I need these pdf files: aug31.p16.pdf aug31.p24.pdf aug31.p34x.pdf aug31.p34y.pdf aug31.sin.pdf aug31.cos.pdf aug31.proof.pdf aug31.sinxoverx.pdf aug31.proof.pdf aug31.sandwich.pdf aug31.coslimit.pdf aug31.ex1.pdf aug31.ex2.pdf aug31.ex3.pdf aug31.ex4.pdf aug31.xint.pdf Thank you ##################################################################### 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.165.09 Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. ##################################################################### Monday. August 31 1.4 Trig Limits Assignment:P77: 8, 12, 16, 24, (x-intercept) (x-intercept) A point P(s)=( Cos(s), Sin(s) ) is located on the unit circle. A second point Q(s)=(1,s) is located on the line x=1. The line L determined by the two points P(s) and Q(s) intersects the x axis at the point (R(s),0). (a) Give the function y = R(s). (b) Graph the function y = R(s). (c) Find the the limit of R(s) as s approaches 0. ##################################################################### Previous Assignment: P72: 16, 24, 30, 34 Page 72 Problem 16: Find the indicated limit or state that it does not exist. 2 x + x 0 Limit -------- = ----- = 0 x-> -1 2 2 x + 1 ##################################################################### Get["font.math"]; f[x_] := (x^2+x)/(x^2+1); p1 = Plot[f[x],{x,-2,2}]; p2 = ListPlot[{{-1,0}},PlotStyle->{RGBColor[1,0,0]}]; p3 = Show[p1,p2, PlotLabel->"P72 p16 (x^2+x)/(x^2+1) at x=-1", PlotRange->All, AxesOrigin->Automatic, AxesLabel->{x,y}, AxesStyle->Directive[Orange,15] ]; Export["/math/www/hentzel/class.165.09/aug31.p16.pdf",p3]; ##################################################################### Page 72 Problem 24 2 (w+2)(x - w - 6) (w+2)(w-3)(w+2) Limit ------------------- = Limit -------------- = Limit w-3 = -5 w-> -2 2 w-> -2 (w+2)(w+2) w->-2 w + 4 w + 4 ##################################################################### Get["font.math"]; f[w_] := (w+2)(w^2 - w - 6)/(w^2 + 4 w + 4); p1 = Plot[f[w],{w,-3,3}]; p2 = Plot[-5,{x,-2,0},PlotStyle->{RGBColor[1,0,0]}]; p3 = ParametricPlot[ {-2,y},{y,f[-2.001],0},PlotStyle->{RGBColor[1,0,0]}]; p4 = ListPlot[{{-2,-5}},PlotStyle->{RGBColor[1,0,0],PointSize[0.02]}]; p5 = Show[p1,p2,p3,p4, PlotLabel->"P72 p24 f[w] = (w+2)(w^2-w-6)/(w^2+4w+4) at w=-2", PlotRange->All, AxesOrigin->Automatic, AxesLabel->{w,y}, AxesStyle->Directive[Orange,15] ]; Export["/math/www/hentzel/class.165.09/aug31.p24.pdf",p5]; ##################################################################### Page 72 Problem 30 Find the limits if Limit f(x) = 3 and Limit g(x) = -1 x->a x->a 3 3 Limit [ f(u) + 3 g(u) ] = ( 3 - 3) = 0 u->a ##################################################################### Page 72 Problem 34 f(x) - f(2) 3 Find Limit -------------- where f(x) = ----- . x->2 x-2 2 x 2 3 3 12 - 3 x --- - --- ---------- 2 4 2 2 f(x)-f(2) x 4 x 3 (4-x ) --------- = ---------- = --------- = ---------- x-2 x-2 x-2 2 4 x (x-2) -3(x+2)(x-2) -3(x+2) = ------------- = ------- 2 2 4 x (x-2) 4 x 3/x^2 - 3/4 Limit ------------- = -12/16 = -3/4. x->2 x-2 ##################################################################### Get["font.math"]; f[x_] := 3/x^2; fp[x_] := (f[x]-f[2])/(x-2); p1 = Plot[ fp[x],{x, 0.627834, 3},PlotRange->All ]; p2 = Plot[ fp[x],{x,-3,-0.477834},PlotRange->All ]; p3 = Plot[-3/4, {x,0,2},PlotStyle->{RGBColor[1,0,0]}]; p4 = ParametricPlot[ {2,y},{y,-3/4,0}, PlotStyle->{RGBColor[1,0,0]}]; p5 = ListPlot[{{2,-3/4}},PlotStyle->{RGBColor[1,0,0],PointSize[0.02]}]; p6 = Show[p1,p2,p3,p4,p5, PlotLabel->"P72 p34 (f[x]-f[2])/(x-2); f[x] = 3/x^2", PlotRange->All, AxesOrigin->Automatic, AxesLabel->{w,y}, AxesStyle->Directive[Orange,18] ]; Export["/math/www/hentzel/class.165.09/aug31.p34x.pdf",p6]; ##################################################################### Get["font.math"]; f[x_] := 3/x^2; p1 = Plot[f[x],{x,0.80,6}]; p2 = Plot[ -3/4(x-2)+f[2],{x,1.0,2.8},PlotStyle->{RGBColor[1,0,0]}]; p3 = ListPlot[{{2,f[2]}},PlotStyle->{RGBColor[1,0,0],PointSize[0.005]}]; p4 = Show[p1,p2,p3, PlotLabel->"P72 p34 Tangent to y = 3/x^2 at (2,3/4)", AspectRatio->Automatic, PlotRange->All, AxesOrigin->{0,0}, AxesLabel->{w,y}, AxesStyle->Directive[Orange,18], TicksStyle->Directive[Red,24] ]; Export["/math/www/hentzel/class.165.09/aug31.p34y.pdf",p4]; ##################################################################### New Material. (1) The Sine and Cosine are continuous. If you look at their graphs, they certainly look continuous. ##################################################################### Get["font.math"]; a = Plot[ Sin[t],{t,-2 Pi,2 Pi},PlotStyle->{Thickness[0.005]}]; b = Show[a, PlotLabel->"y = Sin[x]", PlotRange->All, AspectRatio->Automatic, PlotRange->All, AxesOrigin->Automatic, AxesLabel->{x,y}, AxesStyle->Directive[Orange,15], TicksStyle->Directive[Orange,12] ]; Export["/math/www/hentzel/class.165.09/aug31.sin.pdf",b]; a = Plot[ Cos[t],{t,-2 Pi,2 Pi},PlotStyle->{Thickness[0.005]}]; b = Show[a, PlotLabel->"y = Cos[x]", PlotRange->All, AspectRatio->Automatic, AxesLabel->{x,y}, AxesStyle->Directive[Orange,15], TicksStyle->Directive[Orange,12] ]; Export["/math/www/hentzel/class.165.09/aug31.cos.pdf",b]; ##################################################################### We give a pictorial proof using the essence of the horizontal vertical line approach. | . '''|' ' -.------------ ' | / @ ' | /---.'`----------- . | / .' . | / .' . : |/.' : :------------+------------:-- : | : | ` | ' '. | ' ' . | . ' '''|''' | The proof for the cosine is the same, but you have to use vertical lines since the cosine function is plotted on the horizontal axis. ##################################################################### Get["font.math"]; p1 = ParametricPlot[ {Cos[t],Sin[t]},{t,0,2 Pi}]; w = 1; epsilon = 0.1; p2 = ListPlot[{{Cos[w],Sin[w]}},PlotStyle->{RGBColor[1,0,0],PointSize[0.02]}]; p3 = Plot[ Sin[w]+epsilon ,{x,0,1}, PlotStyle->{RGBColor[0,1,0],Thickness[0.01]}]; p4 = Plot[ Sin[w]-epsilon,{x,0,1}, PlotStyle->{RGBColor[0,1,0],Thickness[0.01]}]; deltaL = Abs[ArcSin[Sin[w]+epsilon]-w]; deltaR = Abs[ArcSin[Sin[w]-epsilon]-w]; delta = Min[deltaL,deltaR]; p5 = ParametricPlot[ t {Cos[w+delta],Sin[w+delta]},{t,0,1}, PlotStyle->{RGBColor[0,0,1],Thickness[0.01]}]; p6 = ParametricPlot[ t {Cos[w-delta],Sin[w-delta]},{t,0,1}, PlotStyle->{RGBColor[0,0,1],Thickness[0.01]}]; p7 = ParametricPlot[ t {Cos[w],Sin[w]},{t,0,1}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p8 = Show[p1,p2,p3,p4,p5,p6,p7, PlotLabel->"Given epsilon(green); There exists delta (blue)", AspectRatio->Automatic]; Export["/math/www/hentzel/class.165.09/aug31.proof.pdf",p8]; ##################################################################### There is one limit that is the most important. Sin[x] Lim ---------- = 1 x->0 x ##################################################################### Get["font.math"]; a = Plot[ Sin[x]/x,{x,-4Pi,4Pi},PlotStyle->{Thickness[0.004]}]; b = ListPlot[ {{0,1}},PlotStyle->{RGBColor[1,0,0],PointSize[0.011]}]; c = Show[a,b,PlotLabel->"y = Sin[x]/x",PlotRange->All]; Export["/math/www/hentzel/class.165.09/aug31.sinxoverx.pdf",c]; (* The Proof Diagram: *) p1 = ParametricPlot[ {Cos[t],Sin[t]},{t,0,2 Pi}, PlotStyle->{Thickness[0.01]}]; theta = 1.0; p2 = ParametricPlot[ t{1,Tan[theta]},{t,0,1}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p3 = ParametricPlot[ {1,y},{y,0,Tan[theta]}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p4 = ParametricPlot[ t {Cos[theta],Sin[theta]}+(1-t){1,0},{t,0,1}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p5 = Show[p1,p2,p3,p4, PlotLabel->"The Three areas", AspectRatio->Automatic, PlotRange->All ]; Export["/math/www/hentzel/class.165.09/aug31.proof.pdf",p5]; ##################################################################### This limit is so important that we really nead to examine a proof carefully. /| / | / | ..... / | . ' | '/. | . | /\ `. | | / \ .| . | / \ | |/ t \ | :_________/________\|___ : | ' . | ' `. | ' 2 2 ` . | . ' x + y = 1 ``|`` The Area of the small triangle is less than the area of the sector and the area of the sector is less than the area of the outer triangle. ##################################################################### Small Triangle <= Sector <= Big Triangle 1/2 Sin[t] <= 1/2 t <= 1/2 Tan[t] Sin[t] <= t <= Tan[t] 1 <= t/Sin[t] <= 1/Cos[t] (* inversion switches the sign of the inequality. 1 >= Sin[t]/t >= Cos[t] The limit of Cos[t] = 1 and Sin[t]/t is sandwiched between Cos[t] and 1. Therefore the limit of Sin[t]/t = 1. ##################################################################### Get["font.math"]; p1 = Plot[Sin[x]/x,{x,-Pi/2, Pi/2},PlotStyle->{Thickness[0.004]}]; p2 = Plot[1,{x,-Pi/2,Pi/2 },PlotStyle->{RGBColor[1,0,0],Thickness[0.004]}]; p3 = Plot[Cos[x],{x,-Pi/2,Pi/2},PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; p4 = Show[p1,p2,p3,PlotLabel->"Sandwich,y=Sin[x]/x,y=1,y=Cos[x]", AspectRatio->Automatic,PlotRange->All]; Export["/math/www/hentzel/class.165.09/aug31.sandwich.pdf",p4]; ##################################################################### Examples: In a Quonset hut, the walls are nearly vertical. Up close, the earth is flat. ##################################################################### The secondary limit is 1-Cos[t] Limit ----------- = 0 t->0 t What does this say about the lengths of lines on the unit circle? What does this day about distances on the earth? ##################################################################### Proof: 1-Cos[t] (1-Cos[t])(1+Cos[t]) Limit --------- = Limit -------------------- t->0 t t->0 t (1+Cos[t]) 2 1-Cos [t] = Limit --------------- t->0 t (1+Cos[t] 2 Sin [t] = Limit ------------ t->0 t(1+Cos[t]) Sin[t] Sin[t] = Limit ------ * ----------- t->0 t 1+Cos[t] Sin[t] Sin[t] = Limit ------ * Limit --------- t->0 t t->0 1+Cos[t] = 1 * 0 = 0 ##################################################################### Get["font.math"]; a = Plot[ (1-Cos[x])/x,{x,-4Pi,4Pi},PlotStyle->{Thickness[0.004]}]; b = ListPlot[ {{0,0}},PlotStyle->{RGBColor[1,0,0],PointSize[0.02]}]; c = Show[a,b,PlotLabel->"y = (1-Cos[t])/t"]; Export["/math/www/hentzel/class.165.09/aug31.coslimit.pdf",c]; ##################################################################### Various Applications. 1-Cos[t] Limit -------------- t-> 0 2 t ##################################################################### Get["font.math"]; f[x_] := (1-Cos[x])/x^2; a = Plot[ f[x],{x, -2 Pi, -0.01},PlotStyle->{Thickness[0.004]},PlotRange->All]; b = Plot[ f[x],{x, 0.01, 2 Pi},PlotStyle->{Thickness[0.004]},PlotRange->All]; c = ListPlot[{{0,1/2}},PlotStyle->{RGBColor[1,0,0],PointSize[0.005]}]; d = Show[a,b,c, PlotLabel->"y = (1-Cos[x])/x^2)", PlotRange->All, AspectRatio->Automatic, AxesOrigin->{0,0}, AxesLabel->{x,y}, AxesStyle->Directive[Orange,4 ], TicksStyle->Directive[Orange,4 ] ]; Export["/math/www/hentzel/class.165.09/aug31.ex1.pdf",d]; ##################################################################### Sin[3x] Limit --------- t->0 x ##################################################################### Get["font.math"]; f[x_] := Sin[3x]/x; a = Plot[ f[x],{x,-5,5},PlotStyle->{Thickness[0.004]},PlotRange->All]; b = ListPlot[ {{0,3}}, PlotStyle->{RGBColor[1,0,0],PointSize[0.02]},PlotRange->All]; c = Show[a,b,PlotLabel->"y = Sin[3x]/x",PlotRange->All]; Export["/math/www/hentzel/class.165.09/aug31.ex2.pdf",c]; ##################################################################### 1-Cos[t] Limit --------- t->0 2 Sin [t] ##################################################################### Get["font.math"]; f[x_] := (1-Cos[x])/Sin[x]^2; a = Plot[ f[x],{x,-5 Pi/10,5 Pi/10},PlotStyle->{Thickness[0.004]}, PlotRange->All]; b = ListPlot[ {{0,1/2}},PlotStyle->{RGBColor[1,0,0],PointSize[0.02]}, PlotRange->All]; c = Show[a,b, PlotLabel->"y = (1-Cos[x])/Sin[x]^2", PlotRange->All, AspectRatio->Automatic, AxesLabel->{x,y}, AxesOrigin->{0,0}, AxesStyle->Directive[Orange,15], TicksStyle->Directive[Orange,12] ]; Export["/math/www/hentzel/class.165.09/aug31.ex3.pdf",c]; ##################################################################### Sin[4x] Limit ------------ t->0 Tan[x] ##################################################################### Get["font.math"]; f[x_] := Sin[4x]/Tan[x]; a = Plot[ f[x],{x,-Pi/2,Pi/2},PlotStyle->{Thickness[0.004]}]; b = ListPlot[ {{0,4}},PlotStyle->{RGBColor[1,0,0],PointSize[0.02]}]; c = Show[a,b,PlotLabel->"y = Sin[4x]/Tan[x]",PlotRange->All]; Export["/math/www/hentzel/class.165.09/aug31.ex4.pdf",c]; ##################################################################### ...Thinking..... Imagine the tangent to the curve y = Sin[x]? Where is the tangent line horizontal? What is the slope of the tangent line at the origin? ##################################################################### Get["font.math"]; s = 3/4; R[s_] := 1-s(1-Cos[s])/(s-Sin[s]); p1 = ParametricPlot[{Cos[t],Sin[t]},{t,s,2 Pi}, PlotStyle->{Thickness[0.004]}]; p2 = ParametricPlot[{Cos[t],Sin[t]},{t,0,s}, PlotStyle->{RGBColor[1,0,0],Thickness[0.004]}]; p3 = ParametricPlot[{1,y},{y,0,s}, PlotStyle->{RGBColor[1,0,0],Thickness[0.004]}]; p4 = Plot[ (s-Sin[s])/(1-Cos[s])(x-Cos[s])+Sin[s],{x,-2.1,1.1}, PlotStyle->{RGBColor[1,0,1],Thickness[0.004]}]; p5 = ListPlot[{ {Cos[s],Sin[s]},{1,s}, {R[s],0}}, PlotStyle->{RGBColor[0,1,1],PointSize[0.011]}]; p6 = Show[p1,p2,p3,p4,p5,PlotLabel->"X-Intercept,s=3/4", AspectRatio->Automatic, PlotRange->All]; Export["/math/www/hentzel/class.165.09/aug31.xint.pdf",p6]; #####################################################################