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# Class Note for MATH 1432 at UH 2

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This 8 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 28 views.

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Date Created: 02/06/15

Lecture 14Section 96 Curves Given Parametrically J iwen He 1 Parametrized curve 1 1 Parametrized curve Parametrized curve lt xr yr X Parametrized curve A parametrized Curve is a path in the Qty plane traced out by the point 95t7 as the parameter t ranges over an interval I CxtyttEI I Examples 1 o The graph of a function y at at 6 I7 is a curve C that is parametrized by Itt7 ytft7 tEI o The graph of a polar equation r MO 0 6 I7 is a curve C that is parametrized by the functions 95t rcost pt cost7 yt rsint pt sint7 t E I 1 2 Examples Example Line Segment Wk 3 6 127 37 1 2 10 27 47 Line Segment y 25 ac 6 13 I Set act t then yt 2t t 6 13 I Set act t 1 then yt 2t 2 t 6 02 I Set act 3 7t then yt 6 7 2t t 6 02 I Set act 3 7 4t then yt 6 7 8t t 6 0 12 gtltV 0 Set 5675 2 cost then yt 4 2cost t E 04 We parametrize the line segment in different ways and interpret each parametriza tion as the motion of a particle With the parameter 15 being time Example Parabola O 1t 7139 gl zo x 1 y2 xt sin2 I yt cos r re 0 7r Parabola Arc w 1 y2 1 g y g 1 0 Set yt t then 5615 1 752 t E 1 1 i changing the domain to all real 15 gives us the Whole parabola 0 Set yt cost then 5675 1 cos2 t t E 0 7T i changing the domain to all real 15 does not give us any more of the parabola Example Spiral of Archimedes Fm i7r I 4 4 PolaraXIs J m 7 0 27a anquot 5 5 Km 1 7 7 1 2 3 3 5 5 ra 620 spiral of Archimedes Spiral of Archimedes 7 9 9 Z 0 o The curve is a nonending spiral Here it is shown in detail from 9 0 to 9271 o The parametric representation is 5675 tcost yt tsint t2 0 Example Limagons Q quot r lc056 2 limagon with an inner loop r3COSt9 z oos r10056 convex limagon cardioid Iimagon with a dimple Limagons Snails 7 a bcos0 The parametric representation is xt a boost cost yt a boos t sint t E 0 2 Example Petal Curves 7r l 6 1 7T 1 rsin39 rCOS40 Petal Curves Flowers 7 acos n0 7 asin 710 The parametric representations are xt a cosnt cost yt a cosnt sin 75 t E 0 2w xt a sinnt cost yt a sinnt sin 75 t E 0 2w 2 Locus 21 Circles naming n ID Art n InI1 2 r 2 r4sin0 r4c039 Center 0 at 00 gt x2y2 a2 gt39ra gt tE027T xtacost ytasint Center 0 at 061 gt 962 y a 2 a2 gt39r392asin t 2asintcostasin2t gt t E 07 7T yt 2a s1nts1nt a1 cos 275 Another parametric representation is by translation gt 7SE027T xtacost ytasinta CenterOata0gtx a y a2gt39r2acos 7T 377 9675 2acostcosta1cos2t 27 2 7 yt2acostsintasin2t Another parametric representation is by translation gt 756 027T 9675 acosta yt asint 22 Ellipses x gt t6 y V O b Px i a a b J Cl C C a X F1 c O F2c O X 0 b dPF1zPF2k A ellipse is the set of points P in a plane that the sum of Whose distances from two xed points the foci F1 and F2 separated by a distance 2c is a given positive constant 2a E P2 ldP F1 dPF2 2a With F1 at c 0 and F2 at c 0 and setting I V a2 2 x2 y2 Ellipses Cosine and Sine Pxy a a b J a O C C a O X F1 c 0 F2c 0 V O b The ellipse can also be given by a simple parametric form analogous to that of a circle but With the 96 and y coordinates having different scalings 96 a cost y b sint t6 027r Note that cos2 t sin2 75 1 dP F1c P F2k 23 Hyperbolas Hyperbolas A hyperbola is the set of points P in a plane that the difference of Whose distances from two xed points the foci F1 and F2 separated by a distance 2c is a given positive constant 2a H 13 ldP F1 dP F2l 2a With F1 at c O and F2 at c O and setting I V c2 a2 we have 332 yZ Hyperbolas Hyperbolic Cosine and Hyperbolic Sine Jquot A f 1 area of hyperbolic sector 2 r The right branch of a hyperbola can be parametrized by a a cosht y b sinht 756 00 oo The left branch can be parametrized by a a cosht y b sinht 756 00 oo Note that cosht eat I et sinht e7t 6 7 and cosh2 t sinh2t 1 l 2 Hyperbolas Other Parametric Representation x2y21 Pcosh t sinh t in I I 0 10 Q 2 2 L V 1 area of hyperbolic sector it Another parametric representation for the right branch of the hyperbola is 76 a sect y b tant t E 7r27r2 Parametric equations for the left branch is xz asect yzbtant t 7r27r2 24 Lemniscates Lemniscates Ribbons r2 02 cos 26 r2 4 cos 26 A lemm39scate is the set of points P in a plane that the product of Whose distances from two xed points the foci F1 and F2 a distance 20 away is the constant 2 5 RPdPF1dPF2lc2 With F1 at 700 and F2 at 007 2 I2 f 252 I2 7 y2 Switching to polar coordinates gives 2 2 II 35 T726 cos2t9t9 44U4742 The parametric equations for the lemniscate With a2 20 is I acos 7 yasintcst7 7607270 ls1nt ls1nt Outline Contents 1 Parametrized curve 1 1 1 Parametrized curve i i i i i i i i i i i i i i i i i i i i i i i i i i 1 12 Examples i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 2 2 Locus 4 2 1 Circles i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 4 2 2 Ellipses i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 5 2 3 Hyperbolas i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 6 2 4 Lemniscates i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 7

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