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. Monday April 21 10.6 Polar Graphs Assignment: Page 545: 9,15,21,29,45 Main Idea: More polar graphs Key Words: polar coordinates, limacon, cardioid, hyperbolic spiral, spiral Archimedes, rose, lemniscate, circle Goal: Learn to tease out the appearance of a polar graph. ------------------------------------------------------------------ Previous assignment: Page 541 22 Find the Cartesian equations of the graphs of the given polar equations 2 r - 6 r Cos[t] - 4 r Sin[t] + 9 = 0 2 2 x + y - 6 x - 4 y + 9 = 0 2 2 x - 6 x + y - 4 y = -9 2 2 x - 6 x + 9 + y - 4 y + 4 = -9 + 9 + 4 2 2 (x-3) + (y-2) = 4 It is a circle centered at (3,2) with radius 2. 6 Cos[t] + 4 Sin[t] Sqrt[36+16]( 6/Sqrt[52] Cos[t] + 4/Sqrt[52] Sin[t] ) Sqrt[36+16]( Cos[w] Cos[t] + Sin[w] Sin[t]) Sqrt[52] Cos[w-t] w = ArcCos[6/Sqrt[52]] = ArcCos[3/Sqrt[13]] Sqrt[52] Cos[w-t] = 6 Cos[w-t] = 6/Sqrt[52] t-w = ArcCos[6/Sqrt[52]] t = 2 ArcCos[6/Sqrt[52]] w = 2 ArcCos[6/Sqrt[52]] rp[t_] = (( 6 Cos[t]+4 Sin[t]) + Sqrt[ (-6 Cos[t]-4 Sin[t])^2 -36])/2; rm[t_] = (( 6 Cos[t]+4 Sin[t]) - Sqrt[ (-6 Cos[t]-4 Sin[t])^2 -36])/2; h[t_] := (-6 Cos[t]-4 Sin[t])^2 -36; ---------------------------------------------------- Get["font.math"]; w = 2 ArcCos[6/Sqrt[52]] p[r_,t_] := {r Cos[t], r Sin[t]}; rp[t_] = (( 6 Cos[t]+4 Sin[t]) + Sqrt[ (-6 Cos[t]-4 Sin[t])^2 -36])/2; rm[t_] = (( 6 Cos[t]+4 Sin[t]) - Sqrt[ (-6 Cos[t]-4 Sin[t])^2 -36])/2; P1 = ParametricPlot[ {3,2} + 2{Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.02 ]}]; P2 = ParametricPlot[ p[rp[t],t],{t,0,w},PlotStyle->{Thickness[0.008]}]; P3 = ParametricPlot[ p[rp[t],t],{t,Pi,Pi+w},PlotStyle->{Thickness[0.008]}]; P4 = ParametricPlot[ p[rm[t],t],{t,0,w},PlotStyle->{Thickness[0.008]}]; P5 = ParametricPlot[ p[rm[t],t],{t,Pi,Pi+w},PlotStyle->{Thickness[0.008]}]; P6 = Show[P1,P2,P3,P4,P5, PlotLabel->"P541 p22 r^2-6 r Cos[t]-4 r Sin[t] + 9 = 0", PlotRange->All, AspectRatio->Automatic]; Display["p22.ps",P6]; -------------------------------------------------------------- Page 541 24 Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. theta = 2 Pi/3 It is a straignt line with Slope -Sqrt[3]. y = -Sqrt[3] x. ---------------------------------------------------------------- Get["font.math"]; p[r_,t_] := {r Cos[t], r Sin[t]}; P1 = ParametricPlot[p[r,2 Pi/3],{r,-3, 3}, PlotStyle->{RGBColor[1,0,0],Thickness[0.02]}]; P2 = Plot[ -Sqrt[3] x,{x,-2,2},PlotStyle->{Thickness[0.01]}]; P3 = Show[P1,P2,PlotLabel->"P541 p24 theta = 2 Pi/3",AspectRatio->Automatic, PlotRange->All]; Display["p24.ps",P3]; --------------------------------------------------------- Page 541 28 Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. r = -4 Cos[theta] It is a circle in the 3rd and 4th quadrants, tangent to the y axis at (0,0) with radius 2. ------------------------------------------------------------ Get["font.math"]; p[r_,t_] := {r Cos[t], r Sin[t]}; P1 = ParametricPlot[p[-4 Cos[t],t],{t,-Pi, Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.02]}]; P2 = Plot[0,{x,-4,4}]; P3 = Show[P1,P2,PlotLabel->"P541 p28 r = -4 Cos[theta]", AspectRatio->Automatic,PlotRange->All]; Display["p28.ps",P3]; -------------------------------------------------------------- Page 541 Problem 32 6 r = --------- 4-Cos[t] d e 3/2 r = ------------ = -------------- 1-e Cos[t] 1-1/4 Cos[t] e = 1/4 3/2 = 1/4 d d=6 Ellipse when t = 0 r = 6/3 = 2 when t = Pi r = 6/5 a = 1/2 (2 + 6/5) = 8/5 c+a = 2; c = 2/5; a-c = 6/5; eccentricity = c/a = (2/5)/(8/5) = 1/4. center is at (-6/5+8/5,0) = (2/5,0) Should be the same as 2 2 (x-2/5) y ----------- + ---------------- = 1 2 2 2 (8/5) (8/5) -(2/5) 2 2 (x-2/5) y ----------- + ---------------- = 1 2 60 (8/5) -------- 25 ParametricPlot[ {8/5 Cos[t]+2/5 ,Sqrt[60/25] Sin[t]},{t,0,2 Pi}]; ............................................................. Get["font.math"]; p1 = ParametricPlot[ 6/(4-Cos[t]){Cos[t],Sin[t]},{t,0,2 Pi}]; p2 = ParametricPlot[ {2/5,0} + 8/5 {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0]}]; p3 = Show[p1,p2,PlotLabel->"P541 p32 black = 6/(4-Cos[theta]),red = circle", AspectRatio->Automatic,PlotRange->All]; Display["p32.ps",p3]; ................................................................. New Material Somewhat: The basic curve is the circle: r = Cos[theta] and r = Sin[theta]. Remember that it traces the curve twice, once with positive r and once with negative r. a. Graph: r = Cos[theta] b. Graph: r = Cos[2 theta] c. Graph: r = Cos[3 theta] 2 d. Graph: r = Cos[theta] e. Graph: r = 1 + Cos[theta] f. Graph: r = 1/2 + Cos[theta] g. Graph: r = 2 + Cos[theta] h. Graph: r = 3 + 3 Sin[theta] .................................................. Get["font.math"]; (* a. Graph: r = Cos[theta] *) f[t_] := Cos[t]; p1 = ParametricPlot[ f[t] {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p2 = Show[p1,PlotLabel->"(a) r = Cos[theta]: Circle", PlotRange->All,AspectRatio->Automatic]; Display["a.ps",p2]; (* b. Graph: r = Cos[2 theta] *) f[t_] := Cos[2 t]; p1 = ParametricPlot[ f[t] {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p2 = Show[p1,PlotLabel->"(b) r = Cos[2 theta]: 4-leaf rose", PlotRange->All,AspectRatio->Automatic]; Display["b.ps",p2]; (* c. Graph: r = Cos[3 theta] *) f[t_] := Cos[3 t]; p1 = ParametricPlot[ f[t] {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p2 = Show[p1,PlotLabel->"(c) r = Cos[3 theta]: 3-leaf rose", PlotRange->All,AspectRatio->Automatic]; Display["c.ps",p2]; (* d. Graph: r^2 = Cos[theta] *) f[t_] := Sqrt[ Cos[t] ]; p1 = ParametricPlot[ f[t] {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p2 = ParametricPlot[-f[t] {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[0,1,0],Thickness[0.01]}]; p3 = Show[p1,p2,PlotLabel->"(d) r^2 = Cos[ theta]: green = negative r", PlotRange->All,AspectRatio->Automatic]; Display["d.ps",p3]; ................................................... (* e. Graph: r = 1 + Cos[theta] *) f[t_] := 1 + Cos[ t]; p1 = ParametricPlot[ f[t] {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p2 = Show[p1,PlotLabel->"(e) r = 1+Cos[theta]: Cardioid", PlotRange->All,AspectRatio->Automatic]; Display["e.ps",p2]; ................................................... (* f. Graph: r = 1/2 + Cos[theta] *) f[t_] := 1/2 + Cos[t]; p1 = ParametricPlot[ f[t] {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p2 = Show[p1,PlotLabel->"(f) r = 1/2+Cos[theta]: Limacon", PlotRange->All,AspectRatio->Automatic]; Display["f.ps",p2]; ...................................................... (* g. Graph: r = 2 + Cos[theta] *) f[t_] := 2 + Cos[t]; p1 = ParametricPlot[ f[t] {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p2 = Show[p1,PlotLabel->"(g) r = 2+Cos[theta]: Limacon", PlotRange->All,AspectRatio->Automatic]; Display["g.ps",p2]; ...................................................... Get["font.math"]; f[t_] := 3 + 3 Sin[t]; P1 = ParametricPlot[ f[t] {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = Show[P1,PlotLabel->"r = 3 + 3 Sin[theta]", PlotRange->All,AspectRatio->Automatic]; Display["cardioidx.ps",P2]; p[r_,t_] := {r Cos[t], r Sin[t]}; w[n_] := ParametricPlot[ p[r,n Pi/6],{r,0,3+3 Sin[n Pi/6]+0.001}, PlotStyle->{RGBColor[0,1,0],Thickness[0.01]}]; H = Table[w[n],{n,1,12}] P1 = ParametricPlot[ f[t] {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = Show[H,P1,PlotLabel->"r = 3 + 3 Sin[theta]", PlotRange->All,AspectRatio->Automatic]; Display["cardioidy.ps",P2];