(* Content-type: application/mathematica *)

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

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     31607,        949]
NotebookOptionsPosition[     27568,        815]
NotebookOutlinePosition[     27927,        831]
CellTagsIndexPosition[     27884,        828]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Examples using functions", "Subsubtitle",
 CellChangeTimes->{{3.416577557006275*^9, 3.416577591693158*^9}, 
   3.416654642129732*^9},
 Background->RGBColor[0, 1, 0]],

Cell[TextData[StyleBox["Joe Sloan   April 7, 2008",
 FontSize->10]], "Subsubtitle",
 CellChangeTimes->{{3.416577567351078*^9, 3.416577576877472*^9}}],

Cell[CellGroupData[{

Cell["Simple functions", "Subsubsection",
 CellChangeTimes->{{3.416577662847951*^9, 3.416577665766445*^9}, 
   3.416654713307067*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"P", "[", 
   RowBox[{"p0_", ",", " ", "r_", ",", " ", "t_"}], "]"}], " ", ":=", " ", 
  RowBox[{"p0", " ", "*", " ", 
   RowBox[{"Exp", "[", 
    RowBox[{"r", " ", "*", " ", "t"}], "]"}]}]}]], "Input",
 CellChangeTimes->{{3.4165771051479607`*^9, 3.416577120388797*^9}, {
  3.4165771574736423`*^9, 3.416577209610443*^9}}],

Cell["Functions ease use when doing repetitive calculations", "Text",
 CellChangeTimes->{{3.416654718690641*^9, 3.416654732691285*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"P", "[", 
  RowBox[{"1000", ",", " ", ".1", ",", " ", "2"}], "]"}]], "Input",
 CellChangeTimes->{{3.416577214961321*^9, 3.4165772286419773`*^9}}],

Cell[BoxData["1221.40275816017`"], "Output",
 CellChangeTimes->{3.416577229960679*^9, 3.416655246569488*^9, 
  3.416656525707217*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"P", "[", 
  RowBox[{"1000", ",", " ", ".1", ",", " ", "4"}], "]"}]], "Input",
 CellChangeTimes->{{3.416577214961321*^9, 3.416577247300802*^9}}],

Cell[BoxData["1491.8246976412704`"], "Output",
 CellChangeTimes->{3.416577248246811*^9, 3.416655246612979*^9, 
  3.416656525752129*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"P", "[", 
  RowBox[{"1000", ",", " ", ".1", ",", " ", "6"}], "]"}]], "Input",
 CellChangeTimes->{{3.416577214961321*^9, 3.41657725314075*^9}}],

Cell[BoxData["1822.118800390509`"], "Output",
 CellChangeTimes->{3.4165772537818727`*^9, 3.41665524665285*^9, 
  3.416656525787698*^9}]
}, Open  ]],

Cell["Here is an example using a loop.", "Text",
 CellChangeTimes->{{3.416654917981347*^9, 3.416654930627777*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Do", "[", 
  RowBox[{
   RowBox[{"Print", "[", 
    RowBox[{"P", "[", 
     RowBox[{"1000", ",", " ", ".1", ",", " ", "t"}], "]"}], "]"}], ",", " ", 
   
   RowBox[{"{", 
    RowBox[{"t", ",", " ", "0", ",", " ", "2", ",", " ", ".2"}], "}"}]}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.416577263442134*^9, 3.41657733847262*^9}, 
   3.416656492118594*^9}],

Cell[CellGroupData[{

Cell[BoxData["1000.`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.416656525815775*^9}],

Cell[BoxData["1020.2013400267558`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.416656525818027*^9}],

Cell[BoxData["1040.8107741923882`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.4166565258211317`*^9}],

Cell[BoxData["1061.8365465453596`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.416656525824244*^9}],

Cell[BoxData["1083.2870676749587`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.4166565258273277`*^9}],

Cell[BoxData["1105.1709180756477`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.416656525852015*^9}],

Cell[BoxData["1127.4968515793757`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.416656525908523*^9}],

Cell[BoxData["1150.2737988572273`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.41665652593548*^9}],

Cell[BoxData["1173.5108709918102`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.4166565259623613`*^9}],

Cell[BoxData["1197.2173631218102`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.4166565259904957`*^9}],

Cell[BoxData["1221.40275816017`"], "Print",
 CellChangeTimes->{3.416655246693139*^9, 3.4166564947007017`*^9, 
  3.4166565260148582`*^9}]
}, Open  ]]
}, Open  ]],

Cell["\<\
You can repackage a function to set specific variables.  Here we know the \
initial population and rate and only want to calculate the population at \
different times.\
\>", "Text",
 CellChangeTimes->{{3.416654757691312*^9, 3.416654806447981*^9}, {
  3.416654864266086*^9, 3.41665488576309*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"f1", "[", "t_", "]"}], " ", ":=", " ", 
  RowBox[{"P", "[", 
   RowBox[{"1000", ",", " ", ".1", ",", " ", "t"}], "]"}]}]], "Input",
 CellChangeTimes->{{3.416577386803783*^9, 3.416577433268199*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"f1", "[", "1", "]"}]], "Input",
 CellChangeTimes->{{3.4165774418039513`*^9, 3.41657744581999*^9}}],

Cell[BoxData["1105.1709180756477`"], "Output",
 CellChangeTimes->{3.416577446529358*^9, 3.4166552469742393`*^9, 
  3.416656526107581*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"f1", "[", "2", "]"}]], "Input",
 CellChangeTimes->{{3.4165774473507433`*^9, 3.4165774497559853`*^9}}],

Cell[BoxData["1221.40275816017`"], "Output",
 CellChangeTimes->{3.416577450222321*^9, 3.416655247001696*^9, 
  3.416656526136327*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"f1", "[", "3", "]"}]], "Input",
 CellChangeTimes->{{3.416577450856583*^9, 3.416577454364056*^9}}],

Cell[BoxData["1349.858807576003`"], "Output",
 CellChangeTimes->{3.416577454862514*^9, 3.416655247041757*^9, 
  3.4166565261763067`*^9}]
}, Open  ]],

Cell["Some more functions.", "Text",
 CellChangeTimes->{{3.41665494251818*^9, 3.416654948396295*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"f2", "[", "t_", "]"}], " ", ":=", " ", 
  RowBox[{"P", "[", 
   RowBox[{"1000", ",", " ", ".2", ",", " ", "t"}], "]"}]}]], "Input",
 CellChangeTimes->{{3.416577386803783*^9, 3.416577433268199*^9}, {
  3.4165774683748293`*^9, 3.416577471052786*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"dP", "[", "P_", "]"}], " ", ":=", " ", 
  RowBox[{
   RowBox[{"-", ".0004"}], " ", "*", " ", "P"}]}]], "Input",
 CellChangeTimes->{{3.4165775030182447`*^9, 3.416577530580942*^9}}],

Cell["\<\
Although we interpret the following as a derivative, Mathematica just sees it \
as another function.\
\>", "Text",
 CellChangeTimes->{{3.416654972739423*^9, 3.416655017688265*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"dP", "[", 
   RowBox[{"P_", ",", " ", "M_", ",", " ", "r_"}], "]"}], " ", ":=", " ", 
  RowBox[{"r", " ", "*", "P", "*", 
   RowBox[{"(", 
    RowBox[{"1", " ", "-", " ", 
     RowBox[{"P", "/", "M"}]}], ")"}]}]}]], "Input",
 CellChangeTimes->{{3.416577682673902*^9, 3.416577711640107*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"dP", "[", 
  RowBox[{"100", ",", " ", "1000", ",", " ", ".1"}], "]"}]], "Input",
 CellChangeTimes->{{3.416577749474307*^9, 3.41657776247318*^9}}],

Cell[BoxData["9.`"], "Output",
 CellChangeTimes->{3.416577763572886*^9, 3.416655247151794*^9, 
  3.416656526248423*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"dP", "[", 
  RowBox[{"1005", ",", " ", "1000", ",", " ", ".1"}], "]"}]], "Input",
 CellChangeTimes->{{3.416577767581653*^9, 3.41657777740945*^9}}],

Cell[BoxData[
 RowBox[{"-", "0.5025000000000001`"}]], "Output",
 CellChangeTimes->{3.416577780402608*^9, 3.4166552472054873`*^9, 
  3.416656526283802*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"dP", "[", 
  RowBox[{"1000", ",", "1000", ",", " ", ".5"}], "]"}]], "Input",
 CellChangeTimes->{{3.416577783381459*^9, 3.416577789489357*^9}}],

Cell[BoxData["0"], "Output",
 CellChangeTimes->{3.416577791986763*^9, 3.416655247259696*^9, 
  3.416656526323749*^9}]
}, Open  ]],

Cell["\<\
We can pass function as arguments.  Here is a function for the Euler method \
that uses take a function as the second argument.  The function is applied in \
the body of the do loop.  Note the use of paren' s and semicolons to sequence \
operations.  Note the use of NumberForm and spaces and tabs with in the Print \
command to control formatting.\
\>", "Text",
 CellChangeTimes->{{3.416655036802576*^9, 3.4166551713966427`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Euler1", "[", 
    RowBox[{"p0_", ",", " ", "dP_", ",", " ", "dt_", ",", " ", "tend_"}], 
    "]"}], " ", ":=", "\[IndentingNewLine]", 
   RowBox[{"(", 
    RowBox[{
     RowBox[{"p", " ", "=", " ", "p0"}], ";", "\[IndentingNewLine]", 
     RowBox[{"Do", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"p", " ", "=", " ", 
         RowBox[{"p", " ", "+", " ", 
          RowBox[{
           RowBox[{"dP", "[", "p", "]"}], "*", "dt"}]}]}], ";", 
        "\[IndentingNewLine]", 
        RowBox[{"Print", "[", 
         RowBox[{
         "\"\<Time \>\"", ",", " ", "t", ",", " ", "\[IndentingNewLine]", 
          "\"\<\\tPopulation \>\"", ",", " ", 
          RowBox[{"NumberForm", "[", 
           RowBox[{"p", ",", " ", "10"}], "]"}]}], "]"}]}], ",", 
       "\[IndentingNewLine]", 
       RowBox[{"{", 
        RowBox[{"t", ",", " ", "dt", ",", " ", "tend", ",", " ", "dt"}], 
        "}"}]}], "]"}]}], ")"}]}], "\[IndentingNewLine]"}]], "Input",
 CellChangeTimes->{{3.416577824888781*^9, 3.416577830426427*^9}, {
   3.416577969588328*^9, 3.416578017878983*^9}, {3.4165781117460327`*^9, 
   3.416578161546715*^9}, {3.4165782145492764`*^9, 3.416578264068478*^9}, 
   3.416578306752207*^9, {3.4165784350519238`*^9, 3.4165784368019*^9}, {
   3.416578497830043*^9, 3.416578527756625*^9}, {3.416578776789263*^9, 
   3.416578820443974*^9}, {3.41657889641483*^9, 3.416578916207344*^9}, {
   3.41665616359719*^9, 3.416656170420961*^9}}],

Cell["Here is a 10 % growth function to use as an example.", "Text",
 CellChangeTimes->{{3.416656021571094*^9, 3.416656040294248*^9}, {
  3.416656084288636*^9, 3.41665608831043*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"f1", "[", "p_", "]"}], " ", ":=", " ", 
  RowBox[{"0.1", " ", "*", " ", "p"}]}]], "Input",
 CellChangeTimes->{{3.416578361381503*^9, 3.416578389591178*^9}, 
   3.416655226347872*^9}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Euler1", "[", 
  RowBox[{"100", ",", " ", "f1", ",", " ", "0.005", ",", "  ", "0.040"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.416578397155601*^9, 3.4165784259360437`*^9}, 
   3.4165786876469183`*^9, 3.416655239613804*^9}],

Cell[CellGroupData[{

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.005`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.05\"\>",
     100.05,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.005, "\tPopulation ", 
   NumberForm[100.05, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.4166552473352747`*^9, 3.41665618220322*^9, 
  3.416656526395936*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.01`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.100025\"\>",
     100.100025,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.01, "\tPopulation ", 
   NumberForm[100.100025, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.4166552473352747`*^9, 3.41665618220322*^9, 
  3.416656526403678*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.015`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.150075\"\>",
     100.1500750125,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.015, "\tPopulation ", 
   NumberForm[100.1500750125, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.4166552473352747`*^9, 3.41665618220322*^9, 
  3.4166565264074707`*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.02`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.2001501\"\>",
     100.20015005000624`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.02, "\tPopulation ", 
   NumberForm[100.20015005000624`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.4166552473352747`*^9, 3.41665618220322*^9, 
  3.41665652641121*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.025`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.2502501\"\>",
     100.25025012503124`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.025, "\tPopulation ", 
   NumberForm[100.25025012503124`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.4166552473352747`*^9, 3.41665618220322*^9, 
  3.416656526414947*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.030000000000000002`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.3003753\"\>",
     100.30037525009377`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.030000000000000002`, "\tPopulation ", 
   NumberForm[100.30037525009377`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.4166552473352747`*^9, 3.41665618220322*^9, 
  3.416656526418715*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.034999999999999996`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.3505254\"\>",
     100.35052543771882`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.034999999999999996`, "\tPopulation ", 
   NumberForm[100.35052543771882`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.4166552473352747`*^9, 3.41665618220322*^9, 
  3.416656526422491*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.04`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.4007007\"\>",
     100.40070070043768`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.04, "\tPopulation ", 
   NumberForm[100.40070070043768`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.4166552473352747`*^9, 3.41665618220322*^9, 
  3.416656526426271*^9}]
}, Open  ]]
}, Open  ]],

Cell["\<\
In the next example, we repackage a function so that it takes only one \
variable.\
\>", "Text",
 CellChangeTimes->{{3.4166552690951777`*^9, 3.4166553028986588`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"f2", "[", "p_", "]"}], " ", ":=", " ", 
  RowBox[{"dP", "[", 
   RowBox[{"p", ",", " ", "1000", ",", " ", "0.1"}], "]"}]}]], "Input",
 CellChangeTimes->{{3.4165786553711357`*^9, 3.416578672958646*^9}, 
   3.4165787413858213`*^9}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Euler1", "[", 
  RowBox[{"100", ",", " ", "f2", ",", " ", "0.005", ",", "  ", "0.040"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.416578397155601*^9, 3.4165784259360437`*^9}, 
   3.4165786876469183`*^9, {3.416656373314999*^9, 3.4166563842403173`*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.005`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.045\"\>",
     100.045,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.005, "\tPopulation ", 
   NumberForm[100.045, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.416578427195016*^9, 3.416578534245591*^9, 3.416578689082613*^9, {
   3.4165787226410637`*^9, 3.416578750507038*^9}, 3.416655247403986*^9, 
   3.4166564116515923`*^9, 3.416656526477331*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.01`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.090018\"\>",
     100.0900179989875,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.01, "\tPopulation ", 
   NumberForm[100.0900179989875, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.416578427195016*^9, 3.416578534245591*^9, 3.416578689082613*^9, {
   3.4165787226410637`*^9, 3.416578750507038*^9}, 3.416655247403986*^9, 
   3.4166564116515923`*^9, 3.41665652647952*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.015`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.135054\"\>",
     100.13505400213548`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.015, "\tPopulation ", 
   NumberForm[100.13505400213548`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.416578427195016*^9, 3.416578534245591*^9, 3.416578689082613*^9, {
   3.4165787226410637`*^9, 3.416578750507038*^9}, 3.416655247403986*^9, 
   3.4166564116515923`*^9, 3.416656526483333*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.02`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.180108\"\>",
     100.18010801461654`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.02, "\tPopulation ", 
   NumberForm[100.18010801461654`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.416578427195016*^9, 3.416578534245591*^9, 3.416578689082613*^9, {
   3.4165787226410637`*^9, 3.416578750507038*^9}, 3.416655247403986*^9, 
   3.4166564116515923`*^9, 3.416656526487111*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.025`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.22518\"\>",
     100.22518004160294`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.025, "\tPopulation ", 
   NumberForm[100.22518004160294`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.416578427195016*^9, 3.416578534245591*^9, 3.416578689082613*^9, {
   3.4165787226410637`*^9, 3.416578750507038*^9}, 3.416655247403986*^9, 
   3.4166564116515923`*^9, 3.416656526490844*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.030000000000000002`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.2702701\"\>",
     100.27027008826656`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.030000000000000002`, "\tPopulation ", 
   NumberForm[100.27027008826656`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.416578427195016*^9, 3.416578534245591*^9, 3.416578689082613*^9, {
   3.4165787226410637`*^9, 3.416578750507038*^9}, 3.416655247403986*^9, 
   3.4166564116515923`*^9, 3.4166565264945307`*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.034999999999999996`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.3153782\"\>",
     100.3153781597789,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.034999999999999996`, "\tPopulation ", 
   NumberForm[100.3153781597789, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.416578427195016*^9, 3.416578534245591*^9, 3.416578689082613*^9, {
   3.4165787226410637`*^9, 3.416578750507038*^9}, 3.416655247403986*^9, 
   3.4166564116515923`*^9, 3.416656526498234*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.04`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.3605043\"\>",
     100.36050426131112`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.04, "\tPopulation ", 
   NumberForm[100.36050426131112`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.416578427195016*^9, 3.416578534245591*^9, 3.416578689082613*^9, {
   3.4165787226410637`*^9, 3.416578750507038*^9}, 3.416655247403986*^9, 
   3.4166564116515923`*^9, 3.416656526501966*^9}]
}, Open  ]]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"PredCorr", "[", 
   RowBox[{"p0_", ",", " ", "dP_", ",", " ", "dt_", ",", " ", "tend_"}], 
   "]"}], " ", ":=", "\[IndentingNewLine]", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"p", " ", "=", " ", "p0"}], ";", "\[IndentingNewLine]", 
    RowBox[{"Do", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"pInter", " ", "=", " ", 
        RowBox[{"p", " ", "+", " ", 
         RowBox[{
          RowBox[{"dP", "[", "p", "]"}], "*", "dt"}]}]}], ";", 
       "\[IndentingNewLine]", 
       RowBox[{"p", " ", "=", " ", 
        RowBox[{"p", " ", "+", " ", 
         RowBox[{
          RowBox[{"dP", "[", 
           RowBox[{
            RowBox[{"(", 
             RowBox[{"p", "+", "pInter"}], ")"}], "/", "2.0"}], "]"}], "*", 
          "dt"}]}]}], ";", " ", "\[IndentingNewLine]", 
       RowBox[{"Print", "[", 
        RowBox[{
        "\"\<Time \>\"", ",", " ", "t", ",", " ", "\[IndentingNewLine]", 
         "\"\<\\tPopulation \>\"", ",", " ", 
         RowBox[{"NumberForm", "[", 
          RowBox[{"p", ",", " ", "10"}], "]"}]}], "]"}]}], ",", 
      "\[IndentingNewLine]", 
      RowBox[{"{", 
       RowBox[{"t", ",", " ", "dt", ",", " ", "tend", ",", " ", "dt"}], 
       "}"}]}], "]"}]}], ")"}]}]], "Input",
 CellChangeTimes->{{3.416578866791285*^9, 3.416578869517501*^9}, {
   3.41657893390376*^9, 3.41657893480764*^9}, {3.4166562049610043`*^9, 
   3.416656209158807*^9}, 3.416656454471727*^9}],

Cell["Here is a shot at predictor - corrector.", "Text",
 CellChangeTimes->{{3.416655474140587*^9, 3.416655485843856*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"PredCorr", "[", 
  RowBox[{"100", ",", " ", "f1", ",", " ", "0.005", ",", "  ", "0.040"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.416578397155601*^9, 3.4165784259360437`*^9}, 
   3.4165786876469183`*^9, {3.4166553290282927`*^9, 3.416655330814745*^9}, 
   3.416656392473757*^9}],

Cell[CellGroupData[{

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.005`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.0500125\"\>",
     100.0500125,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.005, "\tPopulation ", 
   NumberForm[100.0500125, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.416655334719173*^9, 3.416656216481185*^9, 
  3.416656394945485*^9, 3.416656526557088*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.01`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.10005\"\>",
     100.10005001250155`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.01, "\tPopulation ", 
   NumberForm[100.10005001250155`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.416655334719173*^9, 3.416656216481185*^9, 
  3.416656394945485*^9, 3.416656526559169*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.015`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.1501126\"\>",
     100.15011255001406`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.015, "\tPopulation ", 
   NumberForm[100.15011255001406`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.416655334719173*^9, 3.416656216481185*^9, 
  3.416656394945485*^9, 3.416656526562896*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.02`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.2002001\"\>",
     100.20020012505313`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.02, "\tPopulation ", 
   NumberForm[100.20020012505313`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.416655334719173*^9, 3.416656216481185*^9, 
  3.416656394945485*^9, 3.416656526566516*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.025`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.2503128\"\>",
     100.25031275014067`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.025, "\tPopulation ", 
   NumberForm[100.25031275014067`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.416655334719173*^9, 3.416656216481185*^9, 
  3.416656394945485*^9, 3.416656526570105*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.030000000000000002`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.3004504\"\>",
     100.30045043780483`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.030000000000000002`, "\tPopulation ", 
   NumberForm[100.30045043780483`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.416655334719173*^9, 3.416656216481185*^9, 
  3.416656394945485*^9, 3.4166565265737543`*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.034999999999999996`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.3506132\"\>",
     100.35061320058003`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.034999999999999996`, "\tPopulation ", 
   NumberForm[100.35061320058003`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.416655334719173*^9, 3.416656216481185*^9, 
  3.416656394945485*^9, 3.416656526577344*^9}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Time \"\>", "\[InvisibleSpace]", "0.04`", 
   "\[InvisibleSpace]", "\<\"\\tPopulation \"\>", "\[InvisibleSpace]", 
   TagBox[
    InterpretationBox["\<\"100.4008011\"\>",
     100.40080105100697`,
     AutoDelete->True],
    NumberForm[#, 10]& ]}],
  SequenceForm["Time ", 0.04, "\tPopulation ", 
   NumberForm[100.40080105100697`, 10]],
  Editable->False]], "Print",
 CellChangeTimes->{3.416655334719173*^9, 3.416656216481185*^9, 
  3.416656394945485*^9, 3.416656526580954*^9}]
}, Open  ]]
}, Open  ]],

Cell["Exact answer.", "Text",
 CellChangeTimes->{{3.4166564260709133`*^9, 3.416656430598085*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"100", "*", 
  RowBox[{"Exp", "[", 
   RowBox[{"0.04", "*", ".1"}], "]"}]}]], "Input",
 CellChangeTimes->{{3.416656224632962*^9, 3.416656255916621*^9}}],

Cell[BoxData["100.40080106773419`"], "Output",
 CellChangeTimes->{3.416656257433243*^9, 3.416656526618946*^9}]
}, Open  ]]
}, Open  ]]
}, Open  ]]
},
WindowSize->{665, 661},
WindowMargins->{{102, Automatic}, {Automatic, 52}},
Magnification->1.25,
FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (April 20, 2007)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[590, 23, 171, 3, 52, "Subsubtitle"],
Cell[764, 28, 149, 2, 24, "Subsubtitle"],
Cell[CellGroupData[{
Cell[938, 34, 133, 2, 31, "Subsubsection"],
Cell[1074, 38, 355, 8, 33, "Input"],
Cell[1432, 48, 135, 1, 32, "Text"],
Cell[CellGroupData[{
Cell[1592, 53, 169, 3, 33, "Input"],
Cell[1764, 58, 133, 2, 33, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1934, 65, 167, 3, 33, "Input"],
Cell[2104, 70, 135, 2, 33, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[2276, 77, 166, 3, 33, "Input"],
Cell[2445, 82, 135, 2, 33, "Output"]
}, Open  ]],
Cell[2595, 87, 114, 1, 32, "Text"],
Cell[CellGroupData[{
Cell[2734, 92, 379, 11, 33, "Input"],
Cell[CellGroupData[{
Cell[3138, 107, 123, 2, 25, "Print"],
Cell[3264, 111, 136, 2, 25, "Print"],
Cell[3403, 115, 138, 2, 25, "Print"],
Cell[3544, 119, 136, 2, 25, "Print"],
Cell[3683, 123, 138, 2, 25, "Print"],
Cell[3824, 127, 136, 2, 25, "Print"],
Cell[3963, 131, 136, 2, 25, "Print"],
Cell[4102, 135, 135, 2, 25, "Print"],
Cell[4240, 139, 138, 2, 25, "Print"],
Cell[4381, 143, 138, 2, 25, "Print"],
Cell[4522, 147, 136, 2, 25, "Print"]
}, Open  ]]
}, Open  ]],
Cell[4685, 153, 305, 6, 51, "Text"],
Cell[4993, 161, 232, 5, 33, "Input"],
Cell[CellGroupData[{
Cell[5250, 170, 122, 2, 33, "Input"],
Cell[5375, 174, 137, 2, 33, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[5549, 181, 125, 2, 33, "Input"],
Cell[5677, 185, 133, 2, 33, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[5847, 192, 121, 2, 33, "Input"],
Cell[5971, 196, 136, 2, 33, "Output"]
}, Open  ]],
Cell[6122, 201, 101, 1, 32, "Text"],
Cell[6226, 204, 283, 6, 33, "Input"],
Cell[6512, 212, 214, 5, 33, "Input"],
Cell[6729, 219, 190, 4, 51, "Text"],
Cell[6922, 225, 325, 8, 33, "Input"],
Cell[CellGroupData[{
Cell[7272, 237, 169, 3, 33, "Input"],
Cell[7444, 242, 119, 2, 33, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[7600, 249, 170, 3, 33, "Input"],
Cell[7773, 254, 154, 3, 33, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[7964, 262, 166, 3, 33, "Input"],
Cell[8133, 267, 117, 2, 33, "Output"]
}, Open  ]],
Cell[8265, 272, 440, 7, 89, "Text"],
Cell[8708, 281, 1482, 33, 148, "Input"],
Cell[10193, 316, 182, 2, 32, "Text"],
Cell[10378, 320, 217, 5, 33, "Input"],
Cell[CellGroupData[{
Cell[10620, 329, 250, 5, 33, "Input"],
Cell[CellGroupData[{
Cell[10895, 338, 477, 13, 25, "Print"],
Cell[11375, 353, 487, 13, 25, "Print"],
Cell[11865, 368, 499, 13, 25, "Print"],
Cell[12367, 383, 505, 13, 25, "Print"],
Cell[12875, 398, 508, 13, 25, "Print"],
Cell[13386, 413, 539, 13, 25, "Print"],
Cell[13928, 428, 539, 13, 25, "Print"],
Cell[14470, 443, 506, 13, 25, "Print"]
}, Open  ]]
}, Open  ]],
Cell[15003, 460, 176, 4, 32, "Text"],
Cell[15182, 466, 264, 6, 33, "Input"],
Cell[CellGroupData[{
Cell[15471, 476, 276, 5, 33, "Input"],
Cell[CellGroupData[{
Cell[15772, 485, 603, 15, 25, "Print"],
Cell[16378, 502, 623, 15, 25, "Print"],
Cell[17004, 519, 630, 15, 25, "Print"],
Cell[17637, 536, 628, 15, 25, "Print"],
Cell[18268, 553, 629, 15, 25, "Print"],
Cell[18900, 570, 664, 15, 25, "Print"],
Cell[19567, 587, 658, 15, 25, "Print"],
Cell[20228, 604, 629, 15, 25, "Print"]
}, Open  ]]
}, Open  ]],
Cell[20884, 623, 1435, 36, 148, "Input"],
Cell[22322, 661, 122, 1, 32, "Text"],
Cell[CellGroupData[{
Cell[22469, 666, 304, 6, 33, "Input"],
Cell[CellGroupData[{
Cell[22798, 676, 513, 13, 25, "Print"],
Cell[23314, 691, 525, 13, 25, "Print"],
Cell[23842, 706, 529, 13, 25, "Print"],
Cell[24374, 721, 527, 13, 25, "Print"],
Cell[24904, 736, 529, 13, 25, "Print"],
Cell[25436, 751, 562, 13, 25, "Print"],
Cell[26001, 766, 560, 13, 25, "Print"],
Cell[26564, 781, 527, 13, 25, "Print"]
}, Open  ]]
}, Open  ]],
Cell[27118, 798, 97, 1, 32, "Text"],
Cell[CellGroupData[{
Cell[27240, 803, 175, 4, 33, "Input"],
Cell[27418, 809, 110, 1, 33, "Output"]
}, Open  ]]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)