Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.08 Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. Wednesday, April 16 10.4 Parametric Representation of Curves in the Plane Assignment: Page 534 11,28,46,56 Main idea: Specify the x and y coordinates separately. Key Words: Paramteric, (f(t),g(t)), x = f(t), y = g(t) goal: Learn to find slopes when the curve is given parametrically. Previous Assignment: April 9 p505: 29-42 ======================================================================= New Material: Suppose x = f(t) and y = g(t). We can plot the points ( f(t),g(t) ) and get a curve. For example: (a) { Sin[3 t], Sin[7 t] }, {t,0,2 Pi} Sin[t] Cos[t] (b) { e, ,e } {t,-10,10}]; ----------------------------------------------------------- If we plot the points for t in [a,b], then t=a determines the initial and t=b determines the final point and we think of the curve as being directed from a to b. A simple curve does not cross itself. A closed curve starts and stops in the same point. A line segment is simple and not closed. A circle is simple and closed A figure eight is closed and not simple. _ _ _ / \ / \ / \ \ / \ / \ / X X X / \ / \ / \ / \_/ \_/ \ Is neither simple nor closed. Page 530 Example 1. Eliminate the parameter in 2 x = t + 2 t y = t - 3 -2 <= t <= 3 t = y+3 2 2 x = (y+3) + 2 (y+3) = y + 8 y + 15. 2 x+1 = (y+4) ---------------------------------------------------------- Get["font.math"]; P1 = ParametricPlot[ {Sin[3 t], Sin[7 t]},{t, -Pi, Pi}, PlotStyle->{Thickness[0.01]}]; P2 = ParametricPlot[ {Sin[3 t], Sin[7 t]}, {t,-10 Pi,10 Pi}, PlotStyle->{RGBColor[1,0,0]},PlotPoints->2000]; P3 = Show[P1,P2,PlotLabel->"Ex a, { Sin[3 t], Sin[7 t] }",PlotRange->All, AspectRatio->Automatic]; Display["exa.ps",P3]; P4 = ParametricPlot[ {E^Sin[t],E^Cos[t]} ,{t,0,2 Pi}]; P5 = Show[P4,PlotLabel->"Ex b, { E^Sin[t], E^Cos[t]}",PlotRange->All, AspectRatio->Automatic]; Display["exb.ps",P5]; P1 = ParametricPlot[ { t^2 + 2 t, t-3}, {t,-2,3}, PlotStyle->{Thickness[0.01]}]; P2 = ParametricPlot[ { (y+4)^2-1,y},{y,-8.5,1/2}, PlotStyle->{RGBColor[1,0,0]},PlotPoints->2000]; P3 = ListPlot[{ {0,-5} },PlotStyle->{RGBColor[1,0,0],PointSize[0.02]}]; P4 = Show[P1,P2,P3,PlotLabel->"P530 Ex 1 { t^2 + 2 t, t-3} red=start ", PlotRange->All, AspectRatio->Automatic]; Display["ex1.ps",P4]; ------------------------------------------------------------------------ Theoretically, you can eliminate the parameter, but most of the time you do not want to. The parametric form is far easier to work with. Page 531 Example 3 The same graph can be parametrized in different ways. 2 (a) ( Sqrt[1-t ], t ) -1 <= t <= 1 (b) ( Cos[t], Sin[t] ) -Pi/2 <= t <= Pi/2 / 2 \ / 1-t 2t \ (c) ( -------, ------- ) -1 <= t <= 1 \ 2 2 / \ 1+t 1+t / --------------------------------------------------------------- Get["font.math"]; P1 = ParametricPlot[ { Sqrt[1-t^2], t }, {t,-1,1}, PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; P2 = ParametricPlot[ { Cos[t],Sin[t] }, {t,-Pi/2,Pi/2}, PlotStyle->{RGBColor[0,1,0],Thickness[0.015]}]; P3 = ParametricPlot[ { (1-t^2)/(1+t^2),2t/(1+t^2)}, {t,-1,1}, PlotStyle->{RGBColor[0,0,1],Thickness[0.03]}]; P4 = Show[P3,P2,P1,PlotLabel->"P531 Ex3", PlotRange->All,AspectRatio->Automatic]; Display["ex3.ps",P4]; ------------------------------------------------------------------------ Page 531 the parametric Equations of the cycloid. ---- .|\ . .d| \ . . |__. . . c |b . .................. .|... <--------- a -------> Find these values: a = ___________________________________________________ b = ___________________________________________________ c = ___________________________________________________ d = ___________________________________________________ ------------------------------------------------------------- Get["font.math"]; r = 2; t = 1.2; P1 = ListPlot[ {{0,0},{ r t, 0}, { r t, r}, { r t, r}-r{Sin[t],Cos[t]} }, PlotJoined->True,PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = ParametricPlot[ { r t,r}+2{Cos[x],Sin[x]},{x,0,2 Pi}, PlotStyle->{RGBColor[0,1,0],Thickness[0.008]}]; P3 = Show[P1,P2,PlotLabel->"P531 Ex 5 Cycloid",AspectRatio->Automatic, PlotRange->All]; Display["ex5.ps",P3]; ---------------------------------------------------------------------- Page 532 Finding the derivative. { f(t), g(t) }, / g (t) Then dy/dx = --------- / f (t) delta y --------- delta y delta t dy/dt Proof: ---------- = --------------- = --------- delta x delta x dx/dt --------- delta t / / | g (t+dt) g (t) 2 | -/------ - --/--- d y | f (t+dt) f (t) ---- = | ------------------ 2 | f(t+dt) - f(t) dx |______ dt --> 0 / / | g (t+dt) g (t) | -/------ - --/--- | f (t+dt) f (t) d / dy \ 2 | --------------------- --- ( ---- ) d y | dt dt \ dx / ---- = | ------------------ = ---------------- 2 | f(t+dt) - f(t) dx dx |______ -------------- ---- dt --> 0 dt dt Page 533 Example 6 { 5 Cos[t], 4 Sin[t] } Find dy/dx when t = Pi/6. 4 Cos[t] dy/dx ----------- = -(4/5) Ctn[t] -5 Sin[t] 2 2 2 4/5 Csc [t] 3 d y/dx = ------------- = -(4/25) Csc [t] -5 Sin[t] At t = Pi/6 dy/dx = -4/5 Ctn[Pi/6] = - 4 Sqrt[3]/5 2 2 3 d y/dx = -(4/25) Csc [Pi/6] = -32/25 ------------------------------------------------------------- Get["font.math"]; P1 = ParametricPlot[ { 5 Cos[t], 4 Sin[t] }, {t,0,3}, PlotStyle->{Thickness[0.01]} ]; P2 = ParametricPlot[{ 5 Cos[Pi/6], 4 Sin[Pi/6] } + t{-5 Sin[Pi/6],4 Cos[Pi/6]}, {t,-1,1},PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P3 = ListPlot[ {{ 5 Cos[Pi/6], 4 Sin[Pi/6] }},PlotStyle->{PointSize[0.02]}]; P4 = Show[P1,P2,P3,PlotLabel->"P533 Ex6 { 5 Cos[t], 4 Sin[t] }", PlotRange->All, AspectRatio->Automatic]; Display["ex6x.ps",P4]; x[t_] := 5 Cos[t]; y[t_] := 4 Sin[t]; xp[t_] = D[x[t],t]; yp[t_] = D[y[t],t]; xpp[t_] = D[xp[t],t]; ypp[t_] = D[yp[t],t]; k[t_] := (xp[t] ypp[t]-yp[t] xpp[t])/ (xp[t]^2 + yp[t]^2)^(3/2); c[t_] := ParametricPlot[ {x[t],y[t]}+1/k[t]{-yp[t],+xp[t]}/(yp[t]^2+xp[t]^2)^(1/2)+1/k[t]{Cos[w],Sin[w]}, {w,0,2 Pi},PlotStyle->{RGBColor[1,0,0]}]; P5 = c[Pi/3]; P6 = Show[ P1, P2, P3, P5, PlotLabel->"P533 Ex 6 { 5 Cos[t], 4 Sin[t] }", PlotRange->All, AspectRatio->Automatic]; Display["ex6y.ps",P6]; ----------------------------------------------------------------------- Page 533 Example 7: x=3 Evaluate (a) INT y dx x=1 x=3 2 (b) INT x y dx x=1 2 Where x = 2 t -1 and y = t + 2 _ 3 _ t=2 x=3 t=2 2 | t | INT y dx = INT (t + 2) 2 dt = 2 | ---- + 2 t | = 26/3 x=1 t=1 |_ 3 _| t=1 x=3 2 t=2 2 2 INT x y dx = INT (2t-1)(t +2) 2 dt x=1 t=1 t=2 5 4 3 2 1304 = 2 INT 2 t - t + 8 t - 4 t + 8 t - 4 dt = ------- t=1 15 ------------------------------------------------------------------ Page 533 Example 8. Find the area under one arch of a cycloid and the length of the arch x = a(t-Sin[t]) dx/dt = a(1-Cos[t]) y = a(1-Cos[t]) dy/dt = a Sin[t] Arc Length: t=2 Pi 2 2 2 S = INT Sqrt[ a (1 - 2 Cos[t] + Cos[t] + Sin[t] ) dt t=0 t=2 Pi 2 S = INT Sqrt[ a (2 - 2 Cos[t] ) dt t=0 _ _ t=2 Pi | 1 - Cos[t] | S = INT a 2 Sqrt| ------------ | dt t=0 |_ 2 _| t=2 Pi S = INT a 2 Sin(t/2) dt t=0 _ _ t = 2 Pi | - Cos[t/2] | 2 a | -------------- | |_ 1/2 _| t = 0 4 a (-Cos[Pi] + Cos[0] ) = 8 a. <=== Arc Length. Area: t = 2 Pi A = INT y dx t=0 t = 2 Pi A = INT a(1-Cos[t]) a(1-Cos[t] dt t = 0 t = 2 Pi 2 2 A = INT a (1 - 2 Cos[t] + Cos [t] ) dt t = 0 t = 2 Pi 2 1 + Cos[2t] A = INT a (1 - 2 Cos[t] + ------------ ) dt t = 0 2 _ _ t = 2 Pi 2 | | A = a | t - 2 Sin[t] + t/2 + Sin[2t]/4 | |_ _| t = 0 2 2 A = a ( 2 Pi +Pi ) = 3 Pi a -----------------------------------------------------------------------