coordinates Polar Coordinates Relation to Rectanr Circles in Polar Coordinates r22 r 2 r4sin6 r 4 0056 Circles in Polar Coordinates In rectangular coordinates In polar coordinates X2y2za2 r23 x2 l y a2a2 X2 l y a2a2 gt X2y22ay gt r22arsin6 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 6 19 Polar Coordinates Polar Coordinates Relation to Rectanr Circles in Polar Coordinates r22 r 2 r4sin6 r 4 0056 Circles in Polar Coordinates In rectangular coordinates In polar coordinates X2 y2 32 r 2 a X2 l y a2a2 r2asin0 X2 l y a2a2 gt X2y22ay gt r22arsin6 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 6 19 Polar Coordinates Polar Coordinates Relation to Rectanr Circles in Polar Coordinates r22 r 2 r4sin6 r 4 0056 Circles in Polar Coordinates In rectangular coordinates In polar coordinates X2 y2 32 r 2 a X2 l y a2a2 r2asin0 X a2y2a Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 6 19 Polar Coordinates Polar Coordinates Relation to Rectanr Circles in Polar Coordinates r22 r 2 r4sin6 r 4 0056 Circles in Polar Coordinates In rectangular coordinates In polar coordinates X2 y2 32 r 2 a X2 l y a2a2 r2asin0 X a2y2a X a2 y2a2 gt X2y22ax gt r22arc050 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 6 19 Polar Coordinates Polar Coordinates Relation to Rectanr Circles in Polar Coordinates r22 r 2 r4sin6 r 4 0056 Circles in Polar Coordinates In rectangular coordinates In polar coordinates X2 y2 a2 r a X2 l y a2a2 r2asin0 X a2 y2a2 r2acos X a2 y2a2 gt X2y22ax gt r22arc050 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 6 19 Polar Coordinates i v Lines in Polar Coordinates y y y y y 2 y x y x l X l X X X 1 5 0 7r 6 7r 4 4 r 2e09 1 220500 Lines in Polar Coordinates In rectangular coordinates In polar coordinates Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 7 19 Polar Coordinates i v Lines in Polar Coordinates y y y y y 2 y x y x l X l X X X 1 5 0 7r 6 7r 4 4 r 2e09 1 220500 Lines in Polar Coordinates In rectangular coordinates In polar coordinates y mX Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 7 19 Polar Coordinates i v Lines in Polar Coordinates y y y y y 2 y x y x l X l X X X 1 5 0 7r 6 7r 4 4 r 2e09 1 220500 Lines in Polar Coordinates In rectangular coordinates In polar coordinates y mX y y mx gt m tanQ m X Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 Polar Coordinates i v Lines in Polar Coordinates y y y y y 2 y x y x l X l X X X 1 5 0 7r 6 7r 4 4 r 2e09 1 220500 Lines in Polar Coordinates In rectangular coordinates In polar coordinates ymx 6ozwithoztan1m y y mx gt m tanQ m X Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 Polar Coordinates i v Lines in Polar Coordinates r 2e06 rZCsce Lines in Polar Coordinates In rectangular coordinates In polar coordinates ymx 6ozwithoztan1m X a Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 7 19 Polar Coordinates i v Lines in Polar Coordinates r 2e06 rZCsce Lines in Polar Coordinates In rectangular coordinates In polar coordinates ymx 6ozwithoztan1m X a Xa rcosdza rasec0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 7 19 Polar Coordinates Lines in Polar Coordinates r 2e06 rZCsce Lines in Polar Coordinates In rectangular coordinates In polar coordinates ymx 6ozwithoztan1m Xa rasec0 Xa rcosdza rasec0 February 21 2008 7 19 Math 1432 Section 26626 Lecture 12 Jiwen He University of Houston Polar Coordinates Lines in Polar Coordinates 5 r 2e09 1 220500 Lines in Polar Coordinates In rectangular coordinates In polar coordinates ymx 6ozwithoztan1m X a r asec0 y a Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 7 19 Polar Coordinates i v Lines in Polar Coordinates 5 r 2e09 1 220500 Lines in Polar Coordinates In rectangular coordinates In polar coordinates ymx 6ozwithoztan1m X a r asec0 y 5 ya rsin0a racsc0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 7 19 Polar Coordinates i v Lines in Polar Coordinates 5 r 2e09 1 220500 Lines in Polar Coordinates In rectangular coordinates In polar coordinates ymx 6ozwithoztan1m X a r asec0 y a r acsc0 ya rsin0a racsc0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 7 19 Polar Coordinates Polar Coordinates Relation to Rectanular Coordinates Squot mme Symmetry lr 7r t9 symmetry about the x axis symmetry about the yaxis symmetry about the origin 3 Lemniscate ribbon r2 cos 26 10 10 x r2 cos 26 lemniscate Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 8 19 Polar Coordinates Polar Coordinates Relation to Rectanular Coordinates Squot mme Symmetry lr 7t 9 symmetry about the x axis symmetry about the yaxis symmetry about the origin 3 Lemniscate ribbon r2 cos 26 1 O 1 O 1 1 10 x COS26 COS 29 COS 29 r2 cos 26 lemniscate Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 8 19 Polar Coordinates quot i i v Polar Coordinates Relation to Rectanular Coordinates Squot mme Symmetry lr 7t 9 symmetry about the x axis symmetry about the yaxis symmetry about the origin Lemniscate ribbon r2 10 M x cos2 9 cos 29 cos 26 gt if r6 E graph then r 6 E graph r2cos26 lemniscate 14 February 21 2008 8 19 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 Polar Coordinates 139 quot I I39 II Polar Coordinates Relation to Rectanular Coordinates Squot mme Symmetry Ill yil lquot A l7139 9 I I9 9 39 rig I I I gt l x x x l r l 6 r lr n9 symmetry about the x axis symmetry about the yaxis symmetry about the origin Lemniscate ribbon r2 10 M x cos2 9 cos 29 cos 26 gt if r6 E graph then r 6 E graph zzcosgg symmetric about the X axis lemniscate 14 February 21 2008 8 19 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 Polar Coordinates Polar Coordinates Relation to Rectanular Coordinates Squot mme Symmetry 1139 it 9 symmetry about the x axis symmetry about the y axis symmetry about the origin 3 Lemniscate ribbon r2 cos 26 HO 1 0 COS27T 9 COS27T 29 cos 26 17z 10 x r2 cos 26 lemniscate Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 8 19 Polar Coordinates quot I I39 II Polar Coordinates Relation to Rectanular Coordinates Squot mme Symmetry VII l A ll II I II 7r Ir f 1 F I I I l x I I 3 if 9 0 lr n9 symmetry about the x axis symmetry about the y axis symmetry about the origin 3 Lemniscate ribbon r2 cos 26 HO 1 0 COS27T 9 COS27T 29 cos 26 M 1 0 x gt if r6 E graph then r 7T 9 E graph lezrniafe6 XI Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 8 19 Polar Coordinates 139 quot i i v Polar Coordinates Relation to Rectanular Coordinates Squot mme Symmetry lr it 9 symmetry about the x axis symmetry about the y axis symmetry about the origin 3 Lemniscate ribbon r2 cos 26 HO 1 0 COS27T 9 COS27T 29 cos 26 M 1 0 x gt if r6 E graph then r 7T 9 E graph lezrniafe6 symmetric about the y axis I Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 8 19 Polar Coordinates Polar Coordinates Relation to Rectanular Coordinates Squot mme Symmetry 1139 it 9 symmetry about the x axis symmetry about the y axis symmetry about the origin 3 Lemniscate ribbon r2 cos 26 HO 1 0 COS27T 9 COS27T 26 cos 26 17z 10 x r2 cos 26 lemniscate Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 8 19 Polar Coordinates 139 quot i i v Polar Coordinates Relation to Rectanular Coordinates Squot mme Symmetry lr it 9 symmetry about the x axis symmetry about the y axis symmetry about the origin 3 Lemniscate ribbon r2 cos 26 HO 1 0 COS27T 9 COS27T 26 cos 26 M 1 0 x gt if r6 E graph then r 7139 9 E graph lrniafe6 XI Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 8 19 Polar Coordinates 139 quot i i v Polar Coordinates Relation to Rectanular Coordinates Squot mme Symmetry lr it 9 symmetry about the x axis symmetry about the y axis symmetry about the origin 3 Lemniscate ribbon r2 cos 26 HO 1 0 COS27T 9 COS27T 26 cos 26 M 1 0 x gt if r6 E graph then r 7139 9 E graph lezrniafe6 gt symmetric about the origin I Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 8 19 Polar Coordinates Relation to Rectanular Coordinates 57 mm Lemniscates Ribbons r2 asin 26 r2 acos 26 Polar Coordinates r24sin 26 r24cos 26 Lemniscate r2 asin 20 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 29018 9 19 Polar Coordinates Relation to Rectanular Coordinates 57 mm Lemniscates Ribbons r2 asin 26 r2 acos 26 Polar Coordinates r24sin 26 r24cos 26 Lemniscate r2 asin 20 sin27r 6 sin27r 20 sin 26 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 29018 9 19 Polar Coordinates Polar Coordinates Relation to Rectanular Coordinates 57 mm Lemniscates Ribhons r2 asin 26 r2 acos 26 r24sin 26 r24cos 26 Lemniscate r2 asin 20 sin27r 6 sin27r 20 sin 26 gt if r0 E graph then r7r 0 E graph Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 29018 9 19 Polar Coordinates Relation to Rectanular Coordinates 57 mm Lemniscates Ribbons r2 asin 26 r2 acos 26 Polar Coordinates r24sin 26 r24cos 26 Lemniscate r2 asin 20 sin27r 6 sin27r 20 sin 26 gt if r0 E graph then r7r 0 E graph gt symmetric about the origin 3911 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 29018 rz i ssym mm 9mg byrzgtlt 5 awn 2 mews mug mm mama Graphing in Polar Coordinates Slral Limacons Flowers Intersections Spiral of Archimedes r 9 9 0 l Polar aXIs rza 620 spiral of Archimedes The curve is a nonending spiral Here it is shown in detail from 9Oto 2w mi Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 11 19 r Graphing in Polar Coordinates S Limagon Snail r 1 2c056 V 377 39 V polar axis 0 S SS 71 Polar axis 0 polar axis 0393377 0363277 polar axis 1 OSHSEIT 9 0 774 773 772 273 374 77 574 473 372 573 774 277 77 1 041 0 1 2 241 3 241 2 1 0 041 1 Ill Jiwen He University of Houston Math 1432 Section 26625 Lecture 12 February 21 2008 12 19 Graphing in Polar Coordinates S Limagon Snail r 1 2cos 9 quot39 polaraxis 37r 39 V polar axis Oses n Osesn Polar axis 0 polar axis 0393ng 03632 9 0 714 753 762 2713 37T4 7T 57T4 4753 37T2 SIT3 7JT4 27139 I 1 041 0 1 2 241 3 241 2 1 O 041 1 o r O at 9 7r gm lrl is a local maximum at 9 07r 27r Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 in 1219 Graphing in Polar Coordinates S Limacon Snail r 1 2c056 V 39 olar aXIs 3 7 V polar aXlS P 1 OSQSEIT OSQSK Polar axis 0 polar axis 03632 5 OSQSEIT 9 0 774 773 772 273 374 77 574 473 372 573 774 277 7 1 041 0 1 2 241 3 241 2 1 0 041 1 o r O at 9 7r gm lrl is a local maximum at 9 O7r27r 0 Sketch in 4 stages 07 7T7T 71quot 7T27T in Jiwen He University of Houston Math 1432 Section 26625 Lecture 12 February 21 2008 12 19 Graphing in Polar Coordinates S Limacon Snail r 1 2c056 V 39 polar aXIs 3 7 V polar aXlS 1 OSQSEIT polar axis 0 S SS 71 U polar axis 03632 5 OSQSEIT 9 0 774 773 772 273 374 77 574 473 372 573 774 277 77 1 041 0 1 2 241 3 241 2 1 0 041 1 o r O at 9 7r gm lrl is a local maximum at 9 O7r27r 0 Sketch in 4 stages 07 7T7T 71quot 7T27T o cos 9 c058 gt if r9 E graph then r7 9 E graph gt symmetric about the X axis Math 1432 Section 26626 Lecture 12 February 21 2008 12 19 Jiwen He University of Houston Graphing in Polar Coordinates Limagons Snails r a bcosQ r3cose r cose r10056 rlcose convex limagon cardioid limagon with limagon with a dimple an inner loop The general shape of the curve depends on the relative magnitudes of lal and Ill Jiwen He University of Houston Math 1432 SectiOn 26626 Lecture 12 February 21 2008 13 19 Graphing in Polar Coordinates Siral Limacons Flowers Intersections Cardioids Heart Shaped r 1 it cos 9 r 1 i sin 9 LDEW r1c050 r1sin0 r1 c050 rzl sine Each change c056 gt sin6 gt cos gt sin6 represents a counterclockwise rotation by 7r radians Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 14 19 Graphing in Polar Coordinates Siral Limacons Flowers Intersections Cardioids Heart Shaped r 1 it cos 9 r 1 i sin 9 LDEW r1c050 r1sin0 r1 c050 rzl sine Each change c056 gt sin6 gt cos gt sin6 represents a counterclockwise rotation by 7r radians o Rotation by 7r r 21 cos6 7r 1 sin 6 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 14 19 Graphing in Polar Coordinates Siral Limaconis Flowers Intersections Cardioids Heart Shaped r 1 it cos 9 r 1 i sin 9 LDEW r1c030 r1sin0 r1 c050 rzl sine Each change c056 gt sin6 gt cos gt sin6 represents a counterclockwise rotation by 7r radians o Rotation by 7r r 21 cos6 7r 1 sin 6 o Rotation by 7r r 1 sin6 7r 1 c056 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 14 19 Graphing in Polar Coordinates Siral Limaconis Flowers Intersections Cardioids Heart Shaped r 1 it cos 9 r 1 i sin 9 LDEW r1c030 r1sin0 r1 c050 rzl sine Each change c056 gt sin6 gt cos gt sin6 represents a counterclockwise rotation by 7r radians o Rotation by 7r r 21 cos6 7r 1 sin 6 o Rotation by 7r r 1 sin6 7r 1 c056 o Rotation by 7r r 1 cos6 7r 1 sin 6 1H Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 14 19 3 mm W mquot hygm 2 MM b m z my a we 2 9 mg in web 2 Graphing in Polar Coordinates Petal Curve r cos 26 1 9Z7Z39 Osesin 0363 OSQSEJI Olt0lt7r 0303 Osesgzr 0363 Oltes2zr Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 16 19 Graphing in Polar Coordinates Petal Curve r cos 26 1 9Z7Z39 OSQSi 39 0363 OSQSEJI Olt0lt7r 030337 osesgzz Osaggn oses2zz 37T57T77T or Oat6 4444 r is a local maximum at 6 0 gm 377K271 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 Graphing in Polar Coordinates Petal Curve r cos 26 1 9Z7Z39 f OSQSi 39 0363 OSQSEJI Olt0lt7r 030337 osesgzz Osaggn oses2zz 37T57T77T or Oat6 4444 r is a local maximum at 6 0 gm 3772271 0 Sketch the curve in 8 stages Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 16 19 Graphing in Polar Coordinates Petal Curve r cos 26 030337 osesgzz Osaggn oses2zz 1 37T 57T 77T or Oat6 4444 3 r IS a local maXImum at 6 0 gm 772271 0 Sketch the curve in 8 stages 0 cos2 6 cos 26 cos27r i 6 cos 26 In gt symmetric about the X axis the y axis and the origin Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 rsin30 rcos40 Jiwen He University of Houston Math 1432 Set2f r1 261626 Lecture 12 February 21 2008 17 19 rsin30 rCOS46 o If n is odd there are n petals Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 17 19 rsin30 rCOS46 o If n is odd there are n petals o If n is even there are 2n petals m Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 vi Graphing in Polar Coordinates Siral Limacons Flowers Intersections Intersections r a1 cos 6 and r a1 cos 6 1 19 27 la gm O a polar axis r a1 c056 r a1 C089 3 a E7 la in o a Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 18 19 iv Graphing in Polar Coordinates Siral Limacons Flowers Intersections Intersections r 31 cos 6 and r 31 cos 6 1 19 27 la gm O a polar axis r a1 c056 r a1 C0819 3 3 637 a 7r O a o r 31 cos and r 31cos gt r 3 3nd c056 0 gt r 3 3nd 9 g mr gt 3 g MT 6 intersection gt n even 3 g MT 2 3 n odd 3 g MT 2 3 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 18 19 quot iv Graphing in Polar Coordinates Siral Limacons Flowers Intersections Intersections r 31 cos 6 and r 31 cos 6 1 19 27 la gm O a polar axis r a1 c056 r a1 C0819 a E7 la in o a o r 31 cos and r 31cos gt r 3 and c056 0 gt r 3 and 9 g mr gt 3 g MT 6 intersection gt n even 3 g MT 2 3 n odd 3 g MT 2 3 0 Two intersection points 3 03 and 3 O 3 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 18 19 39 Graphing in Polar Coordinates Siral Limacons Flowers Intersections Intersections r 31 cos 6 and r 31 cos 6 1 19 27 a gm O a polar axis r a1 c056 r a1 C089 3 a E7 a in o a 0 r 31 c056 and r 31 cosd gt r 3 and c056 0 gt r 3 and 9 g mr gt 3 g MT 6 intersection gt n even 3 g MT 2 3 n odd 3 g MT 2 3 377T 0 Two intersection points 3 03 and 3 O 3 o The intersection third point the origin but the two cardioids pass through the origin at different times H 1819 Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008 a My Damn My mdnms 4 Rehimn w medium aurdmziz a Symmat39y mm w my aurdmziz 4 5w Wm was merainrs Lecture 12 Section 93 Polar Coordinates Section 94 Graphing in Polar Coordinates Jiwen He Department of Mathematics University of Houston jiwenhe mathuhedu httpmathuhedumjiwenheMath1432 l l 2m2 j gain 39 4 4 I i Polar axis 7r7r 0 4 272 27x ma mm 11 Eva ramp 620 siral of Archimedes Jiwen He University of Houston Math 1432 Section 26626 Lecture 12 February 21 2008

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over $500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.