(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 89925, 2553]*) (*NotebookOutlinePosition[ 90712, 2580]*) (* CellTagsIndexPosition[ 90668, 2576]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " kasutamine" }], "Title"], Cell["1. Sissejuhatus", "Subtitle"], Cell[TextData[{ StyleBox["Programmi ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" avanemisel pole programmi tuum - ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"], StyleBox["Kernel", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" - vaikimisi k\[ADoubleDot]ivitatud. ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"], StyleBox["Kernel", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" on programmi see osa, mis tegelikult teostab arvutusi. ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" abil v\[OTilde]ib mitte ainult arvutada, vaid ka kujundada \ dokumente. Peale selle, programmi tuum v\[OTilde]ib t\[ODoubleDot]\ \[ODoubleDot]tada hoopis teises arvutis, mis asub samas arvutiv\[OTilde]rgus. \ Seda k\[OTilde]ike on v\[OTilde]imalik konfigureerida, vaikimisi k\ \[ADoubleDot]ivitatakse tuum siiski samas arvutis esimese arvutuse k\ \[ADoubleDot]ivitamisel.", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"], StyleBox["\n", CharacterEncoding->"WindowsANSI"], StyleBox["Arvutuse teostamiseks tuleb p\[ADoubleDot]rast arvutuse \ sisestamist vajutada ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"], StyleBox["Shift+Enter", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[". N\[ADoubleDot]iteks:", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"], StyleBox["\n", CharacterEncoding->"WindowsANSI"], StyleBox["2+4", FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox[" Shift+Enter.", FontFamily->"Courier New", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox["\n", CharacterEncoding->"WindowsANSI"], StyleBox["Ekraanile ilmub:", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"] }], "Text", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ \(2 + 4\)], "Input"], Cell[BoxData[ \(6\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["2. Aritmeetika, algebra.", "Subtitle"], Cell["2.1. Aritmeetika. T\[ADoubleDot]psed ja ligikaudsed tehted.", \ "Subsubtitle"], Cell["Tavalistes arvutustes kasutatavad tehted:", "Text"], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"\t\t ", StyleBox[" ", FontFamily->"Courier New"], StyleBox[" ", FontFamily->"Courier New", FontSlant->"Italic"]}]], RowBox[{ RowBox[{ StyleBox["x", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["+", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["y", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["+", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], RowBox[{ StyleBox["z", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["\t\t ", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox["liitmine", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"]}]}], StyleBox["\n", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[ RowBox[{ StyleBox["\t", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], "\t"}]], RowBox[{ RowBox[{ StyleBox[\(-x\), FormatType->StandardForm, FontFamily->"Courier New", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[ RowBox[{ StyleBox["\t ", FormatType->StandardForm, FontFamily->"Courier New", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"]}]], StyleBox["lahutamine", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"]}], StyleBox[",", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox["miinus", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[",", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[\(neg . \ arv\), FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"]}], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox["\n", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[ RowBox[{ StyleBox["\t", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], "\t"}]], RowBox[{ RowBox[{ StyleBox["x", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["y", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["z", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["v\[OTilde]i", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["x", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["*", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["y", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["*", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["z", FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox["\t ", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox["korrutamine", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"]}], StyleBox[",", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[\(korrutusm\[ADoubleDot]rk\ v\[OTilde]i\ t\[UDoubleDot]hik\ \ selle\ asemel\), FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"]}], StyleBox["\n", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[ RowBox[{ StyleBox["\t", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], "\t"}]], RowBox[{ StyleBox[\(x/y\), FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox[ RowBox[{ StyleBox[" ", FormatType->StandardForm, FontFamily->"Courier New", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"]}]], StyleBox["jagamine", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"]}], StyleBox["\n", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[ RowBox[{ StyleBox["\t", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], "\t"}]], StyleBox[\(x^y\), FormatType->StandardForm, FontFamily->"Courier New", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Courier New", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox["ehk", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], \(x\^y\), StyleBox[ RowBox[{ StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"]}]], StyleBox["astendamine", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"]}]}]], "Text", Background->GrayLevel[0.900008]], Cell["\<\ ASCII koodis on astendamism\[ADoubleDot]rk 94, selle saab k\[ADoubleDot]tte \ kui vajutada Alt+94 (94 klaviatuuri paremalt poolelt, toimib ka Alt+\ \[CapitalADoubleDot]). N\[ADoubleDot]ited:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(2.3 + 5.6*2\)], "Input"], Cell[BoxData[ \(13.5`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(2\ 4\ 6\)], "Input"], Cell[BoxData[ \(48\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(2.4\^32\)], "Input"], Cell[BoxData[ \(1.4681138466456628`*^12\)], "Output"] }, Open ]], Cell["\<\ Kuigi Mathematica v\[OTilde]ib leida v\[ADoubleDot]ga pikki arvutustulemusi t\ \[ADoubleDot]pselt, on tavaliselt soovitav need esitada siiski n\[ODoubleDot] \ \[UDoubleDot]mmardatud kujul, k\[UDoubleDot]mne astmete kaudu. \ K\[UDoubleDot]mne astmete kaudu (ligikaudsel kujul) saab arve esitada \ kasutades funktsiooni N[ ]:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(3\^32\)], "Input"], Cell[BoxData[ \(1853020188851841\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(3\^32\ // N\)], "Input"], Cell[BoxData[ \(1.853020188851841`*^15\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[3\^32]\)], "Input"], Cell[BoxData[ \(1.853020188851841`*^15\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[2\^40]\)], "Input"], Cell[BoxData[ \(1.099511627776`*^12\)], "Output"] }, Open ]], Cell["M\[OTilde]ned matemaatilised funktsioonid.", "Text", FontWeight->"Bold", FontSlant->"Italic"], Cell[TextData[{ StyleBox["Sqrt[x] ", FontFamily->"Courier"], StyleBox["ehk", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\@x\)]], "\t\t\t\truutjuur x-st\n", StyleBox["Sqrt[x,n]", FontFamily->"Courier New"], " ", StyleBox["ehk ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\@x\%n\)]], "\t\t\tn-s juur x-st\n", StyleBox["Log[x]", FontFamily->"Courier New"], "\t\t\t\t\tnaturaallogaritm - ln x\n", StyleBox["Log[a,x]", FontFamily->"Courier New"], "\t\t\t\t\tlogaritm alusel a - ", Cell[BoxData[ \(TraditionalForm\`\(log\_a\) x\)]], "\nSin[x], Cos[x], Tan[x], Cot[x]\t\t\ttrigonomeetrilised funktsioonid, \ argument on radiaanides!\nArcSin[x], ArcCos[x], ArcTan[x], ArcCot[x]\t\ trigonomeetrilised p\[ODoubleDot]\[ODoubleDot]rdfunktsioonid \ (arkusfunktsioonid),\n \t\t\t\t\t\tannavad vastuse radiaanides\nn!\t\t\t\t\t\t\ faktoriaal\nAbs[x]\t\t\t\t\t\tabsoluutv\[ADoubleDot]\[ADoubleDot]rtus\n\ Round[x]\t\t\t\t\t\[UDoubleDot]mmardamine\nMod[n,m]\t\t\t\t\tj\[ADoubleDot]\ \[ADoubleDot]k, mis saadakse jagamisel ", Cell[BoxData[ \(TraditionalForm\`n\/m\)]], "\nQuotient[n,m]\t\t\t\t\tjagatise ", Cell[BoxData[ \(TraditionalForm\`n\/m\)]], " t\[ADoubleDot]isosa\nRandom[]\t\t\t\t\tpseudojuhuslik arv 0 ja 1 vahel\n\ Max[x,y,...], Min[x,y,...]\t\t\tarvude (avaldiste) x,y,... maksimum, miinimum\ \nFactorInteger[n]\t\t\t\tt\[ADoubleDot]isarvu n lahutamine algarvude \ korrutiseks\n", StyleBox["avaldis // ", FontSlant->"Italic"], "N ", StyleBox["ehk", FontSlant->"Italic"], " N[", StyleBox["avaldis", FontSlant->"Italic"], "]\t\t", StyleBox["avaldise", FontSlant->"Italic"], " ligikaudne v\[ADoubleDot]\[ADoubleDot]rtus; esitaminek\[UDoubleDot]mne \ astmete kaudu\nN[", StyleBox["avaldis,", FontSlant->"Italic"], "n]\t\t\t\t\t", StyleBox["avaldise", FontSlant->"Italic"], " v\[ADoubleDot]\[ADoubleDot]rtus n numbrikoha (mitte komakoha!) t\ \[ADoubleDot]psusega" }], "Text", Background->GrayLevel[0.900008]], Cell[TextData[{ StyleBox["NB! ", FontSlant->"Italic"], StyleBox["Mathematica's on k\[OTilde]igi funktsioonide argumendid antud \ alati kandilistes sulgudes. Funktsioonide ja protseduuride nimed algavad \ alati suure t\[ADoubleDot]hega. Seet\[OTilde]ttu on m\[OTilde]istlik enda \ defineeritud suurused ja funktsioonid anda v\[ADoubleDot]ikeste algust\ \[ADoubleDot]htedega.", FontWeight->"Plain", FontSlant->"Italic"] }], "Text", FontWeight->"Bold"], Cell[TextData[StyleBox["Konstandid.", FontWeight->"Bold"]], "Text", FontWeight->"Plain", FontSlant->"Italic"], Cell[TextData[{ StyleBox["Pi ", FontWeight->"Bold"], StyleBox["v\[OTilde]i", FontSlant->"Italic"], " \[Pi]\t\t\t\t\t\[Pi] \[TildeTilde] 3,14159\nE ", StyleBox["v\[OTilde]i", FontSlant->"Italic"], " \[ExponentialE]\t\t\t\t\te \[TildeTilde] 2,71828\nDegree ", StyleBox["v\[OTilde]i ", FontSlant->"Italic"], "\[Degree]\t\t\t\tnurgakraadi t\[ADoubleDot]his\nI ", StyleBox["v\[OTilde]i", FontSlant->"Italic"], " \[ImaginaryI]\t\t\t\t\timmaginaar\[UDoubleDot]hik, ", Cell[BoxData[ \(TraditionalForm\`\[ImaginaryI] = \@\(-1\)\)]], "\nInfinity ", StyleBox["v\[OTilde]i \[Infinity]\t\t\t\t ", FontSlant->"Italic"], "l\[OTilde]pmatus" }], "Text", Background->GrayLevel[0.900008]], Cell[CellGroupData[{ Cell["\<\ 2.3. Muutujatele, avaldistele v\[ADoubleDot]\[ADoubleDot]rtuse omistamine, tehted eelnevate arvutustulemustega\ \>", "Subsubtitle"], Cell[TextData[{ "x = v\[ADoubleDot]\[ADoubleDot]rtus\t\t\t", StyleBox["\t\tx-le antakse mingi v\[ADoubleDot]\[ADoubleDot]rtus, see v\ \[OTilde]ib olla arvuline v\[ADoubleDot]\[ADoubleDot]rtus, \n\t\t\t\t\t\ algebraline avaldis v\[OTilde]i mingi pikem funktsioon, arvutus, \ v\[OTilde]rrand jne\n", FontSlant->"Plain"], "x = . v\[OTilde]i ", StyleBox["Clear[x]\t\t\tx-i v\[ADoubleDot]\[ADoubleDot]rtus \ t\[UDoubleDot]histatakse\n", FontSlant->"Plain"], "x := v\[ADoubleDot]\[ADoubleDot]rtus\t", StyleBox["\t\t\t\tx-le antakse mingi v\[ADoubleDot]\[ADoubleDot]rtus, \ sarnane esimese omistamisega\n", FontSlant->"Plain"], "avaldis /. x-> v\[ADoubleDot]\[ADoubleDot]rtus", StyleBox["\t\tannab avaldises ", FontSlant->"Plain"], "x", StyleBox["-le v\[ADoubleDot]\[ADoubleDot]rtuse\n", FontSlant->"Plain"], "avaldis /. ", Cell[BoxData[ \(TraditionalForm\`{x -> x\_v\[ADoubleDot]\[ADoubleDot]rt\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y -> y\_v\[ADoubleDot]\[ADoubleDot]rt\)]], StyleBox["}\tavaldises antakse v\[ADoubleDot]\[ADoubleDot]rtused nii x-le \ kui ka y-le", FontSlant->"Plain"], StyleBox["\t\t\t\t\t", FontSlant->"Plain"] }], "Text", FontSlant->"Italic", Background->GrayLevel[0.900008]], Cell["Defineerime avaldise s j\[ADoubleDot]rgmiselt:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(s = x\^2 + 4.5 x - y\)], "Input"], Cell[BoxData[ \(4.5`\ x + x\^2 - y\)], "Output"] }, Open ]], Cell["\<\ J\[ADoubleDot]rgnevalt n\[ADoubleDot]itame mitmeid erinevaid \ v\[OTilde]imalusi avaldises s sisalduvatele muutujatele v\[ADoubleDot]\ \[ADoubleDot]rtuste omistamiseks.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(s /. \ x \[Rule] 4\)], "Input"], Cell[BoxData[ \(\(\(34.`\)\(\[InvisibleSpace]\)\) - y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(s /. {x \[Rule] 2.3, \ y -> 3}\)], "Input"], Cell[BoxData[ \(12.639999999999999`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(s /. {x \[Rule] y, \ y \[Rule] t}\)], "Input"], Cell[BoxData[ \(\(-t\) + 4.5`\ y + y\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(s /. {x \[Rule] x\^2, y \[Rule] y\^3}\)], "Input"], Cell[BoxData[ \(4.5`\ x\^2 + x\^4 - y\^3\)], "Output"] }, Open ]], Cell[TextData[{ "Viimane on nn rekursiivne v\[ADoubleDot]\[ADoubleDot]rtustamine ", StyleBox["Mathematica's", FontSlant->"Italic"], " ja peaks v\[ADoubleDot]ga tuttav olema neile, kes on programmeerimisega \ kokku puutunud. /. abil v\[ADoubleDot]\[ADoubleDot]rtuste omistamine ja \ arvutus toimib j\[ADoubleDot]rgmiselt: ", StyleBox["esmalt teostatakse arvutus avaldises olevate muutujatega, seej\ \[ADoubleDot]rel omistatakse muutujatele uus v\[ADoubleDot]\[ADoubleDot]rtus \ ning antakse l\[OTilde]plik arvutustulemus. ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(a = 3. ;\)\), "\[IndentingNewLine]", \(\(b = 2.5;\)\), "\[IndentingNewLine]", \(\(c = x - y;\)\), "\[IndentingNewLine]", \(a\^2 + b\^3 - c\/2. \)}], "Input"], Cell[BoxData[ \(\(\(24.625`\)\(\[InvisibleSpace]\)\) - 0.5`\ \((x - y)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Viimases n\[ADoubleDot]ites oleme enne arvutust \ omistanud v\[ADoubleDot]\[ADoubleDot]rtused muutujatele a, b, c. Semikoolon \ rea l\[OTilde]pus eristab erinevaid sisestusi; semikooloniga l\[OTilde]ppeva \ rea arvutuse tulemust ei kuvata ekraanile (ka lihtne v\[ADoubleDot]\ \[ADoubleDot]rtuse omistamine a=3. on siin m\[OTilde]istetud arvutusena), \ samas on iga sellise arvutuse tulemus ikkagi hiljem kasutatav. Semikoolonit \ on vahel m\[OTilde]tekas kasutada ka siis, kui on tegemist pikkade \ arvutustega ning mingi vahetulemus on v\[ADoubleDot]ga pikk, sel juhul v\ \[OTilde]ib kasutada pidevalt viimast arvutust (vahepeal lihtsustades ning \ taandades/koondades erinevaid liikmeid) iga kord semikooloniga sisestuse l\ \[OTilde]pus ning alles viimase sisestus on ilma semikoolonita. Semikooloni \ kasutamine on m\[OTilde]istlik ka arvutim\[ADoubleDot]lu s\[ADoubleDot]\ \[ADoubleDot]stmise seisukohalt. ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"]], "Text"], Cell["\<\ %\t\t\t\t\t\teelmise arvutuse v\[ADoubleDot]ljund (resultaat) %%\t\t\t\t\t\t\[UDoubleDot]le-eelmise arvutuse v\[ADoubleDot]ljund %n\t\t\t\t\t\tn-nda arvutuse v\[ADoubleDot]ljund Out[n]\t\t\t\t\t\tn-nda arvutuse v\[ADoubleDot]ljund\ \>", "Text", Background->GrayLevel[0.900008]], Cell["\<\ Veel m\[OTilde]ned n\[ADoubleDot]ited. Pidage silmas, et kui teie sisestate \ avaldised t\[ADoubleDot]pselt nii nagu alltoodud n\[ADoubleDot]ites, v\ \[OTilde]ite saada t\[ADoubleDot]iesti erineva tulemuse, kui teie arvutuste \ numeratsioon erineb antud n\[ADoubleDot]idete omast:\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 3. S\[UDoubleDot]mbolarvutus. Tuletised, integraalid. Funktsioonid.\ \>", "Subtitle"], Cell["\<\ 3.1. Tuletis, diferentsiaal, integraal, piirv\[ADoubleDot]\[ADoubleDot]rtus, summa.\ \>", "Subsubtitle"], Cell[BoxData[ \(Clear[f]; $Line = 0;\)], "Input"], Cell[TextData[{ "D[f,x] ", StyleBox["v\[OTilde]i", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_x\ f\)]], "\t\t\t\t(osa)tuletis funktsioonist f x j\[ADoubleDot]rgi: ", Cell[BoxData[ \(TraditionalForm\`df\/dx\)]], "ehk ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]f\/\[PartialD]x\)]], "\nD[f,x,y,...] ", StyleBox["v\[OTilde]i ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x, y, z, ... \)f\)]], "\t\t\tmitmekordne (osa)tuletis funktsioonist v\[OTilde]i avaldisest f: ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\^3 f\/\(\[PartialD]x \[PartialD]y \ \[PartialD]z\)\)]], ".\nD[f,{x,n}]\t\t\t\t\tn-kordne tuletis avaldisest f x j\[ADoubleDot]rgi: \ ", Cell[BoxData[ \(TraditionalForm\`\(\(d\^n\) f\)\/dx\^n\)]], "\n", Cell[BoxData[ \(TraditionalForm\`Limit[f, x -> x\_0]\)]], "\t\t\t\tpiirv\[ADoubleDot]\[ADoubleDot]rtus ", Cell[BoxData[ \(TraditionalForm\`lim\_\(x \[Rule] x\_0\)\ f\)]], "\nDt[f]\t\t\t\t\t\tt\[ADoubleDot]isdiferentsiaal\nDt[f,x]\t\t\t\t\t\tt\ \[ADoubleDot]istuletisavaldisest f muutuja x j\[ADoubleDot]rgi\n", Cell[BoxData[ \(TraditionalForm\`\[Integral]f \[DifferentialD]x\)]], " ", StyleBox["v\[OTilde]i", FontSlant->"Italic"], " Integrate[f,x]\t\t\tm\[ADoubleDot]\[ADoubleDot]ramata integraal\n", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_a\%b\( f(x)\) \[DifferentialD]x\)]], " ", StyleBox["v\[OTilde]i", FontSlant->"Italic"], " Integrate[f,{x,a,b}]\t\tm\[ADoubleDot]\[ADoubleDot]ratud integraal\n", Cell[BoxData[ \(TraditionalForm\`\[Integral]\(\[Integral]\(f(x, y)\) \[DifferentialD]x \[DifferentialD]y\)\)]], " ", StyleBox["v\[OTilde]i", FontSlant->"Italic"], " Integrate[f,x,y]\tmitmekordne m\[ADoubleDot]\[ADoubleDot]ramata integraal \ muutujate x ja y j\[ADoubleDot]rgi\n", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_a\%b\(\[Integral]\_c\%d\( f(x, y)\) \[DifferentialD]x \[DifferentialD]y\)\)]], "\t", StyleBox["v\[OTilde]i", FontSlant->"Italic"], " ", "\t\t\tmitmekordne m\[ADoubleDot]\[ADoubleDot]ratud integraal\n\ Integrate[f,{x,a,b},{y,c,d}]\n", Cell[BoxData[ \(TraditionalForm\`Series[f, {x, x\_0, j\[ADoubleDot]rk}]\)]], "\t\t\tTaylori astmerida funktsioonist (avaldisest) f punkti ", Cell[BoxData[ \(TraditionalForm\`x = x\_0\)]], " juures\nNormal[", StyleBox["rida", FontSlant->"Italic"], "]\t\t\t\t\tjuhul, kui ", StyleBox["rida", FontSlant->"Italic"], " on Taylori astemrida, siis see protseduur annab\n\t\t\t\t\t\ttulemuseks \ Taylori rea ilma viimase liikmeta\n", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(i = i\_min\)\%\(i\_max\)f\)]], " ", StyleBox["v\[OTilde]i", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`Sum[f, {i, i\_min, i\_max}]\)]], "\tsumma\n", Cell[BoxData[ \(TraditionalForm\`Sum[ f, {i, \ i\_min, \ i\_max}, {j, j\_min, j\_max}]\)]], "\tmitmekordne summa\n", Cell[BoxData[ \(TraditionalForm\`\[Product]\+\(i = i\_min\)\%\(i\_max\)f\)]], " ", StyleBox["ehk", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`Product[f, {i, i\_min, i\_max}]\)]], "\tkorrutis" }], "Text", Background->GrayLevel[0.900008]], Cell["Tuletised, diferentsiaalid", "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x\ x\^3\)], "Input"], Cell[BoxData[ \(3\ x\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x\^n, x]\)], "Input"], Cell[BoxData[ \(n\ x\^\(\(-1\) + n\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x\^n, n]\)], "Input"], Cell[BoxData[ \(x\^n\ Log[x]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x\ x\^2 Log[x + a]\)], "Input"], Cell[BoxData[ \(2\ x\ Log[a + x]\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Esimene avaldis on tavaline tuletis. Teine n\[ADoubleDot]itab, et \ kui tuletist v\[OTilde]etakse ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" j\[ADoubleDot]rgi, on ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsANSI"], StyleBox[" lihtsalt konstant (tegemist on osatuletisega x \ j\[ADoubleDot]rgi).. Kolmandas arvutuses oleme leidnud osatuletise muutuja n \ j\[ADoubleDot]rgi, Ka viimases arvutuses leitakse osatuletis x \ j\[ADoubleDot]rgi v\[OTilde]ttes a konstandiks.", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_\(x, y\)\((3\ \(x\^2\) y\^5)\)\)], "Input"], Cell[BoxData[ \(30\ x\ y\^4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_\(x, y\)\ 3 \( x\^2\) y\^5\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["Integreerimine.", "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\(\(3 y\)\/\(x\^2 + y\^2\)\) \[DifferentialD]x\)], "Input"], Cell[BoxData[ \(3\ ArcTan[x\/y]\)], "Output"] }, Open ]], Cell["\<\ Antud integraali leidmisel arvestati loomulikult, et y ei s\[OTilde]ltu x-st, \ y on konstant.Koolimatemaatikast on meil teada, et \ m\[ADoubleDot]\[ADoubleDot]ramata integraali leidmisel liidetakse alati otsa \ konstant. Antud programm j\[ADoubleDot]tab selle kasutajate hooleks.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_0\%5 2 \( x\^3\) \[DifferentialD]x\)], "Input"], Cell[BoxData[ \(625\/2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[\(2 y\)\/\(x\^2 + y\^2\), x, y]\)], "Input"], Cell[BoxData[ \(\(-2\)\ x + 2\ y\ ArcTan[x\/y] + x\ Log[x\^2 + y\^2]\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["J\[ADoubleDot]rgmisena vaatame kahekordset m\[ADoubleDot]\ \[ADoubleDot]ratud integraali, kus integreerimispiirkond on \ n\[ADoubleDot]idatud allpool toodud joonisel. See t\[ADoubleDot]hendab, et v\ \[OTilde]ime integreerida funktsiooni ", FontFamily->"Times New Roman", FontColor->GrayLevel[0], CharacterEncoding->"WindowsANSI"], StyleBox["f(x,y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], CharacterEncoding->"WindowsANSI"], StyleBox[") esmalt ", FontFamily->"Times New Roman", FontColor->GrayLevel[0], CharacterEncoding->"WindowsANSI"], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], CharacterEncoding->"WindowsANSI"], StyleBox[" j\[ADoubleDot]rgi 0-st kuni ", FontFamily->"Times New Roman", FontColor->GrayLevel[0], CharacterEncoding->"WindowsANSI"], Cell[BoxData[ \(TraditionalForm\`\(2 x\)\/3\)]], StyleBox["-ni ja siis ", FontFamily->"Times New Roman", FontColor->GrayLevel[0], CharacterEncoding->"WindowsANSI"], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], CharacterEncoding->"WindowsANSI"], StyleBox[" j\[ADoubleDot]rgi 0-st 3-ni:", FontFamily->"Times New Roman", FontColor->GrayLevel[0], CharacterEncoding->"WindowsANSI"] }], "Text"], Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@0003SP0@0006@000000000000005@1003A0000 0000000000188`00^aD00215CDH00040h3P004/2000400000000000000000000E@400=80001J0000 =`00000000000000000000X0000@00000000000000090000400005D1003B0000500000`0000=0000 4P0000`0000100008@0000P0000Q0000200001@0000<00003@0002L0000H00000@000000003oool0 000002D0000<00000@0002H0000L00000P0000000000000000000000000U000030000080000[0000 6000000000000000E@400=80000D0000300000d0000R000030000?oooolD0000300000d0000Q0000 200002D0000<00001`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U0000 3000008000040000;0000840002a0000P@000;<000020000P@000;<000210000/@0002D0000<0000 1`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U000030000080000?0000 50000840002a000000000580001<0@000`000?Coool0000000000000002@0@0000000000009306l0 M@1b06T0I@1b0200CP1U07L000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000005P0000`0000H0000 600000`0000000009@0000`000030000E00005@0001n0000``0008D0003B00000@00000000000000 OP000<<000010000C0000080000000000000000000000000D0000340000600009@0000`000070020 :00000`0000200009P0001`0000200000000000000000000000002D0000<00000P0000@0000/0000 dP000;40003B0000/`000080003B0000/`000=80002a00009@0000`000070020:00000`000020000 9P0001`0000200000000000000000000000002D0000<00000P0000l0000D0000dP000;4000000000 9@0000`0000:0020:00000`000030000DP0004`100030000m?ooo`00000000000000090100000000 00000T<0K`1e0780J@1U0780801>06D0M`0000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000F0000 300001P0000H000030000000000U0000300000<0001D0000E00000000``000040001<00000P0000000000000000000000001@000080002b0000 hP000;<000020000hP000;<0003R0000/P0002D0000<00001`00P2P0000<00000P0002H0000L0000 0P0000000000000000000000000U000030000080000?000050000>80002b0000000002D0000<0000 1`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U00003000008000040000 ;0000?80002b0000lP000;<000020000lP000;<0003b0000/P0002D0000<00001`00P2P0000<0000 0P0002H0000L00000P0000000000000000000000000U000030000080000?000050000?80002b0000 000002D0000<00001`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U0000 3000008000040000;0000081002b00000P400;<0000200000P400;<000020@00/P0002D0000<0000 1`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U000030000080000?0000 50000081002b0000000002D0000<00001`00P2P0000<00000P0002H0000L00000P00000000000000 00000000000U00003000008000040000;00001<1002b00004`400;<0000200004`400;<0000C0@00 /P0002D0000<00001`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U0000 30000080000?0000500001<1002b0000000002D0000<00001`00P2P0000<00000P0002H0000L0000 0P0000000000000000000000000U00003000008000040000;0000200002b000080000;<000020000 80000;<0000P0000/P0002D0000<00001`00P2P0000<00000P0002H0000L00000P00000000000000 00000000000U000030000080000?000050000200002b0000000002D0000<00001`00P2P0000<0000 0P0002H0000L00000P0000000000000000000000000U00003000008000040000;00000l0002b0000 3`000;<0000200003`000;<0000?0000/P0002D0000<00001`00P2P0000<00000P0002H0000L0000 0P0000000000000000000000000U000030000080000?0000500000l0002b0000000002D0000<0000 1`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U00003000008000040000 ;00003<1002b0000<`400;<000020000<`400;<0000c0@00/P0002D0000<00001`00P2P0000<0000 0P0002H0000L00000P0000000000000000000000000U000030000080000?0000500003<1002b0000 000002D0000<00001`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U0000 3000008000040000;00004<1002b0000@`400;<000020000@`400;<000130@00/P0002D0000<0000 1`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U000030000080000?0000 500004<1002b0000000002D0000<00001`00P2P0000<00000P0002H0000L00000P00000000000000 00000000000U00003000008000040000;0000000002c0000D`400;<00002000000000;<0001C0@00 /`0002D0000<00001`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U0000 30000080000?0000500005<1002c0000000002D0000<00001`00P2P0000<00000P0002H0000L0000 0P0000000000000000000000000U00003000008000040000;0000340002>0000<`0008h000020000 <@0008h0000c0000SP0002D0000<00001`00P2P0000<00000P0002H0000L00000P00000000000000 00000000000U000030000080000?0000500003<0002>0000000002D0000<00002P00P2P0000<0000 0`000580001<0@000`000?Coool0000000000000002@0@0000000000009306l0M@1b06T0I@1b0200 CP1U07L0000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000005P0000`0000H0000600000`000000000 9@0000`000030000E0000600000G0000TP0002T0002Q00000@000000000000005`00098000030000 C0000080000000000000000000000000E0000300;P0e00001P0000H0000600009@0000`000070020 :00000`0000200009P0001`0000200000000000000000000000002D0000<00000P0000@0000/0000 <@0006P0000c0000J0000080000a0000J00003<0001X00009@0000`000070020:00000`000020000 9P0001`0000200000000000000000000000002D0000<00000P0000l0000D0000<`0006P000000000 9@0000`0000:0020:00000`000030000DP0004`100030000m?ooo`00000000000000090100000000 00000T<0K`1e0780J@1U0780801>06D0M`0000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000F0000 300001P0000H000030000000000U0000300000<0001D0000E00002D0001/0000;00007/000010000 00000000000U0000K0000040001<00000P0000000000000000000000001@0000<@0000H0000U0000 300000L0080X000030000080000V0000700000800000000000000000000000009@0000`000020000 100002`0000a0000@P0003<0001200000P00034000120000<`000480000U0000300000L0080X0000 30000080000V0000700000800000000000000000000000009@0000`0000200003`0001@0000c0000 @P000000000U0000300000X0080X0000300000<0001B0000C04000<0003doooo0000000000000000 T040000000000002@`1_07D0LP1Y06D0LP0P04h0I@1g000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 000001H0000<0000600001P0000<0000000002D0000<00000`0005@0001P00005`0004H0000Y0000 E@00004000000000000001L0001600000`0004`0000200000000000000000000000005@0000a02h0 =@0000H0000600001P0002D0000<00001`00P2P0000<00000P0002H0000L00000P00000000000000 00000000000U00003000008000040000;0000340000N0000<`0001h000020000<@0001h0000c0000 7P0002D0000<00001`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U0000 30000080000?0000500003<0000N0000000002D0000<00002P00P2P0000<00000`000580001<0@00 0`000?Coool0000000000000002@0@0000000000009306l0M@1b06T0I@1b0200CP1U07L000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000005P0000`0000H0000600000`0000000009@0000`000030000 E00005@0000U00008@0002`0000`00000@000000000000009@00024000010000C000008000000000 0000000000000000D0000380000600009@0000`000070020:00000`0000200009P0001`000020000 0000000000000000000002D0000<00000P0000@0000/0000<@000:`0000b0000[0000080000a0000 [0000380002/00009@0000`000070020:00000`0000200009P0001`0000200000000000000000000 000002D0000<00000P0000l0000D0000`000080000a0000>`000380000k00009@0000`000070020:00000`000020000 9P0001`0000200000000000000000000000002D0000<00000P0000l0000D000000009@0000`000070020:00000`0000200009P0001`0000200000000000000000000 000002D0000<00000P0000l0000D00000000170000k@0003l0003j0000 =`0000L1000^00005@4002H0000S0@0070000301000E0000?`4000`0001;0@001@0002D0000<0000 1`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U000030000080000?0000 500004/10005000000000280000<0000oooooa@0000<00003@000280000<0000oooooa@0000<0000 3@0002D0000<00001`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U0000 3000008000040000;0000281000O00008P400;<0000200008P400;<0000R0@007`00024000080000 9@0000`0000:0020:00000`000030000DP0004`100030000kOooo`00000000000000090100000000 00004U@0J@1]06D0L`0P04h0I@1g0200DP1_06d0H@1^000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000F0000 300001P0000H000030000000000U0000300000<0001D0000E00004<100350000C0400=X000010000 0000000000130@00a@000040001<00000P0000000000000000000000001@0000N00000T0000R0000 30000?oooolD0000300000d0000Q0000200002D0000<00002P00P2P0000<00000`000580001<0@00 0`000>goool0000000000000002@0@0000000000019D06T0K@1U07<0801>06D0M`0P0580K`1]0640 KP000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000005P0000`0000H0000600000`0000000009@0000`000030000 E00005@0000N00005@0002L0000Z00000@000000000000007P0001D000010000C000008000000000 0000000000000000D00007T0000900008P0000`0003ooooo500000`0000=00009@0000`000010020 :00000`0000100009`0001P00001000000000>KViP0000009@0000`0000100009@0000`000070020 :00000`0000200009P0001`0000200001@00000000000000oooo02D0000<00000P0000<0000l0000 <00001h0000R0@00/`0000@0000`0000/`000281000N00008P400;<0000`0000/`0002D0000<0000 1`00P2P0000<00000P0002H0000L00000P0000000000000000000000000U00003000008000040000 ?0000300000N00008P400;<000040000<0000;<0000R0@007P000281002c0000<0000;<0000>0000 50000000000F00005000 \>"], "Graphics", ImageSize->{340, 209}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_0\%3\(\[Integral]\_0\%\(\(2 x\)\/3\)\((x\^2 + y\^2)\) \[DifferentialD]y \[DifferentialD]x\)\)], "Input"], Cell[BoxData[ \(31\/2\)], "Output"] }, Open ]], Cell[TextData[{ "J\[ADoubleDot]rgmised paar n\[ADoubleDot]idet on piirv\[ADoubleDot]\ \[ADoubleDot]rtuste kohta. Esmalt defineerime avaldise, siis leiame piirv\ \[ADoubleDot]\[ADoubleDot]rtuse ning vaatama ka seda, kuidas ", StyleBox["Mathematica", FontSlant->"Italic"], " otsese arvutusega \[UDoubleDot]ritab avaldise \ v\[ADoubleDot]\[ADoubleDot]rtust leida." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(s = Sin[x]\/x\)], "Input"], Cell[BoxData[ \(Sin[x]\/x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[s, x \[Rule] 0]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell["\<\ Seega, avaldisel leidus l\[OTilde]plik piirv\[ADoubleDot]\[ADoubleDot]rtus - \ 1, kuid otsesel arvutusel oleks tulnud jagada nulliga ning tulemus j\ \[ADoubleDot]\[ADoubleDot]nud leidmata.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t = \(\((9 - x\^2)\) 2 x\^3\)\/\((3 x - 9)\)\)], "Input"], Cell[BoxData[ \(\(2\ x\^3\ \((9 - x\^2)\)\)\/\(\(-9\) + 3\ x\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[t, x \[Rule] 3]\)], "Input"], Cell[BoxData[ \(\(-108\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[\[ExponentialE]\^\(-x\) - 1, x \[Rule] \[Infinity]]\)], "Input"], Cell[BoxData[ \(\(-1\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[Tan[x], x \[Rule] \[Pi]\/2]\)], "Input"], Cell[BoxData[ InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["3.2. Funktsioonide defineerimine.", "Subsubtitle"], Cell[BoxData[{ \(\(Clear[s, f];\)\), "\[IndentingNewLine]", \(\($Line = 0;\)\)}], "Input"], Cell[TextData[{ "Lisaks ", StyleBox["Mathematica", FontSlant->"Italic"], "'s defineeritud funktsioonidele ja protseduuridele v\[OTilde]ib neid ka \ ise teha. Defineerime j\[ADoubleDot]rgmise ruutfunktsiooni:" }], "Text"], Cell[BoxData[ \(f[x_] := 3 x\^2 + 2 x + 1\)], "Input"], Cell["\<\ Siin alakriips ('blank') t\[ADoubleDot]hendab, et see funktsioon ehk reegel \ antud funktsiooni leidmiseks kehtib ka siis, kui me edaspidi asendama muutuja \ x muutujaga y:\ \>", "Text"], Cell[BoxData[ \(f[y]\)], "Input"], Cell[BoxData[ \(f[2.5]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(g[x] /. x \[Rule] y\)], "Input"], Cell[BoxData[ \(1 + 2\ y + 3\ y\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[x] /. x \[Rule] 3\)], "Input"], Cell[BoxData[ \(34\)], "Output"] }, Open ]], Cell["\<\ Seega, sellisel, ilma alakriipusta definitsioonil 'kui funktsioon x-st' \ puudub m\[OTilde]te. Selliselt v\[OTilde]ime m\[ADoubleDot]\[ADoubleDot]rata \ ka lihtsa avaldisena (v\[ADoubleDot]hem vaeva edasistes arvutustes). Esmalt t\ \[UDoubleDot]histame g[x] definitsiooni:\ \>", "Text"], Cell[BoxData[ \(Clear[g]\)], "Input"], Cell[BoxData[ \(g := 3 x\^2 + 2 x + 1\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(g /. x \[Rule] 4\)], "Input"], Cell[BoxData[ \(57\)], "Output"] }, Open ]], Cell[TextData[{ "Analoogiliselt v\[OTilde]ib defineerida ka mitme muutuja funktsioone. \ Allpool toome n\[ADoubleDot]itena f\[UDoubleDot]\[UDoubleDot]sikast tuntud \ avaldise teepikkuse m\[ADoubleDot]\[ADoubleDot]ramiseks mingil ajahetkel kui \ funktsiooni ajast t, algkiirusest ", Cell[BoxData[ \(TraditionalForm\`v\_0\)]], " ja kiirendusest a.\nM\[ADoubleDot]rkus: alaindeksite kasutamine \ arvutustes ei ole alati v\[ADoubleDot]ga hea idee!" }], "Text"], Cell[BoxData[ \(Clear[s]\)], "Input"], Cell[BoxData[ \(s[t_, v0_, a_] := v0*t + \(1\/2\) a*t\^2\)], "Input"], Cell[BoxData[ \(s[4, 0.5, 1]\)], "Input"], Cell[BoxData[ \(s[4, 1, 0.5]\)], "Input"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`f[x_] := x\^3\)]], "\t\t\t\tfunktsiooni f(x) defineerimine\n", Cell[BoxData[ \(TraditionalForm\`f[x_, y_] := x\^3 + Sin[2 x + 3]\)]], "\t\tmitme muutuja funktsiooni defineerimine\nClear[f]\t\t\t\tfunktsiooni f \ definitsiooni t\[UDoubleDot]histamine\n?f\t\t\t\t\tfunktsiooni f kuju n\ \[ADoubleDot]itamine (kehtib ka arvuti \n\t\t\t\t\tpoolt defineeritud \n\t\t\t\ \t\tfunktsioonide ja protseduuride kohta\n??f\t\t\t\t\tkogu funktsiooni v\ \[OTilde]i protseduuri f kohta k\[ADoubleDot]iva \n\t\t\t\t\tinfo n\ \[ADoubleDot]itamine\n", Cell[BoxData[ \(TraditionalForm\`f[x] = x\^3\)]], "\t\t\t\tavaldise (ja mitte funktsiooni) f defineerimine" }], "Text", Background->GrayLevel[0.900008]] }, Open ]], Cell[CellGroupData[{ Cell["3.3. V\[OTilde]rrandid.", "Subsubtitle"], Cell[TextData[{ "Alapunktis 2.3 n\[ADoubleDot]itasime, kuidas omistada v\[ADoubleDot]\ \[ADoubleDot]rtusi muutujatele. N\[ADoubleDot]iteks ", StyleBox["x = y", FontSlant->"Italic"], " on lihtsalt muutujale ", StyleBox["x", FontSlant->"Italic"], " muutuja ", StyleBox["y", FontSlant->"Italic"], " v\[ADoubleDot]\[ADoubleDot]rtuse omistamine. Selline omistamine \ deklareerib m\[OTilde]lemad suurused v\[OTilde]rdseks, kuid \[UDoubleDot]he t\ \[ADoubleDot]psustusega. Kui anname ", StyleBox["y", FontSlant->"Italic"], "-le mingi v\[ADoubleDot]\[ADoubleDot]rtuse, saab ka muutuja ", StyleBox["x", FontSlant->"Italic"], " automaatselt sama v\[ADoubleDot]\[ADoubleDot]rtuse, kuid muutujale ", StyleBox["x", FontSlant->"Italic"], " v\[ADoubleDot]\[ADoubleDot]rtuse omistamisega, \ j\[ADoubleDot]\[ADoubleDot]b ", StyleBox["y", FontSlant->"Italic"], " v\[ADoubleDot]\[ADoubleDot]rtus muutumatuks ning sellest hetkest alates \ ei olen enam ", StyleBox["x", FontSlant->"Italic"], " ", StyleBox["y", FontSlant->"Italic"], "-ga v\[OTilde]rdne. ", StyleBox["x == y", FontSlant->"Italic"], " aga kontrollib, kas ", StyleBox["x", FontSlant->"Italic"], " on ", StyleBox["y", FontSlant->"Italic"], "-ga v\[OTilde]rdne. Kahte v\[OTilde]rdusm\[ADoubleDot]rki kasutatakse ka v\ \[OTilde]rrandite lahendamisel (v\[OTilde]rrandite defineerimisel samuti)." }], "Text", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ \(x = y\)], "Input"], Cell[BoxData[ \(y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x \[Equal] y\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(2 + 4 \[Equal] 6\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(2*5 \[Equal] 11\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[BoxData[ \(x =. \)], "Input"], Cell[TextData[{ "Viimase v\[OTilde]rdusega t\[UDoubleDot]histasime ", StyleBox["x", FontSlant->"Italic"], " v\[ADoubleDot]\[ADoubleDot]rtuse." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x \[Equal] 5\)], "Input"], Cell[BoxData[ \(x == 5\)], "Output"] }, Open ]], Cell[TextData[{ "Kuiv\[OTilde]rd ", StyleBox["x", FontSlant->"Italic"], "-l v\[ADoubleDot]\[ADoubleDot]rtus puudub, ei oska ka programm \ \[ODoubleDot]elda, kas v\[OTilde]rdus on t\[OTilde]ene v\[OTilde]i v\ \[ADoubleDot]\[ADoubleDot]r. \nAvaldis x==5 on tegelikult juba \ v\[OTilde]rrand (mille lahendiks on ilmselt x=5), sest ", StyleBox["x", FontSlant->"Italic"], " v\[ADoubleDot]\[ADoubleDot]rtus on defineerimata.\nProgrammis ", StyleBox["Mathematica ", FontSlant->"Italic"], "(ja ka n\[ADoubleDot]iteks Maple's) on tihti tavaks anda mingile \ avaldisele v\[ADoubleDot]\[ADoubleDot]rtuseks v\[OTilde]rrand ise." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq = x\^2 - 4 x - 7 \[Equal] 0\)], "Input"], Cell[BoxData[ \(\(-7\) - 4\ x + x\^2 == 0\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["x = y", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox[" \t\t\t\t\tomistab x-le v\[ADoubleDot]\[ADoubleDot]rtuse y\n", Background->GrayLevel[0.900008]], StyleBox["x == y", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox[" \t\t\t\t\tkontrollib, kas ", Background->GrayLevel[0.900008]], StyleBox["x", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox[" ja ", Background->GrayLevel[0.900008]], StyleBox["y", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox[" on v\[OTilde]rdsed\n", Background->GrayLevel[0.900008]], StyleBox["x != y", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox["\t\t\t\t\tkontrollib, kas ", Background->GrayLevel[0.900008]], StyleBox["x", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox[" ei v\[OTilde]rdu ", Background->GrayLevel[0.900008]], StyleBox["y-", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox["ga (kui x ja y ei ole v\[OTilde]rdsed, \n\t\t\t\t\ton vastus \ True)\n", Background->GrayLevel[0.900008]], StyleBox["x > y", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox["\t\t\t\t\tkontrollib, kas ", Background->GrayLevel[0.900008]], StyleBox["x", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox[" on suurem, kui ", Background->GrayLevel[0.900008]], StyleBox["y", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox["\n", Background->GrayLevel[0.900008]], StyleBox["x < y", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox["\t\t\t\t\tkontrollib, kas ", Background->GrayLevel[0.900008]], StyleBox["x", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox[" on v\[ADoubleDot]iksem kui ", Background->GrayLevel[0.900008]], StyleBox["y", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox["\n", Background->GrayLevel[0.900008]], StyleBox["x <= y", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox["\t\t\t\t\tv\[ADoubleDot]iksem v\[OTilde]i v\[OTilde]rdne\n", Background->GrayLevel[0.900008]], StyleBox["x >= y", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox["\t\t\t\t\tsuurem v\[OTilde]i v\[OTilde]rdne\n", Background->GrayLevel[0.900008]], StyleBox["x == y == z", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox["\t\t\t\tkas k\[OTilde]ik on omavahel v\[OTilde]rdsed\n", Background->GrayLevel[0.900008]], StyleBox["x != y != z", FontSlant->"Italic", Background->GrayLevel[0.900008]], StyleBox["\t\t\t\tkas k\[OTilde]ik on omavahel mittev\[OTilde]rdsed", Background->GrayLevel[0.900008]] }], "Text", Background->GrayLevel[0.900008]], Cell[TextData[{ "\[CapitalUDoubleDot]laltoodud avaldisi saab eriti h\[ADoubleDot]sti \ kasutada keerulisemata \[UDoubleDot]lesannet lahendamisel, \ n\[ADoubleDot]iteks programmeerimisel ", StyleBox["Mathematica", FontSlant->"Italic"], "s, kasutades laused While-, Do, For, If." }], "Text"], Cell[TextData[StyleBox["V\[OTilde]rrandite lahendamine.", FontWeight->"Bold", FontSlant->"Italic"]], "Text"], Cell[TextData[{ "Solve[v", StyleBox["asak pool == parem pool, x", FontSlant->"Italic"], "]\tv\[OTilde]rrandi lahendamine ", StyleBox["x", FontSlant->"Italic"], " suhtes, tulemuseks on \n\t\t\t\t\t\ton reegilite hulk (list)\nx /. lahend\ \t\t\t\t\treeglite listi kasutatakse x v\[ADoubleDot]\[ADoubleDot]rtuste \ saamiseks\navaldis /. lahend\t\t\t\treeglite listi kasutatakse avaldisele v\ \[ADoubleDot]\[ADoubleDot]rtuse andmiseks" }], "Text", Background->GrayLevel[0.900008]], Cell["\<\ Solve abil saab lahendada algebralisi v\[OTilde]rrandeid, milles muutuja aste \ on maksimaalselt 4. Kui k\[OTilde]rgeim aste on \[UDoubleDot]le nelja, ei \ pruugi t\[ADoubleDot]pset lahendit antud protseduur leida.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[eq, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 2 - \@11}, {x \[Rule] 2 + \@11}}\)], "Output"] }, Open ]], Cell["\<\ Kuiv\[OTilde]rd v\[OTilde]rrandis olid k\[OTilde]ik arvud \ t\[ADoubleDot]isarvud, siis antakse tulemus t\[ADoubleDot]pselt. Ligikaudse \ lahendi saamiseks oleks kordajatest v\[ADoubleDot]hemalt \[UDoubleDot]ks \ tulnud ette anda reaalarvuna (nt 7.0) v\[OTilde]i teha nii nagu \ j\[ADoubleDot]rgnevas n\[ADoubleDot]ites:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[%]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(-1.3166247903553998`\)}, {x \[Rule] 5.3166247903554`}}\)], "Output"] }, Open ]], Cell["\<\ Viimane avaldis on reeglite hulk (list). Ainult arvude hulga saame j\ \[ADoubleDot]rgmiselt:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lahendid = x\ /. \ %28\)], "Input"], Cell[BoxData[ \({\(-1.3166247903553998`\), 5.3166247903554`}\)], "Output"] }, Open ]], Cell[TextData[{ "Kui n\[UDoubleDot]\[UDoubleDot]d soovime omistada esimese v\[ADoubleDot]\ \[ADoubleDot]rtuse ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], "-le, teise ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], "-le, tuleks teha j\[ADoubleDot]rgmiselt:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x\_0 = lahendid[\([1]\)]\)], "Input"], Cell[BoxData[ \(\(-1.3166247903553998`\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x\_1 = lahendid[\([2]\)]\)], "Input"], Cell[BoxData[ \(5.3166247903554`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\@x\ /. \ %28\)], "Input"], Cell[BoxData[ \({\(\(0.`\)\(\[InvisibleSpace]\)\) + 1.1474427176793618`\ \[ImaginaryI], 2.3057807333646014`}\)], "Output"] }, Open ]], Cell[TextData[{ "Kuiv\[OTilde]rd Out[9] oli meil list, milles x-l oli kaks v\[ADoubleDot]\ \[ADoubleDot]rtust, siis ka ", Cell[BoxData[ \(TraditionalForm\`\@x\)]], " protseduuri rakendamine andis tulemuseks listi, milles on kaks liiget. Ja \ kuiv\[OTilde]rd esimene v\[ADoubleDot]\[ADoubleDot]rtus oli negatiivne, siis \ sellest ruutjuure v\[OTilde]tmisel saame tulemuseks loomulikult \ kompleksarvu." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[x\^5 - 4 x\^4 + 9 \[Equal] 0, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] Root[9 - 4\ #1\^4 + #1\^5 &, 1]}, {x \[Rule] Root[9 - 4\ #1\^4 + #1\^5 &, 2]}, {x \[Rule] Root[9 - 4\ #1\^4 + #1\^5 &, 3]}, {x \[Rule] Root[9 - 4\ #1\^4 + #1\^5 &, 4]}, {x \[Rule] Root[9 - 4\ #1\^4 + #1\^5 &, 5]}}\)], "Output"] }, Open ]], Cell["\<\ Nagu n\[ADoubleDot]he, ei suutnud antud juhul Solve-protseduur anda t\ \[ADoubleDot]pset lahendit. K\[UDoubleDot]ll saame ligikaudse: \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[x\^5 - 4 x\^4 + 9.0 \[Equal] 0, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(-1.1497767610903715`\)}, {x \[Rule] \ \(-0.08619637221601012`\) - 1.2025654913272121`\ \[ImaginaryI]}, {x \[Rule] \ \(-0.08619637221601012`\) + 1.2025654913272121`\ \[ImaginaryI]}, {x \[Rule] 1.3586376003599896`}, {x \[Rule] 3.9635319051624025`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[Sin[x] \[Equal] a, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] ArcSin[a]}}\)], "Output"] }, Open ]], Cell["\<\ Antud juhul kasutatakse siinuse p\[ODoubleDot]\[ODoubleDot]rdfunktsiooni \ arcsin, seet\[OTilde]ttu on siin ka hoiatus,et et m\[OTilde]ned lahendid v\ \[OTilde]ivad j\[ADoubleDot]\[ADoubleDot]da leidmata. Antud v\[OTilde]rrandil \ on tegelikult l\[OTilde]pmata hulk lahendeid, mis erinevad \ \[UDoubleDot]ksteisest 2 \[Pi] n v\[OTilde]rra, n on siin \ t\[ADoubleDot]isarv. \ \>", "Text", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[Sin[x] \[Equal] x, x]\)], "Input"], Cell[BoxData[ \(Solve[Sin[x] == x, x]\)], "Output"] }, Open ]], Cell[TextData[StyleBox["See t\[ADoubleDot]hendab, et \[UDoubleDot]laltoodud v\ \[OTilde]rrand sisaldab transendentseid funktsioone ning algebraliselt on \ seda v\[OTilde]rrandit v\[OTilde]imatu lahendada. Selliste transendetnsete v\ \[OTilde]rrandeid lahendatakse numbriliselt ning nende lahendamiseks on \ funktsioon FindRoot, mida k\[ADoubleDot]sitleme hiljem.", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"]], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`Solve[{vp\_1 \[Equal] pp\_1, vp\_2 \[Equal] pp\_2, ... }, {x\_1, x\_2, \ ... }]\)]], "\t\tv\[OTilde]rrandis\[UDoubleDot]steemi \n\t\t\t\t\t\t\t\tlahendamine ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], ", ..suhtes\n", Cell[BoxData[ \(TraditionalForm\`NSolve[{vp\_1 \[Equal] pp\_1, \ vp\_2 \[Equal] pp\_2, \ ... }, {x\_1, x\_2, \ ... }]\)]], " \t\tv\[OTilde]rrandis\[UDoubleDot]steemi numbriline \t\n\t\t\t\t\t\t\t\t\ lahendamine ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], " ..suhtes\n", Cell[BoxData[ \(TraditionalForm\`\(\(Eliminate[{vp\_1 \[Equal] pp\_1, \ vp\_2 \[Equal] pp\_2, \ ... }, {x\_1, x\_2, \ ... }]\)\(\ \)\)\)]], " \tv\[OTilde]rrandis\[UDoubleDot]steemist muutujate ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], " \n\t\t\t\t\t\t\t\tellimineerimine\nReduce[{", Cell[BoxData[ \(TraditionalForm\`{vp\_1 \[Equal] pp\_1, \ vp\_2 \[Equal] pp\_2, \ ... }, {x\_1, x\_2, \ ... }\)]], "] \tannab lihtsustatud v\[OTilde]rrandite s\[UDoubleDot]steemi koos\n\t\t\t\ \t\t\t\t\tv\[OTilde]imalike lahendustega\n", Cell[BoxData[ \(TraditionalForm\`FindRoot[vp \[Equal] pp, \ {x, x\_0}]\)]], "\t\t\t\t\tleiab numbriliselt lahendi v\[OTilde]rrandile, alustades ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], "-st" }], "Text", Background->GrayLevel[0.900008]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{2 x - 4 y + 3 z \[Equal] 0, x - 3 y - z \[Equal] 0, x - 3 z \[Equal] \(-9\)}, {x, y, z}]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(-\(117\/19\)\), y \[Rule] \(-\(45\/19\)\), z \[Rule] 18\/19}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(NSolve[{2 x - 4 y + 3 z \[Equal] 0, x - 3 y - z \[Equal] 0, x - 3 z \[Equal] \(-9\)}, {x, y, z}]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(-6.157894736842104`\), y \[Rule] \(-2.3684210526315788`\), z \[Rule] 0.9473684210526315`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Eliminate[{2 x + 4 y + 3 z \[Equal] 0, x - 3 y \[Equal] z - 2}, z]\)], "Input"], Cell[BoxData[ \(\(-6\) + 5\ y == 5\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Reduce[{2 x + 4 y + 3 z \[Equal] 0, x - 3 y \[Equal] z - 2}, {x, y}]\)], "Input"], Cell[BoxData[ \(x == 1\/10\ \((\(-8\) - 5\ z)\) && y == 1\/10\ \((4 - 5\ z)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Eliminate k\[ADoubleDot]suga saime kahest kolme \ muutujaga v\[OTilde]rrandist \[UDoubleDot]he kahe muutujaga v\[OTilde]rrandi. \ Reduce abiga antud juhul saime kaks v\[OTilde]imalikku uut v\[OTilde]rrandit \ - esimesel juhul elimineerisime y, teisel juhul x. \nKuigi NSolve leiab v\ \[OTilde]rrandite lahendeid numbriliselt, sobib see protseduur siiski vaid v\ \[OTilde]rrandite jaoks, kus muutujad on mingites astmetes \ (pol\[UDoubleDot]noomv\[OTilde]rrandid). Vastasel juhul tuleb kasutada ikkagi \ protseduuri FindRoot, mis leiab nullkohad Newtoni iteratsioonimeetodil, kuid \ selleks tuleb anda piisavalt hea l\[ADoubleDot]htepunkt, st ligikaudne \ nullkoht. Viimast on \[UDoubleDot]sna hea leida joonise abil.", FontFamily->"Times New Roman", CharacterEncoding->"WindowsANSI"]], "Text"], Cell["\<\ Jooniselt on n\[ADoubleDot]ha, et \[UDoubleDot]ks nullkoht on umbes kohal 0, \ teine kohal 2,5. Tuleb kasutada kaks korda protseduuri FindRoot\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindRoot[g[x] \[Equal] 0, {x, 0}]\)], "Input"], Cell[BoxData[ \({x \[Rule] 0.`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FindRoot[g[x] \[Equal] 0, {x, 2.5}]\)], "Input"], Cell[BoxData[ \({x \[Rule] 2.5917110181968916`}\)], "Output"] }, Open ]], Cell[TextData[{ "Seega on kaks nullkohta: ", Cell[BoxData[ \(TraditionalForm\`x\_1 = 0, \ x\_2 \[TildeTilde] 2, 59\)]], "." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["4. Joonised", "Subtitle"], Cell["4.1. Joonised tasandil", "Subsubtitle"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`Plot[f, {x, x\_min, \ x\_max}]\)]], "\t\t\tjoonistab funktsiooni f graafiku ", Cell[BoxData[ \(TraditionalForm\`x\_min\)]], "-st \n\t\t\t\t\t\tkuni ", Cell[BoxData[ \(TraditionalForm\`x\_max\)]], "-ni\n", Cell[BoxData[ \(TraditionalForm\`Plot[{f, g, ... }, {x, x\_min, \ x\_max}]\)]], "\t\tjoonistab mitme funktsiooni graafikud\nShow[joonis1, joonis2]\t\t\t\t\ joonistab erinevad joonised \[UDoubleDot]hele kokku" }], "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\($Line = 0;\)\)], "Input"], Cell[BoxData[ \(Plot[E\^\(\(-0.5\) x\)\ Sin[x], {x, 0, 2 \[Pi]}]\)], "Input"], Cell[BoxData[ \(Plot[{2 x\^2 - 4 x, 2 x}, {x, \(-2\), 4}, AxesLabel \[Rule] {x, y}]\)], "Input"], Cell[BoxData[ \(Show[%1, %2]\)], "Input"], Cell[TextData[{ "Nagu teisest arvutusest n\[ADoubleDot]ha, v\[OTilde]ib Plot-protseduurile \ anda erinevaid lisav\[OTilde]imalusi. T\[ADoubleDot]ieliku info \ v\[OTilde]imalustest saab Help-st v\[OTilde]i kasutades \ Help-k\[ADoubleDot]sku (kehtib k\[OTilde]igi ", StyleBox["Mathematica", FontSlant->"Italic"], " funktsioonide ja protseduuride kohta)." }], "Text"], Cell["M\[OTilde]ned n\[ADoubleDot]ited erinevate v\[OTilde]imaluste \ kasutamise kohta.", "Text"], Cell[BoxData[ \(Plot[{E\^x, x, Log[x]}, {x, \(-3\), 3}, PlotStyle \[Rule] {RGBColor[1, 0, 0], RGBColor[0.5, 0.5, 0], RGBColor[0, 0.5, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotRange \[Rule] {{\(-3\), 3}, {\(-3\), 4}}, AspectRatio \[Rule] Automatic]\)], "Input"], Cell["\<\ Nagu n\[ADoubleDot]ha, andis programm ka teada, et funktsiooni ln(x) (Log[x]) \ ei saa negatiivsete v\[ADoubleDot]\[ADoubleDot]rtuste ja nulli kohal leida. \ Kuiv\[OTilde]rd tegemist on kolme funktsiooni graafikuga, siis on ka \ PlotStyle-s iga funktsiooni jaoks v\[ADoubleDot]rv \[ADoubleDot]ra m\ \[ADoubleDot]\[ADoubleDot]ratud, st PlotStyle's on tegemist listiga.\ \>", "Text"], Cell[BoxData[ \(funkt[x_] := 2*Sin[2 x]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[funkt[x], {x, 0, 12}, TextStyle \[Rule] {FontSlant -> "\", \ FontSize \[Rule] 12}, \[IndentingNewLine]Ticks \[Rule] {{{1.5, "\"}, {3, "\<2a\ \>"}, 8}, Automatic}, Epilog \[Rule] {Text["\", {8.7, 0.8}, {0, 0}]}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["valik\t\t\tvaikv\[ADoubleDot]\[ADoubleDot]rtus\n", FontSlant->"Italic"], "Axes\t\t\tAutomatic \tkas teljed lisada, ntx Axes->No (Yes)\nAxesLabel\t\t\ None\t\tmillised t\[ADoubleDot]hised telgede juurde panna, nt\n\t\t\t\t\t\ AxesLabel->{\"a\",\"b/s\"}\nAxesOrigin\t\tAutomatic\tmillises punktis asub \ telgede ristumiskoht (nullpunkt)\n\t\t\t\t\tnt AxesOrigin->{2,0}\nFrame\t\t\t\ False\t\tkas t\[OTilde]mmata kast \[UDoubleDot]mber joonise True v\[OTilde]i \ False \n\t\t\t\t\tnt Frame->True\nGridLines\t\tNone\t\tabijooned. Automatic \ lisab iga olulisema v\[ADoubleDot]\[ADoubleDot]rtuse kohale joone\nPlotRange\t\ \tAutomatic\tmilline vahemik joonisele pannakse. Tihti joonistab programm\n\t\ \t\t\t\t\"k\[OTilde]ige huvitavama\" piirkonna, mitte k\[OTilde]ike, \ vastupidine oleks\n\t\t\t\t\tPlotRange->All" }], "Text", Background->GrayLevel[0.900008]], Cell["K\[OTilde]ikv\[OTilde]imalikke valikuid saab leida nn 'help-k\ \[ADoubleDot]su abil:", "Text"], Cell[CellGroupData[{ Cell["4.2. Joonised ruumis.", "Subsubtitle"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`Plot3D[ f, {x, x\_min, x\_max}, {y, y\_min, y\_max}]\)]], "\tteeb kolmem\[OTilde]\[OTilde]tmelise joonise kui funktsiooni kahest \n\t\ \t\t\t\t\tmuutujast x ja y" }], "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(Plot3D[x\^2 + 2 y\^2, {x, \(-2\), 2}, {y, \(-2\), 2}]\)], "Input"], Cell[BoxData[ \(Plot3D[2 y\ Sin[x\ y], {x, \(-1\), \[Pi]}, {y, \(-1\), 2}, AxesLabel \[Rule] {x, y, z}, \[IndentingNewLine]ViewPoint \[Rule] {1.6, \(-1.5\), 2}, Mesh -> False, PlotPoints \[Rule] 25, PlotLabel \[Rule] "\"]\)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 5. Diferentsiaalv\[OTilde]rrandite lahendamine. Integraalide ja diferentsiaalv\[OTilde]rrandite numbriline lahendamine.\ \>", "Subtitle"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(\(DSolve[{vor1\_vas \[Equal] vor1\_par, \ vor2\_vas \[Equal] vor2\_par, ... }, \ {f\_1[x], f\_2[x], ... }, x]\)\(\ \)\)\)]], "\n\t\t\t\tdiferentsiaalv\[OTilde]rrandite s\[UDoubleDot]steemi lahendamine\ \nN", Cell[BoxData[ \(TraditionalForm\`\(\(DSolve[{vor1\_vas \[Equal] vor1\_par, \ vor2\_vas \[Equal] vor2\_par, ... }, \ {f\_1, f\_2, ... }, {x, x\_1, x\_2}]\)\(\ \)\)\)]], "\n\t\t\t\tdiferentsiaalv\[OTilde]rrandite s\[UDoubleDot]steemi numbriline \ lahendamine l\[OTilde]igus ", Cell[BoxData[ \(TraditionalForm\`\([x\_1, x\_2]\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`NIntegrate[f[x], {x, x\_1, x\_2}]\)]], " \t\tnumbriline integreerimine\t\t" }], "Text", Background->GrayLevel[0.900008]], Cell[TextData[{ "Vaatame n\[ADoubleDot]itena j\[ADoubleDot]rgmist \ diferentsiaalv\[OTilde]rrandit: ", Cell[BoxData[ \(TraditionalForm\`\(\(d\^2\) \(y(x)\)\)\/dx\^2 = a\ \(y(x)\) + 1\)]], ", mis on teist j\[ADoubleDot]rku diferentsiaalv\[OTilde]rrand. Selle \ lahendamine:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[\[PartialD]\_x y[x] \[Equal] a*y[x] + 1, y[x], x]\)], "Input"], Cell[BoxData[ \({{y[ x] \[Rule] \(-\(1\/a\)\) + \[ExponentialE]\^\(a\ x\)\ C[ 1]}}\)], "Output"] }, Open ]], Cell["Selle t\[ADoubleDot]histusviisiga analoogiline on \ j\[ADoubleDot]rgmine:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[\(y'\)[x] \[Equal] a*y[x] + 1, y[x], x]\)], "Input"], Cell[BoxData[ \({{y[ x] \[Rule] \(-\(1\/a\)\) + \[ExponentialE]\^\(a\ x\)\ C[ 1]}}\)], "Output"] }, Open ]], Cell["\<\ Nagu n\[ADoubleDot]ha, on siin sees ka kaks konstanti, mis m\[ADoubleDot]\ \[ADoubleDot]ratakse lisatingimustest. Anname ette ka lisatingimused ning vaatame, milline n\[ADoubleDot]eb v\ \[ADoubleDot]lja lahendi graafik.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lahend = DSolve[{\[PartialD]\_x y[x] \[Equal] a*y[x] + 1, y[1] == 2\/a}, y[x], x]\)], "Input"], Cell[BoxData[ \({{y[ x] \[Rule] \(\[ExponentialE]\^\(-a\)\ \((\(-\[ExponentialE]\^a\) \ + 3\ \[ExponentialE]\^\(a\ x\))\)\)\/a}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(lahend[\([1]\)]\)[\([1]\)]\)[\([2]\)]\)], "Input"], Cell[BoxData[ \(\(\[ExponentialE]\^\(-a\)\ \((\(-\[ExponentialE]\^a\) + 3\ \ \[ExponentialE]\^\(a\ x\))\)\)\/a\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[\(\(lahend[\([1]\)]\)[\([1]\)]\)[\([2]\)]\ /. a \[Rule] 3, {x, \(-1\), 2}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ \[CapitalUDoubleDot]laltoodud t\[ADoubleDot]histus lahend[[1]][[1]][[2]] \ annab parasjagu kahekordsetes loogelistes sulgudes oleva avaldise teise \ liikme, st funktsiooni. (Proovige j\[ADoubleDot]rgi, mis on lahend[[1]], \ lahend[[1]][[1]] ja lahend[[1]][[1]][[1]] ). Loomulikult oleksime v\[OTilde]inud lihtsalt defineerida uue funktsiooni \ (cut-paste abil) ning siis selle graafiku \[UDoubleDot]les joonistada juhul, \ kui a=5 n\[ADoubleDot]iteks:\ \>", "Text"], Cell[BoxData[ \(u[x_] := \(\[ExponentialE]\^\(-a\)\ \((\(-\[ExponentialE]\^a\) + 3\ \ \[ExponentialE]\^\(a\ x\))\)\)\/a\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[u[x] /. a \[Rule] 5, {x, \(-2\), 1}, PlotRange \[Rule] All]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Teise n\[ADoubleDot]itena vaatame sumbuvat v\[OTilde]nkumist kirjeldavat v\ \[OTilde]rrandit koos algtingimustega. See on teist j\[ADoubleDot]rku \ lineaarne konstantsete kordajatega diferentsiaalv\[OTilde]rrand: y''(x)+0.01y'(x)+4y(x)==0.2 cos (3x), y'(0)=2, y(0)=10\ \>", "Text"], Cell[BoxData[ \(lahendus = DSolve[{\(y''\)[x] + 0.15 \( y'\)[x] + 4 y[x] \[Equal] 5 Cos[3 x], \[IndentingNewLine]\(y'\)[0] \[Equal] 2, y[0] \[Equal] 10}, y[x], x]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[\(\(lahendus[\([1]\)]\)[\([1]\)]\)[\([2]\)], {x, 0, 70}, PlotPoints \[Rule] 100, PlotRange \[Rule] All]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Kuigi lahendis on kompleksarvud sees, on lahend tegelikult reaalne. I ees \ olevad kordajad on arvutust\[ADoubleDot]psuse piiril. Nagu oli \ n\[ADoubleDot]ha, arvestas programm seda funktsiooni graafiku joonistamisel. \ Kuid selleks, et antud juhul neist neist lahti saada, v\[OTilde]ib kasutada \ funktsiooni Chop:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lahend = Chop[%20]\)], "Input"], Cell[BoxData[ \({{y[ x] \[Rule] \(-1.2421551317942252`\)\ \[ExponentialE]\^\(\(-0.075`\ \)\ x\)\ \((\(-8.849108135915191`\)\ Cos[1.9985932552673142`\ x] + 1.`\ \[ExponentialE]\^\(0.075`\ x\)\ Cos[ 1.0014067447326858`\ x]\ Cos[1.9985932552673142`\ x] - 0.20141610541330285`\ \[ExponentialE]\^\(0.075`\ x\)\ Cos[ 1.9985932552673142`\ x]\ Cos[4.9985932552673145`\ x] - 0.0748946423563588`\ \[ExponentialE]\^\(0.075`\ x\)\ Cos[ 1.9985932552673142`\ x]\ Sin[1.0014067447326858`\ x] - 1.029809493000439`\ Sin[1.9985932552673142`\ x] - 0.0748946423563588`\ \[ExponentialE]\^\(0.075`\ x\)\ Cos[ 1.0014067447326858`\ x]\ Sin[1.9985932552673142`\ x] - 0.0030220918435560655`\ \[ExponentialE]\^\(0.075`\ x\)\ Cos[ 4.9985932552673145`\ x]\ Sin[1.9985932552673142`\ x] - 1.`\ \[ExponentialE]\^\(0.075`\ x\)\ Sin[ 1.0014067447326858`\ x]\ Sin[1.9985932552673142`\ x] + 0.0030220918435560655`\ \[ExponentialE]\^\(0.075`\ x\)\ Cos[ 1.9985932552673142`\ x]\ Sin[4.9985932552673145`\ x] - 0.20141610541330288`\ \[ExponentialE]\^\(0.075`\ x\)\ Sin[ 1.9985932552673142`\ x]\ Sin[ 4.9985932552673145`\ x])\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "Chop[avaldis]\t\t\t\tviskab \[ADoubleDot]ra liikmed, mis on \ v\[ADoubleDot]iksemad kui ", Cell[BoxData[ \(TraditionalForm\`10\^\(-10\)\)]], "\nChop[avaldis, dx]\t\t\tviskab \[ADoubleDot]ra liikmed, mis on v\ \[ADoubleDot]iksemad kui dx\t" }], "Text", Background->GrayLevel[0.900008]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[\(\(lahend[\([1]\)]\)[\([1]\)]\)[\([2]\)], {x, 0, 70}, PlotPoints \[Rule] 100, PlotRange \[Rule] All]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[BoxData[ \(Clear[x, u, t, y]\)], "Input"], Cell["\<\ Tihti pole v\[OTilde]imalik leida diferentsiaalv\[OTilde]rrandi v\[OTilde]i v\ \[OTilde]rrandite s\[UDoubleDot]steemi lahendit \ anal\[UDoubleDot]\[UDoubleDot]tiliselt. Sel juhul tuleb alati ette anda ka algtingimused. Tulemuseks saame funktsiooni, mis on \ esitatud interpoleeritud funktsioonina. Viimast saame omakorda \ l\[ADoubleDot]hemalt uurida graafiku abil.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(funkt = NDSolve[{\(u'\)[t] \[Equal] 2 - \((u[t])\)\^2 + Sin[t], u[0] \[Equal] 2}, u, {t, 0, 15}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"u", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 15.`}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[ Evaluate[u[t] /. \(funkt[\([1]\)]\)[\([1]\)]], {t, 0, 15}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ \[CapitalUDoubleDot]laltoodud protseduuris Evaluate[u[t]/.funkt[[1]][[1]]] \ ongi k\[ADoubleDot]sklus, mille kohaselt programm v\[OTilde]tab funktsiooni \ u[t] v\[ADoubleDot]\[ADoubleDot]rtuse nii nagu diferensiaalv\[OTilde]rrandi \ lahendamisel leitud.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Evaluate[u[3] /. \(funkt[\([1]\)]\)[\([1]\)]\ ]\)], "Input"], Cell[BoxData[ \(1.5524819569378314`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(2*Sin[4]*Evaluate[u[4] /. \(funkt[\([1]\)]\)[\([1]\)]\ ]\)], "Input"], Cell[BoxData[ \(\(-1.883023830775053`\)\)], "Output"] }, Open ]], Cell["\<\ V\[OTilde]ime lahendada ka mitmest diferentsiaalv\[OTilde]rrandist koosneva s\ \[UDoubleDot]steemi.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol = NDSolve[{\(x'\)[t] \[Equal] 2 \((x[t])\)\^2 - 0.1 \((y[t])\)\^2, \[IndentingNewLine]\(y'\)[ t] \[Equal] \(-x[t]\) + y[t], x[0] \[Equal] \(-2\), y[0] \[Equal] 1}, {x, y}, {t, 0, 10}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 10.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"y", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 10.`}}, "<>"]\), False, Editable->False]}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(sol[\([1]\)]\)[\([1]\)]\)], "Input"], Cell[BoxData[ RowBox[{"x", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 10.`}}, "<>"]\), False, Editable->False]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[ Evaluate[x[t] /. \(sol[\([1]\)]\)[\([1]\)]\ ], {t, 0, 10}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[ Evaluate[y[t] /. \(sol[\([1]\)]\)[\([2]\)]\ ], {t, 0, 10}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ M\[OTilde]ned n\[ADoubleDot]ited integraali numbrilise arvutuse kohta. Esmalt \ - kuidas oleks tulemus tavakujul ja NIntegrate abil.\ \>", "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 922}}, WindowToolbars->"EditBar", WindowSize->{851, 597}, WindowMargins->{{2, Automatic}, {Automatic, 5}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->1.25, StyleDefinitions -> "Classic.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 96, 4, 98, "Title"], Cell[1875, 59, 35, 0, 77, "Subtitle"], Cell[1913, 61, 2405, 66, 177, "Text"], Cell[CellGroupData[{ Cell[4343, 131, 38, 1, 45, "Input"], Cell[4384, 134, 35, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4456, 140, 44, 0, 77, "Subtitle"], Cell[4503, 142, 84, 1, 72, "Subsubtitle"], Cell[4590, 145, 57, 0, 38, "Text"], Cell[4650, 147, 11423, 308, 126, "Text"], Cell[16076, 457, 214, 5, 84, "Text"], Cell[CellGroupData[{ Cell[16315, 466, 44, 1, 45, "Input"], Cell[16362, 469, 39, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16438, 475, 40, 1, 45, "Input"], Cell[16481, 478, 36, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16554, 484, 40, 1, 46, "Input"], Cell[16597, 487, 57, 1, 40, "Output"] }, Open ]], Cell[16669, 491, 348, 6, 84, "Text"], Cell[CellGroupData[{ Cell[17042, 501, 38, 1, 46, "Input"], Cell[17083, 504, 50, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17170, 510, 45, 1, 46, "Input"], Cell[17218, 513, 56, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17311, 519, 41, 1, 46, "Input"], Cell[17355, 522, 56, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17448, 528, 41, 1, 46, "Input"], Cell[17492, 531, 53, 1, 40, "Output"] }, Open ]], Cell[17560, 535, 103, 2, 39, "Text"], Cell[17666, 539, 2084, 63, 417, "Text"], Cell[19753, 604, 467, 11, 85, "Text"], Cell[20223, 617, 115, 3, 39, "Text"], Cell[20341, 622, 754, 22, 138, "Text"], Cell[CellGroupData[{ Cell[21120, 648, 142, 3, 94, "Subsubtitle"], Cell[21265, 653, 1274, 34, 183, "Text"], Cell[22542, 689, 63, 0, 38, "Text"], Cell[CellGroupData[{ Cell[22630, 693, 54, 1, 46, "Input"], Cell[22687, 696, 52, 1, 40, "Output"] }, Open ]], Cell[22754, 700, 192, 4, 38, "Text"], Cell[CellGroupData[{ Cell[22971, 708, 51, 1, 45, "Input"], Cell[23025, 711, 71, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[23133, 717, 63, 1, 45, "Input"], Cell[23199, 720, 53, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[23289, 726, 66, 1, 45, "Input"], Cell[23358, 729, 57, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[23452, 735, 70, 1, 46, "Input"], Cell[23525, 738, 58, 1, 40, "Output"] }, Open ]], Cell[23598, 742, 623, 12, 84, "Text"], Cell[CellGroupData[{ Cell[24246, 758, 195, 4, 120, "Input"], Cell[24444, 764, 90, 1, 40, "Output"] }, Open ]], Cell[24549, 768, 1015, 14, 176, "Text"], Cell[25567, 784, 289, 6, 114, "Text"], Cell[25859, 792, 304, 5, 61, "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[26212, 803, 96, 3, 114, "Subtitle"], Cell[26311, 808, 115, 3, 94, "Subsubtitle"], Cell[26429, 813, 53, 1, 45, "Input"], Cell[26485, 816, 3381, 100, 426, "Text"], Cell[29869, 918, 64, 1, 38, "Text"], Cell[CellGroupData[{ Cell[29958, 923, 53, 1, 46, "Input"], Cell[30014, 926, 41, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[30092, 932, 43, 1, 45, "Input"], Cell[30138, 935, 54, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[30229, 941, 43, 1, 45, "Input"], Cell[30275, 944, 46, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[30358, 950, 65, 1, 46, "Input"], Cell[30426, 953, 50, 1, 40, "Output"] }, Open ]], Cell[30491, 957, 885, 22, 84, "Text"], Cell[CellGroupData[{ Cell[31401, 983, 76, 1, 46, "Input"], Cell[31480, 986, 45, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[31562, 992, 72, 1, 46, "Input"], Cell[31637, 995, 35, 1, 40, "Output"] }, Open ]], Cell[31687, 999, 53, 1, 38, "Text"], Cell[CellGroupData[{ Cell[31765, 1004, 91, 1, 62, "Input"], Cell[31859, 1007, 49, 1, 54, "Output"] }, Open ]], Cell[31923, 1011, 304, 5, 61, "Text"], Cell[CellGroupData[{ Cell[32252, 1020, 80, 1, 64, "Input"], Cell[32335, 1023, 40, 1, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[32412, 1029, 75, 1, 62, "Input"], Cell[32490, 1032, 86, 1, 54, "Output"] }, Open ]], Cell[32591, 1036, 1426, 41, 63, "Text"], Cell[34020, 1079, 19848, 248, 282, 19756, 245, "GraphicsData", "Metafile", \ "Graphics"], Cell[CellGroupData[{ Cell[53893, 1331, 153, 2, 69, "Input"], Cell[54049, 1335, 39, 1, 54, "Output"] }, Open ]], Cell[54103, 1339, 383, 8, 61, "Text"], Cell[CellGroupData[{ Cell[54511, 1351, 46, 1, 59, "Input"], Cell[54560, 1354, 43, 1, 55, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[54640, 1360, 54, 1, 45, "Input"], Cell[54697, 1363, 35, 1, 40, "Output"] }, Open ]], Cell[54747, 1367, 212, 4, 61, "Text"], Cell[CellGroupData[{ Cell[54984, 1375, 79, 1, 65, "Input"], Cell[55066, 1378, 80, 1, 58, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[55183, 1384, 54, 1, 45, "Input"], Cell[55240, 1387, 42, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[55319, 1393, 90, 1, 45, "Input"], Cell[55412, 1396, 40, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[55489, 1402, 66, 1, 56, "Input"], Cell[55558, 1405, 94, 2, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[55689, 1412, 56, 0, 72, "Subsubtitle"], Cell[55748, 1414, 99, 2, 66, "Input"], Cell[55850, 1418, 230, 6, 61, "Text"], Cell[56083, 1426, 60, 1, 46, "Input"], Cell[56146, 1429, 196, 4, 61, "Text"], Cell[56345, 1435, 37, 1, 45, "Input"], Cell[56385, 1438, 39, 1, 45, "Input"], Cell[CellGroupData[{ Cell[56449, 1443, 52, 1, 45, "Input"], Cell[56504, 1446, 52, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[56593, 1452, 52, 1, 45, "Input"], Cell[56648, 1455, 36, 1, 40, "Output"] }, Open ]], Cell[56699, 1459, 296, 5, 61, "Text"], Cell[56998, 1466, 41, 1, 45, "Input"], Cell[57042, 1469, 56, 1, 46, "Input"], Cell[CellGroupData[{ Cell[57123, 1474, 49, 1, 45, "Input"], Cell[57175, 1477, 36, 1, 40, "Output"] }, Open ]], Cell[57226, 1481, 468, 9, 84, "Text"], Cell[57697, 1492, 41, 1, 45, "Input"], Cell[57741, 1495, 73, 1, 59, "Input"], Cell[57817, 1498, 45, 1, 45, "Input"], Cell[57865, 1501, 45, 1, 45, "Input"], Cell[57913, 1504, 768, 16, 229, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[58718, 1525, 46, 0, 72, "Subsubtitle"], Cell[58767, 1527, 1491, 46, 130, "Text"], Cell[CellGroupData[{ Cell[60283, 1577, 38, 1, 45, "Input"], Cell[60324, 1580, 35, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[60396, 1586, 45, 1, 45, "Input"], Cell[60444, 1589, 38, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[60519, 1595, 49, 1, 45, "Input"], Cell[60571, 1598, 38, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[60646, 1604, 48, 1, 45, "Input"], Cell[60697, 1607, 39, 1, 40, "Output"] }, Open ]], Cell[60751, 1611, 38, 1, 45, "Input"], Cell[60792, 1614, 164, 5, 38, "Text"], Cell[CellGroupData[{ Cell[60981, 1623, 45, 1, 45, "Input"], Cell[61029, 1626, 40, 1, 40, "Output"] }, Open ]], Cell[61084, 1630, 654, 15, 84, "Text"], Cell[CellGroupData[{ Cell[61763, 1649, 64, 1, 46, "Input"], Cell[61830, 1652, 59, 1, 40, "Output"] }, Open ]], Cell[61904, 1656, 2898, 88, 252, "Text"], Cell[64805, 1746, 299, 7, 61, "Text"], Cell[65107, 1755, 112, 2, 39, "Text"], Cell[65222, 1759, 496, 12, 114, "Text"], Cell[65721, 1773, 238, 4, 61, "Text"], Cell[CellGroupData[{ Cell[65984, 1781, 45, 1, 45, "Input"], Cell[66032, 1784, 78, 1, 44, "Output"] }, Open ]], Cell[66125, 1788, 344, 6, 61, "Text"], Cell[CellGroupData[{ Cell[66494, 1798, 37, 1, 45, "Input"], Cell[66534, 1801, 113, 2, 40, "Output"] }, Open ]], Cell[66662, 1806, 116, 3, 38, "Text"], Cell[CellGroupData[{ Cell[66803, 1813, 56, 1, 45, "Input"], Cell[66862, 1816, 78, 1, 40, "Output"] }, Open ]], Cell[66955, 1820, 290, 9, 38, "Text"], Cell[CellGroupData[{ Cell[67270, 1833, 57, 1, 45, "Input"], Cell[67330, 1836, 58, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[67425, 1842, 57, 1, 45, "Input"], Cell[67485, 1845, 50, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[67572, 1851, 47, 1, 48, "Input"], Cell[67622, 1854, 133, 2, 40, "Output"] }, Open ]], Cell[67770, 1859, 425, 9, 84, "Text"], Cell[CellGroupData[{ Cell[68220, 1872, 72, 1, 46, "Input"], Cell[68295, 1875, 303, 5, 82, "Output"] }, Open ]], Cell[68613, 1883, 158, 3, 38, "Text"], Cell[CellGroupData[{ Cell[68796, 1890, 74, 1, 46, "Input"], Cell[68873, 1893, 318, 5, 61, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[69228, 1903, 60, 1, 45, "Input"], Cell[69291, 1906, 57, 1, 40, "Output"] }, Open ]], Cell[69363, 1910, 444, 9, 61, "Text"], Cell[CellGroupData[{ Cell[69832, 1923, 60, 1, 45, "Input"], Cell[69895, 1926, 55, 1, 40, "Output"] }, Open ]], Cell[69965, 1930, 444, 6, 84, "Text"], Cell[70412, 1938, 1635, 45, 229, "Text"], Cell[CellGroupData[{ Cell[72072, 1987, 142, 2, 45, "Input"], Cell[72217, 1991, 118, 2, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[72372, 1998, 143, 2, 45, "Input"], Cell[72518, 2002, 156, 3, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[72711, 2010, 110, 2, 45, "Input"], Cell[72824, 2014, 55, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[72916, 2020, 114, 2, 45, "Input"], Cell[73033, 2024, 102, 2, 54, "Output"] }, Open ]], Cell[73150, 2029, 821, 11, 153, "Text"], Cell[73974, 2042, 166, 3, 38, "Text"], Cell[CellGroupData[{ Cell[74165, 2049, 66, 1, 45, "Input"], Cell[74234, 2052, 49, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[74320, 2058, 68, 1, 45, "Input"], Cell[74391, 2061, 65, 1, 40, "Output"] }, Open ]], Cell[74471, 2065, 147, 5, 38, "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[74667, 2076, 31, 0, 77, "Subtitle"], Cell[74701, 2078, 45, 0, 72, "Subsubtitle"], Cell[74749, 2080, 550, 15, 114, "Text"], Cell[75302, 2097, 47, 1, 45, "Input"], Cell[75352, 2100, 82, 1, 46, "Input"], Cell[75437, 2103, 110, 2, 46, "Input"], Cell[75550, 2107, 45, 1, 45, "Input"], Cell[75598, 2110, 374, 8, 61, "Text"], Cell[75975, 2120, 97, 1, 38, "Text"], Cell[76075, 2123, 302, 5, 108, "Input"], Cell[76380, 2130, 391, 6, 84, "Text"], Cell[76774, 2138, 57, 1, 45, "Input"], Cell[CellGroupData[{ Cell[76856, 2143, 309, 6, 87, "Input"], Cell[77168, 2151, 130, 3, 40, "Output"] }, Open ]], Cell[77313, 2157, 893, 14, 298, "Text"], Cell[78209, 2173, 100, 1, 38, "Text"], Cell[CellGroupData[{ Cell[78334, 2178, 44, 0, 72, "Subsubtitle"], Cell[78381, 2180, 276, 7, 68, "Text"], Cell[78660, 2189, 87, 1, 46, "Input"], Cell[78750, 2192, 294, 5, 87, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[79093, 2203, 147, 4, 151, "Subtitle"], Cell[79243, 2209, 836, 20, 132, "Text"], Cell[80082, 2231, 297, 7, 66, "Text"], Cell[CellGroupData[{ Cell[80404, 2242, 89, 1, 45, "Input"], Cell[80496, 2245, 128, 3, 54, "Output"] }, Open ]], Cell[80639, 2251, 89, 1, 38, "Text"], Cell[CellGroupData[{ Cell[80753, 2256, 79, 1, 45, "Input"], Cell[80835, 2259, 128, 3, 54, "Output"] }, Open ]], Cell[80978, 2265, 244, 5, 61, "Text"], Cell[CellGroupData[{ Cell[81247, 2274, 130, 3, 59, "Input"], Cell[81380, 2279, 158, 3, 55, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[81575, 2287, 74, 1, 45, "Input"], Cell[81652, 2290, 128, 2, 55, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[81817, 2297, 122, 2, 45, "Input"], Cell[81942, 2301, 130, 3, 40, "Output"] }, Open ]], Cell[82087, 2307, 475, 8, 107, "Text"], Cell[82565, 2317, 136, 2, 60, "Input"], Cell[CellGroupData[{ Cell[82726, 2323, 104, 2, 45, "Input"], Cell[82833, 2327, 130, 3, 40, "Output"] }, Open ]], Cell[82978, 2333, 291, 5, 84, "Text"], Cell[83272, 2340, 213, 4, 66, "Input"], Cell[CellGroupData[{ Cell[83510, 2348, 148, 2, 66, "Input"], Cell[83661, 2352, 130, 3, 40, "Output"] }, Open ]], Cell[83806, 2358, 339, 6, 84, "Text"], Cell[CellGroupData[{ Cell[84170, 2368, 51, 1, 45, "Input"], Cell[84224, 2371, 1451, 21, 187, "Output"] }, Open ]], Cell[85690, 2395, 315, 8, 68, "Text"], Cell[CellGroupData[{ Cell[86030, 2407, 146, 2, 45, "Input"], Cell[86179, 2411, 130, 3, 40, "Output"] }, Open ]], Cell[86324, 2417, 50, 1, 45, "Input"], Cell[86377, 2420, 390, 7, 84, "Text"], Cell[CellGroupData[{ Cell[86792, 2431, 147, 3, 46, "Input"], Cell[86942, 2436, 225, 6, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[87204, 2447, 103, 2, 45, "Input"], Cell[87310, 2451, 130, 3, 40, "Output"] }, Open ]], Cell[87455, 2457, 277, 5, 61, "Text"], Cell[CellGroupData[{ Cell[87757, 2466, 80, 1, 45, "Input"], Cell[87840, 2469, 53, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[87930, 2475, 89, 1, 45, "Input"], Cell[88022, 2478, 57, 1, 40, "Output"] }, Open ]], Cell[88094, 2482, 123, 3, 38, "Text"], Cell[CellGroupData[{ Cell[88242, 2489, 265, 5, 67, "Input"], Cell[88510, 2496, 415, 11, 61, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[88962, 2512, 58, 1, 45, "Input"], Cell[89023, 2515, 157, 4, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[89217, 2524, 103, 2, 45, "Input"], Cell[89323, 2528, 130, 3, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[89490, 2536, 103, 2, 45, "Input"], Cell[89596, 2540, 130, 3, 40, "Output"] }, Open ]], Cell[89741, 2546, 156, 3, 38, "Text"] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)