(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 7.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     64588,       1977]
NotebookOptionsPosition[     60123,       1853]
NotebookOutlinePosition[     60594,       1872]
CellTagsIndexPosition[     60551,       1869]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell["6. Vaja", "Title",
 CellChangeTimes->{{3.449887566859375*^9, 3.449887568796875*^9}, {
  3.44988767521875*^9, 3.449887675515625*^9}, {3.4498877125625*^9, 
  3.449887746859375*^9}, {3.44988778396875*^9, 3.449887785171875*^9}},
 TextAlignment->Left,
 FontFamily->"Arial",
 FontSize->36,
 FontWeight->"Bold"],

Cell["Implicitno podane krivulje\t", "Subtitle",
 CellChangeTimes->{{3.44988782228125*^9, 3.449887835359375*^9}, 
   3.449896483921875*^9},
 TextAlignment->Left,
 FontFamily->"Arial",
 FontSlant->"Plain",
 FontVariations->{"CompatibilityType"->0},
 FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
 "Nekatere slike krivulj smo do sedaj \:017ee spoznali, ko smo risali grafe \
funkcij y=f(x). Te krivulje so bile v ravnini podane eksplicitno, poznamo pa \
tudi krivulje, katerih predpis je podan implicitno (ne moremo izraziti y). Te \
krivulje ri\[SHacek]emo z ukazom ",
 StyleBox["ContourPlot", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.449887815921875*^9, 3.44988781853125*^9}, {
  3.44988785121875*^9, 3.449887852*^9}, {3.449887936828125*^9, 
  3.4498879491875*^9}},
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[TextData[{
 "1.a) Parabole v ravnini so podane z ena\[CHacek]bami ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SuperscriptBox["y", "2"], "=", 
     RowBox[{"2", "px"}]}], ",", " ", 
    RowBox[{"kjer", " ", "je"}]}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 " p polparameter parabole. Z ukazom ",
 StyleBox["ContourPlot ", "Input"],
 "nari\[SHacek]i parabolo ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    SuperscriptBox["y", "2"], "=", 
    RowBox[{"2", "x"}]}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 "."
}], "Text",
 CellChangeTimes->{{3.44988796159375*^9, 3.44988800171875*^9}, {
  3.44988848290625*^9, 3.44988851909375*^9}, {3.449888612390625*^9, 
  3.449888624953125*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"ContourPlot", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"y", "^", "2"}], "\[Equal]", 
    RowBox[{"2", "x"}]}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "1"}], ",", "4"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
   RowBox[{"Axes", "\[Rule]", "True"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449887924328125*^9, 3.449887927671875*^9}, {
  3.44988800721875*^9, 3.449888031125*^9}, {3.449888065859375*^9, 
  3.44988806903125*^9}, {3.449888228796875*^9, 3.449888236015625*^9}, {
  3.449888267*^9, 3.449888267765625*^9}}],

Cell[TextData[{
 "Z ukazom ",
 StyleBox["Manipulate", "Input"],
 " lahko interaktivno izbiramo polparameter p dane parabole. "
}], "Text",
 CellChangeTimes->{{3.449887815921875*^9, 3.44988781853125*^9}, {
  3.44988785121875*^9, 3.449887852*^9}, {3.449887936828125*^9, 
  3.4498879491875*^9}, {3.44988834459375*^9, 3.4498884206875*^9}},
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[BoxData[
 RowBox[{"Manipulate", "[", 
  RowBox[{
   RowBox[{"ContourPlot", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"y", "^", "2"}], "\[Equal]", 
      RowBox[{"2", "*", "p", "*", "x"}]}], ",", 
     RowBox[{"{", 
      RowBox[{"x", ",", 
       RowBox[{"-", "1"}], ",", "4"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"y", ",", 
       RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
     RowBox[{"Axes", "\[Rule]", "True"}]}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"p", ",", "1", ",", "\"\<Polparameter p\>\""}], "}"}], ",", "0",
      ",", "1"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449887924328125*^9, 3.449887927671875*^9}, {
  3.44988800721875*^9, 3.449888031125*^9}, {3.449888065859375*^9, 
  3.44988806903125*^9}, {3.449888105875*^9, 3.4498881466875*^9}, {
  3.449888244859375*^9, 3.44988827628125*^9}, {3.449888441890625*^9, 
  3.449888467265625*^9}}],

Cell[TextData[{
 "1.b) Hiperbole v ravnini so podane z ena\[CHacek]bami ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     FractionBox[
      SuperscriptBox["x", "2"], 
      SuperscriptBox["a", "2"]], "-", 
     FractionBox[
      SuperscriptBox["y", "2"], 
      SuperscriptBox["b", "2"]]}], "=", "1"}], TraditionalForm]]],
 ", kjer je a glavna polos hiperbole in b stranska polos hiperbole."
}], "Text",
 CellChangeTimes->{{3.449888628984375*^9, 3.44988866584375*^9}, {
  3.449888989203125*^9, 3.449889003796875*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"Manipulate", "[", 
  RowBox[{
   RowBox[{"ContourPlot", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"x", "^", "2"}], "/", 
        RowBox[{"a", "^", "2"}]}], "-", 
       RowBox[{
        RowBox[{"y", "^", "2"}], "/", 
        RowBox[{"b", "^", "2"}]}]}], "\[Equal]", "1"}], ",", 
     RowBox[{"{", 
      RowBox[{"x", ",", 
       RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"y", ",", 
       RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", 
     RowBox[{"Axes", "\[Rule]", "True"}]}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "1", ",", "\"\<Glavna polos a\>\""}], "}"}], ",", "0",
      ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"b", ",", "1", ",", "\"\<Stranska polos b\>\""}], "}"}], ",", 
     "0", ",", "1"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449887924328125*^9, 3.449887927671875*^9}, {
  3.44988800721875*^9, 3.449888031125*^9}, {3.449888065859375*^9, 
  3.44988806903125*^9}, {3.449888105875*^9, 3.4498881466875*^9}, {
  3.449888244859375*^9, 3.44988827628125*^9}, {3.449888441890625*^9, 
  3.449888467265625*^9}, {3.449888680515625*^9, 3.44988876640625*^9}, {
  3.4498888859375*^9, 3.449888895640625*^9}, {3.449888927359375*^9, 
  3.449888930890625*^9}}],

Cell[" Parametri\[CHacek]no podane krivulje\t", "Subtitle",
 CellChangeTimes->{3.449888975078125*^9, 3.449896480921875*^9},
 TextAlignment->Left,
 FontFamily->"Arial",
 FontSlant->"Plain",
 FontVariations->{"CompatibilityType"->0},
 FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
 "Parametri\[CHacek]na krivulja je slika vektorske funkcije ene spremenljivke \
(ki jo ponavadi ozna\[CHacek]imo s t kot 'time'=\[CHacek]as) in ima dve ali \
tri komponente. Te krivulje ri\[SHacek]emo z ukazom ",
 StyleBox["ParametricPlot", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.449889167015625*^9, 3.44988932290625*^9}},
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[TextData[{
 "2.a) Elipsa s polosema a in b ima parametri\[CHacek]no obliko \
x(t)=acos(t),y(t)=bsin(t). Nari\[SHacek]i elipso s polosema a=2 in b=1 z \
ukazom ",
 StyleBox["ParametricPlot", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.449892965*^9, 3.449893042375*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"ParametricPlot", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"2", 
      RowBox[{"Cos", "[", "t", "]"}]}], ",", 
     RowBox[{"Sin", "[", "t", "]"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "0", ",", 
     RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449892955390625*^9, 3.44989295665625*^9}, 
   3.449893046109375*^9}],

Cell["\<\
Parametri\[CHacek]no podano krivuljo si pogosto predstavljamo kot gibanje to\
\[CHacek]ko po ravnini ali v prostoru.\
\>", "Text",
 CellChangeTimes->{{3.449889167015625*^9, 3.44988932290625*^9}, {
  3.44989433615625*^9, 3.449894370671875*^9}, {3.449924686890625*^9, 
  3.4499246879375*^9}},
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[BoxData[
 RowBox[{"Manipulate", "[", 
  RowBox[{
   RowBox[{"Animate", "[", 
    RowBox[{
     RowBox[{"Graphics", "[", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"RGBColor", "[", 
          RowBox[{"1", ",", "0", ",", "0"}], "]"}], ",", 
         RowBox[{"Line", "[", 
          RowBox[{"Table", "[", 
           RowBox[{
            RowBox[{"{", 
             RowBox[{
              RowBox[{"a", " ", 
               RowBox[{"Cos", "[", "t", "]"}]}], ",", 
              RowBox[{"b", " ", 
               RowBox[{"Sin", "[", "t", "]"}]}]}], "}"}], ",", 
            RowBox[{"{", 
             RowBox[{"t", ",", "0", ",", 
              RowBox[{"2", "Pi"}], ",", 
              RowBox[{"Pi", "/", "50"}]}], "}"}]}], "]"}], "]"}], ",", 
         RowBox[{"RGBColor", "[", 
          RowBox[{"0", ",", "0", ",", "0"}], "]"}], ",", 
         RowBox[{"PointSize", "[", "0.03", "]"}], ",", 
         RowBox[{"Point", "[", 
          RowBox[{"{", 
           RowBox[{
            RowBox[{"a", " ", 
             RowBox[{"Cos", "[", "T", "]"}]}], ",", 
            RowBox[{"b", " ", 
             RowBox[{"Sin", "[", "T", "]"}]}]}], "}"}], "]"}], ",", 
         RowBox[{"RGBColor", "[", 
          RowBox[{"0", ",", "0", ",", "1"}], "]"}], ",", 
         RowBox[{"If", "[", 
          RowBox[{"hitrost", ",", 
           RowBox[{"Table", "[", 
            RowBox[{
             RowBox[{"{", 
              RowBox[{"Arrow", "[", 
               RowBox[{"{", 
                RowBox[{
                 RowBox[{"{", 
                  RowBox[{
                   RowBox[{"a", " ", 
                    RowBox[{"Cos", "[", 
                    RowBox[{"k", " ", 
                    RowBox[{"Pi", "/", "10"}]}], "]"}]}], ",", 
                   RowBox[{"b", " ", 
                    RowBox[{"Sin", "[", 
                    RowBox[{"k", " ", 
                    RowBox[{"Pi", "/", "10"}]}], "]"}]}]}], "}"}], ",", 
                 RowBox[{"{", 
                  RowBox[{
                   RowBox[{
                    RowBox[{"a", " ", 
                    RowBox[{"Cos", "[", 
                    RowBox[{"k", " ", 
                    RowBox[{"Pi", "/", "10"}]}], "]"}]}], "-", 
                    RowBox[{
                    RowBox[{"1", "/", "4"}], " ", "a", " ", 
                    RowBox[{"Sin", "[", 
                    RowBox[{"k", " ", 
                    RowBox[{"Pi", "/", "10"}]}], "]"}]}]}], ",", 
                   RowBox[{
                    RowBox[{"b", " ", 
                    RowBox[{"Sin", "[", 
                    RowBox[{"k", " ", 
                    RowBox[{"Pi", "/", "10"}]}], "]"}]}], "+", 
                    RowBox[{
                    RowBox[{"1", "/", "4"}], " ", "b", " ", 
                    RowBox[{"Cos", "[", 
                    RowBox[{"k", " ", 
                    RowBox[{"Pi", " ", "/", "10"}]}], "]"}]}]}]}], "}"}]}], 
                "}"}], "]"}], "}"}], ",", 
             RowBox[{"{", 
              RowBox[{"k", ",", "20"}], "}"}]}], "]"}], ",", 
           RowBox[{"{", "}"}]}], "]"}]}], "}"}], ",", 
       RowBox[{"ImageSize", "\[Rule]", 
        RowBox[{"{", 
         RowBox[{"512", ",", "384"}], "}"}]}]}], "]"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"T", ",", "0", ",", "\"\<Vrednost parametra t\>\""}], "}"}], 
       ",", "0", ",", 
       RowBox[{"2", "\[Pi]"}]}], "}"}], ",", 
     RowBox[{"AnimationRunning", "\[Rule]", "False"}]}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "2", ",", "\"\<Velikost polosi x\>\""}], "}"}], ",", 
     RowBox[{"Table", "[", 
      RowBox[{"j", ",", 
       RowBox[{"{", 
        RowBox[{"j", ",", "1", ",", "3"}], "}"}]}], "]"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"b", ",", "1", ",", "\"\<Velikost polosi y\>\""}], "}"}], ",", 
     RowBox[{"Table", "[", 
      RowBox[{"k", ",", 
       RowBox[{"{", 
        RowBox[{"k", ",", "1", ",", "3"}], "}"}]}], "]"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
      "hitrost", ",", "False", ",", 
       "\"\<Prika\:017ei vektorje hitrosti\>\""}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"True", ",", "False"}], "}"}]}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449850366859375*^9, 3.44985045303125*^9}, {
  3.449850492828125*^9, 3.449850565578125*^9}, {3.449850638328125*^9, 
  3.44985073584375*^9}, {3.449850766203125*^9, 3.44985083665625*^9}, {
  3.449850871171875*^9, 3.449850944625*^9}, {3.44985119365625*^9, 
  3.44985129003125*^9}, {3.449851358359375*^9, 3.449851365390625*^9}, {
  3.4498925266875*^9, 3.449892530328125*^9}, {3.449892895078125*^9, 
  3.449892898296875*^9}, {3.449893519828125*^9, 3.44989363828125*^9}, {
  3.449893691453125*^9, 3.449893868375*^9}, {3.4498939653125*^9, 
  3.449894002703125*^9}, {3.4498940565625*^9, 3.449894104578125*^9}, {
  3.449894149859375*^9, 3.449894163171875*^9}, {3.449894214578125*^9, 
  3.449894221828125*^9}, {3.449924516265625*^9, 3.44992458796875*^9}, {
  3.449924637*^9, 3.449924640515625*^9}}],

Cell[TextData[{
 "2.b) Epicikloida je krivulja, ki jo opi\[SHacek]e to\[CHacek]ka \
kro\:017enice, ki se kotali po drugi kro\:017enici zunaj nje. Njena parametri\
\[CHacek]na ena\[CHacek]ba je x=(",
 StyleBox["R+r",
  FontSlant->"Italic"],
 ")cos(t)-rcos(",
 Cell[BoxData[
  FormBox[
   StyleBox[
    FractionBox[
     RowBox[{"R", "+", "r"}], "r"],
    FontSize->16], TraditionalForm]]],
 "t), y=(",
 StyleBox["R+r",
  FontSlant->"Italic"],
 ")sin(t)-rsin(",
 Cell[BoxData[
  FormBox[
   StyleBox[
    FractionBox[
     RowBox[{"R", "+", "r"}], "r"],
    FontSize->16], TraditionalForm]]],
 "t), kjer so ",
 StyleBox["R",
  FontSlant->"Italic"],
 "=polmer mirujo\[CHacek]e kro\:017enice, ",
 StyleBox["r",
  FontSlant->"Italic"],
 "=polmer kotale\[CHacek]e kro\:017enice in t=polarni kot. Nari\[SHacek]i \
epicikloido za primera ",
 Cell[BoxData[
  FormBox[
   RowBox[{"R", "=", 
    RowBox[{
     RowBox[{"1", " ", "in", " ", "r"}], "=", 
     FractionBox["1", "4"]}]}], TraditionalForm]]],
 ",1! "
}], "Text",
 CellChangeTimes->{{3.4498472304375*^9, 3.44984725315625*^9}, {
  3.449894456109375*^9, 3.4498944849375*^9}, {3.44989462753125*^9, 
  3.449894638171875*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.714855, 0.906264, 0.718761]],

Cell[BoxData[
 RowBox[{"R", "=", "1"}]], "Input",
 CellChangeTimes->{{3.44992314334375*^9, 3.449923144015625*^9}}],

Cell[BoxData[
 RowBox[{"r", "=", 
  RowBox[{"1", "/", "4"}]}]], "Input",
 CellChangeTimes->{{3.44992314896875*^9, 3.449923150765625*^9}}],

Cell[BoxData[
 RowBox[{"ParametricPlot", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"R", "+", "r"}], ")"}], 
       RowBox[{"Cos", "[", "t", "]"}]}], "-", 
      RowBox[{"r", " ", 
       RowBox[{"Cos", "[", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"R", "+", "r"}], ")"}], 
         RowBox[{"t", "/", "r"}]}], "]"}]}]}], ",", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"R", "+", "r"}], ")"}], 
       RowBox[{"Sin", "[", "t", "]"}]}], "-", 
      RowBox[{"r", " ", 
       RowBox[{"Sin", "[", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"R", "+", "r"}], ")"}], 
         RowBox[{"t", "/", "r"}]}], "]"}]}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "0", ",", 
     RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.4498472180625*^9, 3.44984722034375*^9}, {
  3.449923157375*^9, 3.449923157984375*^9}}],

Cell[BoxData[
 RowBox[{"r", "=", "1"}]], "Input",
 CellChangeTimes->{{3.44989464771875*^9, 3.44989464953125*^9}}],

Cell[BoxData[
 RowBox[{"ParametricPlot", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"R", "+", "r"}], ")"}], 
       RowBox[{"Cos", "[", "t", "]"}]}], "-", 
      RowBox[{"r", " ", 
       RowBox[{"Cos", "[", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"R", "+", "r"}], ")"}], 
         RowBox[{"t", "/", "r"}]}], "]"}]}]}], ",", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"R", "+", "r"}], ")"}], 
       RowBox[{"Sin", "[", "t", "]"}]}], "-", 
      RowBox[{"r", " ", 
       RowBox[{"Sin", "[", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"R", "+", "r"}], ")"}], 
         RowBox[{"t", "/", "r"}]}], "]"}]}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "0", ",", 
     RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input"],

Cell[TextData[{
 "Poglejmo si \[SHacek]e pridru\:017eene animacije za R=1 in nekatera \
razmerja ",
 Cell[BoxData[
  FormBox[
   FractionBox["r", "R"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 "."
}], "Text",
 CellChangeTimes->{{3.449889167015625*^9, 3.44988932290625*^9}, {
  3.44989433615625*^9, 3.449894370671875*^9}, {3.44989458440625*^9, 
  3.44989459684375*^9}, {3.449894690046875*^9, 3.449894728109375*^9}},
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[BoxData[
 RowBox[{"Manipulate", "[", 
  RowBox[{
   RowBox[{"Animate", "[", 
    RowBox[{
     RowBox[{"Graphics", "[", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"Thickness", "[", "0.003", "]"}], ",", 
         RowBox[{"PointSize", "[", "0.013", "]"}], ",", 
         RowBox[{"RGBColor", "[", 
          RowBox[{"1", ",", "0", ",", "0"}], "]"}], ",", 
         RowBox[{"Circle", "[", 
          RowBox[{
           RowBox[{"{", 
            RowBox[{"0", ",", "0"}], "}"}], ",", "1"}], "]"}], ",", 
         RowBox[{"RGBColor", "[", 
          RowBox[{"0", ",", "0", ",", "1"}], "]"}], ",", 
         RowBox[{"Line", "[", 
          RowBox[{"Table", "[", 
           RowBox[{
            RowBox[{"{", 
             RowBox[{
              RowBox[{
               RowBox[{
                RowBox[{"(", 
                 RowBox[{"1", "+", "r"}], ")"}], 
                RowBox[{"Cos", "[", "t", "]"}]}], "-", 
               RowBox[{"r", " ", 
                RowBox[{"Cos", "[", 
                 RowBox[{
                  RowBox[{"(", 
                   RowBox[{"1", "+", "r"}], ")"}], 
                  RowBox[{"t", "/", "r"}]}], "]"}]}]}], ",", 
              RowBox[{
               RowBox[{
                RowBox[{"(", 
                 RowBox[{"1", "+", "r"}], ")"}], 
                RowBox[{"Sin", "[", "t", "]"}]}], "-", 
               RowBox[{"r", " ", 
                RowBox[{"Sin", "[", 
                 RowBox[{
                  RowBox[{"(", 
                   RowBox[{"1", "+", "r"}], ")"}], 
                  RowBox[{"t", "/", "r"}]}], "]"}]}]}]}], "}"}], ",", 
            RowBox[{"{", 
             RowBox[{"t", ",", "0", ",", 
              RowBox[{"2", "Pi"}], ",", 
              RowBox[{"Pi", "/", "50"}]}], "}"}]}], "]"}], "]"}], ",", 
         "\[IndentingNewLine]", 
         RowBox[{"RGBColor", "[", 
          RowBox[{"0", ",", "0.8", ",", "0"}], "]"}], ",", 
         RowBox[{"Rotate", "[", 
          RowBox[{
           RowBox[{"{", 
            RowBox[{"Translate", "[", "\[IndentingNewLine]", 
             RowBox[{
              RowBox[{"{", 
               RowBox[{
                RowBox[{"Circle", "[", 
                 RowBox[{
                  RowBox[{"{", 
                   RowBox[{"0", ",", "0"}], "}"}], ",", "r"}], "]"}], ",", 
                "\[IndentingNewLine]", 
                RowBox[{"RGBColor", "[", 
                 RowBox[{"0", ",", "0", ",", "0"}], "]"}], ",", 
                RowBox[{"Point", "[", 
                 RowBox[{"{", 
                  RowBox[{
                   RowBox[{"-", "r"}], ",", "0"}], "}"}], "]"}]}], "}"}], ",", 
              RowBox[{
               RowBox[{"(", 
                RowBox[{"1", "+", "r"}], ")"}], 
               RowBox[{"{", 
                RowBox[{
                 RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", 
                 RowBox[{"Sin", "[", "\[Theta]", "]"}]}], "}"}]}]}], "]"}], 
            "}"}], ",", "\[IndentingNewLine]", 
           RowBox[{"\[Theta]", 
            RowBox[{"(", 
             RowBox[{"1", "+", 
              FractionBox["1", "r"]}], ")"}]}], ",", 
           RowBox[{
            RowBox[{"(", 
             RowBox[{"1", "+", "r"}], ")"}], 
            RowBox[{"{", 
             RowBox[{
              RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", 
              RowBox[{"Sin", "[", "\[Theta]", "]"}]}], "}"}]}]}], "]"}]}], 
        "}"}], ",", 
       RowBox[{"{", "}"}], ",", 
       RowBox[{"PlotRange", "\[Rule]", 
        RowBox[{"{", 
         RowBox[{
          RowBox[{"{", 
           RowBox[{
            RowBox[{"-", 
             RowBox[{"(", 
              RowBox[{"1", "+", 
               RowBox[{"2", "*", "r"}]}], ")"}]}], ",", 
            RowBox[{"1", "+", 
             RowBox[{"2", "*", "r"}]}]}], "}"}], ",", 
          RowBox[{"{", 
           RowBox[{
            RowBox[{"-", 
             RowBox[{"(", 
              RowBox[{"1", "+", 
               RowBox[{"2", "*", "r"}]}], ")"}]}], ",", 
            RowBox[{"1", "+", 
             RowBox[{"2", "*", "r"}]}]}], "}"}]}], "}"}]}], ",", 
       RowBox[{"ImageSize", "\[Rule]", 
        RowBox[{"{", 
         RowBox[{"500", ",", "375"}], "}"}]}]}], "]"}], ",", 
     "\[IndentingNewLine]", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"\[Theta]", ",", "0", ",", "\"\<Vrednost parametra t\>\""}], 
        "}"}], ",", "0", ",", 
       RowBox[{"2", "\[Pi]"}]}], "}"}], ",", 
     RowBox[{"AnimationRunning", "\[Rule]", "False"}]}], "]"}], ",", 
   "\[IndentingNewLine]", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"r", ",", 
       RowBox[{"1", "/", "4"}], ",", "\"\<Razmerje polmerov\>\""}], "}"}], 
     ",", 
     RowBox[{"Table", "[", 
      RowBox[{
       RowBox[{"1", "/", "j"}], ",", 
       RowBox[{"{", 
        RowBox[{"j", ",", "1", ",", "4"}], "}"}]}], "]"}]}], "}"}]}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.449924695375*^9, 3.449924715796875*^9}}],

Cell["\<\
2.c)Izra\[CHacek]unaj plo\[SHacek]\[CHacek]ino lika, ki ga oklepata zgornji \
krivulji. \
\>", "Text",
 CellChangeTimes->{{3.44989479328125*^9, 3.449894804125*^9}},
 FontFamily->"Courier New",
 Background->RGBColor[0.714855, 0.906264, 0.718761]],

Cell[BoxData[
 RowBox[{
  RowBox[{"x", "[", "t_", "]"}], ":=", 
  RowBox[{
   RowBox[{
    RowBox[{"(", 
     RowBox[{"R", "+", "r"}], ")"}], 
    RowBox[{"Cos", "[", "t", "]"}]}], "-", 
   RowBox[{"r", " ", 
    RowBox[{"Cos", "[", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"R", "+", "r"}], ")"}], 
      RowBox[{"t", "/", "r"}]}], "]"}]}]}]}]], "Input"],

Cell[BoxData[
 RowBox[{
  RowBox[{"y", "[", "t_", "]"}], ":=", 
  RowBox[{
   RowBox[{
    RowBox[{"(", 
     RowBox[{"R", "+", "r"}], ")"}], 
    RowBox[{"Sin", "[", "t", "]"}]}], "-", 
   RowBox[{"r", " ", 
    RowBox[{"Sin", "[", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"R", "+", "r"}], ")"}], 
      RowBox[{"t", "/", "r"}]}], "]"}]}]}]}]], "Input"],

Cell[BoxData[
 RowBox[{"r", "=", 
  RowBox[{"1", "/", "4"}]}]], "Input",
 CellChangeTimes->{{3.449894863875*^9, 3.449894865734375*^9}}],

Cell[BoxData[
 RowBox[{"ParametricPlot", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"a", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"R", "+", "r"}], ")"}], 
         RowBox[{"Cos", "[", "t", "]"}]}], "-", 
        RowBox[{"r", " ", 
         RowBox[{"Cos", "[", 
          RowBox[{
           RowBox[{"(", 
            RowBox[{"R", "+", "r"}], ")"}], 
           RowBox[{"t", "/", "r"}]}], "]"}]}]}], ")"}]}], ",", 
     RowBox[{"a", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"R", "+", "r"}], ")"}], 
         RowBox[{"Sin", "[", "t", "]"}]}], "-", 
        RowBox[{"r", " ", 
         RowBox[{"Sin", "[", 
          RowBox[{
           RowBox[{"(", 
            RowBox[{"R", "+", "r"}], ")"}], 
           RowBox[{"t", "/", "r"}]}], "]"}]}]}], ")"}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"a", ",", "0", ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "0", ",", 
     RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.44985205259375*^9, 3.44985208553125*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"1", "/", "2"}], 
  RowBox[{"Integrate", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"x", "[", "t", "]"}], 
      RowBox[{
       RowBox[{"y", "'"}], "[", "t", "]"}]}], "-", 
     RowBox[{
      RowBox[{
       RowBox[{"x", "'"}], "[", "t", "]"}], 
      RowBox[{"y", "[", "t", "]"}]}]}], ",", 
    RowBox[{"{", 
     RowBox[{"t", ",", "0", ",", 
      RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.449894775796875*^9, 3.449894778078125*^9}, {
  3.4498948248125*^9, 3.4498948269375*^9}, {3.449894895734375*^9, 
  3.44989491465625*^9}}],

Cell[BoxData[
 RowBox[{"r", "=", "1"}]], "Input",
 CellChangeTimes->{{3.4498949204375*^9, 3.449894921984375*^9}}],

Cell[BoxData[
 RowBox[{"ParametricPlot", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"a", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"R", "+", "r"}], ")"}], 
         RowBox[{"Cos", "[", "t", "]"}]}], "-", 
        RowBox[{"r", " ", 
         RowBox[{"Cos", "[", 
          RowBox[{
           RowBox[{"(", 
            RowBox[{"R", "+", "r"}], ")"}], 
           RowBox[{"t", "/", "r"}]}], "]"}]}]}], ")"}]}], ",", 
     RowBox[{"a", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"R", "+", "r"}], ")"}], 
         RowBox[{"Sin", "[", "t", "]"}]}], "-", 
        RowBox[{"r", " ", 
         RowBox[{"Sin", "[", 
          RowBox[{
           RowBox[{"(", 
            RowBox[{"R", "+", "r"}], ")"}], 
           RowBox[{"t", "/", "r"}]}], "]"}]}]}], ")"}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"a", ",", "0", ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "0", ",", 
     RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.44985205259375*^9, 3.44985208553125*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"1", "/", "2"}], 
  RowBox[{"Integrate", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"x", "[", "t", "]"}], 
      RowBox[{
       RowBox[{"y", "'"}], "[", "t", "]"}]}], "-", 
     RowBox[{
      RowBox[{
       RowBox[{"x", "'"}], "[", "t", "]"}], 
      RowBox[{"y", "[", "t", "]"}]}]}], ",", 
    RowBox[{"{", 
     RowBox[{"t", ",", "0", ",", 
      RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.449894775796875*^9, 3.449894778078125*^9}, {
  3.4498948248125*^9, 3.4498948269375*^9}, {3.449894895734375*^9, 
  3.44989491465625*^9}}],

Cell[TextData[{
 "2.d) Nari\[SHacek]i vija\[CHacek]nico r(t)=( cos(t), sin(t), ",
 Cell[BoxData[
  FormBox[
   FractionBox["t", 
    RowBox[{"2", "\[Pi]"}]], TraditionalForm]],
  FormatType->"TraditionalForm"],
 ")  za vrednosti t med -4\[Pi] in 4\[Pi] s pomo\[CHacek]jo ukaza  ",
 StyleBox["ParametricPlot3D", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.449894993734375*^9, 3.449895156484375*^9}, {
  3.449895193703125*^9, 3.449895196671875*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"Clear", "[", "r", "]"}]], "Input"],

Cell[BoxData[
 RowBox[{
  RowBox[{"r", "[", "t_", "]"}], ":=", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"Cos", "[", "t", "]"}], ",", 
    RowBox[{"Sin", "[", "t", "]"}], ",", 
    RowBox[{"t", "/", 
     RowBox[{"(", 
      RowBox[{"2", "Pi"}], ")"}]}]}], "}"}]}]], "Input",
 CellChangeTimes->{{3.4498496650625*^9, 3.4498496785*^9}, {3.4498497203125*^9,
   3.4498497224375*^9}}],

Cell[BoxData[
 RowBox[{"krivulja", "=", 
  RowBox[{"ParametricPlot3D", "[", 
   RowBox[{
    RowBox[{"r", "[", "t", "]"}], ",", 
    RowBox[{"{", 
     RowBox[{"t", ",", 
      RowBox[{
       RowBox[{"-", "4"}], "Pi"}], ",", 
      RowBox[{"4", "Pi"}]}], "}"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.449849733453125*^9, 3.449849735328125*^9}}],

Cell["\<\
2.e) Izra\[CHacek]unaj dol\:017eino enega zavoja vija\[CHacek]nice.\
\>", "Text",
 CellChangeTimes->{{3.449895247125*^9, 3.449895256375*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"Integrate", "[", 
  RowBox[{
   RowBox[{"Sqrt", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"r", "'"}], "[", "t", "]"}], ".", 
     RowBox[{
      RowBox[{"r", "'"}], "[", "t", "]"}]}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "0", ",", 
     RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input"],

Cell["Krivulje v polarnih koordinatah", "Subtitle",
 CellChangeTimes->{
  3.4498952653125*^9, {3.449896466296875*^9, 3.449896471828125*^9}},
 TextAlignment->Left,
 FontFamily->"Arial",
 FontSlant->"Plain",
 FontVariations->{"CompatibilityType"->0},
 FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
 "Doslej smo krivulje podajali le v kartezi\[CHacek]nih korrdinatah (x,y), \
sedaj pa bomo predstavili tudi polarne koordinate (r,\[CurlyPhi]), kjer je r \
oddaljenost to\[CHacek]ke od izhodi\[SHacek]\[CHacek]a, \[CurlyPhi] pa kot, \
ki ga oklepata pozitivni del osi x in daljica, ki povezuje to\[CHacek]ko z \
izhodi\[SHacek]\[CHacek]em. Za prehod med polarnimi in kartezi\[CHacek]nimi \
koordinatami veljata zvezi:\nx = r cos(\[CurlyPhi])\tr =",
 Cell[BoxData[
  FormBox[
   SqrtBox[
    RowBox[{
     SuperscriptBox["x", "2"], "+", 
     SuperscriptBox["y", "2"]}]], TraditionalForm]]],
 "\ny = r sin(\[CurlyPhi]) \t\[CurlyPhi] = arctg(",
 Cell[BoxData[
  FormBox[
   FractionBox["y", "x"], TraditionalForm]]],
 ")"
}], "Text",
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[TextData[{
 "Krivulje v polarnih koordinatah, ki so podane z zvezo r = f(\[CurlyPhi]), \
za \[Alpha]\[LessEqual]\[CurlyPhi]\[LessEqual]\[Beta], ri\[SHacek]emo z \
ukazom ",
 StyleBox["PolarPlot", "Input"],
 "."
}], "Text",
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell["\<\
3.a) Sto\:017enice s koordinatnim izhodi\[SHacek]\[CHacek]em v gori\[SHacek]\
\[CHacek]u lahko podamo tudi v polarni obliki  r(\[Phi])=p/(1+\[Epsilon]cos(\
\[Phi]))  za 0\[LessEqual]\[CurlyPhi]\[LessEqual]2\[Pi]. \
\>", "Text",
 CellChangeTimes->{{3.44989545821875*^9, 3.449895459453125*^9}, {
  3.44989586509375*^9, 3.44989601690625*^9}, {3.449896108953125*^9, 
  3.44989610925*^9}, {3.44990855928125*^9, 3.44990856640625*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"Manipulate", "[", 
  RowBox[{
   RowBox[{"PolarPlot", "[", 
    RowBox[{
     RowBox[{"p", "/", 
      RowBox[{"(", 
       RowBox[{"1", "+", 
        RowBox[{"e", " ", 
         RowBox[{"Cos", "[", "t", "]"}]}]}], ")"}]}], ",", 
     RowBox[{"{", 
      RowBox[{"t", ",", "0", ",", 
       RowBox[{"2", "Pi"}]}], "}"}], ",", 
     RowBox[{"PlotRange", "\[Rule]", "5"}]}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
      "e", ",", "0", ",", "\"\<Ekscentri\[CHacek]nost sto\:017enice e\>\""}], 
      "}"}], ",", "0", ",", "2"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"p", ",", "1", ",", "\"\<Parameter p\>\""}], "}"}], ",", "0", 
     ",", "2"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449895665734375*^9, 3.44989569059375*^9}, {
  3.4498957490625*^9, 3.44989578659375*^9}, {3.44989613584375*^9, 
  3.4498962368125*^9}, {3.449896305546875*^9, 3.449896324703125*^9}}],

Cell[TextData[{
 "3.b) Izra\[CHacek]unaj plo\[SHacek]\[CHacek]ino elipse ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"r", 
     RowBox[{"(", "\[Phi]", ")"}]}], "=", 
    RowBox[{"1", "/", 
     RowBox[{"(", 
      RowBox[{"1", "+", "0.5"}]}]}]}], TraditionalForm]]],
 "cos(\[Phi])). "
}], "Text",
 CellChangeTimes->{{3.44989545821875*^9, 3.449895459453125*^9}, {
  3.44989586509375*^9, 3.44989601690625*^9}, {3.449896108953125*^9, 
  3.44989610925*^9}, {3.44990855928125*^9, 3.44990856640625*^9}, {
  3.44990860359375*^9, 3.449908665703125*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"ParametricPlot", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"r", " ", 
      RowBox[{"1", "/", 
       RowBox[{"(", 
        RowBox[{"1", "+", 
         RowBox[{"0.5", " ", 
          RowBox[{"Cos", "[", "t", "]"}]}]}], ")"}]}], " ", 
      RowBox[{"Cos", "[", "t", "]"}]}], ",", 
     RowBox[{"r", " ", 
      RowBox[{"1", "/", 
       RowBox[{"(", 
        RowBox[{"1", "+", 
         RowBox[{"0.5", " ", 
          RowBox[{"Cos", "[", "t", "]"}]}]}], ")"}]}], 
      RowBox[{"Sin", "[", "t", "]"}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "0", ",", 
     RowBox[{"2", "Pi"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"r", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.44992253359375*^9, 3.449922624390625*^9}, {
  3.4499227126875*^9, 3.4499227761875*^9}, {3.449922807625*^9, 
  3.44992281503125*^9}, {3.449922851015625*^9, 3.449922932359375*^9}, {
  3.4499229690625*^9, 3.449922984890625*^9}}],

Cell[BoxData[
 RowBox[{"S", "=", 
  RowBox[{
   RowBox[{"1", "/", "2"}], 
   RowBox[{"Integrate", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"1", "/", 
        RowBox[{"(", 
         RowBox[{"1", "+", 
          RowBox[{
           RowBox[{"1", "/", "2"}], " ", 
           RowBox[{"Cos", "[", "t", "]"}]}]}], ")"}]}], ")"}], "^", "2"}], 
     ",", 
     RowBox[{"{", 
      RowBox[{"t", ",", "0", ",", 
       RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]}]], "Input",
 CellChangeTimes->{{3.44990851740625*^9, 3.449908537109375*^9}, {
  3.449908687921875*^9, 3.449908729*^9}, {3.4499229975*^9, 
  3.44992307590625*^9}}],

Cell[CellGroupData[{

Cell["Grafi funkcij dveh spremenljivk", "Subtitle",
 CellChangeTimes->{{3.44989645871875*^9, 3.449896474875*^9}},
 TextAlignment->Left,
 FontFamily->"Arial",
 FontSlant->"Plain",
 FontVariations->{"CompatibilityType"->0},
 FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
 "Sedaj se bomo ukvarjali s funkcijami dveh spremenljivk, ki so podane s \
predpisom f(x,y); a\[LessEqual]x\[LessEqual]b, c\[LessEqual]y\[LessEqual]d. \
Grafi teh funkcij so ploskve v prostoru in jih lahko u\[CHacek]inkovito \
predstavimo s funkcijo ",
 StyleBox["Plot3D", "Input"]
}], "Text",
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[TextData[{
 "4. a) Nari\[SHacek]i grafa funkcij f(x,y)=sin(x)+y in ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"f", 
     RowBox[{"(", 
      RowBox[{"x", ",", "y"}], ")"}]}], "=", 
    SuperscriptBox["y", "2"]}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 "."
}], "Text",
 CellChangeTimes->{{3.449898902078125*^9, 3.449898970140625*^9}, {
  3.449899041609375*^9, 3.4498990429375*^9}, {3.449899080328125*^9, 
  3.449899168015625*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"Sin", "[", "x", "]"}], "+", "y"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{
      RowBox[{"-", "4"}], "Pi"}], ",", 
     RowBox[{"4", "Pi"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "5"}], ",", "5"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449847432015625*^9, 3.449847503171875*^9}, {
  3.449848026609375*^9, 3.44984802684375*^9}, {3.449848195546875*^9, 
  3.4498482758125*^9}}],

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{"y", "^", "2"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "5"}], ",", "5"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449847432015625*^9, 3.449847503171875*^9}, {
   3.449848026609375*^9, 3.44984807540625*^9}, {3.449848144375*^9, 
   3.449848181078125*^9}, 3.44984831940625*^9}],

Cell[TextData[{
 "4. b) Nari\[SHacek]i sto\:017eec, ki je podan z ena\[CHacek]bo z = ",
 Cell[BoxData[
  FormBox[
   SqrtBox[
    RowBox[{
     SuperscriptBox["x", "2"], "+", 
     SuperscriptBox["y", "2"]}]], TraditionalForm]]],
 "in paraboloid, ki je podan z ena\[CHacek]bo ",
 Cell[BoxData[
  FormBox[
   RowBox[{"z", "=", 
    RowBox[{
     SuperscriptBox["x", "2"], "+", 
     RowBox[{
      SuperscriptBox["y", "2"], "."}]}]}], TraditionalForm]]]
}], "Text",
 CellChangeTimes->{{3.44984917478125*^9, 3.449849198921875*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{"Sqrt", "[", 
    RowBox[{
     RowBox[{"x", "^", "2"}], "+", 
     RowBox[{"y", "^", "2"}]}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449849233171875*^9, 3.44984925728125*^9}}],

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"x", "^", "2"}], "+", 
    RowBox[{"y", "^", "2"}]}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449847366359375*^9, 3.44984736953125*^9}, {
  3.449847662484375*^9, 3.449847665*^9}, {3.449849206515625*^9, 
  3.449849210640625*^9}}],

Cell[TextData[{
 "Slike nekaterih ploskev (predvsem tistih, ki so sestavljene iz \
kro\:017enic) lep\[SHacek]e nari\[SHacek]emo z ukazom ",
 StyleBox["ParametricPlot3D", "Input"],
 "."
}], "Text",
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell["\<\
4. c) Nari\[SHacek]i sto\:017eec in paraboloid \[SHacek]e v \
parametri\[CHacek]ni obliki.\
\>", "Text",
 CellChangeTimes->{{3.44984934575*^9, 3.44984935328125*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"ParametricPlot3D", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"r", " ", 
      RowBox[{"Cos", "[", "t", "]"}]}], ",", 
     RowBox[{"r", " ", 
      RowBox[{"Sin", "[", "t", "]"}]}], ",", " ", "r"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "0", ",", 
     RowBox[{"2", "Pi"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"r", ",", "0", ",", "2"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.44984768409375*^9, 3.449847722078125*^9}, {
  3.44984936690625*^9, 3.449849382296875*^9}}],

Cell[BoxData[
 RowBox[{"ParametricPlot3D", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"r", " ", 
      RowBox[{"Cos", "[", "t", "]"}]}], ",", 
     RowBox[{"r", " ", 
      RowBox[{"Sin", "[", "t", "]"}]}], ",", " ", 
     RowBox[{"r", "^", "2"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "0", ",", 
     RowBox[{"2", "Pi"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"r", ",", "0", ",", "2"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.44984768409375*^9, 3.449847722078125*^9}}],

Cell["\<\
Pogosto nas zanimajo nivojnice neke funkcije. To so krivuje, ki jih dobimo, \
\[CHacek]e graf funkcije presekamo z ravnino z=a, za neko vrednost \
a\[Element]R.\
\>", "Text",
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[TextData[{
 "4. d) Poi\[SHacek]\[CHacek]i nivojnice paraboloida ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"f", 
     RowBox[{"(", 
      RowBox[{"x", ",", "y"}], ")"}]}], "=", 
    SuperscriptBox["x", "2"]}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 "+ ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["y", "2"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 " in pa funkcije ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"f", 
     RowBox[{"(", 
      RowBox[{"x", ",", "y"}], ")"}]}], "=", 
    SuperscriptBox["y", "2"]}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 " z ukazom ",
 StyleBox["ContourPlot", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.449849413890625*^9, 3.44984949634375*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"ContourPlot", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"x", "^", "2"}], "+", " ", 
    RowBox[{"y", "^", "2"}]}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
   RowBox[{"Contours", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"0", ",", 
      RowBox[{"1", "/", "5"}], ",", 
      RowBox[{"1", "/", "4"}], ",", 
      RowBox[{"1", "/", "3"}], ",", 
      RowBox[{"1", "/", "2"}], ",", "1", ",", "2", ",", "3", ",", "4", ",", 
      "5", ",", "6", ",", "7", ",", "8"}], "}"}]}]}], "]"}]], "Input",
 CellChangeTimes->{{3.44984758125*^9, 3.44984761453125*^9}, {
  3.44984788440625*^9, 3.4498479300625*^9}}],

Cell[BoxData[
 RowBox[{"ContourPlot", "[", 
  RowBox[{
   RowBox[{"y", "^", "2"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", 
   RowBox[{"Contours", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"0", ",", 
      RowBox[{"1", "/", "4"}], ",", 
      RowBox[{"1", "/", "3"}], ",", 
      RowBox[{"1", "/", "2"}], ",", "1", ",", 
      RowBox[{"3", "/", "2"}], ",", "2", ",", 
      RowBox[{"5", "/", "2"}], ",", "3", ",", 
      RowBox[{"7", "/", "2"}], ",", "4"}], "}"}]}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449847834*^9, 3.44984786859375*^9}, {
  3.449847940046875*^9, 3.449848005859375*^9}, {3.449848343484375*^9, 
  3.44984836703125*^9}, {3.4498488461875*^9, 3.449848894078125*^9}, {
  3.449848934828125*^9, 3.449848937125*^9}}]
}, Open  ]],

Cell["Ekstremi funkcij dveh spremenljivk", "Subtitle",
 TextAlignment->Left,
 FontFamily->"Arial",
 FontSlant->"Plain",
 FontVariations->{"CompatibilityType"->0},
 FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
 "Pri funkcijah dveh spremenljivk obravnavamo parcialna odvoda, ki ju izra\
\[CHacek]unamo z ukazoma ",
 StyleBox["D[f[x,y],x] ", "Input"],
 "in  ",
 StyleBox["D[f[x,y],y]", "Input"],
 ". "
}], "Text",
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[TextData[{
 "5. a) Za funkcijo ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"f", 
     RowBox[{"(", 
      RowBox[{"x", ",", "y"}], ")"}]}], "=", 
    SuperscriptBox["x", "2"]}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 " izra\[CHacek]unaj parcialna odvoda ",
 Cell[BoxData[
  FormBox[
   FractionBox[
    RowBox[{"\[PartialD]", " ", "f"}], 
    RowBox[{"\[PartialD]", " ", "x"}]], TraditionalForm]]],
 " in ",
 Cell[BoxData[
  FormBox[
   FractionBox[
    RowBox[{"\[PartialD]", " ", "f"}], 
    RowBox[{"\[PartialD]", " ", "y"}]], TraditionalForm]]],
 "."
}], "Text",
 CellChangeTimes->{{3.44989939725*^9, 3.449899417515625*^9}, {
  3.449899732140625*^9, 3.449899741734375*^9}, {3.449903178078125*^9, 
  3.449903184125*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"Clear", "[", "f", "]"}]], "Input",
 CellChangeTimes->{{3.449901893328125*^9, 3.449901896515625*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "y_"}], "]"}], ":=", " ", 
  RowBox[{"x", "^", "2"}]}]], "Input",
 CellChangeTimes->{{3.449899608*^9, 3.449899608390625*^9}, {
   3.449899705359375*^9, 3.449899709609375*^9}, {3.449901995953125*^9, 
   3.44990201278125*^9}, 3.4499030093125*^9, {3.4499030714375*^9, 
   3.44990307353125*^9}, {3.449903139234375*^9, 3.449903139359375*^9}}],

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", "y"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449901857625*^9, 3.449901880578125*^9}, {
  3.44990202359375*^9, 3.449902059328125*^9}, {3.44990217528125*^9, 
  3.449902196046875*^9}, {3.44990263565625*^9, 3.449902693828125*^9}, {
  3.449902775515625*^9, 3.449902802109375*^9}, {3.449903089421875*^9, 
  3.4499031139375*^9}, {3.44990314753125*^9, 3.449903163390625*^9}}],

Cell[BoxData[{
 RowBox[{"D", "[", 
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", "y"}], "]"}], ",", "x"}], "]"}], "\[IndentingNewLine]", 
 RowBox[{"D", "[", 
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", "y"}], "]"}], ",", "y"}], "]"}]}], "Input",
 CellChangeTimes->{{3.4498994288125*^9, 3.449899430625*^9}}],

Cell["\<\
5. b) Poi\[SHacek]\[CHacek]i ena\[CHacek]bo tangentne ravnine na graf \
funkcije f v to\[CHacek]ki T(1,2) in jo nari\[SHacek]i na isto sliko skupaj s \
funkcijo.\
\>", "Text",
 CellChangeTimes->{{3.4498997570625*^9, 3.4498997585625*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{
  RowBox[{"TangentnaRavnina", "[", 
   RowBox[{"f_", ",", "a_", ",", "b_", ",", "x_", ",", "y_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"a", ",", "b"}], "]"}], "+", 
   RowBox[{
    RowBox[{"(", 
     RowBox[{
      RowBox[{"D", "[", 
       RowBox[{
        RowBox[{"f", "[", 
         RowBox[{"x", ",", "y"}], "]"}], ",", "x"}], "]"}], "/.", 
      RowBox[{"{", 
       RowBox[{
        RowBox[{"x", "\[Rule]", "a"}], ",", 
        RowBox[{"y", "\[Rule]", "b"}]}], "}"}]}], ")"}], 
    RowBox[{"(", 
     RowBox[{"x", "-", "a"}], ")"}]}], "+", 
   RowBox[{
    RowBox[{"(", 
     RowBox[{
      RowBox[{"D", "[", 
       RowBox[{
        RowBox[{"f", "[", 
         RowBox[{"x", ",", "y"}], "]"}], ",", "y"}], "]"}], "/.", 
      RowBox[{"{", 
       RowBox[{
        RowBox[{"x", "\[Rule]", "a"}], ",", 
        RowBox[{"y", "\[Rule]", "b"}]}], "}"}]}], ")"}], 
    RowBox[{"(", 
     RowBox[{"y", "-", "b"}], ")"}]}]}]}]], "Input",
 CellChangeTimes->{{3.449900147703125*^9, 3.449900150921875*^9}, {
  3.4499004789375*^9, 3.449900528*^9}, {3.44990056875*^9, 
  3.4499005690625*^9}, {3.449902429609375*^9, 3.449902430171875*^9}}],

Cell[BoxData[
 RowBox[{"TangentnaRavnina", "[", 
  RowBox[{"f", ",", "1", ",", "2", ",", "x", ",", "y"}], "]"}]], "Input",
 CellChangeTimes->{{3.4499022224375*^9, 3.44990224984375*^9}, {
  3.44990239740625*^9, 3.4499024320625*^9}, {3.44990320584375*^9, 
  3.449903206625*^9}}],

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", "y"}], "]"}], ",", 
     RowBox[{"1", "+", 
      RowBox[{"2", 
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", "x"}], ")"}]}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449901857625*^9, 3.449901880578125*^9}, {
  3.44990202359375*^9, 3.449902059328125*^9}, {3.44990217528125*^9, 
  3.449902196046875*^9}, {3.449902478265625*^9, 3.449902481796875*^9}, {
  3.44990287153125*^9, 3.449902895421875*^9}, {3.4499030281875*^9, 
  3.44990303915625*^9}, {3.449903210484375*^9, 3.449903226625*^9}}],

Cell["\<\
S parcialnimi odvodi funkcije pa tudi i\[SHacek]\[CHacek]emo kandidate za \
stacionarne to\[CHacek]ke, to so to\[CHacek]ke, kjer sta oba parcialna odvoda \
enaka ni\[CHacek]. Tangentna ravnina je v teh to\[CHacek]kah vzporedna x-y \
ravnini.\
\>", "Text",
 CellChangeTimes->{
  3.44990326034375*^9, {3.4499034889375*^9, 3.4499035183125*^9}},
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[TextData[{
 "5. c) Poi\[SHacek]\[CHacek]i stacionarne to\[CHacek]ke funkcije ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"f", 
     RowBox[{"(", 
      RowBox[{"x", ",", "y"}], ")"}]}], "=", 
    RowBox[{
     SuperscriptBox["x", "2"], "+", 
     SuperscriptBox["y", "2"]}]}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 "."
}], "Text",
 CellChangeTimes->{{3.4499032965*^9, 3.44990332790625*^9}, {
  3.44990345815625*^9, 3.449903476703125*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"Clear", "[", "f", "]"}]], "Input",
 CellChangeTimes->{{3.449903428*^9, 3.4499034298125*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "y_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"x", " ", "^", "2"}], "+", 
   RowBox[{"y", "^", "2"}], "+", "1"}]}]], "Input",
 CellChangeTimes->{{3.44990327009375*^9, 3.449903272546875*^9}, {
  3.44990336878125*^9, 3.44990337671875*^9}, {3.449903424796875*^9, 
  3.449903426015625*^9}, {3.44990355215625*^9, 3.44990355234375*^9}}],

Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"D", "[", 
       RowBox[{
        RowBox[{"f", "[", 
         RowBox[{"x", ",", "y"}], "]"}], ",", "x"}], "]"}], "\[Equal]", "0"}],
      ",", 
     RowBox[{
      RowBox[{"D", "[", 
       RowBox[{
        RowBox[{"f", "[", 
         RowBox[{"x", ",", "y"}], "]"}], ",", "y"}], "]"}], "\[Equal]", 
      "0"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "y"}], "}"}]}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"TangentnaRavnina", "[", 
  RowBox[{"f", ",", "0", ",", "0", ",", "x", ",", "y"}], "]"}]], "Input",
 CellChangeTimes->{{3.4499022224375*^9, 3.44990224984375*^9}, {
  3.44990239740625*^9, 3.4499024320625*^9}, {3.44990320584375*^9, 
  3.449903206625*^9}, {3.449903535328125*^9, 3.449903536015625*^9}}],

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", "y"}], "]"}], ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "3"}], ",", "3"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.4499035411875*^9, 3.449903558140625*^9}}],

Cell[TextData[{
 "Globalne ekstreme funkcij dveh spremenljivk i\[SHacek]\[CHacek]emo z \
ukazoma  ",
 StyleBox["Minimize ", "Input"],
 "in ",
 StyleBox["Maximize", "Input"],
 ". "
}], "Text",
 CellChangeTimes->{{3.4499042908125*^9, 3.449904331453125*^9}},
 FontFamily->"Arial",
 Background->GrayLevel[0.900008]],

Cell[TextData[{
 "5. d) Poi\[SHacek]\[CHacek]i globalni minimum in globalni maksimum funkcije \
",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"f", 
     RowBox[{"(", 
      RowBox[{"x", ",", "y"}], ")"}]}], "=", 
    RowBox[{
     RowBox[{"5", 
      SuperscriptBox["x", "2"]}], "-", 
     RowBox[{"3", "xy"}], "+", 
     RowBox[{"4", 
      SuperscriptBox["y", "2"]}]}]}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 "+3x na kvadratu z ogli\[SHacek]\[CHacek]i A(-1,-1), B(1,-1), C(1,1) in \
D(-1,1). "
}], "Text",
 CellChangeTimes->{{3.4499041916875*^9, 3.44990427178125*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"Clear", "[", "f", "]"}]], "Input",
 CellChangeTimes->{{3.449903617390625*^9, 3.449903628125*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "y_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"5", 
    RowBox[{"x", "^", "2"}]}], "-", 
   RowBox[{"3", "x", " ", "y"}], "+", 
   RowBox[{"4", 
    RowBox[{"y", "^", "2"}]}], "+", 
   RowBox[{"3", "x"}]}]}]], "Input",
 CellChangeTimes->{{3.449903631453125*^9, 3.4499036476875*^9}, {
  3.44990371884375*^9, 3.449903753234375*^9}, {3.449904131421875*^9, 
  3.449904177640625*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", "y"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.4499035411875*^9, 3.449903558140625*^9}, {
  3.449903659265625*^9, 3.44990366103125*^9}, {3.449903974640625*^9, 
  3.449903985890625*^9}}],

Cell[BoxData[
 RowBox[{"Maximize", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", "y"}], "]"}], ",", 
     RowBox[{
      RowBox[{"-", "1"}], "\[LessEqual]", "x", "\[LessEqual]", "1"}], ",", 
     RowBox[{
      RowBox[{"-", "1"}], "\[LessEqual]", "y", "\[LessEqual]", "1"}]}], "}"}],
    ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "y"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449903927203125*^9, 3.449903965265625*^9}, {
  3.449904013921875*^9, 3.449904061328125*^9}}],

Cell[BoxData[
 RowBox[{"Minimize", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", "y"}], "]"}], ",", 
     RowBox[{
      RowBox[{"-", "1"}], "\[LessEqual]", "x", "\[LessEqual]", "1"}], ",", 
     RowBox[{
      RowBox[{"-", "1"}], "\[LessEqual]", "y", "\[LessEqual]", "1"}]}], "}"}],
    ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "y"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449903927203125*^9, 3.449903965265625*^9}, {
  3.449904013921875*^9, 3.449904074484375*^9}}]
}, Open  ]],

Cell["Integrali funkcij dveh spremenljivk", "Subtitle",
 CellChangeTimes->{{3.449904436375*^9, 3.449904440046875*^9}},
 TextAlignment->Left,
 FontFamily->"Arial",
 FontSlant->"Plain",
 FontVariations->{"CompatibilityType"->0},
 FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
 "6.a) Integriraj funkcijo f(x)=",
 Cell[BoxData[
  FormBox[
   RowBox[{"2", "-", 
    SuperscriptBox["x", "2"], "-", 
    SuperscriptBox["y", "2"]}], TraditionalForm]]],
 " po pravokotniku -1\[LessEqual]x\[LessEqual]1, -1\[LessEqual]y\[LessEqual]1."
}], "Text",
 CellChangeTimes->{{3.44990485246875*^9, 3.449904915125*^9}, {3.449904965*^9, 
  3.449904968234375*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[BoxData[
 RowBox[{"RegionPlot3D", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"2", "-", 
     RowBox[{"x", "^", "2"}], "+", 
     RowBox[{"-", 
      RowBox[{"y", "^", "2"}]}]}], ">", "z"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"z", ",", "0", ",", "2"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.4499045471875*^9, 3.449904577625*^9}, {
  3.449904609703125*^9, 3.44990475090625*^9}, {3.449904828734375*^9, 
  3.44990483140625*^9}},
 CellID->1579825],

Cell[BoxData[
 RowBox[{"Integrate", "[", 
  RowBox[{
   RowBox[{"Integrate", "[", 
    RowBox[{
     RowBox[{"2", "-", 
      RowBox[{"x", "^", "2"}], "-", 
      RowBox[{"y", "^", "2"}]}], ",", 
     RowBox[{"{", 
      RowBox[{"x", ",", 
       RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449904939390625*^9, 3.44990495353125*^9}}],

Cell[TextData[{
 "6.b) Integriraj funkcijo ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"f", " ", 
     RowBox[{"(", "x", ")"}]}], " ", "=", 
    RowBox[{"1", "-", "x", "-", "y", " "}]}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 " po trikotniku omejenem s premico y =1- x in pa x ter y osjo."
}], "Text",
 CellChangeTimes->{{3.449904978671875*^9, 3.449904993046875*^9}, {
  3.449905408828125*^9, 3.44990541040625*^9}, {3.4499055245*^9, 
  3.44990553171875*^9}, {3.449905622125*^9, 3.4499056358125*^9}, {
  3.449905801*^9, 3.449905821765625*^9}},
 FontFamily->"Arial",
 Background->RGBColor[0.717647, 0.909804, 0.721569]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{"1", "-", "x", "-", "y"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", "0", ",", "1"}], "}"}], ",", 
   RowBox[{"AxesOrigin", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"0", ",", "0", ",", "0"}], "}"}]}]}], "]"}]], "Input",
 CellChangeTimes->{{3.449905276875*^9, 3.449905304765625*^9}, {
  3.4499054580625*^9, 3.4499054795625*^9}, {3.449905539375*^9, 
  3.449905597609375*^9}}],

Cell[BoxData[
 RowBox[{"RegionPlot3D", "[", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"1", "-", "x", "-", "y"}], ">", "z"}], "&&", 
    RowBox[{"y", "\[LessEqual]", 
     RowBox[{"1", "-", "x"}]}], "&&", 
    RowBox[{"y", ">", "0"}], "&&", 
    RowBox[{"x", ">", "0"}]}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", "0", ",", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"z", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.4499045471875*^9, 3.449904577625*^9}, {
  3.449904609703125*^9, 3.44990475090625*^9}, {3.449904828734375*^9, 
  3.44990483140625*^9}, {3.449905022*^9, 3.44990508525*^9}, {
  3.449905124046875*^9, 3.449905127109375*^9}, {3.449905220765625*^9, 
  3.449905224234375*^9}, {3.44990548746875*^9, 3.449905492390625*^9}, {
  3.449905642421875*^9, 3.449905663375*^9}},
 CellID->13992]
}, Open  ]],

Cell[BoxData[
 RowBox[{"Integrate", "[", 
  RowBox[{
   RowBox[{"Integrate", "[", 
    RowBox[{
     RowBox[{"1", "-", "x", "-", "y"}], ",", 
     RowBox[{"{", 
      RowBox[{"y", ",", "0", ",", 
       RowBox[{"1", "-", "x"}]}], "}"}]}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.44990571040625*^9, 3.449905752421875*^9}}],

Cell[TextData[StyleBox["Naloge",
 FontSlant->"Plain",
 FontVariations->{"CompatibilityType"->0},
 FontColor->RGBColor[1, 0, 0]]], "Subtitle",
 CellChangeTimes->{3.449905784265625*^9},
 TextAlignment->Left,
 FontFamily->"Arial"],

Cell[TextData[{
 "1.a) Nari\[SHacek]i parametri\[CHacek]no podano krivuljo ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"r", "(", "t", ")"}], "=", 
    RowBox[{"(", 
     RowBox[{
      SuperscriptBox["t", "2"], ",", 
      RowBox[{"t", "-", 
       SuperscriptBox["t", "3"]}]}]}]}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 ") za t\[Element][-2,2].\nb) Izra\[CHacek]unaj plo\[SHacek]\[CHacek]ino \
lika, ki ga opi\[SHacek]e krivulja na intervalu  t\[Element][-1,1].\nc) Izra\
\[CHacek]unaj dol\:017eino krivulje na intervalu t\[Element][-2,2]."
}], "Text",
 CellChangeTimes->{{3.44990657078125*^9, 3.449906734515625*^9}, {
   3.4499069005625*^9, 3.449906900671875*^9}, 3.449907719046875*^9},
 FontFamily->"Arial",
 Background->RGBColor[1, 0.745098, 0.490196]],

Cell["\<\
2.a) Nari\[SHacek]i krivuljo r=cos(3\[CurlyPhi]), ki je podana v polarnih \
koordinatah za \[Phi]\[Element][0,2\[Pi]]. 
b) Izra\[CHacek]unaj plo\[SHacek]\[CHacek]ino lika, ki ga krivulja omejuje.\
\>", "Text",
 CellChangeTimes->{{3.44990684409375*^9, 3.449906911609375*^9}, 
   3.449907717328125*^9},
 FontFamily->"Arial",
 Background->RGBColor[1, 0.745098, 0.490196]],

Cell[TextData[{
 "3.a) Nari\[SHacek]i graf funkcije f(x,y) = ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["x", "2"], TraditionalForm]]],
 "+ 3x y",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["", "2"], TraditionalForm]]],
 " + ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["y", "2"], TraditionalForm]]],
 " + ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["y", "3"], TraditionalForm]]],
 ".\nb) Poi\[SHacek]\[CHacek]i njene stacionarne to\[CHacek]ke.\nc) Poi\
\[SHacek]\[CHacek]i globalni minimum in globalni maksimum funkcije f na \
pravokotniku omejenim s to\[CHacek]kami A(-2,-2), B(3,-2), C(3,2) in D(-2,2)."
}], "Text",
 CellChangeTimes->{{3.449907578171875*^9, 3.449907663796875*^9}, 
   3.449907715703125*^9},
 FontFamily->"Arial",
 Background->RGBColor[1, 0.745098, 0.490196]],

Cell["\<\
4.a)  Integriraj funkcijo f(x,y)=1+sin(x) po pravokotniku -\[Pi]\[LessEqual]x\
\[LessEqual]\[Pi], -1\[LessEqual]y\[LessEqual]1.
b) Integriraj funkcijo f(x,y)=xy po trikotniku omejenem s premico y =1+ x in \
pa x ter y osjo.\
\>", "Text",
 CellChangeTimes->{{3.449907684109375*^9, 3.449907844890625*^9}},
 FontFamily->"Arial",
 Background->RGBColor[1, 0.745098, 0.490196]]
},
CellGrouping->Manual,
WindowSize->{1272, 903},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
DockedCells->FEPrivate`FrontEndResource[
 "FEExpressions", "CompatibilityToolbar"],
ShowSelection->True,
FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (November 10, 2008)",
StyleDefinitions->"Classic.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[545, 20, 310, 7, 53, "Title"],
Cell[858, 29, 278, 7, 59, "Subtitle"],
Cell[1139, 38, 550, 12, 53, "Text"],
Cell[1692, 52, 798, 25, 36, "Text"],
Cell[2493, 79, 627, 16, 36, "Input"],
Cell[3123, 97, 391, 9, 34, "Text"],
Cell[3517, 108, 923, 24, 36, "Input"],
Cell[4443, 134, 603, 17, 47, "Text"],
Cell[5049, 153, 1364, 36, 36, "Input"],
Cell[6416, 191, 262, 6, 59, "Subtitle"],
Cell[6681, 199, 404, 9, 34, "Text"],
Cell[7088, 210, 353, 9, 34, "Text"],
Cell[7444, 221, 399, 12, 36, "Input"],
Cell[7846, 235, 356, 8, 33, "Text"],
Cell[8205, 245, 5102, 125, 117, "Input"],
Cell[13310, 372, 1245, 43, 73, "Text"],
Cell[14558, 417, 114, 2, 36, "Input"],
Cell[14675, 421, 137, 3, 36, "Input"],
Cell[14815, 426, 963, 31, 36, "Input"],
Cell[15781, 459, 113, 2, 36, "Input"],
Cell[15897, 463, 854, 29, 36, "Input"],
Cell[16754, 494, 483, 13, 40, "Text"],
Cell[17240, 509, 4965, 135, 247, "Input"],
Cell[22208, 646, 255, 6, 33, "Text"],
Cell[22466, 654, 363, 13, 36, "Input"],
Cell[22832, 669, 363, 13, 36, "Input"],
Cell[23198, 684, 135, 3, 36, "Input"],
Cell[23336, 689, 1127, 36, 36, "Input"],
Cell[24466, 727, 611, 19, 36, "Input"],
Cell[25080, 748, 113, 2, 36, "Input"],
Cell[25196, 752, 1127, 36, 36, "Input"],
Cell[26326, 790, 611, 19, 36, "Input"],
Cell[26940, 811, 527, 14, 44, "Text"],
Cell[27470, 827, 58, 1, 36, "Input"],
Cell[27531, 830, 378, 11, 36, "Input"],
Cell[27912, 843, 349, 10, 36, "Input"],
Cell[28264, 855, 226, 5, 33, "Text"],
Cell[28493, 862, 329, 11, 36, "Input"],
Cell[28825, 875, 279, 7, 59, "Subtitle"],
Cell[29107, 884, 802, 20, 106, "Text"],
Cell[29912, 906, 283, 8, 34, "Text"],
Cell[30198, 916, 513, 9, 33, "Text"],
Cell[30714, 927, 972, 27, 36, "Input"],
Cell[31689, 956, 629, 17, 33, "Text"],
Cell[32321, 975, 980, 27, 36, "Input"],
Cell[33304, 1004, 639, 20, 36, "Input"],
Cell[CellGroupData[{
Cell[33968, 1028, 252, 6, 59, "Subtitle"],
Cell[34223, 1036, 365, 8, 34, "Text"],
Cell[34591, 1046, 534, 16, 35, "Text"],
Cell[35128, 1064, 514, 15, 36, "Input"],
Cell[35645, 1081, 459, 12, 36, "Input"],
Cell[36107, 1095, 604, 19, 42, "Text"],
Cell[36714, 1116, 426, 13, 36, "Input"],
Cell[37143, 1131, 483, 14, 36, "Input"],
Cell[37629, 1147, 252, 7, 34, "Text"],
Cell[37884, 1156, 250, 6, 33, "Text"],
Cell[38137, 1164, 539, 15, 36, "Input"],
Cell[38679, 1181, 517, 15, 36, "Input"],
Cell[39199, 1198, 240, 6, 33, "Text"],
Cell[39442, 1206, 826, 30, 36, "Text"],
Cell[40271, 1238, 766, 21, 36, "Input"],
Cell[41040, 1261, 877, 22, 36, "Input"]
}, Open  ]],
Cell[41932, 1286, 193, 5, 59, "Subtitle"],
Cell[42128, 1293, 273, 9, 34, "Text"],
Cell[42404, 1304, 827, 28, 45, "Text"],
Cell[43234, 1334, 124, 2, 36, "Input"],
Cell[43361, 1338, 404, 8, 36, "Input"],
Cell[43768, 1348, 651, 15, 36, "Input"],
Cell[44422, 1365, 326, 9, 57, "Input"],
Cell[44751, 1376, 322, 7, 33, "Text"],
Cell[45076, 1385, 1177, 35, 36, "Input"],
Cell[46256, 1422, 276, 5, 36, "Input"],
Cell[46535, 1429, 831, 22, 36, "Input"],
Cell[47369, 1453, 407, 9, 33, "Text"],
Cell[47779, 1464, 543, 17, 35, "Text"],
Cell[48325, 1483, 116, 2, 36, "Input"],
Cell[48444, 1487, 395, 9, 36, "Input"],
Cell[48842, 1498, 506, 18, 36, "Input"],
Cell[49351, 1518, 322, 5, 36, "Input"],
Cell[49676, 1525, 423, 13, 36, "Input"],
Cell[50102, 1540, 311, 10, 34, "Text"],
Cell[50416, 1552, 669, 21, 35, "Text"],
Cell[51088, 1575, 121, 2, 36, "Input"],
Cell[51212, 1579, 444, 13, 36, "Input"],
Cell[CellGroupData[{
Cell[51681, 1596, 467, 13, 36, "Input"],
Cell[52151, 1611, 529, 15, 36, "Input"],
Cell[52683, 1628, 529, 15, 36, "Input"]
}, Open  ]],
Cell[53227, 1646, 257, 6, 59, "Subtitle"],
Cell[53487, 1654, 458, 12, 35, "Text"],
Cell[53948, 1668, 635, 19, 36, "Input",
 CellID->1579825],
Cell[54586, 1689, 468, 14, 36, "Input"],
Cell[55057, 1705, 641, 16, 33, "Text"],
Cell[CellGroupData[{
Cell[55723, 1725, 508, 13, 36, "Input"],
Cell[56234, 1740, 903, 22, 36, "Input",
 CellID->13992]
}, Open  ]],
Cell[57152, 1765, 398, 11, 36, "Input"],
Cell[57553, 1778, 227, 6, 59, "Subtitle"],
Cell[57783, 1786, 781, 19, 75, "Text"],
Cell[58567, 1807, 378, 8, 52, "Text"],
Cell[58948, 1817, 787, 24, 73, "Text"],
Cell[59738, 1843, 381, 8, 52, "Text"]
}
]
*)

(* End of internal cache information *)