Lesson 10. Parametric Curves


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1 Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find the slope of tngent to the curve, or the re enclosed by the curve. In clculus this problem cn be solved if the curve is expressed by pir of prmetric equtions: x = f (, y = g(, where t is the prmeter with the rnge of t b in generl. Ech vlue of the prmeter t determines point (, y) ( f (, g( ) ( f ( ), g( ) ) nd the terminl point is ( f ( b), g( b) ). As t vries, the point ( y) x =. The initil point is x, will trce out curve tht is clled prmetric curve. In mny pplictions of prmetric curves, t stnds for time, but does not lwys necessrily represent time. In this section we will lern how to sketch the prmetric curves using Mthemtic. Exmple1 Sketch nd identify the curve given by the prmetric equtions x = t t, y = t + 1 for t. Ech vlue of the prmetert gives point on the curve, which cn be evluted by: ^,,,,, Note tht in the first brcket listed two prmetric functions, nd in the second brcket,,, mens t runs from  to in steps of 1. We cn find the following points on the prmetric curve,,,,,,,,,,,,, They correspond to the set of t vlues, 1, 0, 1,,, in order. So this curve will strt from the point ( 8, 1) nd end t the point ( 8, 5). We cn plot this curve using PrmetricPlot. ^,,,, 1
2 The curve trced out my be prbol. This cn be confirmed by eliminting the prmeter t s follows. We obtin t = y 1 from the second eqution nd substitute into the first eqution. It gives ( y 1) ( y 1) = y + x = t t = y nd so the curve represented by the given prmetric equtions is prbol tht mtches the Crtesin eqution x = y y +. The rnge of t vlue will ffect the grph prmetric curve significntly. Look t the exmple gin by modifying the rnge of t vlues to 0 t. Exmple Sketch nd identify the curve given by the prmetric equtions x = t t, y = t + 1 for 0 t. We evlute the points on the curve by running t from 0 to in steps of 1,,,,,, nd these points on curve will be:,,,,,,,,,. The two points 8, 1,,0 from Exmple 1 re missing becuse the vlues of prmeter t = nd t = 1re removed from the originl list. Agin we cn plot the curve using PrmetricPlot. 5 ^,,,, 6 8
3 Exmple Wht curve is represented by the prmetric equtions 0 t π. x = cos t, y = sin t for,,,, The grph represented by the pir of prmetric equtions x = cos t, y = sin t is obviously unit circle centered t the origin. Menwhile it is not hrd to relize tht by the Pythgoren Theorem x + y = cos t + sin t = 1, its Crtesin eqution is truly unit circle Since t vlue runs from 0 to π in this cse, the circle will go round the origin exct one revolution. If the rnge of t vlue is modified, the shpe of prmetric curve will chnge. Check out the next exmple. Exmple Wht curve is represented by the prmetric equtions x = cos t, y = sin t for 0 t π?,,,,,,
4 We cn see tht the lower hlf of the unit circle is gone becuse the rnge of t vlue defined over the intervl 0 t π which represents the upper hlf of circle sketched in Qudrnts 1 nd. (B) The Cycloid The cycloid is one of the most interesting prmetric curves tht we should know. If circle hs rdius r nd rolls long stright line (the xxis), then point P on the circumference of the circle will trce out the curve of the cycloid, which cn be represented by the prmetric equtions: x = r( θ sinθ ), = r( 1 cosθ ) y for < θ < Note tht the cycloid cn be nicely nd netly represented by the prmetric equtions, but it is very complicted to be expressed by Crtesin eqution. Exmple5 Plot the cycloid given by x = ( θ sinθ ), y = (1 cosθ ) for π θ π.,,,, In this cse it might be problem when you try to type in the Greek letterθ. Here it is the direction of where to find the letterθ from the Clssroom Assistnt. To lern more bout the cycloid, plese visit the site to visulize the formtion of the cycloid by trcing movement of fixed point on the circumference of circle.
5 (C) Tngents of Prmetric Curve For prmetric curve given by the equtions: x = f (, y = g(, nd t b, the tngent t ny t vlue cn be clculted by dy dx dy g '( = dt = if f '( 0. dx f '( dt Exmple6 Suppose curve C is defined by the prmetric eqution x = t, y = t t. (1) Plot the curve, () find the eqution(s) of the tngent line(s) to the curve t the point (, 0). () Plot the tngent line(s) t the point (, 0). We will work it out step by step. (1),,,, We observe tht there re two curves tht pss through the point (, 0), nd correspondingly there re two tngent lines pssing through this point. Since (, 0) is Crtesin point, nmely x = nd y = 0. We need to find out the corresponding vlue of the prmeter t. Type in, to find the t vlues:, 5
6 Type in, to find the t vlues: 0,,. Tking the solutions of t in common from the two sets, we hve t = or t =. dy g '( () Now let s clculte the first derivtive function by =, which leds to the slopes of the dx f '( tngent lines. We will run the following commnds to find the slopes of tngents for t = nd t =. _ : Shift + Enter _ : Shift Enter Type in to get Type in to get Type in to get Thus we found tht the slopes of the two tngent lines t the point (, 0) re nd. Then by the PointSlope Form, we cn formulte the equtions of the two tngent lines tht pss through the point (, 0) s: y = ( x ) nd y = ( x ). () We will plot these two tngent line(s) t the point (, 0) in the sme viewing box of the curve.,,,,,,,,,, 6
7 y= (x) H, 0L y=  (x) (D) The Are Under Prmetric Curve If the curve is trced out once by the prmetric equtions x = f (, y = g( for t b, then we cn clculte the re under the curve using the following formul: A = g( f '( dt. It is very fst to find the re under prmetric curve by Mthemtic. Exmple7 Find the re under one rch of the cycloid x = r( θ sinθ ), = r( 1 cosθ ),,,, b y for r = It is esy to relize tht one rch of the cycloid is trced out by running t from 0 to π. We cn find the re under one rch by pplying the Are Formul A = g( f '( dt. b 7
8 _ : _ :,,, Therefore when r =, the re under one rch of the cycloid equls 1 π. (E) Arc Length of Prmetric Curve Suppose the curve C is described by the prmetric equtions x = f (, y = g( for t b. If f ' nd g ' re both continuous on [, b] nd C is trversed exctly once s t increses from to b, then the length of C cn clculted by b dx dy b L = + dt, or L = ( f t ) ( g t ) dt dt dt '( ) + '( ) Exmple9 Find the length of one rch of the cycloid x = r( θ sinθ ), = r( 1 cosθ ) y for r =. Like we did in the previous exmple, first we define the two prmetric equtions, nd then pply the Arc Length Formul L ( f '( ) + ( g '( ) dt b =. _ : _ :,,, Therefore when r =, the length of one rch of the cycloid equls 8 r = 16. Return to List of Lessons 8
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