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# Class Note for MATH 1432 at UH 3

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Date Created: 02/06/15
Lecture 14 Section 96 Curves Given Parametrically Jiwen He Department of Mathematics University of Houston jiwenhe mathuhedu httpmathuhedumjiwenheMath1432 3 Mi yt Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 1 15 Parametrized curve 15 Para metrized curve Para metrized curve xt yf Jiwen He University of Houston Math 1432 5661M 12552153 Le zm r e 1394 328 2098 2 Parametrized curve 115 Lsci Para metrized curve Para metrized curve A parametrized Curve is a path in the xy plane traced out by the point Xtyt as the parameter t ranges over an interval I x c ltxlttgtylttgt t e I xt yf Jiwen He University of Houston Math 5661mm Lecwre 1394 1287 2098 2 Parametrized curve 115 Lsci Para metrized curve Para metrized curve A parametrized Curve is a path in the xy plane traced out by the point Xtyt as the parameter t ranges over an interval I x c ltxlttgtylttgt t e I 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 xt yf Jiwen He University of Houston Math 5661mm Lecwre 1394 1287 2098 2 Parametrized curve 115 LJ Para metrized curve Para metrized curve A parametrized Curve is a path in the xy plane traced out by the point Xtyt as the parameter t ranges over an interval I x C Xtyt t E I 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 graph of a polar equation r 06 6 E I is a curve C that is parametrized by the functions Xtrcostptcost ytrsintptsint tel xt yf Jiwen He University of Houston Math 5661mm Lecwre 1394 1287 2098 2 1 2 3 6 Parametrized curve a Example Line Segment t7t37t t O 27 47 Jiwen He University of Houston Line Segment y 2X X 6 13 Math 1432 Section 25626 Lecture14 February 28 2008 315 Parametrized curve a Example Line Segment Line Segment y 2X X 6 13 Izmg 0 Set Xt t then yt 2t t e 13 y 3 6 1 2 tzO 27 47 Jiwen He University of Houston Math 1432 Section 25626 Lecture14 February 28 2008 3 Parametrized curve a Example Line Segment Line Segment y 2X X 6 13 y 3396 Izmg 0 Set Xt t then yt 2t t e 13 0 Set Xt t 1 then yt 21 2 t e 02 12 tO27r47t Jiwen He University of Houston Math 1432 Section 25626 Lecture14 February 28 2008 3 Parametrized curve a L Example Line Segment y 3 6 1 2 7T37Z39 t O 27 47 Jiwen He University of Houston Line Segment y 2X X 6 13 0 Set Xt t then yt 2t t e 13 0 Set Xt t 1 then yt 2t 2 t e 02 o Set Xt 3 t then yt 6 2t 1 e 02 Math 1432 Section 25626 Lecture14 February 28 2008 3 15 Parametrized curve a L Example Line Segment y 3 6 1 2 7T37Z39 t O 27 47 Jiwen He University of Houston Line Segment y 2X X 6 13 0 Set Xt t then yt 2t t e 13 0 Set Xt t 1 then yt 2t 2 t e 02 o Set Xt 3 t then yt 6 2t 1 e 02 Math 1432 Section 25626 Lecture14 February 28 2008 3 15 Parametrized curve a L Example Line Segment y 3 6 1 2 7T37Z39 t O 27 47 Jiwen He University of Houston Line Segment y 2X X 6 13 0 Set Xt t then yt 2t t e 13 0 Set Xt t 1 then yt 2t 2 t e 02 o Set Xt 3 t then yt 6 2t 1 e 02 9 Set Xt 3 41 then yt 6 81 t e 012 Math 1432 Section 25626 Lecture14 February 28 2008 3 15 Parametrized curve Ll W 7 I 1 Example Line Segment y 3 6 1 2 7T37Z39 t O 27 47 Jiwen He University of Houston Line Segment y 2X X 6 13 0 Set Xt t then yt 2t t e 13 0 Set Xt t 1 then yt 21 2 t e 02 o Set Xt 3 t then yt 6 2t 1 e 02 9 Set Xt 3 41 then yt 6 81 t e 012 0 Set Xt 2 cos t then yt 4 2cos t t E 047T Math 1432 Section 25626 Lecture14 February 28 2008 315 Parametrized curve x t Example Line Segment Line Segment y 2X X 6 13 y 3396 Izmg 0 Set Xt t then yt 2t t e 13 0 Set Xt t 1 then yt 21 2 t e 02 1 2 o Set Xt 3 t then yt 6 2t 10 27 47 t E 0 Set Xt 3 41 then yt 6 81 x t e 012 0 Set Xt 2 cos t then yt 4 2cos t t E 047T We parametrize the line segment in different ways and interpret each parametrization as the motion of a particle with the H parameter t being time Jiwen He University of Houston Math 1432 Section 26626 Lecture14 February 28 2008 3 15 Parametrized curve I 1 Example Parabola O 1r 7r x 1 y2 xt sin2 I yt cos I re 0 7r Parabola Arc X 1 y2 Jiwen He University of Houston Math 1432 Secticn 2662396 Lecture 14 February 28 2008 4 15 Parametrized curve 1 i e 1 Example Parabola O 1r 7r x 1 y2 xt sin2 I yt cos I e 0 7r Parabola Arc X 1 y2 1 g y g 1 0 Set yt t then Xt 1 t2 t E 11gt changing the domain to all real t gives us the whole parabola Jiwen He University of Houston Math 1432 Section 2662396 Lectnre 14 February 28 2008 4 15 Parametrized curve 1 i e l Example Parabola O 1r 7r x 1 y2 xt sin2 I yt cos I e 0 7r Parabola Arc X 1 y2 1 g y g 1 0 Set yt t then Xt 1 t2 t E 11gt changing the domain to all real t gives us the whole parabola 0 Set yt cos t then Xt 1 cos2 t t E 07T gt changing the domain to all real t does not give us any more of H the parabola Jiwen He University of Houston Math 1432 Section 2662396 Lectnre 14 February 28 2008 4 15 gma 4 4 E i Polar axis mm 0 J 271 2719 g g 7 7 7r 7r gm git 4 4 J rza 620 spiral of Archimedes Spiral of Archimedes r 9 9 Z O Jiwen He University of Houston Math 1432 Section 25626 Lecture 14 February 28 20018 5 15 Parametrized curve r V Paramem zed curve 397 Example Spiral of Archimedes l l 2 7 2 7 3 3 Z71 1 1 1 l 1 1 lPolar axis 7 7r 0 X BaZ 39 g g 7 7 7r 7r gm git 4 4 J rza 620 spiral of Archimedes Spiral of Archimedes r 9 9 Z O 0 The curve is a nonending spiral Here it is shown in detail from90to 27r Jiwen He University of Houston Math 1432 Section 25626 Lecture 14 February 28 20018 5 15 Parametrized curve a V Parametlized curve Exam les Example Spiral of Archimedes em a a L 39 4 4 a POMraXB mm 0 A 2 2 x 5 5 173175 1r 17 4 4 Bit rza 620 spnalofArcthedes Spiral of Archimedes r 9 9 Z O 0 The curve is a nonending spiral Here it is shown in detail from 90to 27t o The parametric representation is Xt tcos t yt tsin t t 2 0 ll Jiwen He University of Houston Math 1432 Section 25626 Lecture 14 February 28 20018 5 15 Parametrized curve V Parametrized curve les Example Limacons 3 1 r3cos6 r cose r10056 r COS6 convex limacon cardioid limagon with limacon with a dimple an inner loop Limacons Snails r a l bcos6 The parametric representation is Xt a l bcos t cos t yt a bcos t sin t t 6 0277 Jiwen He University of Houston Math 1432 Section 26526 Lecture 14 February 28 2008 6 15 Para metrized curve Example Petal Curves rsin 36 rcos46 Petal Curves Flowers r acos m9 r asin n9 The parametric representations are Xt a cosnt cos t yt a cosnt sin t t E 0 27r xt asinnt cos t yt asinnt sin 15 t E 0 27Tl H Jiwen He University of Houston Math 1432 Section 26626 Lecture M February 28 2008 i erbolas Lem n iscates r22 r2 r4sin9 1 2 40050 Jiwen He University of Houston Math 1432 39 Semi n 26626 Lecture 14 February 28 20138 8 15 w r i 39 erbolas Lemniscates Circles C P d r22 r2 r4sin9 1 2 40050 Center 0 at 0 0 gt X2 y2 a2 Jiwen He University of Houston Math 1432 39 Semi n 26626 Lecture 14 February 28 20138 8 15 w r i 39 erbolas Lemniscates Circles C P d r22 r2 r4sin9 1 2 40050 CenterOat 00gtX2 y2a2 gtra Jiwen He University of Houston Math 1432 39 Semi n 26626 Lecture 14 February 28 20138 8 15 Circles Ellises H erbolas Lernniscates r2 r2 r4sin9 142 40050 Center 0 at 00 gt X2 l y2 a2 gt ra gt t 027r Xtacost ytasint Jiwen He University of Houston Math 1432 39 Semi n 25626 Lecture 14 February 28 20138 8 1 5 Locus Circles Ellises H erbolas Lernniscates Circles C P d r2 r2 r4sin9 142 40050 Center 0 at 00 gt X2 l y2 a2 gt ra gt t 027r Xtacost ytasint Center 0 at 0 a gt X2 y a2 a2 Jiwen He University of Houston Math 1432 39 Semi n 25626 Lecture 14 February 28 20138 8 1 5 Circles Ellises H erbolas Lernniscates r2 r2 r4sin9 142 40050 Center 0 at 00 gt X2 l y2 a2 gt ra gt t 027r Xtacost ytasint Center 0 at 0a gt X2 l y a2 a2 gt r 2asin6 Jiwen He University of Houston Math 1432 39 Semi n 25626 Lecture 14 February 28 20138 8 1 5 Locus ercl es Ellises H erbolas Lernniscates CirclesCPd P a m Kb w w r22 r2 r4sin9 CenterOat 00gtX2 y2a2 gtra r 4 cos 6 gt t 027r Xtacost ytasint Center 0 at 0a gt X2 l y a2 a2 gt r 2asin0 Xt 2asin tcos t asin 2t gt t E 07T yt 2aSIn tSIn t al cos 2t Jiwen He University of Houston Math 1432 39 561mm LectUre 14 February is 20138 a 15 Locus Circles Ellises H erbolas Lernniscates Circles C dP 2 ia m Kb w w r2 r2 r4sin9 142 40056 CenterOat 00gtX2 y2a2 gtra gt t 027r Xtacost ytasint Center 0 at 0a gt X2 l y a2 a2 gt r 2asin0 Xt 2asin tcos t asin 2t yt 2aSIn tSIn t al cos 2t Another parametric representation is by translation gt t 027r Xtacost ytasinta in Jiwen He University of Houston Math 1432 39 Semion Lecture 14 F hruary 2839 20138 8 15 Locus Circles Ellises H erbolas Lernniscates Circles C P d r2 r2 r4sin9 142 40050 Center 0 at 00 gt X2 l y2 a2 gt ra gt t 027r Xtacost ytasint Center 0 at a 0 gt X a2 y2 a2 Jiwen He University of Houston Math 1432 39 Semi n 26626 Lecture 14 February 28 20138 8 1 5 Circles Ellises H erbolas Lernniscates r2 r2 r4sin9 142 40050 Center 0 at 00 gt X2 l y2 a2 gt ra gt t 027r Xtacost ytasint Center 0 at a0 gt X a2 y2 a2 gt r 2acos Jiwen He University of Houston Math 1432 39 Seml n 26626 Lecture 14 February 28 20138 8 1 5 I Locus Circles Ellises H erbolas Lernniscates Circles C P d r2 r2 r4sin9 142 40056 Center 0 at 00 gt X2 l y2 a2 gt ra gt t 027r Xtacost ytasint Center 0 at a0 gt X a2 y2 a2 gt r 2acos 7T 3 Xt 2a cos tcos t a1 cos2t gt t e 2 2 yt2acostsnntasrn2t Jiwen He University of Houston Math 1432 39 Semion 25626 Lecture 14 F hruary 2839 20138 8 15 Locus Circles Ellises H erbolas Lernniscates Circles C dP 2 ia m Kb w w r2 r2 r4sin9 142 40056 CenterOat 00gtX2 y2a2 gtra gt t 027r Xtacost ytasint Center 0 at a0 gt X a2 y2 a2 gt r 2acos 7T 3 Xt 2acos tcos t a1 cos 2t 5777 Another parametric representation is by translation gt 139 E yt2acostsint2 asin2t gt t 027r Xtacost l a ytasint H Jiwen He University of Houston Math 1432 39 Semion 25626 Lecture 14 F hruary 2839 20138 8 15 H o erbolas Lemn iscates Ellipses a P F1 a P F2 2 k Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 2008 9 1539 Ellipses a P F1 a P F2 2 k A ellipse is the set of points P in a plane that the sum of whose distances from two fixed points the foci F1 and F2 separated by a distance 2c is a given positive constant 2a E 13 ldPF1 dP F2l 2a Jiwen He University of Houston Math 1432 Selttion 216626 Lecture 14 February 2008 9 15 Ellipses a P F1 a P F2 2 k A ellipse is the set of points P in a plane that the sum of whose distances from two fixed points the foci F1 and F2 separated by a distance 2c is a given positive constant 2a E 13 ldPF1 dP F2l 2a With F1 at cO and F2 at cO and setting b Va2 c2 X2 y2 EXyE1 I Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 2008 9 15 Locus 39 if H erbolas Lemniscates Ellipses Cosine and Sine a P F1 a P F2 2 k The ellipse can also be given by a simple parametric form analogous to that of a circle but with the X and y coordinates having different scalings X acost y bsin t tE 027r Note that cos2 t sin2 t 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 Hyperbolas w 0 9 C b a G F2c 0 X X x2 V2 1 a2 b2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 11 15 Hyperbolas x2 V2 1 L12 b2 A hyperbola is the set of points P in a plane that the difference of whose distances from two fixed points the foci F1 and F2 separated by a distance 2c is a given positive constant 2a H P idP F1 dP F2i 2a Jiwen He University of Houston Math 1432 r Section 26626 Lecture 14 February 28 2008 11 Hyperbolas x2 V2 1 L12 b2 A hyperbola is the set of points P in a plane that the difference of whose distances from two fixed points the foci F1 and F2 separated by a distance 2c is a given positive constant 2a H P idP F1 dP F2i 2a With F1 at cO and F2 at cO and setting b V c2 a2 we have Jiwen He University of Houston Math 1432 r Section 26626 Lecture 14 February 28 2008 11 Hyperbolas Hyperbolic Cosine and i J k y w v 4 0 q x2 7 1 Pcosh r sinh t l C b a x F2C O 0 1 O Q Y x2 V2 I 1 Z 17 2 1 area Of hyperbollc sector E He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 12 15 Locus Circles Ellies erboias Leminjscates Hyperbolas Hyperbolic Cosineand Hyperbolic Sine y K y 2 37339 1 Pcosh r sinh t l C l a X 0 1 0 Q Y x2 V2 I 1 Z 17 2 1 area Of hyperbollc sector E The right branch of a hyperbola can be parametrized by X a cosh t y b sinh t t E oooo Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 12 15 r Locus Circles Ellies erboias Leminjscates Hyperbolas Hyperbolic Cosineand Hyperbolic Sine J t y x 73 1 Pcosh r sinh t l c l a X 0 1 O Q Y x2 V2 39 l 7 2 1 area of hyperbolic sector Et The right branch of a hyperbola can be parametrized by X a cosh t y b sinh t t E oooo The left branch can be parametrized by X a cosh t y b sinh t t E oo oo Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 12 15 Locus Circles Ellies Hyperbolas Leminjscates Hyperbolas Hyperbolic Cosineand Hyperbolic Sine Jquot l y x 73 1 Pcosh r sinh t l C l a X 0 1 O Q Y x2 V2 I l 72 1 area of hyperbollc sector E The right branch of a hyperbola can be parametrized by X a cosh t y b sinh t t E 00 oo The left branch can be parametrized by X a cosh t y b sinh t t E oo oo Note that cosh t et et sinh t et et and cosh2 t sinh2 t 1 Math 1432 Section 26626 Lecture 14 FEbruary 28 2008 1215 Jiwen He University of Houston Hyperbolas Other Parametric Rep Jquot l V 39 1 0 3 i 373 1 Pcosh z sinh r I c l b We 0 x 0 1 0 Q 3 x2 1 2 39 l a2 7 2 1 area of hyperbolic sector Et He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 13 15 Locus Circles EiliSE H eribol as eminisca es Hyperbolas Other Parametric Representation Pcosh z sinh I l l I 0 10 Q x 1 2 2 area of hyperbolic sector 2 t 1 b 2 Another parametric representation for the right branch of the hyperbola is Xasect ybtant tE 7t27t2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 13 15 Pcosh z sinh I l l I 0 10 Q x L 4 1 area of hyperbolic sector 2 it Another parametric representation for the right branch of the hyperbola is X a sec t y b tan t t E 7r27r2 Parametric equations for the left branch is X asect ybtant tE 7r27r2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 13 15 Locus r2 a2 cos26 Lemniscates Ribbons 3 1 7r 7r 2 2 r24cos 26 February 28 2008 14 15 Math 1432 Section 262626 Lecture 14 Jiwen He University of Houston 39 39 Locus Lemniscates Ribbons r2 a2 cos 26 A Iemniscate is the set of points P in a plane that the product of whose distances from two fixed points the foci F1 and F2 1 Z 273 a distance 2c away IS the constant c2 2 x 2 RPdPF1dPF2ic2 r24cos 26 Jiwen He University of Houston Math 1432 Section 262526 Lecture 14 February 28 2008 14 15 39 39 Locus Lemniscates Ribbons r2 a2 cos 26 A Iemniscate is the set of points P in a plane that the product of whose distances from two fixed points the foci F1 and F2 z i i a distance 2c away is the constant c2 2 2 2 R PdPF1 dPF2i c With F1 at c0 and F2 at c0 X2 y22 2C2 X2 y2 r24cos 26 Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 14 15 39 39 Locus Lemniscates Ribbons r2 a2 cos 26 A Iemniscate is the set of points P in a plane that the product of whose distances from two fixed points the foci F1 and F2 i i a distance 2c away is the constant c2 2 2 2 R PdPF1 dPF2i c With F1 at c0 and F2 at c0 X2 y22 2C2 X2 y2 r2 2 4 COS 29 Switching to polar coordinates gives 37 57T 7T 7139 r2 2c2 c0528 9 E Z U T Iii Jiwen He University of Houston Math 1432 Section 26626 Lecture 14 February 28 2008 14 15 39 39 Locus Lemniscates Ribbons r2 a2 cos 26 A Iemniscate is the set of points P in a plane that the product of whose distances from two fixed points the foci F1 and F2 xi i75 a distance 2c away is the constant c2 R P dP F1 dP F2i c2 With F1 at c0 and F2 at c0 X2 y22 2C2 X2 y2 r2 2 4 COS 29 Switching to polar coordinates gives 7T 7T 37T 57T Z I U T T r2 2c2 cos 28 9 E The parametric equations for the Iemniscate with a2 2c2 is acost asintcost x y tEO27r F1 1sin2t7 1sin2t Jiwen He University of Houston Math 1432 13ection 26626 Lecture 14 February 28 20038 14 15 a Mmmmm mammmmm Bump y hams des mg 4 WWW a Lawmg quotN a

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