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 28 Chapter review Work the Practice test on April 30 for next class. ------------------------------------------------------------------ Previous assignment: April 20 10.7 Calculus p553: (26) through (52) Even Only Page 553 Problem 26 Eliminate the parameter to obtain the corresponding Cartesian equation. Sketch the given curve. 2 x = 4 t y = 4 t -1 < t < 2 2 2 x/4 = t = (y/4) 2 x/4 = (y/4) 2 x/4 = y /16 2 y = 4 x ---------------------------------------------- Get["font.math"]; P1 = ParametricPlot[ {4 t^2, 4 t},{t,-1, 2}]; P2= Table[ Graphics[ Text[ FontForm[ t ,{"Courier-Bold",26}],{4 t^2, 4 t} ]], {t,-1, 2}]; P3 = Show[P1,P2, PlotLabel->"P553 p26 {4 t^2, 4 t}, {-1 < t < 2}", PlotRange->{{-1,18},{-8,10}}, AspectRatio->Automatic]; Display["p26.ps",P3]; ----------------------------------------------- Page 553 Problem 28 Eliminate the parameter to obtain the corresponding Cartesian equation. Sketch the given curve. x = 2 Sec[t] y = Tan[t] -Pi/2 < t < Pi/2 2 2 (x/2) = y + 1 2 2 x = 4 y + 4 ---------------------------------------------- Get["font.math"]; P1 = ParametricPlot[ {2 Sec[t], Tan[t]},{t,-1.5,1.5}]; h={ -1.5, -1.4, -1.3, -1.2, -1.0, +0.0, +1.0, +1.2, +1.3, +1.4, +1.5}; P2= Table[ Graphics[ Text[ FontForm[h[[i]],{"Courier-Bold",14}],{2 Sec[h[[i]]],Tan[h[[i]]]}]], {i, 1, Length[h]}]; P3 = Show[P1,P2,PlotLabel->"P553 p28 {2 Sec[t], Tan[t]}, {t,-Pi/2,Pi/2}", PlotRange->{{ -1,30},{-15,15}},AspectRatio->Automatic]; Display["p28x.ps",P3]; P4 = ParametricPlot[ t{2,+1},{t,-20,20},PlotStyle->{RGBColor[1,0,0]}]; P5 = ParametricPlot[ t{2,-1},{t,-20,20},PlotStyle->{RGBColor[1,0,0]}]; P6 = ParametricPlot[ {-2 Sec[t],Tan[t]},{t,-1.5,1.5}]; P7 = Show[P1,P2,P4,P5,P6, PlotLabel->"P 553 p28 (x/2)^2 - y^2 = 1", PlotRange->{{-30,30},{-15,15}},AspectRatio->Automatic]; Display["p28y.ps",P7]; ----------------------------------------------- Page 553 Problem 30 Find the equations of the tangent line at t=0. -t t x = 3 e y = 1/2 e -t dx/dt = -3 e t dy/dt = 1/2 e t 1/2 e | dy/dx = ------- dy/dx | = -1/6 -t |t=0 -3 e ---------------------------------------------- Get["font.math"]; P1 = ParametricPlot[{3 E^(-t),1/2 E^t},{t,-1.5,2.5}, PlotStyle->{Thickness[0.005]}]; P2 = ParametricPlot[ {3,1/2}+t{1,-1/6},{t,-3,3.0}, PlotStyle->{RGBColor[1,0,0], Thickness[0.005]}]; P3 = ListPlot[{{3,1/2}},PlotStyle->{RGBColor[0,1,0],PointSize[0.01]}]; P4 = Show[P1,P2,P3,PlotLabel->"P553 p30 {3 E^(-t), 1/2 E^t},{t,-2,2}", PlotRange->{{0,6},{0,5}},AspectRatio->Automatic]; Display["p30.ps",P4]; ----------------------------------------------- Page 553 Problem 32 Find the length of the curve x = Cos[t] + t Sin[t] y = Sin[t] - t Cos[t] from 0 to 2 Pi. dx/dt = -Sin[t] + Sin[t] + t Cos[t] = t Cos[t] dy/dt = Cos[t] - Cos[t] + t Sin[t] = t Sin[t] t=2Pi S = INT Sqrt[ t^2 Cos[t]^2 + t^2 Sin[t]^2 ] dt t=0 2 t=2Pi | t | t=2 Pi 2 S = INT t dt = | --- | = 2 Pi t=0 | 2 | t=0 ------------------------------------------------- Get["font.math"]; p[r_,t_] := r{Cos[t],Sin[t]}; f[t_] := Cos[t]+t Sin[t]; g[t_] := Sin[t]-t Cos[t]; P1 = ParametricPlot[ {f[t], g[t]} ,{t, 0,2 Pi},PlotStyle->{RGBColor[1,0,0], Thickness[0.01]}]; P2= Table[ Graphics[ Text[ FontForm[t,{"Courier-Bold",14}],{f[Sqrt[2 t]],g[Sqrt[2 t]] }]], {t, 0, 2 Pi^2}]; P3 = Show[P1,P2,PlotLabel->"P553 p32 {Cos[t]+t Sin[t], Sin[t]-t Cos[t]}", AspectRatio->Automatic,PlotRange->All]; Display["p32.ps",P3] -------------------------------------------------------------- Page 553 Problem 34 Analyze the given polar equation and sketch its graph. 5 r = ------------ Sin[theta] ------------------------------------------------------------ Get["font.math"]; p[r_,t_] := r{Cos[t],Sin[t]}; P1 = ParametricPlot[ p[5/Sin[t],t],{t, 0.3, Pi/2 -0.01 }, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = ParametricPlot[ p[5/Sin[t],t],{t,-Pi/2+0.01, -0.3}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P3 = Plot[0,{x,-10,10}]; P4 = ParametricPlot[{0,y},{y,-8,8}]; P5 = Show[P1,P2,P3,P4,PlotLabel->"P553 p34 r = 5/Sin[theta]", AspectRatio->Automatic,PlotRange->All]; Display["p34.ps",P5]; ---------------------------------------------------- Page 553 Problem 36 Analyze the given polar equation and sketch its graph. 3 r = ------------ Cos[theta] -------------------------------------------------------------- Get["font.math"]; p[r_,t_] := r{Cos[t],Sin[t]}; P1 = ParametricPlot[ p[3/Cos[t],t],{t, -Pi/3,Pi/3 }, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = Plot[0,{x,-3,3}]; P3 = ParametricPlot[{0,y},{y,-5,5}]; P4 = Show[P1,P2,P3,PlotLabel->"P553 p36 r = 3/Cos[theta]", AspectRatio->Automatic,PlotRange->All]; Display["p36.ps",P4]; ------------------------------------------------------------------ Page 553 Problem 38 Analyze the given polar equation and sketch its graph. r = 5 - 5 Cos[theta] -------------------------------------------------------------- Get["font.math"]; p[r_,t_] := r{Cos[t],Sin[t]}; P1 = ParametricPlot[ p[ 5 - 5 Cos[t],t],{t, -Pi, Pi }, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = Show[P1,PlotLabel->"P553 p38 r = 5 - 5 Cos[theta]", AspectRatio->Automatic,PlotRange->All]; Display["p38.ps",P2]; ------------------------------------------------------------------ Page 553 Problem 40 Analyze the given polar equation and sketch its graph. r = 2 - 3 Cos[theta] ------------------------------------------------------------- Get["font.math"]; p[r_,t_] := r{Cos[t],Sin[t]}; P1 = ParametricPlot[ p[ 2 - 3 Cos[t],t],{t, -Pi, Pi }, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = Show[P1,PlotLabel->"P553 p40 r = 2 - 3 Cos[theta]", AspectRatio->Automatic,PlotRange->All]; Display["p40.ps",P2]; ------------------------------------------------------------- Page 553 Problem 42 Analyze the given polar equation and sketch its graph. r = 4 Sin[3 theta] ------------------------------------------------------------- Get["font.math"]; <{RGBColor[1,0,0],Thickness[0.005]}]; P2 = Table[ TextListPlot[{{4 Sin[3 t/3] Cos[t/3], 4 Sin[3 t/3] Sin[t/3], t}}], {t, 0, 18}]; P3 = Show[P1,P2,PlotLabel->"P553 p42 r = 4 Sin[3 theta]", AspectRatio->Automatic,PlotRange->All]; Display["p42.ps",P3]; ------------------------------------------------------------- Page 553 Problem 44 Analyze the given polar equation and sketch its graph. r = - theta theta >= 0 ------------------------------------------------------------- Get["font.math"]; <{RGBColor[1,0,0],Thickness[0.005]}]; P2 = Table[ TextListPlot[{ {-t Cos[t],-t Sin[t], t} }], {t, 0, 6}]; P3 = Show[P1,P2,PlotLabel->"P553 p44 r = -theta ", AspectRatio->Automatic,PlotRange->All]; Display["p44.ps",P3]; ----------------------------------------------------------------- Page 553 Problem 46 2 Find a Cartesian equation of the graph r Cos[2 theta ] = 9 and then sketch the graph. 2 2 r ( 2 Cos [theta] -1) = 9 2 2 2 2 x - x - y = 9 2 2 x - y = 9 ------------------------------------------------------------- Get["font.math"]; p[r_,t_] := r{Cos[t],Sin[t]}; P1 = ParametricPlot[ p[ Sqrt[ 9/Cos[2 t]], t ],{t, -Pi/4.5, Pi/4.5 }, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = ParametricPlot[ p[-Sqrt[ 9/Cos[2 t]], t ],{t, -Pi/4.5, Pi/4.5 }, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P3 = Plot[{x,-x},{x,-5,5}]; P4 = Show[P1,P2,P3,PlotLabel->"P553 p46 r^2 Cos[2 theta] = 9 ", AspectRatio->Automatic,PlotRange->All]; Display["p46.ps",P4]; ------------------------------------------------------------- Page 553 Problem 48 Sketch the graphs of r = 5 Sin[theta] and r = 2 + Sin[theta] and Find their points of intersection. 5 Sin[theta] = 2 + Sin[theta] 4 Sin[theta] = 2 Sin[theta] = 1/2 theta = Pi/6, 5 Pi/6 ---------------------------------------------------------------------- Get["font.math"]; p[r_,t_] := r{Cos[t],Sin[t]}; P1 = ParametricPlot[ p[ 5 Sin[t], t ],{t, -Pi, Pi }, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = ParametricPlot[ p[ 2 + Sin[t], t ],{t, -Pi, Pi }, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P3 = ListPlot[ { p[5 Sin[Pi/6],Pi/6], p[5 Sin[5 Pi/6], 5 Pi/6]}, PlotStyle->{ PointSize[0.02]}]; P4 = Show[P1,P2,P3, PlotLabel->"P553 p48 r = 5 Sin[theta] and r = 2 + Sin[theta] ", AspectRatio->Automatic,PlotRange->All]; Display["p48.ps",P4]; ----------------------------------------------------------------- Page 553 Problem 50 Find the area of the region that is outside the limacon r = 2 + Sin[theta] and inside the circle r = 5 Sin[theta] t=5 Pi/6 2 2 INT 1/2 (5 Sin[t]) - 1/2 (2+Sin[t]) t=Pi/6 t=5 Pi/6 2 INT -2 - 2 Sin[t] + 12 Sin [t] dt t=Pi/6 t=5 Pi/6 INT -2 - 2 Sin[t] + 6(1-Cos[2t]) dt t=Pi/6 t=5 Pi/6 INT +4 - 2 Sin[t] - 6 Cos[2t]) dt t=Pi/6 | | t = 5 Pi/6 | 4 t +2 Cos[t] - 3 Sin[2 t] | | | t = Pi/6 10 Pi/3 -2 Sqrt[3]/2 +3 Sqrt[3]/2 - ( 2 Pi/3 + 2 Sqrt[3]/2 - 3 Sqrt[3]/2) 8 Pi Sqrt[3] + ---- 3 ------------------------------------------------------------- Get["font.math"]; p[r_,t_] := r{Cos[t],Sin[t]}; P1 = ParametricPlot[ p[ 5 Sin[t], t ],{t, -Pi, Pi }, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = ParametricPlot[ p[ 2 + Sin[t], t ],{t, -Pi, Pi }, PlotStyle->{RGBColor[0,1,0],Thickness[0.01]}]; P3 = ListPlot[ { p[5 Sin[Pi/6],Pi/6], p[5 Sin[5 Pi/6], 5 Pi/6]}, PlotStyle->{ PointSize[0.02]}]; P4 = Show[P1,P2,P3,PlotLabel-> "P553 p50 Outside green: r=2+Sin[theta]; Inside red: r = 5 Sin[theta] ", AspectRatio->Automatic,PlotRange->All]; Display["p50.ps",P4]; ------------------------------------------------------------------- Page 553 Problem 52 Match each polar equation with its graph. (a) r = 1 - 2 Sin[theta] Sin[theta] (b) r = 1 + ---------- 2 (c) r = 1 + 2 Cos[theta] Cos[theta] (d) r = 1 + ------------ 2 ---------------------------------------------------------------- Get["font.math"]; p[r_,t_] := r{Cos[t],Sin[t]}; P1 = ParametricPlot[ p[ 1 - 2 Sin[t] , t ],{t, -Pi, Pi }, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; P2 = ParametricPlot[ p[ 1 + (1/2) Sin[t] , t ],{t, -Pi, Pi }, PlotStyle->{RGBColor[0,1,0],Thickness[0.01]}]; P3 = ParametricPlot[ p[ 1 + 2 Cos[t] , t ],{t, -Pi, Pi }, PlotStyle->{RGBColor[0,0,1],Thickness[0.01]}]; P4 = ParametricPlot[ p[ 1 + (1/2) Cos[t] , t ],{t, -Pi, Pi }, PlotStyle->{RGBColor[1,1,0],Thickness[0.01]}]; P5 = Show[P1,P2,P3,P4,PlotLabel-> "P553 p52 r=1-2 Sin[t]; r=1+1/2 Sin[t]; r=1+2 Cos[t]; r=1+1/2 Cos[t] ", AspectRatio->Automatic,PlotRange->All]; Display["p52.ps",P5]; ---------------------------------------------------------