Constante matematice:

http://www.youtube.com/playlist?list=PL6EFB5A3AFA1D8398&feature=plcp

Se deschide Wolfram Mathematica 7 for students. Din meniul principal se acceseaz opiunea Palettes ( Basic Math Assistant. Pentru grafice se selecteaz opiunea Basic Commands ( 2D sau 3DFuncia Solve[expresie, variabile, domeniu]Ex:Solve[x^2 + a x + 1 == 0, x]

Shift+Enter

Aflarea primei soluii:Solve[x^2 + a x + 1 == 0, x][[1]]Shift+Enter

Obs: n Mathematica, funcia exponenial e(x) se scrie Exp[x], iar constanta e = 2,71828183 se scrie E. Aceast scriere este recunoscut i n www.wolframalpha.comEx:Solve[Log[x]==Exp[x]-5, x]

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In acest caz se apeleaz funcia FindRoot. Soluia va fi cutat n vecintatea unui punct:

FindRoot[Log[x]==Exp[x]-5, {x, 3}]

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Plot[{Log[x], Exp[x]-5}, {x, 0, 2.5}]

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Din grafic se vede c mai exist o soluie n vecintatea lui 0

FindRoot[Log[x]==Exp[x]-5, {x, 0.5}]

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Ex:Gsirea soluiei n vecintatea unui punct:

FindRoot[Sin[x] + Exp[x], {x, 0}]Shift+Enter

Plot[Sin[x]+Exp[x],{x, -1, 1}]Shift+Enter

Ex:Rezolvarea unui sistem de ecuaii neliniar:FindRoot[{Exp[x - 2] == y, y^2 == x}, {{x, 1}, {y, 1}}]Shift+Enter

ContourPlot[{Exp[x-2]==y, y^2==x}, {x, -2, 6}, {y, -1, 15}]Shift+Enter

Din grafic se observ c mai exist o soluie n vecintatea punctului (2, 2)FindRoot[{Exp[x - 2] == y, y^2 == x}, {{x, 2}, {y, 2}}]Shift+Enter

Ex:Solve[a x + y == 7 && b x - y == 1, {x, y}]

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Folosirea ultimului rezultat se face prin operatorul % astfel:Ex: fie ultimul rezultat ( Out[4]= Solve[% == 0, x]

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Out[5]= conversia soluiei ecuatiei =0 n format numeric: N[%]

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Obs: pentru afiarea soluiilor direct n format numeric se poate folosi funcia NSolve[expresie, variabile]Ex:NSolve[x^5 - 2 x + 3 == 0, x]

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radical: Sqrt[x] sau Ctrl+2 si apoi x scriere exponent: Ctrl+6 i apoi pentru coborrea cursorului: Ctrl+Space

Ex: Sqrt[{2, 3, 4, 5}]Shift+Enter

Ex:Sqrt[2]

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funcia Simplify

Ex:fie ultimul rezultat ( Out[6]= Simplify[%]

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Grafice de funciiPlot[f, {x, xmin, xmax}]

afiarea unei singure funcii pe grafic:

Plot[Sin[x], {x, 0, 6 Pi}]Shift+Enter

afiarea mai multor funcii pe acelai grafic: Plot[{f1,f2,...fn}, { x, xmin, xmax }]Plot[{Sin[x], Sin[2 x], Sin[3 x]}, {x, 0, 2 Pi}]Shift+Enter

colorarea ariei de sub grafic:

Ex:Plot[2 Sin[x] + x, {x, 0, 15}, Filling -> Bottom]Shift+Enter

ContourPlot[f, {x, xmin, xmax}, {y, ymin, ymax}]

Ex:ContourPlot[Cos[x] + Cos[y], {x, 0, 4 Pi}, {y, 0, 4 Pi}]Shift+Enter

ContourPlot3D[f, {x, xmin, xmax}, {y, ymin, ymax}, {z, zmin, zmax}]Ex:ContourPlot3D[x^3 + y^2 - z^2 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]Shift+Enter

Ex:ContourPlot[Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3}]Shift+Enter

DensityPlot[f, {x, xmin, xmax}, {y, ymin, ymax}]

Ex:DensityPlot[Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3}]

Ex:DensityPlot[Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3}, Mesh -> 10]

Ex:Plot3D[x*E^(-(x^2+y^2))+(x^2+y^2)/20,{x,-2,2},{y,-2,2}]

Shift+Enter

Funcia ManipulateEx:Manipulate[Plot3D[Sin[n x y],{x,0,3},{y,0,3}],{n,1,4}]Shift+Enter

Ex:Manipulate[Plot[Sin[n x],{x,0,2Pi}],{n,1,10}]Shift+Enter

Optimizare fr restricii= 26

x >=3, y >= 4folosind Minimize.

Minimize[{x + 2 y, -5 x + y == 7 && x + y >= 26 && x >= 3 && y >= 4}, {x, y}] Shift+Enter

Ex: - aceeai problem folosind Nminimize

NMinimize[{x + 2 y, -5 x + y == 7 && x + y >= 26 && x >= 3 && y >= 4}, {x, y}] Shift+Enter

Ex: - aceeai problem folosind LinearProgramming

LinearProgramming[{1, 2}, {{-5, 1}, {1, 1}}, {{7, 0}, {26, 1}}, {{3, Infinity}, {4, Infinity}}]Shift+Enter

Optimizare local numeric neliniar

Ex:

Minimize[{x - y, -3 x^2 + 2 x y - y^2 >= -1}, {x, y}]Shift+Enter

Ex: - aceeai problem rezolvat numeric

NMinimize[{x - y, -3 x^2 + 2 x y - y^2 >= -1}, {x, y}]Shift+Enter

Funcia FindMinimum gsete minimul local. n acest caz, minimul local gsit este de asemenea minim global.

Ex:FindMinimum[{x - y, -3 x^2 + 2 x y - y^2 >= -1}, {x, y}]Shift+Enter

Ex:

folosind FindMinimumContourPlot[-100/((x-1)^2+(y-1)^2+1)-200/((x+1)^2+(y+2)^2+1),{x,-3,2},{y,-3,2}] Shift+Enter

ContourPlot[-100/((x-1)^2+(y-1)^2+1)-200/((x+1)^2+(y+2)^2+1),{x,-3,2},{y,-3,2},RegionFunction->Function[{x,y,z},(x^2+y^2>3)]]Shift+Enter

(poriunea alb e dat de restricia: x^2+y^2>3)

FindMinimum[{-100/((x-1)^2+(y-1)^2+1)-200/((x+1)^2+(y+2)^2+1),x^2+y^2>3},{x,y}] Shift+Enter

Minim global cutarea minimului pornind de la x=2 i y implicit:

FindMinimum[{-100/((x-1)^2+(y-1)^2+1)-200/((x+1)^2+(y+2)^2+1),x^2+y^2>3},{{x,2},y}]Shift+Enter. n acest caz funcia FindMinimum a gsit un minim local. Acest rezultat satisface condiia (restricia) problemei.ContourPlot[- (100/((x - 1)^2 + (y - 1)^2 + 1)) - 200/((x + 1)^2 + (y + 2)^2 + 1), {x, -3, 2}, {y, -3, 2}, RegionFunction -> (#1^2 + #2^2 > 3 &), Contours -> 10, Epilog -> ({Red, PointSize[.02],

Text["global minimum", {-.995, -2.092}], Point[{-.995, -1.992}],

Text["local minimum", {0.5304, 1.2191}], Point[{1.2304, 1.2191}]}), ContourLabels -> True]Shift+Enter

- o vizualizare 3D:

Show[{Plot3D[- (100/((x - 1)^2 + (y - 1)^2 + 1)) - 200/((x + 1)^2 + (y + 2)^2 + 1), {x, -4, 3}, {y, -4, 3}, RegionFunction -> (#1^2 + #2^2 > 3 &), PlotRange -> All], Graphics3D[{Red, PointSize[.02], Text["Global minimum", {-.995, -2.092, -230}], Point[{-.995, -2.092, -207}], Text["Local minimum", {0.5304, 1.2191, -93.4}], Point[{1.23, 1.22, -103.}]}]}]Shift+Enter

Exemplu 1 Cu05-Ex.01.

reprezentarea grafic a funciei pe intervalele determinate de cele 3 restricii

ContourPlot[x^2+z^2+x*y+2x, {x, -1.25, 0},{y, 0, 1.25}]

Shift+Enter

ContourPlot[x^2+y^2+x*y+2x,{x,-1.25,0},{y,0,1.25},RegionFunction->Function[{x,y,z},x^2+y^2-1.5