### 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

## 17

## 0

## Popular in Course

## Popular in Department

This 92 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 17 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

Lecture 15 Section 97 Tangents to Curves Given Parametrically Jiwen He Department of Mathematics University of Houston jiwenhe mathuhedu httpmathuhedumjiwenheMath1432 I b 5 one tangent two tangents no tangent m Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 1 14 Tangents Tangents to Parametrized curves Tangent line I secant line xt0 h yt0 h tangent line X00 3200 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Martzh 394 2008 2 14 Tangents L Tanents Examles Tangents to Parametrized curves C Let C xtyt t e I secantlme For a time to E I assume X t0 75 O xt0 h yt0 h tangent line X00 3200 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Maren 394 2008 2 14 Tangents L Tanents Examles Tangents to Parametrized curves Tangent ine Let C xtyt t e I secantlme For a time to E I assume X t0 75 O The slope of the curve at time to is xtO7yt0 h to mob y tangent line X X00 3200 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Maren 394 2008 2 14 Tangents L Tanents Examles Tangents to Parametrized curves Tangent ine Let C xtyt t e I secantlme For a time to E I assume X t0 75 O The slope of the curve at time to is xtO7yt0 h to mob y tangent line X X00 3200 Yt h Yt muo Iiino Xt h Xt Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Maren 394 2008 2 14 Tangents L Tanents Examles Tangents to Parametrized curves C Let C xtyt t e I secantlme For a time to E I assume X t0 75 O The slope of the curve at time to is xtO7yt0 h to mob y tangent line X X00 3200 m Yt0 h yto Yt0 h H150 h gtO h Xt0 Xt0 397an h H ma 2 Am Xt0 h Xt0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Mareh 394 2008 2 14 Tangents L Tanents Examles Tangents to Parametrized curves C Let C xtyt t e I secantlme For a time to E I assume X t0 75 O The slope of the curve at time to is xtOhytO h to mob y tangent line X X00 3200 m Yt0 h yto Yt0 h H150 h gtO h yto Xt0 h Xt0 X t0 lian h H ma 2 Am Xt0 h Xt0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Mareh 394 2008 2 14 Tangents L Tanents Examles Tangents to Parametrized curves Tangent line C Let C xtyt t e I For a time to E I assume X t0 75 O The slope of the curve at time to is secant line xt0 h yt0 h y to mt0 Xt tangentline O xz0yz0 The equation of the tangent line is Xt0y Yo yt0X X0 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Mareh 394 2008 2 14 Tangents L Tanents Examles Tangents to Parametrized curves Tangent line C Let C xtyt t e I secantlme For a time to E I assume X t0 75 O The slope of the curve at time to is xl0 h yto h y mt0 t tangentline X 0 xz0yz0 The equation of the tangent line is Xt0y Yo yt0X X0 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Mareh 394 2008 2 14 Tangents L Tanents Examles Tangents to Parametrized curves Tangent line C Let C xtyt t e I secantlme For a time to E I assume X t0 75 O The slope of the curve at time to is xl0 h yto h y mt0 t tangentline X 0 xt0yt0 The equation of the tangent line is Xt0y Yo yt0X X0 0 W150 y Yo X X0 gt Xt0y Yo yt0X X0 Xto Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Mareh 394 2008 2 14 Tangents L Tanents Examles Tangents to Parametrized curves Tangent line C Let C xtyt t e I secantlme For a time to E I assume X t0 75 O The slope of the curve at time to is x00 h Jto h y mt0 t tangentline X 0 xtOytO The equation of the tangent line is Xt0y Yo yt0X X0 0 Definition 0 The curve has a vertical tangent if X t0 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Mareh 394 2008 2 14 Tangents L Tanents Examles Tangents to Parametrized curves Tangent line C Let C xtyt t e I secantlme For a time to E I assume X t0 75 O The slope of the curve at time to is x00 h Jto h y mt0 t tangentline X 0 Worm The equation of the tangent line is Xt0y Yo yt0X X0 0 Definition 0 The curve has a vertical tangent if X t0 0 0 The curve has a horizontal tangent if y t0 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 Mareh 394 2008 2 14 Tangents it 1 Tan ents Examles Tangents to Parametrized curves Tangent line C Let C xtyt t e I secantlme For a time to E I assume X t0 75 O The slope of the curve at time to is Xto h JIo h y mt0 t tangentline X 0 WWW The equation of the tangent line is Xt0y Yo yt0X X0 0 J o The graph of a function y fx X E I is a curve C that is parametrized by Xt t yt ft t E I ll Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 394 2008 2 14 Tangents it 1 Tan ents Examles Tangents to Parametrized curves Tangent line C Let C xtyt t e I secantlme For a time to E I assume X t0 75 O The slope of the curve at time to is Xto h JIo h y mt0 t tangentline X 0 Worm The equation of the tangent line is Xt0y Yo yt0X X0 0 o The graph of a function y fX X E I is a curve C that is parametrized by Xt t yt ft t E I o The slope of the curve at time to is y t0 dy m 0 Xt0 dX X0 X0 X0 0 ll Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 394 2008 2 14 Tangents L 1 Tan ents Examles Velocity and Speed Along a Plae Curve Parametrization by the Motion secant line X00 I l yt0 tangent line X00 yZ0 Velocity and Speed Along a Plane Curve Jiwen He University of Houston 7 Math 1432 Section 26626 Lecture 15 Mareh 394 2008 L 3 14 Tangents L 1 Tan ents Examles Velocity and Speed Along a Plae Curve Parametrization by the Motion secant line 0 Imaging an object moving along xtO1yt0h the curve C tangent line X00 yZ0 Velocity and Speed Along a Plane Curve Jiwen He University of Houston 7 Math 1432 Section 26626 Lecture 15 Mareh 394 2008 L 3 14 Tangents L 1 Tan ents Examles Velocity and Speed Along a Plane Curve Parametrization by the Motion secant line 0 Imaging an object moving along the curve C o Let rt xtyt the position of the object at time t xt0 h yt0 h tangent line X00 yZ0 Velocity and Speed Along a Plane Curve Jiwen He University of Houston 7 Math 1432 Section 26626 Lecture 15 Mareh 394 2008 L 3 14 Tangents L 1 Tan ents Examles Velocity and Speed Along a Plane Curve Parametrization by the Motion secant line 0 Imaging an object moving along the curve C o Let rt xtyt the position of the object at time t xt0 h yt0 h tangent line X00 yZ0 Velocity and Speed Along a Plane Curve 0 The velocity of the object at time t is var at x39ty t Jiwen He University of Houston 7 Math 1432 Section 26626 Lecture 15 March 394 2008 L 3 14 Tangents L 1 Tan ents Examles Velocity and Speed Along a Plane Curv Parametrization by the Motion secant line 0 Imaging an object moving along the curve C o Let rt xtyt the position of the object at time t xt0 h yt0 h tangent line X00 yZ0 Velocity and Speed Along a Plane Curve 0 The velocity of the object at time t is vmmowmwml 0 The speed of the object at time t is woawm km mmt Jiwen He University of Houston 7 Math 1432 Section 26626 Lecture 15 March 394 2008 L 3 14 Tangents L 1 Tan ents Examles Velocity and Speed Along a Plae Curve Parametrization by the Motion secant line 0 Imaging an object moving along the curve C o Let rt xtyt the position of the object at time t xt0 h yt0 h tangent line X00 yZ0 Velocity and Speed Along a Plane Curve 0 The velocity of the object at time t is vmmowmwml 0 The speed of the object at time t is 2 2 WU MWVVWl NM o The instanteneous direction of motion gives the unit tangent vector T TU M l Vt Jiwen He University of Houston 7 Math 1432 Section 26626 Lecture 15 March 394 2008 L 3 14 Tangents Example Line Segment t7r37r 1 2 I O 27 47 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta ngents Example Line Segment 127 37 0 Set Xt t then yt 21 t 6 13 1 2 I O 27 47 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 21 t 6 13 1 2 I O 27 47 At time t 6 13 0 The position rt xtyt t2t Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 1 2 I O 27 47 At time t 6 13 0 The position rt xtyt t2t o The velocity vt X ty t 12 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 1 2 I O 27 47 At time t 6 13 0 The position rt xtyt t2t o The velocity vt X ty t 12 0 The speed vt HvtH X t2 yt2 V3 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 1 2 I O 27 47 At time t 6 13 0 The position rt xtyt t2t o The velocity vt X ty t 12 0 The speed vt HvtH X t2 yt2 V3 0 The unit tangent vector Tt i512 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 1 2 I O 27 47 At time t 6 13 0 The position rt xtyt t2t o The velocity vt X ty t 12 0 The speed vt llvtll X t2 yt2 V3 0 The unit tangent vector Tt i512 o The slope mt E3 2 H Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 1 2 I O 27 47 At time t 6 13 0 The position rt xtyt t2t o The velocity vt X ty t 12 0 The speed vt llvtll X t2 yt2 V3 0 The unit tangent vector Tt i512 o The slope mt E3 2 H o The tangent line y 2X Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 10 27 47 At time t e 0 2 o The position rt xtyt 3 156 2t Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 10 27 47 At time t e 0 2 o The position rt xtyt 3 156 2t 0 The velocity vt X ty t 1 2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 10 27 47 At time t e 0 2 o The position rt xtyt 3 156 2t 0 The velocity vt X ty t 1 2 o The speed vt iiinu X t2 Mn2 V3 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 10 27 47 At time t e 0 2 o The position rt xtyt 3 156 2t 0 The velocity vt X ty t 1 2 o The speed vt iiinu X t2 Mn2 V3 o The unit tangent vector Tt Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 10 27 47 At time t e 0 2 o The position rt xtyt 3 156 2t 0 The velocity vt X ty t 1 2 o The speed vt iiinil X t2 Mn2 V3 0 The slope mt E3 2 H o The unit tangent vector Tt Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 127 37 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 10 27 47 At time t e 0 2 o The position rt xtyt 3 156 2t 0 The velocity vt X ty t 1 2 o The speed vt iiinil X t2 Mn2 V3 0 The slope mt E3 2 H o The unit tangent vector Tt o The tangent line y 2X Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 120 0 Set Xt 2 cos t then yt 4 2COS 15 t E 047t j At time t E 047t o The position rt Xtyt 2 cos t 4 2 cos t Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 120 0 Set Xt 2 cos t then yt 4 2COS 15 t E 047t At time t E 047t o The position rt Xtyt 2 cos t 4 2 cos t o The velocity vt X ty t sin t 2sin t Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts Example Line Segment 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 120 0 Set Xt 2 cos t then yt 4 2COS 15 t E 047t At time t E 047t o The position rt Xtyt 2 cos t 4 2 cos t o The velocity vt X ty t sin t 2sin t o The speed vt HvtH X t2 yt2 sin t Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts n Example Line Segment 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 120 0 Set Xt 2 cos t then yt 4 2COS 15 t E 047t At time t E 047t o The position rt Xtyt 2 cos t 4 2 cos t o The velocity vt X ty t sin t 2sin t o The speed vt HvtH X t2 yt2 sin t o The unit tangent vector Tt i512 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts n Example Line Segment 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 120 0 Set Xt 2 cos t then yt 4 2COS 15 t E 047t At time t E 047t o The position rt Xtyt 2 cos t 4 2 cos t o The velocity vt X ty t sin t 2sin t o The speed vt HvtH X t2 yt2 sin t o The unit tangent vector Tt i512 o The slope mt E3 2 m Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta n ge nts n Example Line Segment 0 Set Xt t then yt 2t t 6 13 12 0 Set Xt 3 t then yt 6 2t t e 0 2 120 0 Set Xt 2 cos t then yt 4 2COS 15 t E 047t At time t E 047t o The position rt Xtyt 2 cos t 4 2 cos t o The velocity vt X ty t sin t 2sin t o The speed vt HvtH X t2 yt2 sin t o The unit tangent vector Tt i512 o The slope mt E3 2 m o The tangent line y 2X Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 4 14 Ta ngents i r Example Parabola Arc X I 0 1t 7T x 1 y2 xt sin2 t yt cos re 0 7r Point of the Vertical Tangent rt0 1 O Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 Ta ngents i r Example Parabola Arc X O1tO I O 1t 7239 x 1 y2 Xt sin2 t yt cos re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 t2 yt t t 6 11 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 Ta ngents i r Example Parabola Arc X O1tO I O 1t 7239 x 1 y2 Xt sin2 t yt cos re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 t2 yt t t 6 11 0 rt0 Xtoyt0 1 t3 to I gt to I Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 Ta ngents i r Example Parabola Arc X O1tO I O 1t 7239 x 1 y2 Xt sin2 t yt cos I re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 t2 yt t t 6 11 0 rt0 Xtoyt0 1 t3 to I gt to I 0 Velocity vt0 x t0y t0 2150 1 01 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 y y O t O O 1t7239 x 1 y2 Xt sin2 t y1 cos I re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 t2 yt t t 6 11 0 rt0 Xtoyt0 1 t3 to I gt to I 0 Velocity vt0 x t0y t0 2150 1 01 0 Speed m0 Hvt0H X t02 yt02 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 y y O t O O 1t7239 x 1 y2 Xt sin2 t yt cos re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 t2 yt t t 6 11 0 rt0 Xtoyt0 1 t3 to I gt to I 0 Velocity vt0 x t0y t0 2150 1 01 0 Speed m0 Hvt0H X t02 yt02 1 0 Unit tangent vector Tt0 O 1 H Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 O1tO I O 1t 7239 x 1 y2 Xt sin2 t yt cos re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 t2 yt t t 6 11 0 rt0 Xtoyt0 1 t3 to I gt to I 0 Velocity vt0 x t0y t0 2150 1 01 0 Speed m0 Hvt0H X t02 yt02 1 50 01 Vt0 0 Unit tangent vector Tt0 o Tangent ine X 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 514 Ta ngents 1 r Example Parabola Arc X I 0 1t 7T x 1 y2 xt sin2 t yt cos re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 cos2 t yt cos t t E 07139 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 Ta ngents i r Example Parabola Arc X O1tO I 0 1t 7T x 1 y2 xt sin2 t yt cos re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 cos2 t yt cos t t E 07139 0 rt0 1 cos2 t0cos to 10 gt to Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 Ta ngents i r Example Parabola Arc X O1tO I O 1t 7239 x 1 y2 xt sin2 t yt cos re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 cos2 t yt cos t t E 07139 0 rt0 1 cos2 t0cos to 10 gt to O Vt0 Xtoylt0 2 sin 21390 sin to I O 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 Ta ngents i r Example Parabola Arc X O1tO I O 1t 7239 x 1 y2 xt sin2 t yt cos re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 cos2 t yt cos t t E 07139 0 rt0 1 cos2 t0cos to 10 gt to O Vt0 Xtoylt0 2 sin 21390 sin to I O 1 0 Speed m0 Hvt0H AX102 yt02 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 Ta ngents i r Example Parabola Arc X O1tO I O 1t 7239 x 1 y2 xt sin2 t yt cos re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 cos2 t yt cos t t E 07r o rt0 1 cos2 t0cos to 10 gt to O Vt0 Xtoylt0 2 sin 21390 sin to I O 1 0 Speed m0 Hvt0H AX102 yt02 1 0 Unit tangent vector Tt0 V050 O 1 H Vt0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 Ta ngents i r Example Parabola Arc X y y O t O x x I O 1t7239 x 1 y2 xt sin2 t yt cos re 0 7 Point of the Vertical Tangent rt0 1 O Curve Xt 1 cos2 t yt cos t t E 07r o rt0 1 cos2 t0cos to 10 gt to O Vt0 Xtoylt0 2 sin 21390 sin to I O 1 0 Speed m0 Hvt0H AX102 yt02 1 0 Unit tangent vector Tt0 V050 O 1 H Vt0 o Tangent line X 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 5 14 Slope of the Spiral of Archimedes r 6 at 60 Jiwen He University of Houston Math 1432 Section 26626 Lecture 1539 March 4 2008 6 14 Slope of the Spiral of Archimedes r 6 at 60 g 0 r090 I X90y90 I 90 COS 9090 sin 90 I 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 1539 March 4 2008 6 14 Slope of the Spiral of Archimedes r 6 at 60 g 0 r090 I X90y90 I 90 COS 9090 sin 90 I 0 o v60 r 6 0 cos 90 80 sin 905in 9080 cos 60 g 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 1539 March 4 2008 6 14 Slope of the Spiral of Archimedes r 6 at 60 g 0 r090 I X90y90 I 90 COS 9090 sin 90 I 0 o v60 r 6 0 cos 90 80 sin 905in 9080 cos 60 g 1 Y 90 2 7T o Slope m6 0 Mao Jiwen He University of Houston Math 1432 Section 26626 Lecture 1539 March 4 2008 6 14 Slope of the Spiral of Archimedes r 6 at 60 g 0 r090 I X90y90 I 90 COS 9090 sin 90 I 0 o v60 r 6 0 cos 90 80 sin 905in 9080 cos 60 g 1 o Slope m6 0 o Tangent line at 90 y g X llll Jiwen He University of Houston Math 1432 Section 26626 Lecture 1539 March 4 2008 6 14 1733zosi 52 ardnd mm zhm2zonwnh2nmnabop 2 yzzsmam ardnd mm zhm2zonwnh2nmnabop V m Ta ngents Example Limaeon a 2 polar axis 1 51 2 3 27r Point of Vertical Tangent for Limaeon Snail r 1 c056 Jiwen He University of Houston Math 1432 Sectiran 26626 Lecture 15 March 4 20018 8 14 Tangents I39w39 v Example Limaeon i5 2 polar axis 1 57 2 3 27r Point of Vertical Tangent for Limaeon Snail r 1 c056 o rt Xtyt 1 cos t cos t 1 cos t sin t Jiwen He University of Houston Math 1432 Sectian 26626 Lecture 15 March 4 20018 8 14 Tangents I39w39 v Example Limaeon i5 2 polar axis 1 57 2 3 27r Point of Vertical Tangent for Limaeon Snail r 1 c056 o rt Xtyt 1 cos t cos t 1 cos t sin t o vt r t 2 cost 1sin t 1 cos t12cos Jiwen He University of Houston Math 1432 Sectian 26626 Lecture 15 March 4 20018 8 14 Tangents I39w39 v Example Limaeon i5 2 polar axis 1 57 2 3 27r Point of Vertical Tangent for Limaeon Snail r 1 c056 o rt Xtyt 1 cos t cos t 1 cos t sin t o vt r t 2 cost 1sin t 1 cos t12cos 0 Set X t 0 cost or sint0 then t 7T Jiwen He University of Houston Math 1432 Sectian 26626 Lecture 15 March 4 20018 Ta ngents i 1H1 Example Limaeon 27r polar axis 151 23 Point of Vertical Tangent for Limaeon Snail r 1 c056 o rt Xtyt 1 cos t cos t 1 cos t sin t o vt r t 2 cost 1sin t 1 cos t12cos 0 Set X t 0 cost or sint 0 then t 7T 5 o Tangent line is vertical at rt 20 ll Jiwen He University of Houston Math 1432 Sectiran 266126 Lecture 15 March 4 20018 8 14 r sin 36 r cos 49 Tangent Lines at the Origin r sin 38 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 914 rsin 36 rzcos 49 Tangent Lines at the Origin r sin 38 o The curve passes through the origin when r sin 38 0 ie 2 atr90 o rt Xtyt sin 3tcos t sin 3tsin t 00 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 9 14 rsin 36 rzcos 49 Tangent Lines at the Origin r sin 38 o The curve passes through the origin when r sin 38 0 ie at 9 0 g o rt Xtyt sin 3tcos t sin 3tsin t 00 0 vt r t 3 cos 3tcos t sin3tsin t3cos3tsin t sin 3tcos t 30 3 g Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 914 rsin 36 rzcos 49 Tangent Lines at the Origin r sin 38 o The curve passes through the origin when r sin 38 O ie at 9 07 g o rt Xtyt sin 3tcos t sin 3tsin t 00 0 vt r t 3 cos 3tcos t sin3tsin t3cos3tsin t sinstcosw 30 lt 7 gt7 37 o Slope mt 5 8 07 3 3 Jiwen He University of Houston Math 1432 Section 266 Lecture 15 March 4 2008 914 rsin 36 rzcos 49 Tangent Lines at the Origin r sin 38 o The curve passes through the origin when r sin 38 0 ie at 9 0 g o rt Xtyt sin 3tcos t sin 3tsin t 00 0 vt r t 3 cos 3tcos t sin3tsin t3cos3tsin t sin3tcos t 30 3 o Slope mt 5 8 0 3 3 o Tangent line at the origin y 0 y 3X y 2 3X Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 914 r sin 36 r cos 46 Tangent Lines at the Origin r cos 46 o The curve passes through the origin when r cos 46 O ie 37T 57T 77T 97T 117T 137T 157T 7T at9 vsvsvsv Jiwen He University of Houston 8787878 Math 1432 Section 26626 Lecture 15 March 4 2008 914 r cos 46 r sin 36 Tangent Lines at the Origin r cos 46 o The curve passes through the origin when r cos 46 O ie atg37T57T77T97T117Tl37T157T 8787878787 87 87 839 cos4cos t cos 4t sin t O O 7T 0 rt Xt7yt March 4 2008 9 14 Math 1432 Section 26626 Lecture 15 Jiwen He University of Houston rsin 36 rcos 46 Tangent Lines at the Origin r cos 46 o The curve passes through the origin when r cos 46 O ie at9z3wSw7wgwnwl3www 8787878787 87 87 839 o rt Xtyt cos4cos tcos4tsin t 00 0 vt r t 4 sin 4t cos t cos4tsin t 4 sin 4t sin t cos4tcos t 4cos 4sin g 4cos 3 7T4sin 3 Jiwen He University of Houston Math 1432 Section 266 Lecture 15 March 4 2008 914 rsin 36 rCOS 46 Tangent Lines at the Origin r cos 46 o The curve passes through the origin when r cos 46 0 Le atQ 37T57T77T97T117T137T157T 878787878787878 o rt Xtyt cos4cos tcos4tsin t 00 0 vt r t 4 sin 4t cos t cos4tsin t 4 sin 4t sin t cos4tcos t 4cos 4sin g 4cos 3 7T4sin 3 0 Slope mt tan gjtan 3 March 4 2008 Math 1432 Section 26626 Lecture 15 914 Jiwen He University of Houston rsin 36 rcos 46 Tangent Lines at the Origin r cos 46 o The curve passes through the origin when r COS 49 Z 0 Le at9 37T57T77T97r 7T137rl57r 8787878787 87 87 839 o rt Xtyt cos4cos tcos4tsin t 00 0 vt r t 4 sin 4t cos t cos4tsin t 4 sin 4t sin t cos4tcos t 4cos 4sin g 4cos 3 7T4sin 3 0 Slope mt g EB tan gjtan 3 o Tangent line at the origin y tan X y tan 3X H Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 9 14 3 yzzsmzan mehsmg mm gawamzmws a ryezumx zmswe zesmmepm s 2 m b mum z Ame v Locus Circles Elli ses Circles C P dl O la r22 r 2 r4sin0 r 4COSQ Horizontal and Vertical Tangent for Circle Centered at c d Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 11 14 v Locus Circles Elli ses Circles C P dl O la mm WW r22 r 2 r4sin0 r 4COSQ Horizontal and Vertical Tangent for Circle Centered at c d o rt xtyt acos t l c asin t l d t E 0 2n Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 11 14 v Locus Circles Elli ses Circles C P dl O la mm WW r22 r 2 r4sin0 r 4COSQ Horizontal and Vertical Tangent for Circle Centered at c d o rt xtyt acos t l c asin t l d t E 0 2n 0 vt r t asin t acos Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 11 14 v Locus Circles Elli ses Circles C P dl O la mm WW r22 r 2 r4sin0 r 4COSQ Horizontal and Vertical Tangent for Circle Centered at c d o rt xtyt acos t l c asin t l d t E 0 2n 0 vt r t asin t acos 0 Set X t 0 sin t 0 then t 07T set y t 0 cos t 0 then t g 3 Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 v Locus Circles Elli ses Circles C P dl O la r22 r 2 r4sin0 r 4 COSQ Horizontal and Vertical Tangent for Circle Centered at c d o rt xtyt acos t l c asin t l d t E 0 2n 0 vt r t asin t acos 0 Set X t 0 sin t 0 then t 07T set y t 0 cos t 0 then t g 37 0 Tangent line is horizontalat rt C d l a C d a it is vertical at rt c l a d c a d Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 I Locus Circles Ellipses Cosine and Sine Ul 37 1 4 3 4 7 4 3 1 i Horizontal and Vertical Tangent for Ellipse Centered at d e Math 1432 Section 26626 Lecture 15 March 4 2008 12 14 Jiwen He University of Houston I Locus Ellipses Cosine and Sine 3 3 7 31 t Horizontal and Vertical Tangent for Ellipse Centered at d e o rt Xtyt 3 cos t d bsin t e t E 0 2n March 4 2008 12 14 Math 1432 Section 26626 Lecture 15 Jiwen He University of Houston gt Locus Circles Ellises Ellipses Cosine and Sine yll 3 7 31 s Horizontal and Vertical Tangent for Ellipse Centered at d e o rt Xtyt 3 cos t d bsin t e t E 0 2n 0 vt r t a sin t bcos March 4 2008 12 14 Math 1432 Section 26626 Lecture 15 Jiwen He University of Houston gt Locus Circles El lises Ellipses Cosine and Sine ylk 3 7 31 E t Xtyt a cost I d bsin t e t E 0 2n 0 vt r t a sin t bcos O cost0 then tzgj377r March 4 2008 Math 1432 Section 26626 Lecture 15 Jiwen He University of Houston gt Locus Circles Ellises Ellipses Cosine and Sine yll 3 7 14lt mm 4 31 all Horizontal and Vertical Tangent for Ellipse Centered at d e o rt Xtyt a cost I d bsin t e t E 0 2n 0 vt r t a sin t bcos 0 Set X t 0 sin t 0 then t 07T set y t 0 cost 0 then t 1 3 27 7 o Tangent line is horizontalat rt d e a d e a it is vertical at rt d b e d a e H Jiwen He University of Houston Math 1432 Section 26626 Lecture 15 March 4 2008 Locus Circles Ellises Lemniscates Ribbons r2 a2 cos 29 Tangent Lines at the Origin 1 7139 7139 4 4 2 2 r2 4 cos 26 The parametric equations for the lemniscate with a2 2C2 is acost asintcost X tE 027r 1sin2t y 1sin2t Jiwen He University of Houston Math 1432 a Section 26626 Leature 15 March 4 2008 13 14 Locus Circles Eilises r2 a2 cos 26 Tangent Lines at the Origin Lemniscates Ribbons Ir0 Xtyt 00 2 in l t E 77 4 4 27 239 2 2 r24C0526 The parametric equations for the Iemniscate with a2 2C2 IS a cost a sin tcost X 2 y 2 tEO27t 1sm t 1sm t I i March 4 2008 13 14 Math 1432 a Section 26626 Leat ure 15 Jiwen He University of Houston Locus Circles Eilises Lemniscates Ribbons r2 a2 cos 29 Tangent Lines at the Origin I1 Xt7t 070 2 7r l7r t E 3 4 4 27 2 39 V0 Ir 0 37 37 3 2 2 r24cos 26 The parametric equations for the Iemniscate with a2 2C2 is acost asintcost X tE 027r 1sin2t y 1sin2t Jiwen He University of Houston Math 1432 a Section 26626 Lemme 15 March 4 2008 13 14 Locus Circles Eilises Lemniscates Ribbons r2 a2 cos 29 Tangent Lines at the Origin 1 o rlttgt xlttyltt 00 Z 175 t g7 V0 Ir 0 3 3 3 2 2 t o Slope mt Xt 1 1 gt 80 tan 1 1 91 tan 1 1 3T r24cos 26 The parametric equations for the Iemniscate with a2 2C2 is acost asintcost X tE 027r 1sin2t y 1sin2t Jiwen He University of Houston Math 1432 a Section 26626 Lemme 15 March 4 2008 13 14 Locus Circles Eilises Lemniscates Ribbons r2 a2 cos 29 Tangent Lines at the Origin Ir05 Xt7yt 070 2 7r in t E 7T 4 4 27 2 39 Vt Ir 0 3 7 2 2 t o Slope mt Xt 1 1 2 90 tan 11 91 tan 1 1 3T V2 4 C05 29 o Tangent line at the origin y X y X The parametric equations for the Iemniscate with a2 2C2 is acost asintcost X tE 027r 1sin2t y 1sin2t Jiwen He University of Houston Math 1432 a Section 26626 Lemme 15 March 4 2008 13 14 a Tzngms w Mmm m5 Tzngawsw pmmmmmm Bump 6 Laws a des mg W mm w

### 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

#### "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."

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I 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."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### 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.