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, February 27 8.1 Type 0/0 p427:14,16,22,26 Main Idea: Take derivatives of the numerator and denominator separately. / / Key Words: L Hopital s Rule Goal: Learn easy way to take limits. -------------------------------------------------------- Previous assignment: p421 (44),(46),(48),(50) Page 421 Problem 44 Find the volume of the solid generated by revolving the region under the graph of 1 y = -------------- 2 Sqrt[3x - x ] from x=1 to x=2 (a) About the x-axis x=2 2 x=2 1 INT Pi f(x) dx = Pi INT -------- dx = x=1 x=1 2 3x - x 1 A B 1/3 -1/3 -------- = ----- + ------- = -------- + ---------- x(3-x) x x-3 x x-3 2 g(x) = 3x - x / g (x) = 3 - 2 x _ _ x=2 x=2 2 | | INT Pi f(x) dx = Pi | 1/3 ln(x) - 1/3 ln(3-x) | x=1 |_ _| x=1 = Pi ( 1/3 ln(2) + 1/3 ln(2) ) 2 Pi ln(2) = ----------- = 1.45172 3 ------------------------------- Integrate[ Pi /(x(3-x)),{x,1,2}]; Get["font.math"]; f[x_] := 1/Sqrt[3x-x^2]; start = 0.7 Pi; stop = -1.2 Pi; P1 = ParametricPlot3D[ {x,f[x] Cos[t],f[x] Sin[t]},{x,1,2},{t,start,stop}]; P2 = ParametricPlot3D[ {x,f[x] Cos[start],f[x] Sin[start],{RGBColor[1,0,0],Thickness[0.01]}},{x,1,2}]; P3 = ParametricPlot3D[ {x,f[x] Cos[stop],f[x] Sin[stop],{RGBColor[1,0,0],Thickness[0.01]}},{x,1,2}]; P4 = ParametricPlot3D[ {2,f[2] Cos[t],f[2] Sin[t],{RGBColor[1,0,0],Thickness[0.01]}},{t,start,stop}]; P5 = ParametricPlot3D[ {1,f[1] Cos[t],f[1] Sin[t],{RGBColor[1,0,0],Thickness[0.01]}},{t,start,stop}]; P6 = Show[P1,P2,P3,P4,P5,PlotLabel->"P421 P44(a) 1/(3x-x^2) around x axis", AspectRatio->Automatic,PlotRange->All]; Display["p44a.ps",P6]; -------------------------------- (b) About the y-axis x=2 x INT 2 Pi x f(x) dx = 2 Pi INT ---------- dx x=1 2 Sqrt[3x-x ] -1/2( 3 - 2 x ) 3/2 = 2 Pi INT ------------------- dx + 2 Pi INT --------------- dx 2 2 Sqrt[3x - x ] Sqrt[3x - x ] _ _ x=2 | 2 1/2 | | (3x-x ) | dx = - Pi | ---------- | + 3 Pi INT ------------------- |_ 1/2 _| x=1 2 Sqrt[9/4 - (x-3/2) ] _ _ x=2 _ _ | 2 1/2 | | | x=2 | (3x-x ) | | (x-3/2) | = - Pi | ---------- | + 3 Pi ArcSin| ------- | |_ 1/2 _| x=1 |_ 3/2 _| x=1 = -2 Pi (Sqrt[2] - Sqrt[2]) ) + 3 Pi (ArcSin(1/3) - ArcSin(-1/3) ) = 6 Pi ArcSin[1/3] = 6.40577 ---------------------------------------------- Approximation 1/Sqrt[2] ( Pi 2^2 - Pi 1^2) = 6.66432 which is a little too big. --------------------------------------------- Get["font.math"]; f[x_] := Sqrt[1/(3x-x^2)]; Integrate[ 2 Pi x f[x],{x,1,2}]; start = - 0.1 Pi; stop = 1.5 Pi; P1 = ParametricPlot3D[ {r Cos[t], r Sin[t], f[r] }, {t, start, stop}, {r, 1, 2}, PlotPoints->50]; P2 = ParametricPlot3D[ {r Cos[t], r Sin[t], 0 }, {t, start, stop},{r,1,2}]; P3 = ParametricPlot3D[ {1 Cos[t], 1 Sin[t], z }, {t, start, stop},{z,0,f[1]}]; P4 = ParametricPlot3D[ {2 Cos[t], 2 Sin[t], z }, {t, start, stop},{z,0,f[2]}]; P5 = ParametricPlot3D[ {2 Cos[t], 2 Sin[t], 1/Sqrt[2], {RGBColor[1,0,0],Thickness[0.004]} }, {t,start,stop}]; P6 = ParametricPlot3D[ { Cos[t], Sin[t], 1/Sqrt[2], {RGBColor[1,0,0],Thickness[0.004]} }, {t,start,stop}]; P7 = Graphics3D[{RGBColor[1,0,0],Thickness[0.004], Line[{{1 Cos[start],1 Sin[start], f[1]}, {1 Cos[start],1 Sin[start], 0 }, {2 Cos[start],2 Sin[start], 0 }, {2 Cos[start],2 Sin[start], f[1]}}]}]; P8 = Graphics3D[{RGBColor[1,0,0],Thickness[0.004], Line[{{1 Cos[stop ],1 Sin[stop ], f[1]}, {1 Cos[stop ],1 Sin[stop ], 0 }, {2 Cos[stop ],2 Sin[stop ], 0 }, {2 Cos[stop ],2 Sin[stop ], f[1]}}]}]; P9 = ParametricPlot3D[ {r Cos[start], r Sin[start], f[r],{RGBColor[1,0,0],Thickness[0.004]} }, {r, 1, 2}, PlotPoints->50]; P10 = ParametricPlot3D[ {r Cos[stop ], r Sin[stop ], f[r],{RGBColor[1,0,0],Thickness[0.004]} }, {r, 1, 2}, PlotPoints->50]; P11 = Show[P1,P2,P3,P4,P5,P6,P7,P8,P9,P10, PlotLabel->"P421 P44b, Sqrt[1/(3x-x^2)] about the y-axis", PlotRange->All]; Display["p44b.ps",P11]; P12 = Plot[f[x],{x,1,2}]; P13 = Plot[1/Sqrt[2],{x,1,2},PlotStyle->{RGBColor[1,0,0]}]; P14 = ParametricPlot[{1,y},{y,0,f[1]}]; P15 = ParametricPlot[{2,y},{y,0,f[2]}]; P16 = Show[P12,P13,P14,P15,PlotLabel->"P421 P44b: f[x] = Sqrt[3x-x^2]", PlotRange->All, AspectRatio->Automatic]; Display["p44c.ps",P16]; ---------------------------------------------------------------------------- Page 421 Problem 46 The region under the curve 1 y = ------------------ 2 x + 5 x + 6 from x = 0 to x = 3 is rotated about the x-axis. Compute the volume of the solid that is generated. x=3 2 x=3 1 INT Pi y dx = Pi INT ------------- dx x=0 x=0 2 2 (x+2) (x+3) 1 A B C D ------------ = ---------- + -------- + -------- + ----- 2 2 2 2 (x+2) (x+3) x+2 (x+2) (x+3) (x+3) 2 2 2 2 1 = A(x+2) (x+3) + B(x+3) + C(x+2) (x+3) + D(x+2) 2 2 2 2 1 = A(x+2)(x +6x+9) + B(x +6x+9) + C(x +4x+4)(x+3) + D(x +4x+4) 3 2 2 3 2 2 1 = A( x +6x +9x )+B(x +6x+9)+C(x +4x +4x ) +D(x +4x+4) 2 2 2x +12x+18) 3x +12x+12) 3 2 2 3 2 2 1 = A( x +8x +21x+18)+B(x +6x+9)+C(x +7x +16x+12)+D(x +4x+4) 3 x 0 = A + C 2 x 0 = 8 A + B + 7 C + D x 0 = 21 A + 6 B + 16 C + 4 D 1 1 = 18 A + 9 B + 12 C + 4 D A B C D 1 0 1 0 0 8 1 7 1 0 21 6 16 4 0 18 9 12 4 1 1 0 1 0 0 0 1 -1 1 0 0 6 -5 4 0 0 9 -6 4 1 1 0 1 0 0 0 1 -1 1 0 0 0 1 -2 0 0 0 3 -5 1 1 0 0 2 0 0 1 0 -1 0 0 0 1 -2 0 0 0 0 1 1 1 0 0 0 -2 0 1 0 0 1 0 0 1 0 2 0 0 0 1 1 1 -2 1 2 1 ------------ = ---------- + -------- + -------- + ----- 2 2 2 2 (x+2) (x+3) x+2 (x+2) (x+3) (x+3) | | x=3 x=3 dx | -1 -1 | INT -------------- = |-2 Log[x+2] + ----- + 2 Log[x+3] + ------ | x=0 2 2 | (x+2) (x+3) | (x+2) (x+3) | | x=0 = -2 Log[5]-1/5+2 Log[6]-1/6 -(-2 Log[2]-1/2 + 2 Log[3]-1/3) = -2 Log[5]+2 Log[6]+2 Log[2]-2 Log[3] -1/5 -1/6 + 1/2 + 1/3 36 4 -6 -5 +15 +10 = Log[ --------] + ---------- 25 9 30 16 14 = Log[ ---- ] + ---- 25 30 = 0.0203796 ------------------------------------------------------ Integrate[ Pi /(x^2 + 5 x + 6)^2,{x,0,3}] = 0.0640243 Get["font.math"]; f[x_] := 1/(x^2 + 5 x + 6); start = -3 Pi/4; stop = 3 Pi/4; P1 = ParametricPlot3D[ { x, f[x] Cos[t], f[x] Sin[t]}, {t,start, stop},{x,0,3}]; P2 = ParametricPlot3D[ { x,f[x] Cos[start],f[x] Sin[start],{RGBColor[1,0,0],Thickness[0.002]}}, {x,0,3}]; P3 = ParametricPlot3D[ { x,f[x] Cos[stop ],f[x] Sin[stop ],{RGBColor[1,0,0],Thickness[0.002]}}, {x,0,3}]; P4 = ParametricPlot3D[ { 0,f[0] Cos[t],f[0] Sin[t],{RGBColor[1,0,0],Thickness[0.002]}}, {t, start, stop}]; P5 = ParametricPlot3D[ { 3,f[3] Cos[t],f[3] Sin[t],{RGBColor[1,0,0],Thickness[0.002]}}, {t, start, stop}]; P6 = Show[P1,P2,P3,P4,P5,PlotLabel->"P421 P46 y=1/(x^2+5x+6)", AspectRatio->Automatic]; Display["p46.ps",P6]; ------------------------------------------------------- Page 421 Problem 48 Find the volume of the solid created by revolving the region bounded by the x-axis and the curve y = 4 x Sqrt[2-x] about the y-axis. x=2 x=2 2 INT 2 Pi x y dx = 2 Pi INT 4 x Sqrt[2-x] dx x=0 x=0 x=2 2 = 8 Pi INT x Sqrt[2-x] dx x=0 u = Sqrt[2-x] 2 u = 2-x 2 x = 2 - u dx = -2 u du x=2 2 2 x=2 2 4 2 8 Pi INT (2-u ) u (-2 u du) = -16 Pi INT (4 - 4 u + u ) u du x=0 x=0 x=2 2 4 6 -16 Pi INT 4 u - 4 u + u du x=0 u = 0 _ _ x=2 | 3 5 7 | | 4 u u u | -16 Pi | ----- - 4 ---- + ---- | |_ 3 5 7 _| x=0 u=Sqrt[2] +16 Pi ( (4/3 2 Sqrt[2] -4/5 4 Sqrt[2] + 8/7 Sqrt[2] ) 16 Pi ( 8/3 - 16/5 + 8/7) Sqrt[2] 16 Pi ( 8 35 - 16 21 + 8 15)/105 Sqrt[2] 16 Pi ( 280 - 336 + 120)/ 105 Sqrt[2] 16 Pi 64 /105 Sqrt[2] 1024 Pi /105 Sqrt[2] = 43.3287 <==================== -------------------------------------------------------------- Get["font.math"]; f[x_] := 4 x Sqrt[2-x]; Integrate[ 2 Pi x f[x],{x,0,2}]; start = 0; stop = 9 Pi/6; P1 = ParametricPlot3D[ {x Cos[t], x Sin[t], f[x]},{x,0,2},{t,start,stop}]; P2 = ParametricPlot3D[ {2 Cos[t], 2 Sin[t],-0.01,{RGBColor[1,0,0],Thickness[0.002]}}, {t,start,stop}]; t = 0; P3 = ParametricPlot3D[ {x Cos[t], x Sin[t], f[x], {RGBColor[1,0,0],Thickness[0.002]} },{x,0,2}]; t = 9 Pi/6; P4 = ParametricPlot3D[ {x Cos[t], x Sin[t], f[x], {RGBColor[1,0,0],Thickness[0.002]} },{x,0,2}]; P5 = Show[P1,P2,P3,P4,PlotLabel->"P421 P48: y = 4 x Sqrt[2-x] about y axis", PlotRange->All,AspectRatio->Automatic]; Display["p48.ps",P5]; ------------------------------------------------------------------------- Page 421 Problem 50 Find the area of the region bounded by the x-axis, 18 the curve y = ------------ 2 2 x Sqrt[x + 9] and the lines x = Sqrt[3] and x = 3 Sqrt[3]; ------------------------------------------------ Get["font.math"]; f[x_] := 18/ (x^2 Sqrt[x^2+9]); P1 = Plot[ f[x],{x,Sqrt[3], 3 Sqrt[3] }, PlotStyle->{Thickness[0.01]} ]; P2 = ParametricPlot[{Sqrt[3],y},{y,0,f[Sqrt[3]]}, PlotStyle->{Thickness[0.01]}]; P3 = ParametricPlot[{ 3 Sqrt[3],y},{y,0,f[ 3 Sqrt[3]] }, PlotStyle->{Thickness[0.01]}]; P4 = ParametricPlot[{0,y},{y,0,2}, PlotStyle->{Thickness[0.001]}]; P5 = Show[P1,P2,P3,P4,PlotLabel->"P421 P50: y = 18/(x^2 Sqrt[x^2+9])", PlotRange->All, AspectRatio->Automatic]; Display["p50.ps",P5]; --------------------------------------------------------------------- x = 3 Sqrt[3] INT y dx x = Sqrt[3] x =3 Sqrt[3] 18 INT ------------- dx x = Sqrt[3] 2 2 x Sqrt[x + 9] /| / | 2 / | Sqrt[x + 9] / | / | / | / | 3 / | / | / | / t | /___________| x x = 3 Ctn[t] 2 Sqrt[x + 9] = 3 Csc[t] 2 dx = -3 Csc [t] x = 3 Sqrt[3] 18 INT ------------- dx x = Sqrt[3] 2 2 x Sqrt[x + 9] x = 3 Sqrt[3] 18 2 INT ----------------- (-3 Csc [t] dt ) x = Sqrt[3] 2 9 Ctn [t] 3 Csc[t] x = 3 Sqrt[3] 2 INT -2 Tan [t] Csc[t] dt x = Sqrt[3] 2 x = 3 Sqrt[3] Sin [t] -2 INT ---------------- dt x = Sqrt[3] 2 Cos [t] Sin[t] x = 3 Sqrt[3] Sin[t] -2 INT ---------------- dt x = Sqrt[3] 2 Cos [t] _ _ | -1 | x = 3 Sqrt[3] | Cos [t] | -2 | ---------- | |_ (-1)(-1) _| x = Sqrt[3] _ _ | | x = 3 Sqrt[3] -2 | Sec[t] | |_ _| x = Sqrt[3] _ 2 _ | Sqrt[x + 9] | x = 3 Sqrt[3] -2 | ------------ | |_ x _| x = Sqrt[3] Sqrt[36] Sqrt[12] -2 ( -------- - --------- ) 3 Sqrt[3] Sqrt[3] 6 2 Sqrt[3] -2 ( -------- - ---------- ) 3 Sqrt[3] Sqrt[3] 2 -2 ( -------- - 2 ) Sqrt[3] -2( (2/3) Sqrt[3] - 2] ) -(4/3) Sqrt[3] + 4 = 1.6906 --------------------------------------- f[x_] := 18/( x^2 Sqrt[x^2+9]); Integrate[f[x],{x,Sqrt[3],3 Sqrt[3]}] --------------------------------------- / / New Material: L Hopital s rule: / f(x) f (x) Lim ------ = Lim ----- x->a g(x) x->a / g (x) But be careful. There are certain restrictions. (1) f(a)/g(a) has to be 0/0 or infinity/infinity (2) The LHS may exist but not the RHS. Page 423 Example 1 Sin[x] Lim ------- x->0 x -------------------------------------------------------------------- Get["font.math"]; f[x_] := Sin[x]/x; P1 = Plot[f[x],{x,-Pi,Pi}]; P2 = ListPlot[{{0,1}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P3 = Show[P1,P2,PlotLabel->"P423 Ex 1. Sin[x]/x", AspectRatio->Automatic,PlotRange->All]; Display["ex1x.ps",P3]; -------------------------------------------------------------------- 1-Cos[x] Lim ----------- x->0 x -------------------------------------------------------------------- Get["font.math"]; f[x_] := (1-Cos[x])/x; P1 = Plot[f[x],{x,-0.01,+0.01},PlotPoints->1000]; P2 = ListPlot[{{0,0}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P3= Show[P1,P2,PlotLabel->"P423 Ex 1. (1-Cos[x])/x",AspectRatio->Automatic]; Display["ex1y.ps",P3]; p[x_] := ParametricPlot[ (1-t){Cos[x],0}+t{Cos[x],Sin[x]},{t,0,1}]; H = Table[p[x],{x,0,0.1,0.01}]; P4 = ParametricPlot[{Cos[t],Sin[t]},{t,0,0.1},PlotStyle->{RGBColor[1,0,0]}]; P5 = Show[H,P4,AspectRatio->Automatic]; -------------------------------------------------------------------- 2 Page 424 x - 9 Example 2 Lim ---------- x->3 2 x - x - 6 -------------------------------------------------------------------- Get["font.math"]; f[x_] := (x^2-9)/(x^2-x-6); P1 = Plot[f[x],{x,1,4}]; P2 = ListPlot[{{3,6/5}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P3 = ParametricPlot[{3,y},{y,0,6/5},PlotStyle->{RGBColor[1,0,0]}]; P4 = Plot[6/5,{x,1,3},PlotStyle->{RGBColor[1,0,0]}]; P5 = Plot[0,{x,3,6/5}]; P6 = Show[P1,P2,P3,P4,P5,PlotLabel->"P424 Ex 2. (x^2-9)/(x^2-x-6)"]; Display["ex2x.ps",P6]; -------------------------------------------------------------------- 2 x + 3x - 10 Lim ------------ + 2 x->2 x - 4 x + 4 -------------------------------------------------------------------- Get["font.math"]; f[x_] := (x^2+3x-10)/(x^2-4x+4); P1 = Plot[f[x],{x,2.02,3}]; P2 = Plot[0,{x,1,3}]; P3 = ParametricPlot[{2,y},{y,0,f[2.02]},PlotStyle->{RGBColor[1,0,0]}]; P4 = Show[P1,P2,P3,PlotLabel->"P424 Ex 2. (x^2+3x-10)/(x^2-4x+4)"]; Display["ex2y.ps",P4]; -------------------------------------------------------------------- Page 424 Tan[2x] Example 3 Lim -------- x->0 ln[1+x] -------------------------------------------------------------------- Get["font.math"]; f[x_] := Tan[2x]/Log[1+x]; P1 = Plot[f[x],{x,-1/2,1/2}]; P2 = Plot[0,{x,-1/2,1/2}]; P3 = ListPlot[{{0,2}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P4 = Show[P1,P2,P3,PlotLabel->"P424 Ex 3: Tan[2x]/ln[1+x]"]; Display["ex3.ps",P4]; -------------------------------------------------------------------- Page 424 Sin[x]-x Example 4 Lim ---------- x->0 3 x -------------------------------------------------------------------- Get["font.math"]; f[x_] := (Sin[x]-x)/(x^3); P1 = Plot[f[x],{x,-2Pi,2Pi}]; P2 = Plot[0,{x,-Pi,Pi}]; P3 = ListPlot[{{0,-1/6}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P4 = Show[P1,P2,P3,PlotLabel->"P424 Ex 4: (Sin[x]-x)/x^3 ", PlotRange->All]; Display["ex4.ps",P4]; -------------------------------------------------------------------- Page 425 1 - Cos[x] Example 5 Lim --------------- x->0 2 x + 3 x -------------------------------------------------------------------- Get["font.math"]; f[x_] := (1-Cos[x])/(x^2+3x); P1 = Plot[f[x],{x,-2,2}]; P2 = Plot[0,{x,-2,2}]; P3 = ListPlot[{{0,0}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P4 = Show[P1,P2,P3,PlotLabel->"P425 Ex 5: (1-Cos[x])/(x^2+3x) ", PlotRange->All,AspectRatio->Automatic]; Display["ex5.ps",P4]; -------------------------------------------------------------------- -x Page 425 e Example 6 Lim --------- x->Infinity -1 x -------------------------------------------------------------------- Get["font.math"]; f[x_] := E^(-x)/(1/x); g[x_] := E^(-x)/(1/x^2); h[x_] := E^(-x)/(2/x^3); i[x_] := E^(-x)/(6/x^4); P1 = Plot[f[x],{x,0.01,10}]; P2 = Plot[g[x],{x,0.01,10},PlotStyle->{RGBColor[1,0,0]}]; P3 = Plot[h[x],{x,0.01,10},PlotStyle->{RGBColor[0,1,0]}]; P4 = Plot[i[x],{x,0.01,10},PlotStyle->{RGBColor[0,0,1]}]; P5 = Plot[0,{x,0.01,10}]; p6 = Show[P1,P2,P3,P4,P5, PlotLabel->"P425 Ex 6: e^(-x)/(1/x),f'/g',f''/g'',f''',g''' ", PlotRange->All]; Display["ex6.ps",P6]; -------------------------------------------------------------------- Page 427 work problems 1-10. f[x]-f[a] / ----------- f (a) Theory: f[x] f[x] - f[a] x-a ----- = ----------- = ---------- ---> --------- g[x] g[x] - g[a] g[x]-g[a] / ----------- g (a) x-a By graphing: | // . | /f (a) x . | / . / | / . g (a) x |/. -----------+--------------------- | | f(b)-f(a) By algebra: s(x) = f(x)-f(a) - ------------ (g(x)-g(a)) g(b)-g(a) s(a) = s(b) = 0 / / f(b)-f(a) / s (x) = f (x) - ---------- g (x) g(b)-g(a) / So when s (c) = 0, we get / f (c) f(b)-f(a) --------- = -------------- / g (c) g(b)-g(a) Then when f(a) = g(a) = 0 / f(b) f (c) --------- = ------- g(b) / g (c) / f (c) f(b) And if ------ has a limit, then so does -------. / g(b) g (c)