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K tomu vyu\:017eijeme menu ", StyleBox["Cell", FontWeight->"Bold"], ".\n\[FilledSmallCircle] ", StyleBox["P\:0159esnost v\[YAcute]sledk\:016f", FontWeight->"Bold"], " - v\[YAcute]sledky v tomto programu jsou v\:017edy p\:0159esn\[YAcute]mi, \ exaktn\[IAcute]mi v\[YAcute]sledky.\n\t\[EmptySmallCircle] \ p\:0159\[IAcute]klad:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(2^100\)], "Input"], Cell[BoxData[ \(1267650600228229401496703205376\)], "Output"] }, Open ]], Cell[TextData[{ "\t\[EmptySmallCircle] Na\[SHacek]e v\[YAcute]sledky v\[SHacek]ak mohou m\ \[IAcute]t i numericky aproximovan\[YAcute] tvar, nap\:0159. jako p\:0159i \ pou\:017eit\[IAcute] kalkul\[AAcute]toru. K tomu, abychom dos\[AAcute]hli v\ \[YAcute]sledku v tomto tvaru, u\:017eijeme funkce ", StyleBox["//N", FontWeight->"Bold"], " (pop\:0159. 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Shrnut\[IAcute] z\[AAcute]kladn\[IAcute]ch pravidel", "Text", TextAlignment->Center, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", Background->GrayLevel[0.900008]], Cell[TextData[{ "\t\t\[FilledSmallCircle] Argumenty a funkce uv\[AAcute]d\[IAcute]me v \ hranat\[YAcute]ch z\[AAcute]vork\[AAcute]ch.\n", StyleBox["\t\t", FontSize->9], "\n\t\t\[FilledSmallCircle] N\[AAcute]zvy funkc\[IAcute] zad\[AAcute]v\ \[AAcute]me s velk\[YAcute]m p\[IAcute]smenem na za\[CHacek]\[AAcute]tku.\n", StyleBox["\t\t", FontSize->9], "\n\t\t\[FilledSmallCircle] N\[AAcute]zvy prom\:011bnn\[YAcute]ch a n\ \[AAcute]mi definovan\[YAcute]ch funkc\[IAcute] zad\[AAcute]v\[AAcute]me mal\ \[YAcute]mi p\[IAcute]smeny.\n", StyleBox["\t\t", FontSize->9], "\n\t\t\[FilledSmallCircle] K umocn\:011bn\[IAcute] pou\:017e\[IAcute]v\ \[AAcute]me ^ nebo palety n\[AAcute]stroj\:016f ", Cell[BoxData[ \(TraditionalForm\`\[Placeholder]\^\[Placeholder]\)]], ".\n", StyleBox["\t\t", FontSize->9], "\n\t\t\[FilledSmallCircle] ", StyleBox["N\[AAcute]soben\[IAcute] m\:016f\:017ee b\[YAcute]t prezentov\ \[AAcute]no mezerou!", FontWeight->"Bold"], "\n", StyleBox["\t\t", FontSize->9], "\n\t\t\[FilledSmallCircle] K z\[AAcute]pisu desetinn\[EAcute]ho \[CHacek]\ \[IAcute]sla pou\:017e\[IAcute]v\[AAcute]me te\[CHacek]ku.\n", StyleBox["\t\t", FontSize->9], "\n\t\t\[FilledSmallCircle] \[CapitalCHacek]\[IAcute]sla ve v\:011bdeck\ \[EAcute]m z\[AAcute]pisu vypadaj\[IAcute] nap\:0159. takto ", StyleBox["1.4*10^-23", FontWeight->"Bold"], " nebo ", StyleBox["1.4 10^-23", FontWeight->"Bold"], ".\n", StyleBox["\t\t", FontSize->9], "\n\t\t\[FilledSmallCircle] N\[AAcute]pov\:011bdu k jak\[EAcute]koliv \ funkci lze z\[IAcute]skat pomoc\[IAcute] kl\[AAcute]vesy F1.\n\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 1. Diferenci\[AAcute]ln\[IAcute] po\[CHacek]et funkce jedn\[EAcute] \ prom\:011bnn\[EAcute]\ \>", "Section", TextAlignment->Center, CellTags->"LastSlide"], Cell["\<\ 1.1. Pr\[AAcute]ce s prom\:011bnnou. Re\[AAcute]ln\[AAcute] funkce \ re\[AAcute]ln\[EAcute] prom\:011bnn\[EAcute]. Grafy funkc\[IAcute]\ \>", "Text", TextAlignment->Center, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox["1.1.1. Definice prom\:011bnn\[EAcute] a pr\[AAcute]ce \ s prom\:011bnn\[YAcute]mi", FontSize->12]], "Text", CellDingbat->None, TextAlignment->Center, FontSize->14, FontWeight->"Bold", Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] Prom\:011bnnou definujeme pomoc\[IAcute] syntaxe:\t", StyleBox["n\[AAcute]zev_prom\:011bnn\[EAcute] = zvolen\[AAcute]_hodnota", FontWeight->"Bold"], ".", StyleBox["\n\t", FontWeight->"Bold"], "\[EmptySmallCircle]", StyleBox[" ", FontWeight->"Bold"], "p\:0159\[IAcute]klad:" }], "Text", TextAlignment->Left], Cell[CellGroupData[{ Cell[BoxData[ \(x\ = \ 12\)], "Input"], Cell[BoxData[ \(12\)], "Output"] }, Open ]], Cell[TextData[{ "\[FilledSmallCircle] Nyn\[IAcute] m\[AAcute]me v prom\:011bnn\[EAcute] ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], " ulo\:017eenu hodnotu ", StyleBox["12", FontWeight->"Bold"], ". Prove\:010fme v\[YAcute]po\[CHacek]et v\[YAcute]razu ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], ", za prom\:011bnnou ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], " je automaticky dosazena jej\[IAcute] hodnota.\n\t\[EmptySmallCircle]", StyleBox[" ", FontWeight->"Bold"], "p\:0159\[IAcute]klad:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x^2\)], "Input"], Cell[BoxData[ \(144\)], "Output"] }, Open ]], Cell[TextData[{ "\[FilledSmallCircle] Dal\[SHacek]\[IAcute] definov\[AAcute]n\[IAcute] prom\ \:011bnn\[EAcute] se stejn\[YAcute]m n\[AAcute]zvem (v na\[SHacek]em p\:0159\ \[IAcute]pad\:011b ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], ") zp\:016fsob\[IAcute] p\:0159eps\[AAcute]n\[IAcute] \ p\:0159edchoz\[IAcute] hodnoty.\n\t\[EmptySmallCircle]", StyleBox[" ", FontWeight->"Bold"], "p\:0159\[IAcute]klad:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x\ = \ 2\ + \ a\)], "Input"], Cell[BoxData[ \(2 + a\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] Tato nov\[AAcute] hodnota prom\:011bnn\[EAcute] ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], " se pak st\[AAcute]v\[AAcute] platnou pro dal\[SHacek]\[IAcute] po\:017eit\ \[IAcute] dan\[EAcute] prom\:011bnn\[EAcute].\n\t\[EmptySmallCircle]", StyleBox[" ", FontWeight->"Bold"], "p\:0159\[IAcute]klad:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x^2\)], "Input"], Cell[BoxData[ \(\((2 + a)\)\^2\)], "Output"] }, Open ]], Cell[TextData[{ "\[FilledSmallCircle] Pro vymaz\[AAcute]n\[IAcute] definice prom\:011bnn\ \[EAcute] pou\:017eijeme symbolu ", StyleBox["te\[CHacek]ka", FontWeight->"Bold"], ". Prom\:011bnn\[EAcute] ji\:017e nebude p\:0159i\:0159azena \ \:017e\[AAcute]dn\[AAcute] hodnota a bude vystupovat pouze jako symbol." }], "Text"], Cell[BoxData[ \(x\ = \ . \)], "Input"], Cell[TextData[{ "\n\[FilledSmallCircle] V n\[AAcute]sleduj\[IAcute]c\[IAcute]m \ v\[YAcute]razu pak nebude p\:0159i\:0159azena ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], " \:017e\[AAcute]dn\[AAcute] hodnota" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x - 2 + 3\ x\)], "Input"], Cell[BoxData[ \(\(-2\) + 4\ x\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] Chceme-li p\:0159i\:0159adit prom\:011bnn\[EAcute] \ hodnotu pouze v jednom v\[YAcute]razu, u\:017eijeme \ n\[AAcute]sleduj\[IAcute]c\[IAcute] syntaxe:\n\t\t\t\t", StyleBox["v\[YAcute]raz/.x\[Rule]zvolen\[AAcute]_hodnota", FontWeight->"Bold"], ".\n", StyleBox["\t\t\t\t", FontSize->9], "\n\t\[FilledSmallSquare] ", StyleBox["pozn.:", FontSlant->"Italic"], StyleBox[" Mathematica CalcCenter ", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["tuto syntaxi pro p\:0159i\:0159azen\[IAcute] nepodporuje\n\t", FontWeight->"Bold"], "\[EmptySmallCircle]", StyleBox[" ", FontWeight->"Bold"], "p\:0159\[IAcute]klad:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(26 - x^3 /. x \[Rule] 2\)], "Input"], Cell[BoxData[ \(18\)], "Output"] }, Open ]], Cell[TextData[{ "\t\[EmptySmallCircle]", StyleBox[" ", FontWeight->"Bold"], "p\:0159\[IAcute]klad:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(26 - x^3 /. x \[Rule] 2 - a\)], "Input"], Cell[BoxData[ \(26 - \((2 - a)\)\^3\)], "Output"] }, Open ]], Cell[TextData[{ "\[FilledSmallCircle] P\:0159esv\:011bd\[CHacek]me se, \:017ee v ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], " nen\[IAcute] ulo\:017eena \:017e\[AAcute]dn\[AAcute] hodnota, tj. \:017ee \ p\:0159edchoz\[IAcute] definice prom\:011bnn\[EAcute] ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], " byly pouze pro dan\[EAcute] v\[YAcute]razy." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(26 - \ x^3\)], "Input"], Cell[BoxData[ \(26 - x\^3\)], "Output"] }, Open ]], Cell["\<\ \[FilledSmallCircle] Definici m\:016f\:017eeme prov\[EAcute]st i pro v\ \[IAcute]ce promenn\[YAcute]ch. Definici v\[SHacek]ak mus\[IAcute]me uzav\ \:0159\[IAcute]t do slo\:017een\[YAcute]ch z\[AAcute]vorek.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((x + y)\)*\((x - y)\)^2 /. {x \[Rule] 3, y \[Rule] 1 - a}\)], "Input"], Cell[BoxData[ \(\((4 - a)\)\ \((2 + a)\)\^2\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[StyleBox["1.1.2. 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Za t\[IAcute]mto symbolem pak n\[AAcute]sleduje vyj\[AAcute]d\:0159en\ \[IAcute] samotn\[EAcute] funkce.\n", StyleBox["\t", FontWeight->"Bold"], "\[EmptySmallCircle]", StyleBox[" ", FontWeight->"Bold"], "p\:0159\[IAcute]klad:" }], "Text", TextAlignment->Left], Cell[BoxData[ \(f[x_] := x^2\)], "Input"], Cell[TextData[{ "\t\[EmptySmallCircle] ", StyleBox["pozor: ! podtr\:017e\[IAcute]tko za prom\:011bnnou v definici \ funkce nesm\[IAcute]me zapomenout !", FontWeight->"Bold"], "\n\[FilledSmallCircle] Funkci pak vol\[AAcute]me jej\[IAcute]m \ n\[AAcute]zvem a argumentem v hranat\[YAcute]ch z\[AAcute]vork\[AAcute]ch. \ Argumentem funkce m\:016f\:017ee b\[YAcute]t \[CHacek]\[IAcute]slo nebo jak\ \[YAcute]koliv v\[YAcute]raz.\n\t\[EmptySmallCircle]", StyleBox[" ", FontWeight->"Bold"], "p\:0159\[IAcute]klady:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[5]\)], "Input"], Cell[BoxData[ \(25\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f[2 + a]\)], "Input"], Cell[BoxData[ \(\((2 + a)\)\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f[x^2 + 3\ x + 6]\)], "Input"], Cell[BoxData[ \(\((6 + 3\ x + x\^2)\)\^2\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] N\[AAcute]mi definovanou funkci m\:016f\:017eeme pou\ \:017e\[IAcute]t k r\:016fzn\[YAcute]m v\[YAcute]po\[CHacek]t\:016fm v \ kombinaci s jin\[YAcute]mi funkcemi vestav\:011bn\[YAcute]mi v prost\:0159ed\ \[IAcute] ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Sqrt[f[4]]\)], "Input"], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell[TextData[{ "\[FilledSmallCircle] Pro zobrazen\[IAcute] definice n\[AAcute]mi vytvo\ \:0159en\[EAcute] funkce lze vyu\:017eit syntaxe ", StyleBox["?n\[AAcute]zev_funkce", FontWeight->"Bold"], ".", StyleBox["\n\t", FontWeight->"Bold"], "\[EmptySmallCircle]", StyleBox[" ", FontWeight->"Bold"], "p\:0159\[IAcute]klad:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?f\)\)], "Input"], Cell["Global`f", "Print", CellTags->"Info3373221891-2629061"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {\(f[x_] := x\^2\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ "f"], Editable->False]], "Print", CellTags->"Info3373221891-2629061"] }, Open ]], Cell[TextData[{ "\[FilledSmallCircle] Pro vymaz\[AAcute]n\[IAcute] n\[AAcute]mi \ vytvo\:0159en\[EAcute] funkce vyu\:017eijeme v\[YAcute]razu ", StyleBox["Clear[n\[AAcute]zev_funkce]", FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[ \(Clear[f]\)], "Input"], Cell[TextData[{ "\[FilledSmallCircle] Jako v p\:0159\[IAcute]pad\:011b \ prom\:011bnn\[EAcute] i zde lze p\:0159edefinovat n\[AAcute]mi \ vytvo\:0159enou funkci. Vytvo\:0159me se funkci ", StyleBox["mocnina", FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[ \(mocnina[x_] := x^2\)], "Input"], Cell[TextData[{ "\n\[FilledSmallCircle] Za ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], " dosa\:010fme nap\:0159. \[CHacek]\[IAcute]slo 3" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mocnina[3]\)], "Input"], Cell[BoxData[ \(9\)], "Output"] }, Open ]], Cell[TextData[{ "\[FilledSmallCircle] P\:0159edefinujme nyn\[IAcute] funkci ", StyleBox["mocnina", FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[ \(mocnina[x_] := \ x^3\)], "Input"], Cell[TextData[{ "\n\[FilledSmallCircle] A op\:011bt dosa\:010fme za ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], " \[CHacek]\[IAcute]slo 3." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mocnina[3]\)], "Input"], Cell[BoxData[ \(27\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[StyleBox["1.1.3. Grafy funkc\[IAcute]", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\t\[FilledSmallSquare] pozn.: ", StyleBox["Tvorba a \[UAcute]prava graf\:016f v ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Bold"], StyleBox["CalcCenter", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" je velmi zjednodu\[SHacek]en\[AAcute].", FontWeight->"Bold"], "\n\n\[FilledSmallCircle] Pro vykreslen\[IAcute] 2D graf\:016f funkc\ \[IAcute] je v ", StyleBox["Mathematice", FontSlant->"Italic"], " zabudov\[AAcute]na funkce ", StyleBox["Plot", FontWeight->"Bold"], ". Pokus\[IAcute]m se zde vystihnout z\[AAcute]kladn\[IAcute] vlastnosti t\ \[EAcute]to funkce, v\[SHacek]e budu prezentovat na p\:0159\[IAcute]slu\ \[SHacek]n\[YAcute]ch p\:0159\[IAcute]kladech." }], "Text", TextAlignment->Left], Cell[TextData[{ "\[FilledSmallCircle] V argumentu funkce Plot mus\[IAcute] b\[YAcute]t zad\ \[AAcute]na nejd\:0159\[IAcute]ve vykreslovac\[IAcute] funkce a d\[AAcute]le \ ve slo\:017een\[YAcute]ch z\[AAcute]vork\[AAcute]ch prom\:011bnn\[AAcute], \ pro kterou je funkce vykreslov\[AAcute]na, a meze t\[EAcute]to \ nez\[AAcute]visl\[EAcute] prom\:011bnn\[EAcute]. 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Derivace funkce", "Text", TextAlignment->Center, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox["1.3.1. V\[YAcute]po\[CHacek]et derivace", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] V prost\:0159ed\[IAcute] ", StyleBox["Mathematica", FontSlant->"Italic"], " je mo\:017eno derivaci zad\[AAcute]vat v\[IAcute]ce zp\:016fsoby. Pod\ \[IAcute]vejme se nyn\[IAcute] na tyto mo\:017enosti:\n\t\[EmptySmallCircle] \ pomoc\[IAcute] symbolu", StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_\[Placeholder]\)\(\ \)\)\)], FontWeight->"Bold"], "nebo funkce ", StyleBox["D", FontWeight->"Bold"], "\t\t\t", Cell[BoxData[ FormBox[ StyleBox[\(\[PartialD]\_x\ fce\), "Text", FontWeight->"Bold"], TraditionalForm]]], " nebo ", StyleBox["D[fce, x]", FontWeight->"Bold"], "\n\t\[EmptySmallCircle] pomoc\[IAcute] funkce ", StyleBox["Derivative", FontWeight->"Bold"], " \t\t\t\t", StyleBox["Derivative[1][fce][x]", FontWeight->"Bold"], "\n\t\t\[FilledVerySmallSquare] zde v\[SHacek]ak nejd\:0159\[IAcute]ve mus\ \[IAcute]me definovat funkci, kterou chceme derivovat, nap\:0159. ", StyleBox["funkce[x_]:=x^2", FontWeight->"Bold"], ", pak ji\:017e m\:016f\:017eeme pou\:017e\[IAcute]t ", StyleBox["Derivative[1][funkce][x]\n\t\t\t\[EmptyVerySmallSquare] ", FontWeight->"Bold"], "pozn.: \[CHacek]\[IAcute]slo v prvn\[IAcute] z\[AAcute]vorce za slovem \ Derivative znamen\[AAcute] stupe\:0148 derivace, tak\:017ee v na\[SHacek]em p\ \:0159\[IAcute]pad\:011b se jedn\[AAcute] o prvn\[IAcute] derivaci.\n\t\ \[EmptySmallCircle] klasicky pomoc\[IAcute] apostrofu\t\t\t\t", StyleBox["f ' [x]", FontWeight->"Bold"], "\n\t\t\[FilledVerySmallSquare] plat\[IAcute] zde to sam\[EAcute] jako v p\ \:0159edchz\[IAcute]m bod\:011b, nejd\:0159\[IAcute]ve definujeme funkci a \ pak ji zderivujeme, tzn.\n\t\t\t", StyleBox["derivovana_fce[x_]:= x^3+x", FontWeight->"Bold"], "\n\t\t\t", StyleBox["derivovana_fce'[x]", FontWeight->"Bold"] }], "Text"], Cell["", "Text", TextAlignment->Center], Cell["\<\ \t\[EmptySmallCircle] uv\:011b\:010fme si p\[AAcute]r p\:0159\[IAcute]klad\ \:016f na r\:016fzn\[EAcute] z\[AAcute]pisy derivace:\ \>", "Text", TextAlignment->Left], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x\((x^4 - 3 x^2 + 10)\)\)], "Input"], Cell[BoxData[ \(\(-6\)\ x + 4\ x\^3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x^4 - 3 x^2 + 10, \ x]\)], "Input"], Cell[BoxData[ \(\(-6\)\ x + 4\ x\^3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f1[x_] := x^4 - 3 x^2 + 10; \ \(\(Derivative[1]\)[f1]\)[x]\)], "Input"], Cell[BoxData[ \(\(-6\)\ x + 4\ x\^3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f2[x_] := x^4 - 3 x^2 + 10; \ \(f2'\)[x]\)], "Input"], Cell[BoxData[ \(\(-6\)\ x + 4\ x\^3\)], "Output"] }, Open ]], Cell["", "Text"], Cell[TextData[{ "\t\[EmptySmallCircle] derivace v dan\[EAcute]m bod\:011b - vyu\:017eijeme \ funkce ", StyleBox["Derivative", FontWeight->"Bold"], ", m\[IAcute]sto prom\:011bnn\[EAcute], podle kter\[EAcute] derivujeme, nap\ \[IAcute]\[SHacek]eme hodnotu:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fce[x_] := 3\ x^2\ - \ 4\ x; \ \(\(Derivative[1]\)[fce]\)[ 3]\)], "Input"], Cell[BoxData[ \(14\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[StyleBox["1.3.2. P\:0159\[IAcute]klady", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] P\:0159\[IAcute]klady k procvi\[CHacek]en\[IAcute] - \ derivujte n\[AAcute]sleduj\[IAcute]c\[IAcute] funkce (podle prom\:011bnn\ \[EAcute] ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], "):\n\n\t", StyleBox["a", FontWeight->"Bold"], ")\tf1(x) = arctg(x)\n\n\t", StyleBox["b", FontWeight->"Bold"], ")\tf2(x) = argcosh(x)\n\n\t", StyleBox["c", FontWeight->"Bold"], ")\tf3(x) = arcsin(", Cell[BoxData[ \(TraditionalForm\`\(1 - x\^2\)\/\(1 + x\^2\)\)]], ")\t\t\t\t\t(obecn\:011b a pro x = 1)\n\n\t", StyleBox["d", FontWeight->"Bold"], ")\tf4(x) = ", Cell[BoxData[ \(TraditionalForm\`\(ln\ \(cos(x)\)\)\/\(ln\ x\)\)]], "\n\n\t", StyleBox["e", FontWeight->"Bold"], ")\tf5(x) = (1+x)", Cell[BoxData[ \(TraditionalForm\`\(\@\(2 + x\^2\)\) \@\(3 + x\^3\)\%3\)]], "\t\t\t(obecn\:011b a pro x = 0)\n\n\t", StyleBox["f", FontWeight->"Bold"], ")\tf6(x) = exp x + exp(exp x) + exp(exp(exp x))\n\n\t", StyleBox["g", FontWeight->"Bold"], ")\tf7(x) = ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)]], "(1+cotg", Cell[BoxData[ \(TraditionalForm\`x\/2\)]], ")\t\t\t\t\t(obecn\:011b a pro x = \[Pi])" }], "Text"], Cell["", "Text", TextAlignment->Center], Cell[TextData[StyleBox["1.3.3. V\[YAcute]sledky", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] \:0158e\[SHacek]en\[IAcute]:\n\t", StyleBox["a", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f1[x_] := \ ArcTan[x]; \ \(f1'\)[x]\)], "Input"], Cell[BoxData[ \(1\/\(1 + x\^2\)\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["b", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f2[x_] := ArcCosh[x]; \ \(f2'\)[x]\)], "Input"], Cell[BoxData[ \(1\/\(\@\(\(-1\) + x\)\ \@\(1 + x\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["c", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f3[x_] := ArcSin[\((1 - x^2)\)/\((1 + x^2)\)]; \ \(f3'\)[x]\)], "Input"], Cell[BoxData[ \(\(\(-\(\(2\ x\ \((1 - x\^2)\)\)\/\((1 + x\^2)\)\^2\)\) - \(2\ x\)\/\(1 \ + x\^2\)\)\/\@\(1 - \((1 - x\^2)\)\^2\/\((1 + x\^2)\)\^2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(f3'\)[1]\)], "Input"], Cell[BoxData[ \(\(-1\)\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["d", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f4[x_] := Log[Cos[x]]/Log[x]; \ \(f4'\)[x]\)], "Input"], Cell[BoxData[ \(\(-\(Log[Cos[x]]\/\(x\ Log[x]\^2\)\)\) - Tan[x]\/Log[x]\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["e", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f5[x_] := \ \((1 + x)\)\ \@\(2 + x^2\)\ \@\(3 + x^3\)\%3; \ \(f5'\)[ x]\)], "Input"], Cell[BoxData[ \(\(x\^2\ \((1 + x)\)\ \@\(2 + x\^2\)\)\/\((3 + x\^3)\)\^\(2/3\) + \(x\ \ \((1 + x)\)\ \((3 + x\^3)\)\^\(1/3\)\)\/\@\(2 + x\^2\) + \@\(2 + x\^2\)\ \((3 \ + x\^3)\)\^\(1/3\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(f5'\)[0]\)], "Input"], Cell[BoxData[ \(\@2\ 3\^\(1/3\)\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["f", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f6[x_] := \[ExponentialE]\^x + \[ExponentialE]\^\(\[ExponentialE]\^x\) \ + \[ExponentialE]\^\(\[ExponentialE]\^\(\[ExponentialE]\^x\)\); \ \(f6'\)[ x]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^x + \[ExponentialE]\^\(\[ExponentialE]\^x + x\) + \ \[ExponentialE]\^\(\[ExponentialE]\^\(\[ExponentialE]\^x\) + \ \[ExponentialE]\^x + x\)\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["g", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f7[x_] := \[ExponentialE]\^x\ \((1 + Cot[x\/2])\); \ \(f7'\)[ x]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^x\ \((1 + Cot[x\/2])\) - 1\/2\ \[ExponentialE]\^x\ Csc[x\/2]\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(f7'\)[Pi]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\[Pi]\/2\)], "Output"] }, Open ]], Cell[TextData[{ "\t\t\[FilledVerySmallSquare] pozn.: funkce Csc[x] je kosekans a je roven \ Csc[x] = ", Cell[BoxData[ \(TraditionalForm\`1\/Sin[x]\)]], " \n" }], "Text"], Cell["\<\ \ \>", "Text", TextAlignment->Center], Cell["1.4. Diferenci\[AAcute]l funkce", "Text", TextAlignment->Center, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox["1.4.1. V\[YAcute]po\[CHacek]et", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\t\[FilledSmallSquare] pozn.: ", StyleBox["Diferenci\[AAcute]l funkce nen\[IAcute] v ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Bold"], StyleBox["CalcCenter", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" podporov\[AAcute]n.", FontWeight->"Bold"], "\n\[FilledSmallCircle] V prost\:0159ed\[IAcute] ", StyleBox["Mathematica", FontSlant->"Italic"], " lze diferenci\[AAcute]l funkce ur\[CHacek]it pomoc\[IAcute] vestav\:011bn\ \[EAcute] funkce Dt. Syntaxe je n\[AAcute]sleduj\[IAcute]c\[IAcute]:\n\t\ \[EmptySmallCircle] ", StyleBox["Dt [fce]\t", FontWeight->"Bold"], "najde diferenci\[AAcute]l funkce ", StyleBox["fce", FontSlant->"Italic"], ".\n\n\[FilledSmallCircle] Uv\:011b\:010fme si n\:011bkolik \ p\:0159\[IAcute]klad\:016f:\n\t\[EmptySmallCircle] p\:0159\[IAcute]klad 1." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[3\ x + 4]\)], "Input"], Cell[BoxData[ \(3\ Dt[x]\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 2.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[2\ x^2\ - 3\ x + 4]\)], "Input"], Cell[BoxData[ \(\(-3\)\ Dt[x] + 4\ x\ Dt[x]\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 3.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Dt[2\ x^2\ - 3\ x + 4]]\)], "Input"], Cell[BoxData[ \(\((\(-3\) + 4\ x)\)\ Dt[x]\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 4.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[a\ x, \ Constants \[Rule] a]\)], "Input"], Cell[BoxData[ \(a\ Dt[x, Constants \[Rule] {a}]\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] Pokud bychom v p\:0159\[IAcute]kladu 4. nedefinovali \ konstantu ", StyleBox["a", FontSlant->"Italic"], " pomoc\[IAcute] parametru ", StyleBox["Constants", FontWeight->"Bold"], ", ", StyleBox["Mathematica", FontSlant->"Italic"], " by s touto konstantou pracovala jako s prom\:011bnnou z\[AAcute]vislou na \ ", StyleBox["x", FontSlant->"Italic"], ", tj. ", StyleBox["a", FontSlant->"Italic"], " = ", StyleBox["a", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ").\t\n\t\[EmptySmallCircle] p\:0159\[IAcute]klad 5." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[\ a\ x^2 + b\ x + c, \ Constants \[Rule] {a, b, c}]\)], "Input"], Cell[BoxData[ \(b\ Dt[x, Constants \[Rule] {a, b, c}] + 2\ a\ x\ Dt[x, Constants \[Rule] {a, b, c}]\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 6.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Dt[\ a\ x^2 + b\ x + c, \ Constants \[Rule] {a, b, c}]]\)], "Input"], Cell[BoxData[ \(\((b + 2\ a\ x)\)\ Dt[x, Constants \[Rule] {a, b, c}]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Out[108]\ /. \ Dt[x, Constants \[Rule] {a, b, c}] \[Rule] dx\)], "Input"], Cell[BoxData[ \(dx\ \((b + 2\ a\ x)\)\)], "Output"] }, Open ]], Cell["\<\ \ \>", "Text", TextAlignment->Center], Cell["1.5. Derivace vy\[SHacek]\[SHacek]\[IAcute]ch \:0159\[AAcute]d\:016f", \ "Text", TextAlignment->Center, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox["1.5.1. V\[YAcute]po\[CHacek]et", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell["\<\ \[FilledSmallCircle] K zad\[AAcute]v\[AAcute]n\[IAcute] derivace vy\[SHacek]\ \[SHacek]\[IAcute]ch \:0159\[AAcute]d\:016f se vyu\:017e\[IAcute]v\[AAcute] \ stejn\[YAcute]ch funkc\[IAcute] jako v p\:0159\[IAcute]pad\:011b derivace \ prvn\[IAcute]ho \:0159\[AAcute]du, uve\:010fme si nyn\[IAcute] tyto funkce v \ obecn\[EAcute]m tvaru:\ \>", "Text"], Cell[TextData[{ "\t\[EmptySmallCircle] funkce ", StyleBox["D", FontWeight->"Bold"], "\t\t\t\t ", StyleBox["D[fce, {x, n}]", FontWeight->"Bold"], "\n\t\[EmptySmallCircle] ", StyleBox["Derivative", FontWeight->"Bold"], " \t\t\t\t", StyleBox["Derivative[n][fce][x]", FontWeight->"Bold"], "\n\t\[EmptySmallCircle] klasicky pomoc\[IAcute] apostrofu\t\t", StyleBox["f '''' [x]", FontWeight->"Bold"], "\n\t\t\[FilledVerySmallSquare] pozn.: kolik apostrof\:016f, \ tolik\[AAcute]t\[AAcute] derivace se bude po\[CHacek]\[IAcute]tat, ", StyleBox["pozor - neplat\[IAcute] zde tento z\[AAcute]pis ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`f\^\((n)\)\)], FontWeight->"Bold"], StyleBox[" jako n-t\[AAcute] derivace!", FontWeight->"Bold"], "\n\t\[EmptySmallCircle] op\:011bt si uvedeme n\:011bkolik \ p\:0159\[IAcute]klad\:016f:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[x^4 - 3 x^2 + 10, \ {x, \ 3}]\)], "Input"], Cell[BoxData[ \(24\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f1[x_] := x^4 - 3 x^2 + 10; \ \(\(Derivative[3]\)[f1]\)[x]\)], "Input"], Cell[BoxData[ \(24\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f2[x_] := x^4 - 3 x^2 + 10; \ \(f2'''\)[x]\)], "Input"], Cell[BoxData[ \(24\ x\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[StyleBox["1.5.2. P\:0159\[IAcute]klady", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] P\:0159\[IAcute]klady k procvi\[CHacek]en\[IAcute] - \ derivujte n\[AAcute]sleduj\[IAcute]c\[IAcute] funkce (podle prom\:011bnn\ \[EAcute] ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], "), po\:017eadovan\[YAcute] \:0159\[AAcute]d je uveden v z\[AAcute]vork\ \[AAcute]ch:\n\n\t", StyleBox["a", FontWeight->"Bold"], ")\tn1(x) = x", Cell[BoxData[ \(TraditionalForm\`\((2 x - 1)\)\^2\)]], Cell[BoxData[ \(TraditionalForm\`\((x + 3)\)\^3\)]], "\t\t(najd\:011bte 6. a 7. derivaci)\n\n\t", StyleBox["b", FontWeight->"Bold"], ")\tn2(x) = ", Cell[BoxData[ \(TraditionalForm\`\(1 + x\)\/\@\(1 - x\)\)]], "\t\t\t\t(najd\:011bte 3. derivaci)\n\n\t", StyleBox["c", FontWeight->"Bold"], ")\tn3(x) = ", Cell[BoxData[ \(TraditionalForm\`\(sin\^2\) x\)]], "ln x\t\t\t(najd\:011bte 2. derivaci)\n\n\t", StyleBox["d", FontWeight->"Bold"], ")\tn4(x) = ", Cell[BoxData[ \(TraditionalForm\`\(cos(3 x)\)\/\@\(1 - 3 x\)\%3\)]], "\t\t\t(najd\:011bte 3. derivaci)" }], "Text"], Cell["", "Text", TextAlignment->Center], Cell[TextData[StyleBox["1.5.3. V\[YAcute]sledky", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] \:0158e\[SHacek]en\[IAcute]:\n\t", StyleBox["a", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[x\ \((2\ x - 1)\)\^2\ \((x + 3)\)\^3, \ {x, \ 6}]\)], "Input"], Cell[BoxData[ \(2880\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(n1[x_] := x\ \((2\ x - 1)\)\^2\ \((x + 3)\)\^3; \ \(\(Derivative[7]\)[n1]\)[ x]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["b", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(n2[x_] := \ \((1 + x)\)/Sqrt[1 - x]; \ \(\(Derivative[3]\)[n2]\)[ x]\)], "Input"], Cell[BoxData[ \(9\/\(4\ \((1 - x)\)\^\(5/2\)\) + \(15\ \((1 + x)\)\)\/\(8\ \((1 - \ x)\)\^\(7/2\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["c", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[Sin[x]^2\ Log[x], \ {x, 2}]\)], "Input"], Cell[BoxData[ \(\(4\ Cos[x]\ Sin[x]\)\/x - Sin[x]\^2\/x\^2 + Log[x]\ \((2\ Cos[x]\^2 - 2\ Sin[x]\^2)\)\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["d", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(n4[x_] := Cos[3\ x]/\@\(1 - 3\ x\)\%3; \ \(n4'''\)[x]\)], "Input"], Cell[BoxData[ \(\(28\ Cos[3\ x]\)\/\((1 - 3\ x)\)\^\(10/3\) - \(27\ Cos[3\ x]\)\/\((1 - \ 3\ x)\)\^\(4/3\) - \(36\ Sin[3\ x]\)\/\((1 - 3\ x)\)\^\(7/3\) + \(27\ Sin[3\ \ x]\)\/\((1 - 3\ x)\)\^\(1/3\)\)], "Output"] }, Open ]], Cell["\<\ \ \>", "Text", TextAlignment->Center] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 2. Diferenci\[AAcute]ln\[IAcute] po\[CHacek]et funkce dvou a v\[IAcute]ce \ prom\:011bnn\[YAcute]ch\ \>", "Section", TextAlignment->Center, CellTags->"LastSlide"], Cell["\<\ 2.1. Re\[AAcute]ln\[AAcute] funkce v\[IAcute]ce re\[AAcute]ln\[YAcute]ch prom\ \:011bnn\[YAcute]ch\ \>", "Text", TextAlignment->Center, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox["2.1.1.", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] Pro zopakov\[AAcute]n\[IAcute] uve\:010fme, \:017ee \ re\[AAcute]lnou funkci jedn\[EAcute] re\[AAcute]ln\[EAcute] prom\:011bnn\ \[EAcute] definujeme jako:\n\t\t\t\t\t\t\t", StyleBox["f [prom_]:= v\[YAcute]raz", FontWeight->"Bold"], "." }], "Text"], Cell[TextData[{ "\[FilledSmallCircle] Podobn\:011b budeme postupovat i u re\[AAcute]ln\ \[EAcute] funkce v\[IAcute]ce prom\:011bnn\[YAcute]ch, z\[AAcute]pis bude \ vypadat takto:\n\t\t\t\t\t\t\t", StyleBox["funkce [", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`prom\_1\)], FontWeight->"Bold"], StyleBox["_, ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(prom\_2\) _\)], FontWeight->"Bold"], StyleBox[", ..., ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`prom\_n\)], FontWeight->"Bold"], StyleBox["_] := v\[YAcute]raz", FontWeight->"Bold"], "\n\t\[EmptySmallCircle] p\:0159\[IAcute]klad:" }], "Text"], Cell[BoxData[ \(f[x_, y_] := \((x - 2\ y)\)^2\)], "Input"], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad:\t", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[3, \(-1\)]\)], "Input"], Cell[BoxData[ \(25\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad:\t", "Text"], Cell[BoxData[ \(f[x_, \ y_, \ z_, \ w_, \ q_] := \ x^y + z/w - q\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f[2, 3, 4, 2, 7]\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell["\<\ \[FilledSmallCircle] Hodnoty pro v\[IAcute]ce prom\:011bnn\[YAcute]ch m\:016f\ \:017eeme zad\[AAcute]vat ka\:017edou zv\[AAcute]\[SHacek]\:0165 do vstupn\ \[IAcute] bu\:0148ky\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x1 = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x2 = 2\)], "Input"], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x3 = 3\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell[TextData[{ " nebo jen do jedn\[EAcute] vstupn\[IAcute] bu\:0148ky. K odd\:011blen\ \[IAcute] p\:0159\[IAcute]kaz\:016f vyu\:017eijeme symbolu ", StyleBox[";", FontWeight->"Bold"], " (st\:0159edn\[IAcute]k)." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x1 = 1; \ x2 = 2; \ x3 = 3\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell["\<\ \[FilledSmallCircle] Pou\:017eijeme-li st\:0159edn\[IAcute]k za posledn\ \[IAcute]m v\[YAcute]razem, zabr\[AAcute]n\[IAcute]me t\[IAcute]m, aby se n\ \[AAcute]m zobrazilo v\[YAcute]stupn\[IAcute] pole.\ \>", "Text"], Cell[BoxData[ \(x1 = 3; \ x2 = 8; \ x3 = 5;\)], "Input"], Cell["\<\ \[FilledSmallCircle] Chceme-li p\:0159i\:0159adit prom\:011bnn\[YAcute]m \ hodnoty pouze v jednom v\[YAcute]razu, pou\:017eijeme podobn\[EAcute]ho z\ \[AAcute]pisu jako v p\:0159\[IAcute]pad\:011b jedn\[EAcute] prom\:011bnn\ \[EAcute]. Definici v\[SHacek]ak mus\[IAcute]me uzav\:0159\[IAcute]t do slo\ \:017een\[YAcute]ch z\[AAcute]vorek:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((x + y)\)*\((x - y)\)^2 /. {x \[Rule] 3, y \[Rule] 1 - a}\)], "Input"], Cell[BoxData[ \(\((4 - a)\)\ \((2 + a)\)\^2\)], "Output"] }, Open ]], Cell["\<\ \ \>", "Text", TextAlignment->Center], Cell["\<\ 2.2. Parci\[AAcute]ln\[IAcute] derivace prvn\[IAcute]ho a \ vy\[SHacek]\[SHacek]\[IAcute]ch \:0159\[AAcute]d\:016f\ \>", "Text", TextAlignment->Center, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox["2.2.1. V\[YAcute]po\[CHacek]et", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] K v\[YAcute]po\[CHacek]tu parci\[AAcute]ln\[IAcute]ch \ derivac\[IAcute] pou\:017e\[IAcute]v\[AAcute]me stejn\[YAcute]ch funkc\ \[IAcute] jako v p\:0159\[IAcute]pad\:011b derivace re\[AAcute]ln\[EAcute] \ funkce jedn\[EAcute] prom\:011bnn\[EAcute]. Pod\[IAcute]vejme se nyn\[IAcute] \ na obecn\[YAcute] z\[AAcute]pis t\:011bchto funkc\[IAcute]:\n\t\ \[EmptySmallCircle] ", Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_x\)\(\ \)\)\)], FontWeight->"Bold"], StyleBox["fce", FontWeight->"Bold"], " nebo ", StyleBox["D[fce, x]", FontWeight->"Bold"], "\t\t\t\tnajde parci\[AAcute]ln\[IAcute] derivaci funkce ", StyleBox["fce", FontSlant->"Italic"], " podle prom\:011bnn\[EAcute] ", StyleBox["x\n\t", FontSlant->"Italic"], "\[EmptySmallCircle] ", StyleBox["D[fce, {x, n}]", FontWeight->"Bold"], " nebo ", StyleBox["Derivative[n][fce][x]", FontWeight->"Bold"], "\t\tnajde parci\[AAcute]ln\[IAcute] derivaci ", StyleBox["n", FontSlant->"Italic"], "-t\[EAcute]ho \:0159\[AAcute]du funkce ", StyleBox["fce", FontSlant->"Italic"], " podle\n\t\t\t\t\t\t\t\tprom\:011bnn\[EAcute] ", StyleBox["x.", FontSlant->"Italic"], " \n\n\[FilledSmallCircle] Nyn\[IAcute] si uve\:010fm\:011b n\:011bkolik p\ \:0159\[IAcute]klad\:016f pou\:017eit\[IAcute] t\:011bchto funkc\[IAcute]:\n\t\ \[EmptySmallCircle] p\:0159\[IAcute]klad 1." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_z\ \((x^3 - 12\ z^4\ y^3)\)\)], "Input"], Cell[BoxData[ \(\(-48\)\ y\^3\ z\^3\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 2.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[x^3 - 12\ z^4\ y^3, \ y]\)], "Input"], Cell[BoxData[ \(\(-36\)\ y\^2\ z\^4\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 3.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Fz[z_] := \ x^3 - 12\ z^4\ y^3; \ \(\(Derivative[3]\)[Fz]\)[ z]\)], "Input"], Cell[BoxData[ \(\(-288\)\ y\^3\ z\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell["\<\ \[FilledSmallCircle] Nyn\[IAcute] si uve\:010fm\:011b n\:011bkolik p\:0159\ \[IAcute]klad\:016f pou\:017eit\[IAcute] t\:011bchto funkc\[IAcute] (pokra\ \[CHacek]ov\[AAcute]n\[IAcute]): \t\[EmptySmallCircle] p\:0159\[IAcute]klad 4.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_\(x, x\)\ \((x^2*y^2)\)\)], "Input"], Cell[BoxData[ \(2\ y\^2\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 5.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[x^2*y^2, \ x, \ y, \ x]\)], "Input"], Cell[BoxData[ \(4\ y\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 6.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[x^2*y^2, \ {x, \ 2}, \ {y, \ 2}]\)], "Input"], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell[TextData[{ "\t\[EmptySmallCircle] p\:0159\[IAcute]klad 7. - pro \ v\[YAcute]po\[CHacek]et funkce (", Cell[BoxData[ \(TraditionalForm\`\(x\^2\) y\^2\)]], ") v bod\:011b x = 2 mus\[IAcute]me pou\:017e\[IAcute]t n\[AAcute]sleduj\ \[IAcute]c\[IAcute]ho z\[AAcute]pisu:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[x^2*y^2, \ x] /. \(\(x\)\(\[Rule]\)\(2\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(4\ y\^2\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[StyleBox["2.2.2. P\:0159\[IAcute]klady", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] P\:0159\[IAcute]klady k procvi\[CHacek]en\[IAcute] - \ najd\:011bte / vypo\[CHacek]t\:011bte n\[AAcute]sleduj\[IAcute]c\[IAcute] \ parci\[AAcute]ln\[IAcute] derivace:\n\n\t", StyleBox["a", FontWeight->"Bold"], ")\tf(x, y) = 3", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], Cell[BoxData[ FormBox[ StyleBox[\(y + sin(xy\^2)\), FontSlant->"Plain"], TraditionalForm]]], "\t\t\t\t(1. parci\[AAcute]ln\[IAcute] derivaci (p.d.) podle ", StyleBox["x", FontSlant->"Italic"], " a\n\t\t\t\t\t\t\t\t\t 1.p.d. podle ", StyleBox["y", FontSlant->"Italic"], ")\n\n\t", StyleBox["b", FontWeight->"Bold"], ")\tf(x, y) = ", Cell[BoxData[ \(TraditionalForm\`\@x\%3\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "\t\t\t\t\t(1. p.d. podle ", StyleBox["x", FontSlant->"Italic"], " a 1.p.d. podle ", StyleBox["y", FontSlant->"Italic"], ", v bod\:011b c = [1, 5])\n\n\t", StyleBox["c", FontWeight->"Bold"], ")\tf(x, y) = 2", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(-\((x\^2 + y\^2)\)\)\)]], "\t\t\t\t\t(2.p.d. podle ", StyleBox["x", FontSlant->"Italic"], ")\n\t\n\t", StyleBox["d", FontWeight->"Bold"], ")\tf(x, y) = ", Cell[BoxData[ \(TraditionalForm\`\(ln(x\/y)\)\/\(x\^2 - y\^2\)\)]], "\t\t\t\t\t\t(2.p.d. podle ", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ")\n\n\t", StyleBox["e", FontWeight->"Bold"], ")\tf(x, y) = ", StyleBox["x", FontSlant->"Italic"], "-", StyleBox["y", FontSlant->"Italic"], "+", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+2", StyleBox["xy", FontSlant->"Italic"], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`x\^3\)]], "-3", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], StyleBox["y", FontSlant->"Italic"], "-", Cell[BoxData[ \(TraditionalForm\`y\^3\)]], "+", Cell[BoxData[ \(TraditionalForm\`x\^4\)]], "-4", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "\t(v\[SHacek]echny 1., 2. a 3. parci\[AAcute]ln\[IAcute] derivace)\n\n" }], "Text"], Cell["", "Text", TextAlignment->Center], Cell[TextData[StyleBox["2.2.3. V\[YAcute]sledky", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] \:0158e\[SHacek]en\[IAcute]:\n\t", StyleBox["a", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x\((3\ x^2\ y\ + \ Sin[x\ y^2])\)\)], "Input"], Cell[BoxData[ \(6\ x\ y + y\^2\ Cos[x\ y\^2]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_y\((3\ x^2\ y\ + \ Sin[x\ y^2])\)\)], "Input"], Cell[BoxData[ \(3\ x\^2 + 2\ x\ y\ Cos[x\ y\^2]\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["b", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[\@x\%3\ y^2, \ x] /. x \[Rule] 1\)], "Input"], Cell[BoxData[ \(y\^2\/3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[\@x\%3\ y^2, \ y] /. y \[Rule] 5\)], "Input"], Cell[BoxData[ \(10\ x\^\(1/3\)\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["c", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[2\ \[ExponentialE]\^\(-\((x^2 + y^2)\)\), \ {x, \ 2}]\)], "Input"], Cell[BoxData[ \(2\ \((\(-2\)\ \[ExponentialE]\^\(\(-x\^2\) - y\^2\) + 4\ \[ExponentialE]\^\(\(-x\^2\) - y\^2\)\ x\^2)\)\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["d", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[Log[x\/y]\/\(x\^2 - y\^2\), \ x, y]\)], "Input"], Cell[BoxData[ \(\(2\ x\)\/\(y\ \((x\^2 - y\^2)\)\^2\) + \(2\ y\)\/\(x\ \((x\^2 - \ y\^2)\)\^2\) - \(8\ x\ y\ Log[x\/y]\)\/\((x\^2 - y\^2)\)\^3\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["e", FontWeight->"Bold"], ")" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[x - y + x^2 + 2\ x\ y + y^2 + x^3 - 3\ x^2\ y - y^3 + x^4 - 4\ x^2\ y^2, \ x]\)], "Input"], Cell[BoxData[ \(1 + 2\ x + 3\ x\^2 + 4\ x\^3 + 2\ y - 6\ x\ y - 8\ x\ y\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x - y + x^2 + 2\ x\ y + y^2 + x^3 - 3\ x^2\ y - y^3 + x^4 - 4\ x^2\ y^2, \ y]\)], "Input"], Cell[BoxData[ \(\(-1\) + 2\ x - 3\ x\^2 + 2\ y - 8\ x\^2\ y - 3\ y\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x - y + x^2 + 2\ x\ y + y^2 + x^3 - 3\ x^2\ y - y^3 + x^4 - 4\ x^2\ y^2, \ {x, 2}]\)], "Input"], Cell[BoxData[ \(2 + 6\ x + 12\ x\^2 - 6\ y - 8\ y\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x - y + x^2 + 2\ x\ y + y^2 + x^3 - 3\ x^2\ y - y^3 + x^4 - 4\ x^2\ y^2, \ x, y]\)], "Input"], Cell[BoxData[ \(2 - 6\ x - 16\ x\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x - y + x^2 + 2\ x\ y + y^2 + x^3 - 3\ x^2\ y - y^3 + x^4 - 4\ x^2\ y^2, \ {y, 2}]\)], "Input"], Cell[BoxData[ \(2 - 8\ x\^2 - 6\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x - y + x^2 + 2\ x\ y + y^2 + x^3 - 3\ x^2\ y - y^3 + x^4 - 4\ x^2\ y^2, {x, 3}]\)], "Input"], Cell[BoxData[ \(6 + 24\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x - y + x^2 + 2\ x\ y + y^2 + x^3 - 3\ x^2\ y - y^3 + x^4 - 4\ x^2\ y^2, \ {x, 2}, y]\)], "Input"], Cell[BoxData[ \(\(-6\) - 16\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x - y + x^2 + 2\ x\ y + y^2 + x^3 - 3\ x^2\ y - y^3 + x^4 - 4\ x^2\ y^2, x, \ {y, \ 2}]\)], "Input"], Cell[BoxData[ \(\(-16\)\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[x - y + x^2 + 2\ x\ y + y^2 + x^3 - 3\ x^2\ y - y^3 + x^4 - 4\ x^2\ y^2, \ {y, 3}]\)], "Input"], Cell[BoxData[ \(\(-6\)\)], "Output"] }, Open ]], Cell["\<\ \ \>", "Text", TextAlignment->Center], Cell["\<\ 2.3. Tot\[AAcute]ln\[IAcute] diferenci\[AAcute]l prvn\[IAcute]ho a \ vy\[SHacek]\[SHacek]\[IAcute]ch \:0159\[AAcute]d\:016f\ \>", "Text", TextAlignment->Center, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox["2.3.1. V\[YAcute]po\[CHacek]et", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\t\[FilledSmallSquare] pozn.: ", StyleBox["Tot\[AAcute]ln\[IAcute] diferenci\[AAcute]l nen\[IAcute] v ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Bold"], StyleBox["CalcCenter", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" podporov\[AAcute]n.", FontWeight->"Bold"], "\n\[FilledSmallCircle] K ur\[CHacek]en\[IAcute] tot\[AAcute]ln\[IAcute]ho \ diferenci\[AAcute]lu pomoc\[IAcute] programu ", StyleBox["Mathematica", FontSlant->"Italic"], " slou\:017e\[IAcute] funkce ", StyleBox["Dt", FontWeight->"Bold"], "." }], "Text"], Cell[TextData[{ "\[FilledSmallCircle] Syntaxe t\[EAcute]to funkce vypad\[AAcute] \ n\[AAcute]sledovn\:011b:\n\t\[EmptySmallCircle]", StyleBox[" Dt [fce]\t", FontWeight->"Bold"], "\tnajde tot\[AAcute]ln\[IAcute] diferenci\[AAcute]l funkce ", StyleBox["fce", FontSlant->"Italic"], ".\n\t\[EmptySmallCircle] ", StyleBox["Dt [fce, x]", FontWeight->"Bold"], "\t\tzobraz\[IAcute] \[UAcute]plnou derivaci funkce ", StyleBox["fce", FontSlant->"Italic"], " podle prom\:011bnn\[EAcute] ", StyleBox["x", FontSlant->"Italic"], ". \n\t\[EmptySmallCircle] ", StyleBox["Dt [fce, {x, n}]", FontWeight->"Bold"], "\tzobraz\[IAcute] ", StyleBox["n", FontSlant->"Italic"], "-tou \[UAcute]plnou derivaci podle ", StyleBox["x", FontSlant->"Italic"], " ( ", Cell[BoxData[ \(TraditionalForm\`d\^n\)]], StyleBox["fce", FontSlant->"Italic"], " / d ", Cell[BoxData[ \(TraditionalForm\`x\^n\)]], ").\n\n\t\[EmptySmallCircle] p\:0159\[IAcute]klad 1." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[x\ y]\)], "Input"], Cell[BoxData[ \(y\ Dt[x] + x\ Dt[y]\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 2.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[x\ y, \ x]\)], "Input"], Cell[BoxData[ \(y + x\ Dt[y, x]\)], "Output"] }, Open ]], Cell[TextData[{ "\[FilledSmallCircle] Z p\:0159\[IAcute]kladu 2. je vid\:011bt, \:017ee \ prom\:011bnn\[AAcute] ", StyleBox["y", FontSlant->"Italic"], " je fukc\[IAcute] prom\:011bnn\[EAcute] ", StyleBox["x", FontSlant->"Italic"], ", tzn. ", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")." }], "Text"], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 3.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[a\ x\ y, \ x, \ Constants \[Rule] a]\)], "Input"], Cell[BoxData[ \(a\ y + a\ x\ Dt[y, x, Constants \[Rule] {a}]\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] Pomoc\[IAcute] parametru ", StyleBox["Constans", FontWeight->"Bold"], " m\:016f\:017eeme definovat r\:016fzn\[EAcute] konstanty \ vyskytuj\[IAcute]c\[IAcute] se ve v\[YAcute]razu. N\[AAcute]zornou \ uk\[AAcute]zku tvo\:0159\[IAcute] p\:0159\[IAcute]klad 3., ve kter\[EAcute]m \ ", StyleBox["a", FontSlant->"Italic"], " je konstantou, ", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") a ", StyleBox["x", FontSlant->"Italic"], " je nez\[AAcute]visl\[AAcute] prom\:011bnn\[AAcute]." }], "Text"], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 4.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[a\ x\ + b\ y, \ x, \ Constants \[Rule] {a, \ b}]\)], "Input"], Cell[BoxData[ \(a + b\ Dt[y, x, Constants \[Rule] {a, b}]\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 5.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[x^3, \ {x, 2}]\)], "Input"], Cell[BoxData[ \(6\ x\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 6.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[x^3\ y, \ {x, 2}]\)], "Input"], Cell[BoxData[ \(6\ x\ y + 6\ x\^2\ Dt[y, x] + x\^3\ Dt[y, {x, 2}]\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\[FilledSmallCircle] K zamy\[SHacek]len\[IAcute]: jak\[YAcute] je rozd\ \[IAcute]l mezi ", StyleBox["x ", FontSlant->"Italic"], "a ", StyleBox["y", FontSlant->"Italic"], " v p\:0159\[IAcute]kladech 7, 8 a 9 ? \:0158e\[SHacek]en\[IAcute] jist\ \:011b najde ka\:017ed\[YAcute] s\[AAcute]m." }], "Text"], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 7.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[x^3\ y, \ x, \ y]\)], "Input"], Cell[BoxData[ \(3\ x\^2 + 6\ x\ y\ Dt[x, y] + 3\ x\^2\ Dt[x, y]\ Dt[y, x]\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 8.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[x^3\ y, \ x, \ y, \ Constants \[Rule] x]\)], "Input"], Cell[BoxData[ \(3\ x\^2\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 9.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Dt[x^3\ y, \ x, \ y, \ Constants \[Rule] y]\)], "Input"], Cell[BoxData[ \(3\ x\^2 + 6\ x\ y\ Dt[x, y, Constants \[Rule] {y}]\)], "Output"] }, Open ]], Cell["\<\ \ \>", "Text", TextAlignment->Center] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 3. Integr\[AAcute]ln\[IAcute] po\[CHacek]et funkce jedn\[EAcute] prom\:011bnn\ \[EAcute]\ \>", "Section", TextAlignment->Center, CellTags->"LastSlide"], Cell["3.1. Neur\[CHacek]it\[YAcute] integr\[AAcute]l", "Text", TextAlignment->Center, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox["3.1.1. 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Diferenci\[AAcute]ln\[IAcute] rovnice prvn\[IAcute]ho a vy\[SHacek]\ \[SHacek]\[IAcute]ch \:0159\[AAcute]d\:016f\ \>", "Text", TextAlignment->Center, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox["4.1.1. V\[YAcute]po\[CHacek]et", FontWeight->"Bold"]], "Text", TextAlignment->Center, FontSize->12, Background->RGBColor[0.501961, 1, 1]], Cell["", "Text", TextAlignment->Center], Cell[TextData[{ "\t\[FilledSmallSquare] pozn.:", StyleBox["Funkce DSolve nen\[IAcute] v ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Bold"], StyleBox["CalcCenter", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" podporov\[AAcute]na. I p\:0159esto je mo\:017en\[EAcute] \ diferenci\[AAcute]ln\[IAcute] rovnice v ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Bold"], StyleBox["CalcCenter", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" re\[SHacek]it u\:017eit\[IAcute]m funkce SolveODE.", FontWeight->"Bold"], "\n\[FilledSmallCircle] Pro \:0159e\[SHacek]en\[IAcute] \ diferenci\[AAcute]ln\[IAcute]ch rovnic n-t\[EAcute]ho \:0159\[AAcute]du a \ soustavy diferenci\[AAcute]ln\[IAcute]ch rovnic se pou\:017e\[IAcute]v\ \[AAcute] integrovan\[EAcute] funkce ", StyleBox["DSolve", FontWeight->"Bold"], ".\n\[FilledSmallCircle] Uka\:017eme si nyn\[IAcute] jednotliv\[EAcute] z\ \[AAcute]pisy funkce ", StyleBox["DSolve", FontWeight->"Bold"], ":\n\t\[EmptySmallCircle] DSolve[rce, y, x]\t\t\t\t\ \:0159e\[SHacek]\[IAcute] diferenci\[AAcute]ln\[IAcute] rovnici ", StyleBox["rce", FontSlant->"Italic"], " pro z\[AAcute]vislou prom\:011bnnou ", StyleBox["y\n\t\t\t\t\t\t", FontSlant->"Italic"], "s nez\[AAcute]visle prom\:011bnnou ", StyleBox["x\n\t", FontSlant->"Italic"], "\[EmptySmallCircle] DSolve[{rce, y[x0]= =x0, ...}, y, x]\t\t\:0159e\ \[SHacek]\[IAcute] diferenci\[AAcute]ln\[IAcute] rovnici ", StyleBox["rce", FontSlant->"Italic"], " s po\[CHacek]\[AAcute]te\[CHacek]n\[IAcute]mi\n\t\t\t\t\t\t\ podm\[IAcute]nkami y[x0]==x0, ...\n\t\[EmptySmallCircle] DSolve[{rce1, rce2, \ ...}, {y1, y2, ...}, x]\t\:0159e\[SHacek]\[IAcute] soustavu \ diferenci\[AAcute]ln\[IAcute]ch rovnic" }], "Text"], Cell[TextData[{ "\n\[FilledSmallCircle] Syntaxe funkce ", StyleBox["SolveODE", FontWeight->"Bold"], " pro ", StyleBox["Mathematica CalcCenter", FontSlant->"Italic"], ".\n\t\[EmptySmallCircle] SolveODE[rce, y, {x, xmin, xmax}]\t\t\t\tnalezne \ numerick\[EAcute] \:0159e\[SHacek]en\[IAcute] oby\[CHacek]ejn\[EAcute] \ diferenci\[AAcute]ln\[IAcute]\n\t\t\t\t\t\t\t\trovnice ", StyleBox["rce", FontSlant->"Italic"], " pro z\[AAcute]vislou prom\:011bnnou ", StyleBox["y", FontSlant->"Italic"], " s nez\[AAcute]visle\n\t\t\t\t\t\t\t\tprom\:011bnnou ", StyleBox["x", FontSlant->"Italic"], " v mez\[IAcute]ch od ", StyleBox["xmin", FontSlant->"Italic"], " po ", StyleBox["xmax", FontSlant->"Italic"], ".\n\t\n\t\[EmptySmallCircle] SolveODE[rce, y, {x, xmin, xmax}, {t, tmin, \ tmax}]\tnalezne numerick\[EAcute] \:0159e\[SHacek]en\[IAcute] \ parci\[AAcute]ln\[IAcute] diferenci\[AAcute]ln\[IAcute]\n\t\t\t\t\t\t\t\t\ rovnice rce pro z\[AAcute]vislou prom\:011bnnou ", StyleBox["y", FontSlant->"Italic"], " s nez\[AAcute]visle\n\t\t\t\t\t\t\t\tprom\:011bnnou ", StyleBox["x", FontSlant->"Italic"], " v mez\[IAcute]ch od ", StyleBox["xmin", FontSlant->"Italic"], " po ", StyleBox["xmax", FontSlant->"Italic"], " a parametr ", StyleBox["t\n\t\t\t\t\t\t\t\t ", FontSlant->"Italic"], "v mez\[IAcute]ch od ", StyleBox["tmin", FontSlant->"Italic"], " po ", StyleBox["tmax", FontSlant->"Italic"], ".\n\t\n\t\[EmptySmallCircle] SolveODE[rce, {y1, y2, ...}, {x, xmin, xmax}]\ \t\tnalezne numerick\[AAcute] \:0159e\[SHacek]en\[IAcute] pro funkce y1, y2, \ ...\n\t" }], "Text"], Cell[TextData[{ "\[FilledSmallCircle] Uka\:017eme si n\:011bkolik \ p\:0159\[IAcute]klad\:016f pou\:017eit\[IAcute] funkce ", StyleBox["DSolve", FontWeight->"Bold"], ":\n\t\[EmptySmallCircle] p\:0159\[IAcute]klad 1." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[\(y'\)[x] - y[x] \[Equal] 0, \ y[x], x]\)], "Input"], Cell[BoxData[ \({{y[x] \[Rule] \[ExponentialE]\^x\ C[1]}}\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 2.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{\(y'\)[x] - y[x] \[Equal] 0, \ y[0] \[Equal] 1}, \ y[x], x]\)], "Input"], Cell[BoxData[ \({{y[x] \[Rule] \[ExponentialE]\^x}}\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 3.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{\(y''\)[x] + Tan[x]\ \(y'\)[x] \[Equal] Sin[2\ x], \ y[0] \[Equal] 1, \ \(y'\)[0] \[Equal] 1}, y[x], \ x]\)], "Input"], Cell[BoxData[ \({{y[x] \[Rule] 1\/2\ \((2 - 2\ x + 6\ Sin[x] - Sin[2\ x])\)}}\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 4.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[\(y''\)[x] + \(y'\)[x] \[Equal] Sin[x], y[x], \ x]\)], "Input"], Cell[BoxData[ \({{y[x] \[Rule] C[2] + 1\/2\ \((\(-2\)\ \[ExponentialE]\^\(-x\)\ C[1] - Cos[x] - Sin[x])\)}}\)], "Output"] }, Open ]], Cell["\t\[EmptySmallCircle] p\:0159\[IAcute]klad 5.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{\(y'\)[x]\ - 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V tomto p\:0159\[IAcute]pad\:011b pro \ zjednodu\[SHacek]en\[IAcute] v\[YAcute]sledku m\:016f\:017eeme pou\:017e\ \[IAcute]t speci\[AAcute]ln\[IAcute] funkce ", StyleBox["Simplify", FontWeight->"Bold"], ". Jej\[IAcute] p\:0159edpis je n\[AAcute]sleduj\[IAcute]c\[IAcute]:\t", StyleBox["Simplify [v\[YAcute]raz]", FontWeight->"Bold"], ".", StyleBox["\n\n", FontWeight->"Bold"], "\[FilledSmallCircle] Uka\:017eme si pou\:017eit\[IAcute] funkce ", StyleBox["Simplify", FontWeight->"Bold"], " na p\:0159edchoz\[IAcute]m p\:0159\[IAcute]kladu \[CHacek].5, \ z\[AAcute]pis bude vypadat n\[AAcute]sledovn\:011b:\n\t\[EmptySmallCircle] p\ \:0159\[IAcute]klad 6." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ DSolve[{\(y'\)[x]\ - \ z[x]\ - \ x \[Equal] 0, \ \(z'\)[x]\ + \ y[x]\ - 2*x \[Equal] 0}, \ {y[x], \ z[x]}, x]]\)], "Input"], Cell[BoxData[ \({{y[x] \[Rule] 1 + 2\ x + C[1]\ Cos[x] + C[2]\ Sin[x], z[x] \[Rule] 2 - x + C[2]\ Cos[x] - C[1]\ Sin[x]}}\)], "Output"] }, Open ]], Cell["", "Text", TextAlignment->Center], Cell[TextData[StyleBox["4.1.2. 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