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, March 3, 8.3 Infinite Limits p441: 8,14,26,36 Main Idea: Stop and turn around, that is the view you will see on the way back. We stop frequently review the past material. Key Words: Converge, Diverge, Gabriel's Horn Improper integrals. Goal: Learn to integrate all the way to infinity. -------------------------------------------------------- Previous assignment: p432: (12),(24),(41),(44) Page 432 Problem 12 2 2 Lim 3 x Csc [x] x->0 2 3 x x x Lim ------------- = Lim 3 ----- ------ = 3 x->0 2 x->0 Sin[x] Sin[x] Sin [x] -------------------------------------------------------- Get["font.math"]; P1 = Plot[3 x^2 Csc[x]^2,{x,-Pi/3,Pi/3}]; P2 = ListPlot[{{0,3}},PlotStyle->{RGBColor[1,0,0],PointSize[0.04]}]; P3 = Plot[0,{x,-Pi/3,Pi/3}]; P4 = Show[P1,P2,P3,PlotLabel->"P432 p12: 3 x^2 Csc[x]^2", AspectRatio->Automatic]; Display["p12.ps",P4]; -------------------------------------------------------- Page 432 Problem 24 _ _ | 1 | | ---- | | 2 | |_ x _| Lim Cos[x] x->0 ln(Cos[x]) ln(y) = ------------ 2 x -Sin[x] ------------ Cos[x] - 1 Sin[x] Lim ln(y) = lim --------- = lim -------- ------ = -1/2 x->0 x->0 2x x->0 2 Cos[x] x -1/2 y = e = 1/Sqrt[e] <==== Answer ----------------------------------------- Get["font.math"]; f[x_,n_] := {x + 2 n Pi, Cos[x]^((x+2 n Pi)^(-2)) }; P1 = ListPlot[{{0,1/Sqrt[E]}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P2 = ParametricPlot[f[x,-2],{x,-Pi/2,Pi/2}]; P3 = ParametricPlot[f[x,-1],{x,-Pi/2,Pi/2}]; P4 = ParametricPlot[f[x,-0],{x,-Pi/2,Pi/2}]; P5 = ParametricPlot[f[x,+1],{x,-Pi/2,Pi/2}]; P6 = ParametricPlot[f[x,+2],{x,-Pi/2,Pi/2}]; g[x_] := Table[ {x+(1/10)^i, Cos[x+(1/10)^i]^(x+(1/10)^i)^(-2)},{i,1,500}]; P7 = ListPlot[ g[-9 Pi/2],PlotJoined->True ]; P8 = ListPlot[ g[-5 Pi/2],PlotJoined->True ]; P9 = ListPlot[ g[-1 Pi/2],PlotJoined->True ]; P10 = ListPlot[ g[+3 Pi/2],PlotJoined->True ]; P11 = ListPlot[ g[+7 Pi/2],PlotJoined->True ]; h[x_] := Table[ {x-(1/10)^i, Cos[x-(1/10)^i]^(x-(1/10)^i)^(-2)},{i,1,500}]; P12 = ListPlot[ h[-7 Pi/2],PlotJoined->True ]; P13 = ListPlot[ h[-3 Pi/2],PlotJoined->True ]; P14 = ListPlot[ h[+1 Pi/2],PlotJoined->True ]; P15 = ListPlot[ h[+5 Pi/2],PlotJoined->True ]; P16 = ListPlot[ h[+9 Pi/2],PlotJoined->True ]; P17 = Show[P1,P2,P3,P4,P5,P6,P7,P8,P9,P10,P11,P12,P13,P14,P15,P16, PlotLabel->"P432 p24, Cos[x]^(1/x^2)", PlotRange->All]; Display["p24.ps",P17]; ----------------------------------------------- Page 432 Problem 41 ______ (a) Lim \ n/ n->Infinity \/ a ______ (b) Lim \ n/ n->Infinity \/ n | ___ | (c) Lim n | \ n/ | n->Infinity | \/ a - 1 | | ___ | (d) Lim n | \ n/ | n->Infinity | \/ n - 1 | ------------------------------------------------------- (a) 1/x Lim a x->infinity ln(y) = 1/x ln(a) Lim ln(y) = 0 x->infinity 0 y -> e = 1. .......................................... (b) 1/x Lim x x->infinity ln(y) = 1/x ln(x) ln(x) 1/x Lim ------- = Lim -------- = 0 x->Infinity x x->infinity 1 0 y -> e = 1. ------------------------------------------- Get["font.math"]; P1 = Plot[ 2^(1/x),{x,1,100},PlotStyle->{RGBColor[1,0,0]}]; P2 = Plot[ x^(1/x),{x,1,100},PlotStyle->{RGBColor[0,1,0]}]; P3 = Show[P1,P2,PlotLabel->"P432 p41 red(a)= 2^(1/x);green(b)=x^(1/x)"]; Display["p41x.ps",P3]; ---------------------------------------------------------- | ___ | (c) Lim n | \ n/ | n->Infinity | \/ a - 1 | 1/x 1/x a -1 Lim x(a -1) = Lim ------------ x->Infinity x->Infinity 1/x 1/x a (-1/x^2) ln(a) Lim ----------------------- = ln(a) x->Infinity -1/x^2 -------------------------------------------------------- | ___ | (d) Lim n | \ n/ | n->Infinity | \/ n - 1 | 1/x 1/x x - 1 Lim x( x -1 ) = Lim ------- x->Infinity x->Infinity 1/x 1/x (1/x)-1 x (-1/x^2) ln(x) + 1/x x = Lim ------------------------------------ x->Infinity (-1/x^2) 1/x 1/x = Lim x ln(x) - x = +Infinity. x->Infinity ---------------------------------------------- Get["font.math"]; P1 = Plot[ x(2^(1/x) -1),{x,1,10}, PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; P2 = Plot[ x(20^(1/x) -1),{x,1,10}, PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; P3 = Plot[ x(200^(1/x) -1),{x,1,10}, PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; P4 = Show[P1,P2,P3,PlotLabel->"P432 p41(c) x (a^(1/x)-1); a=2,20,200"]; Display["41y.ps",P4]; P5 = Plot[ x(x^(1/x) -1),{x,1,100}, PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; P6 = Plot[ Log[x]-1,{x,1,100}, PlotStyle->{RGBColor[0,1,0],Thickness[0.005]}]; P7 = Plot[ x^(1/x)(Log[x]-1),{x,1,100}, PlotStyle->{RGBColor[0,0,1],Thickness[0.005]}]; P8 = Show[P5,P6,P7, PlotLabel->"P432 p41(d) RGB x(x^(1/x)-1); ln(x)-1; x^(1/x)(ln(x)-1)"]; Display["41z.ps",P8]; ---------------------------------------------------------------- Page 432 Problem 44 Find each limit x x (1/x) (a) Lim ( 1 + 2 ) x->0+ x x (1/x) (b) Lim ( 1 + 2 ) x->0- x x (1/x) (c) Lim ( 1 + 2 ) x->Infinity x x (1/x) (d) Lim ( 1 + 2 ) x->-Infinity ------------------------------------------------------------------ x x (1/x) (a) Lim ( 1 + 2 ) = Infinity x->0+ x x (1/x) (b) Lim ( 1 + 2 ) = 0 x->0- x x (1/x) (c) Lim ( 1 + 2 ) x->Infinity x x ln(1 + 2 ) 2 ln(2) Lim -------------- = Lim ----------- = ln(2) x->Infinity x x->Infinity x 1 + 2 x x (1/x) (c) Lim ( 1 + 2 ) = 2. <==== Answer. x->Infinity x x (1/x) (d) Lim ( 1 + 2 ) x->-Infinity x x ln(1 + 2 ) 0 Lim ------------- = Lim ----------- = 0 x->-Infinity x x->-Infinity -Infinity x x (1/x) 0 Lim ( 1 + 2 ) = e = 1 <== Answer x->-Infinity -------------------------------------------------------- Get["font.math"]; f[x_] := (1^x + 2^x)^(1/x); P1 = Plot[f[x],{x,0.1,1},PlotPoints->20]; P2 = Show[P1,PlotLabel->"P432 p44 (a) (1+2^x)^(1/x)"]; Display["p44a.ps",P2]; P3 = Plot[f[x],{x,-2.0,-0.01},PlotPoints->20]; P4 = Show[P3,PlotLabel->"P432 p44 (b) (1+2^x)^(1/x)"]; Display["p44b.ps",P4]; P5 = Plot[f[x],{x,100,1000},PlotPoints->200]; P6 = Show[P5,PlotLabel->"P432 p44 (c) (1+2^x)^(1/x)"]; Display["p44c.ps",P6]; P7 = Plot[f[x],{x,-100,-1000},PlotPoints->200]; P8 = Show[P7,PlotLabel->"P432 p44 (d) (1+2^x)^(1/x)"]; Display["p44d.ps",P8]; P9 = Plot[f[x],{x,-10,0},PlotPoints->200]; P10 = Plot[f[x],{x,0.5,10},PlotPoints->200]; P11 = Show[P9,P10,PlotLabel->"P432 p44 (abcd) (1+2^x)^(1/x)"]; Display["p44.ps",P11]; ---------------------------------------------------- New Material: 1 Find the area under y = ---- to the right of x=1. 2 x |. | 1 | '. y = ---- | '. 2 | '. x | |' . | |/////' . | |// A(a) ////' . | |////////////////////' | . . ----------+------------------------------------------------- | 1 a We find the area between 1 and a and then see what happens as a approaches infinity. _ _ x=a 1 | | x=a INT ---- dx = | -1/x | = -1/a+1 = 1-1/a x=1 2 |_ _| x=1 x The limit 1-1/a = 1. a->Infinity 1 Thus the area under the curve y = ---- from 1 to infinity is 1. 2 x ------------------------------------------------------------------- Example 1 Page 434. 2 -1 -x INT x e dx -Infinity 2 x=-1 -x INT x e dx x=b _ 2 _ | -x | x=-1 = |-1/2 e | |_ _| x=b 2 -b = -1/2 1/e + 1/2 e 2 -b -1 Limit -1/2 1/e + 1/2 e = -------- = -0.18394 b->-Infinity 2 e Find the area from -Infinity to zero. Answer = -1/2 ---------------------------------------------------- Get["font.math"]; P1 = Plot[ x E^(-x^2),{x,-2,1}]; P2 = Show[P1,PlotLabel->"Example 1 Page 434 y = x e^(-x^2)", PlotRange->All,AspectRatio->Automatic]; Display["ex1x.ps",P2]; f[x_] := x E^(-x^2); P3 = Plot[ (x+2) f[-1],{x,-2,-1},PlotStyle->{RGBColor[1,0,0]}]; P4 = ParametricPlot[{-1,y},{y,0,f[-1]},PlotStyle->{RGBColor[1,0,0]}]; P5 = Plot[f[x],{x,-3,1}]; P6 = Show[P3,P4,P5,PlotLabel->"P434 ex 1; Approximation to area"] ; Display["ex1y.ps",P6]; P7 = Plot[ (x+2) f[-1/Sqrt[2]]/(-1/Sqrt[2]+2), {x,-2,-1/Sqrt[2]},PlotStyle->{RGBColor[1,0,0]}]; P8 = Plot[ f[-1/Sqrt[2]]-(x+1/Sqrt[2]) f[-1/Sqrt[2]] Sqrt[2],{x,-1/Sqrt[2],0}, PlotStyle->{RGBColor[1,0,0]} ]; P9= Show[P1,P7,P8,PlotLabel->"P434 Ex 1 Triangular approximation to the area", AspectRatio->Automatic,PlotRange->All ]; Display["ex1z.ps",P9]; ------------------------------------------------------------------------- Area of smaller triangle is: 1/2 base height = 1/2 1 1/E = 0.18394 Approximate area on (-Infinity, -1) Area of larger triangle is: 1/2 base height = 1/2 2 f[-1/Sqrt[2]] = -0.428882 Approximate on (-Infinity,0) --------------------------------------------------------- Page 435 Example 2. Infinity Find INT Sin[x] dx x=0 _ _ x=a | | x=a INT Sin[x] dx = | -Cos[x] | = - Cos[a] + 1 x=0 |_ _| x=0 Limit 1-Cos[a] does not exist so we say that a->infinity the integral "Diverges". ------------------------------------------------------------------- Work = Force x distance. The force of gravity diminishes as the square of the distance. Thus at a distance x, k f = ----- 2 x What happens at x = 0 ? We just won't go there today. Infinity k Work = INT ----- dx x=R 2 x Is the work necessary to push an object all the way to infinity from the earth's gravity. It also represents the energy transferred to an object that is attracted to the earth from way way out in space. a k | | x=a Work = INT ----- dx = | -k/x | = -k/a + k/R x=R 2 | | x=R x Limit k/R - k/a = k/R Which is a finite number. a->Infinity In fact, If the object weighted 1000 pounds at the Earth's surface k 2 1000 = ---------- so k = 1000 3960 2 3960 And the work done is k/R = 1000 3960 pound miles. The great idea is that the work done is finite. One might expect that! If it were infinite, then any comet picked up on the far edge of space would slam into the earth with an amount of energy approaching infinity. This would destroy everything. Put another way, nothing could escape the universe because any particle anywhere would always be able to pull anything else back to it. The actual numbers is not that interesting, Just how much work is 1000 3960 pounds miles? Would that be the amount of work needed to move Mount Everest to Australia, or pull a sled to the north pole and back 100 times? ------------------------------------------------------------ Example 4 Page 435. x=+Infinity 1 INT ------------ dx x=-Infinity 2 1+x In a situation like this, before we say that the intergal converges, we need both sides to converge. x=0 1 x=a 1 INT ------------ dx + INT ------------ dx x=b 2 x=0 2 1+x 1+x ArcTan[0]-ArcTan[b] + ArcTan[a]-ArcTan[0] -(-Pi/2) + Pi/2 = Pi. <======== Answer. ---------------------------------------------------- Get["font.math"]; P1 = Plot[ 1/(1+x^2),{x,-10,10}]; P2 = Show[P1,PlotLabel->"P435 Ex 4 y = 1/(1+x^2)",PlotRange->All]; Display["ex4.ps",P2]; --------------------------------------------------------- The Probability Density Function 2 1 Infinity -x /2 --------- INT e = 1 Sqrt[2 Pi] -Infinity ------------------------------------------------------------------------ P1 = Plot[ E^(-x^2/2),{x,-3,3}]; P2 = Show[P1,PlotLabel->"Probability Density Function y = e^(-x^2/2)", PlotRange->All,AspectRatio->Automatic]; Display["p.ps",P2]; ------------------------------------------------------------------------ Gabriel's Horn. Rotate 1/x on [1, Infinity) about the x-axis. What is its volume. What is its surface area. _ _ x=Infinity | | x=Infinity (a) Volume = INT Pi (1/x)^2 dx = Pi | -1/x | = Pi x=1 |_ _| x=1 x=Infinity (b) Surface area = INT 2 Pi 1/x Sqrt[1 + (-1/x^2)^2] dx x=1 _ _ x=Infinity | 1 | 2 Pi INT 1/x Sqrt|1 + ---- | dx x=1 | 4 | |_ x _| 4 x=Infinity Sqrt[x + 1] 2 Pi INT ---------------- dx x=1 3 x x=Infinity | | x=Infinity >= 2 Pi INT 1/x dx = 2 Pi | ln(x) | = Infinity. x=1 | | x=1 -------------------------------------------------------------------- Get["font.math"]; start = -10 Pi/9; stop = Pi/2; P1 = ParametricPlot3D[ {x, 1/x Cos[t],1/x Sin[t]},{t,start,stop},{x,1,4}]; P2 = ParametricPlot3D[ {x, 1/x Cos[start],1/x Sin[start],{RGBColor[1,0,0],Thickness[0.002]}}, {x,1,4}]; P3 = ParametricPlot3D[ {x, 1/x Cos[stop ],1/x Sin[stop ],{RGBColor[1,0,0],Thickness[0.002]}}, {x,1,4}]; P4 = ParametricPlot3D[ {1, 1/1 Cos[t],1/1 Sin[t],{RGBColor[1,0,0],Thickness[0.002]}}, {t,start,stop}]; P5 = ParametricPlot3D[ {4, 1/4 Cos[t],1/4 Sin[t],{RGBColor[1,0,0],Thickness[0.002]}}, {t,start,stop}]; P6= Show[P1,P2,P3,P4,P5,PlotLabel->"Gabriel's Horn"]; Display["gab.ps",P6];