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Mathematica I (2001 5 31, 6 7 ) UNIX EDS vncviewer Internet Exploler http://www.efc.sec.eng.shizuoka.ac.jp/admin/pubsoft/ vncviewer.exe : 1

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I Mathematica 1 Mathematica 1.1 vncviewer UNIX EDS kterm mathematica Enter Mathematica 1.2 File Save Untitled-1.nb part1.nb OK.nb 1.3 File Save As 1.2 OK 4

1.2 1.4 File New 1.5 File Open 1.6 Mathematica File Quit 2 Mathematica part1.nb 1+1 SHIFT Enter 3 + 5 5

3 Mathematica 3.1 3 4 3 4 3 4 5 8 2 3 27 N [ ] 5 N 8 N [ 2 ] N[π] 3.2 Solve [x 2 +ax+b==0, x] 6

%11 /. a > 1 %11 Out[11] Out[11]:= a =1 a b %11 /. {a > 1, b > 2} 3.3 Factor [x 3 19x + 30] Factor [x 100 1] Expand [(a + a 2 x+x 2 ) 3 ] 7

3.4 Plot [x 2 +x 2, {x, 5, 4}] x x 2 + x 2 5 x 4 y = x fig1 = % fig2 = Plot [x, {x, 5, 4}] Show [fig1, fig2] fig1 = % fig1 % % y = e x y = log x y = x Plot[{2 x, Log[2, x], x}, {x, 0., 5}, PlotStyle > {Dashing[{0, 0}], Dashing[{0, 0}], Dashing[{0.01, 0.01}]} PlotRange > {{0, 5}, {0, 5}}, AxesLabel > { x, y }] 3.5 << Graphics`ImplicitPlot` ImplicitPlot [x 2 +y 2 == 1, {x, 1, 1}, {y, 1, 1}] ImplicitPlot [x 2 +y 2 == 1, {x, 1, 1}, {y, 1, 1}, AxesOrigin > {0, 0}, AxesLabel > { x, y }] 8

3.6 Limit[Sin[x]/x, x > 0] n k=1 k=1 k 2 ( ) k 1 2 3.7 f[x ]:=x 3 3x d1 = D[f[x], x] d2 = D[f[x], {x, 2}] << Graphics`Legend` Plot[{f[x], d1, d2}, {x, 3, 3}, PlotStyle > {Dashing[{0, 0}], Dashing[{0.01, 0.01}], Dashing[{0.02, 0.02}]}, PlotLegend > { f(x), f (x), f (x) }, LegendPosition > {0.3, 1.2}, AxesLabel > { x, y }] x f(x) f[x ] := a x(1 x) f[x ] x 3.8 Kernel Quit Kernel Local Yes f[x ] := a x(1 x) f[0.5] g[x ]:=x Solve [f[x] == g[x], x] a = 2; Plot [{f[x], g[x]}, {x, 0, 1}] f[1 1 ] a 9

1. C : y = ax(1 x) l : y = x 0 <a 4 (1) 0 x 1 C l a (2) 0 x 1 C l 2 a (3) 1 <x 1 2 C l X a (4) (3) X (5) (4) 1 a 10

II 4 1( ). a x 0 ax 0 a 2 x 0 n a n 1 x 0 Mathematica chu[a, x0 ] := Table [a n 1 x0, {n, 10}] chu[a, x0] chu[2, 1] ListPlot[chu[2, 1]] ListPlot[chu[2, 1], PlotJoined > True] n a n 1 x 0 n x n (1) x n+1 = ax n 5 (2) (1) n +1 n x n+1 = f(x n ), n =0, 1, 2, f(x) ex. (1) f(x) =ax f(x) x 0 (2) n =0, 1, 2, x 0,x 1,x 2, f[x ]:=ax NestList[f, 1, 5] Nest[f, 1, 20] a=0.5; ListPlot[NestList[f, 1, 20], PlotStyle > PointSize[0.02], PlotJoined > True] 11

2. 6 2( ). 18 1 19 ( dn dt = r 1 N ) (3) N. K N t K r (3) N t tmp = NDSolve[{x [t] == 4(1 x[t])x[t], x[0] == 0.1}, x[t], {t, 0, 10}]; Plot[Evaluate[x[t]/.tmp], {t, 0, 5}, PlotRange > {{0, 5}, {0, 2}}] Enter 12

3( ). x y dx = r(1 by)x (4) dt dy =( c + dx)y. dt r = 4; b = 0.5; c = 1; d = 2; tmp = NDSolve[{x [t] == r(1 b y[t])x[t], y [t] == ( c + d x[t])y[t], x[0] == 1, y[0] == 1}, {x[t], y[t]}, {t, 0, 30}]; ParametricPlot[Evaluate[{x[t], y[t]}/.tmp], {t, 0, 30}, AxesOrigin > {0, 0}] Plot[Evaluate[x[t]/.tmp[[1]]], {t, 0, 30}] Plot[Evaluate[y[t]/.tmp[[1]]], {t, 0, 30}] Enter Enter 3.5 3 2.5 2 1.5 1 0.25 0.5 0.75 1 1.25 1.5 1.5 1.25 1 0.75 0.5 0.25 3.5 3 2.5 2 1.5 5 10 15 20 25 30 5 10 15 20 25 30 13

7 2 f(x) x 0 f(x) =αx(1 x), 0 x 0 1 0 <a 4 (5) x n+1 = αx n (1 x n ), 0 <x 0 < 1. (3) [4] 3. x 0 =0.9 a 1 4. α 3 4 toi3[α, xini, n Integer] := ( Enter f[x ]:=α x(1 x); Enter data = Table[{i, Nest[f, xini, i]}, {i, 0, n}]; Enter ten = ListPlot[data, PlotStyle > {Hue[1], PointSize[0.03], } DisplayFunction > Identity]; Enter sen = Graphics[{Hue[0.3], Thickness[0.01], Line[data]}]; Enter Show[ten, sen, PlotRange > {{0, n}, {0, 1}}, AxesLabel > { n, x n }, DisplayFunction > $DisplayFunction]; ) α =1 x 0 =0.5 n =10 toi3[1, 0.5, 10] 14

III 8 f(x) (2) y = f(x) f(x) = 1 2 x Step 1 : y = f(x) x x 0 A 0 = A 0 (x 0,f(x 0 )) = (x 0,x 1 ) Step 1 : y = x y x 1 B 1 = B 1 (x 1,x 1 ) Step 2 : y = f(x) x x 1 A 1 = A 1 (x 1,f(x 1 )) = (x 1,x 2 ) Step 2 : y = x y x 2 B 2 = B 2 (x 2,x 2 ) y y = x B 0 y = f(x) B x 1 = f(x 0 ) 1 A 0 B f(x 1 ) 2 A 1 A 2 x 0 x 2 x 1 x 0 1: f(x) = 1 2 x Step y = x B 1, B 2, B 3, (2) x 0,x 1,x 2, (x 0,x 0 ) B 0 y = x B 0, B 1, B 2, 15

9 {B n } (n =0, 1, 2, ) Mathematica B 0, B 1, B 2, 5. f(x) =mx x 0 =12 m 1 2, 1, 2 3, 3 2, 2 B 0, B 1, B 2, y 12 10 8 6 4 2 2 4 6 8 10 12 2. f(x) = 1 2 x B n n!! x y 350 300 250 200 150 100 50 50 100 150 200 250 300 350 3. f(x) =2x B n n!! x 16

10 f(x) =2x? f(x) =2x y = x X f(x) B n y y =2x y = x y = f(x) X 0 x 4: y =2x f(x) ( 1, 1) r 2 y = x r<1 6. r {B n }? x 0 = 0.11 r = 1, 0, 1, 1, 3, 2 2 2 2 r =0!! r = 1!! 7. r = 2 ( 5 ) x 0 {B n }? x 0 0.1 0.11 0.101 r 1 {B n } = ( ) 17

1 y y = x X =( 2, 2) 3 3 y = f(x) 0 1 2 1 x 5: f(x) =1 2x 1 11 7 (5) x n+1 = ax n (1 x n ), 0 <x 0 < 1 a 0 4 {B n }? Mathematica 1 (I) (IV) (I) 0 <a 1 5 m<1 (II) 1 <a 2 5 m>1 6 0 r<1 (III) 2 <a 3 5 m>1 6 1 <r<0 (IV) 3 <a 4 5 m>1 6 r 1? 8. (I) (IV) a {B n } (IV)!! a a 4 a 3.57 9. a a 4 {B n } x 0 0.1 0.101 18

12 {x n } n 10. 3 4 0 <a<a n {x n }. n {x n } (1 ) 4. (i) 0 <a 1 = 0 (ii) 1 <a 2 = 1+a a (iii) 2 <a 3 = 1+a a (iv) 3 <a 1+ 6 = (a+1) (a+1)(a 3), 2a (v) a =4 = 0 1 (a+1)+ (a+1)(a 3) 2a ( ) (i) (iii) (iv) x 0 = 1+a (v) a 0 1 f(x) =ax(1 x) a {x n } x n 1 0.8 0.6 0.4 0.2 0.5 1 1.5 2 2.5 3 3.5 4 a 6. f(x) =ax(1 x) 19

13 (2D) { xn+1 = y n sin x n bxn r y n+1 = x n + a 7. (2D) (a =3,b=0.3, r=0.3, x 0 =0.1, y 0 =0) [1], - vs -, 1996. [2],, 1996. [3],, 1995. [4], - -,, 1986. [5], - -,, 1991. 20

[6],,, 1993. [7],, 1994. [8], Mathematica, 1997. [9], Mathematica [ ], 1995. [10], Mathematica [ ], 1995. [11], Mathematica, 2000. [12], Mathematica 3.0, 1998. [13], Mathematica, 1994. Mathematica $DisplayFunction, 14 %, 7, 8 /., 7,12, 13 :=, 9, 9, 11, 14 << Graphics, 8, 9 AxesLabel, 8, 9, 14 AxesOrigin, 8, 13 D, 9 Dashing, 8, 9 DisplayFunction, 14 Evaluate, 12, 13 Expand, 7 Factor, 7 Graphics, 14 Hue, 14 Identity, 14 ImplicitPlot, 8 Legend, 9 LegendPosition, 9 Limit, 9 Line, 14 ListPlot, 11, 11, 14 N, 6 NDSolve, 12, 13 Nest, 11, 14 NestList, 11 ParametricPlot, 13 Plot, 8, 9, 12, 13 PlotJoined, 11, 11 PlotLegend, 9 PlotRange, 8, 12, 14 PlotStyle, 8, 9, 11, 14 PointSize, 11, 14 Show, 8, 14 Solve, 6, 9 Table, 11, 14 Thickness, 14 37