(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 9.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 100843, 2722] NotebookOptionsPosition[ 78266, 2173] NotebookOutlinePosition[ 91021, 2452] CellTagsIndexPosition[ 90707, 2441] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Zad\[AAcute]n\[IAcute] p\[RHacek]\[IAcute]klad\[URing]", "Section", CellChangeTimes->{{3.572062462421875*^9, 3.572062470796875*^9}}, CounterAssignments->{{"Chapter", 4}, {"ExerciseNumber", 0}}, CellTags->"Exercises"], Cell[TextData[StyleBox["\[CapitalRHacek]e\[SHacek]en\[IAcute] rovnic", \ "Subsection"]], "ExerciseHeading", CellChangeTimes->{{3.57206247509375*^9, 3.57206248725*^9}}, CellTags->{"Exercises", "Solving Equations Ex"}], Cell[TextData[{ "Ur\[CHacek]et hodnoty ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " jako \[RHacek]e\[SHacek]en\[IAcute] rovnice ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"a", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"b", " ", "x"}], "+", "c"}], "=", "0"}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{{3.5720624973125*^9, 3.572062513765625*^9}}, CellTags->{"Exercises", "Solving Equations Ex"}], Cell[BoxData[ RowBox[{ RowBox[{"Clear", "[", RowBox[{"a", ",", "b", ",", "c"}], "]"}], ";"}]], "Input", CellChangeTimes->{{3.578303900109375*^9, 3.57830391171875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"a", " ", RowBox[{"x", "^", "2"}]}], "+", RowBox[{"b", " ", "x"}], " ", "+", " ", "c"}], "\[Equal]", "0"}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.57830385671875*^9, 3.578303881875*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "b"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", " ", "a", " ", "c"}]}]]}], RowBox[{"2", " ", "a"}]]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "b"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", " ", "a", " ", "c"}]}]]}], RowBox[{"2", " ", "a"}]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.57830388596875*^9, 3.578303913625*^9}}] }, Open ]], Cell[TextData[{ "\n\[CapitalRHacek]e\[SHacek]te nasleduj\[IAcute]c\[IAcute] soustavu rovnice \ pro ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " a ", Cell[BoxData[ FormBox["y", TraditionalForm]]], " a uka\[ZHacek]te, \[ZHacek]e \[RHacek]e\[SHacek]en\[IAcute] vyhovuje \ rovnic\[IAcute]m. Mo\[ZHacek]n\[EAcute] \[RHacek]e\[SHacek]en\[IAcute] je \ substituce pomoc\[IAcute] substitu\[CHacek]n\[IAcute]ho pravidla do \ p\[URing]vodn\[IAcute]ch rovnic a uk\[AAcute]zat, \[ZHacek]e v\[YAcute]sledek \ je True. 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Racion\[AAcute]ln\ \[IAcute] funkce je takov\[AAcute], kter\[AAcute] v \[CHacek]itateli a \ jmenovateli obsahuje polynom. P\[RHacek]eve\[DHacek]te funkci na \ v\[YAcute]raz se spole\[CHacek]n\[YAcute]m jmenovatelem.\ \>", "Exercise", CellChangeTimes->{{3.572062763234375*^9, 3.572062837375*^9}, { 3.578304024828125*^9, 3.578304024859375*^9}, 3.57830413159375*^9, { 3.5783042635*^9, 3.578304283*^9}}, CellTags->{"Exercises", "Rational Functions"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"expr", "=", RowBox[{"Sum", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{"k", "+", "x"}], ")"}]}], ",", RowBox[{"{", RowBox[{"k", ",", "5"}], "}"}]}], "]"}]}]], "Input", CellTags->{"Exercises", "Rational Functions"}], Cell[BoxData[ RowBox[{ FractionBox["1", RowBox[{"1", "+", "x"}]], "+", FractionBox["1", RowBox[{"2", "+", "x"}]], "+", FractionBox["1", RowBox[{"3", "+", "x"}]], "+", FractionBox["1", RowBox[{"4", "+", "x"}]], "+", FractionBox["1", RowBox[{"5", "+", "x"}]]}]], "Output", CellChangeTimes->{3.578303820453125*^9}, CellTags->{"Exercises", "Rational Functions"}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"newexpr", " ", "=", " ", RowBox[{"Together", "[", "expr", "]"}]}]], "Input", CellChangeTimes->{{3.578304335484375*^9, 3.578304346078125*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"274", "+", RowBox[{"450", " ", "x"}], "+", RowBox[{"255", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"60", " ", SuperscriptBox["x", "3"]}], "+", RowBox[{"5", " ", SuperscriptBox["x", "4"]}]}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{"3", "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{"4", "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{"5", "+", "x"}], ")"}]}]]], "Output", CellChangeTimes->{3.5783043469375*^9}] }, Open ]], Cell["", "Exercise", CellChangeTimes->{3.57206286546875*^9}, CellTags->{"Exercises", "Rational Functions"}], Cell[TextData[{ StyleBox["P\[RHacek]eve\[DHacek]te v\[YAcute]sledek zp\[EHacek]t do \ p\[URing]vodn\[IAcute] formy. 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Sp\[IAcute]\[SHacek]e ne\[ZHacek] \ zad\[AAcute]vat jednu hodnotu argumentu na vstup je vhodn\[EHacek]j\[SHacek]\ \[IAcute] pou\[ZHacek]\[IAcute]t p\[RHacek]\[IAcute]stup, p\[RHacek]i kter\ \[EAcute]m je mo\[ZHacek]n\[EAcute] zadat v\[IAcute]ce hodnot parametru \ najednou. " }], "Text", CellChangeTimes->{{3.572065025203125*^9, 3.572065165171875*^9}, { 3.5783047841875*^9, 3.5783047894375*^9}}, CellTags->{"Exercises", "Checking Symbolic Results"}], Cell[TextData[{ "Vyhodno\[THacek]te n\[AAcute]sleduj\[IAcute]c\[IAcute] vstup a \ obdr\[ZHacek]\[IAcute]te hodnotu integr\[AAcute]lu pro hodnotu ", StyleBox["b", "MR"], " rovnou 3. Hodnota parametru ", StyleBox["b", "MR"], " je zad\[AAcute]na pomoc\[IAcute] ", Cell[BoxData[ ButtonBox["Block", BaseStyle->"Link", ButtonData->"paclet:ref/Block"]], "InlineFormula"], "." }], "Exercise", CellChangeTimes->{{3.572065168640625*^9, 3.57206523221875*^9}}, CellTags->{"Exercises", "Checking Symbolic Results"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Block", "[", RowBox[{ RowBox[{"{", RowBox[{"b", "=", "3"}], "}"}], ",", RowBox[{"NIntegrate", "[", RowBox[{ RowBox[{ RowBox[{"Exp", "[", RowBox[{"-", "x"}], "]"}], "/", RowBox[{"(", RowBox[{"1", "+", RowBox[{"b", " ", "x"}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}]}], "]"}]}], "]"}]], "Input", CellTags->{"Exercises", "Checking Symbolic Results"}], Cell[BoxData["0.3856020121393501`"], "Output", CellChangeTimes->{3.578303825953125*^9, 3.57830497259375*^9}, CellTags->{"Exercises", "Checking Symbolic Results"}] }, Open ]], Cell[TextData[{ "Vyhodno\[THacek]te n\[AAcute]sleduj\[IAcute]c\[IAcute] vstup a \ ur\[CHacek]ete \[CHacek]\[IAcute]selnou aproximaci symbolick\[EAcute]ho v\ \[YAcute]sledku pro stejnou hodnotu ", StyleBox["b", "MR"], ". Srovnejte v\[YAcute]sledek s v\[YAcute]stupem \[CHacek]\[IAcute]seln\ \[EAcute] integrace z p\[RHacek]edch\[AAcute]zej\[IAcute]c\[IAcute]ho. 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V\[YAcute]sledek je tak\[EAcute] vr\[AAcute]cen \ v odli\[SHacek]n\[EAcute] form\[EHacek]. 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Jak\[YAcute] efekt m\[AAcute] pou\[ZHacek]it\[IAcute] funkce ", Cell[BoxData[ ButtonBox["Normal", BaseStyle->"Link", ButtonData->"paclet:ref/Normal"]], "InlineFormula"], " ve v\[YAcute]po\[CHacek]tu? Co se stane, pokud aplikaci funkce ", Cell[BoxData[ ButtonBox["Normal", BaseStyle->"Link", ButtonData->"paclet:ref/Normal"]], "InlineFormula"], " vynech\[AAcute]me? 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