(* 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[     43929,       1381]
NotebookOptionsPosition[     39013,       1236]
NotebookOutlinePosition[     39469,       1255]
CellTagsIndexPosition[     39426,       1252]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Lesson 5    Replacement Rules and Solving Equations", "Section"],

Cell[CellGroupData[{

Cell["Replacement Rules", "Subsection",
 CellChangeTimes->{{3.460574460989143*^9, 3.460574467703837*^9}, {
  3.460829025784966*^9, 3.4608290260646276`*^9}}],

Cell[TextData[{
 "We need to know a bit more about assignment statements. First we make sure \
that ",
 StyleBox["x", "Input"],
 ", ",
 StyleBox["y", "Input"],
 " and ",
 StyleBox["z", "Input"],
 " are undefined and then define ",
 StyleBox["y", "Input"],
 ":"
}], "Text",
 CellChangeTimes->{{3.4605744732972383`*^9, 3.460574526520751*^9}, {
   3.4605745740180492`*^9, 3.460574599576582*^9}, {3.460658256947237*^9, 
   3.46065826468664*^9}, 3.460828989376523*^9}],

Cell[BoxData[{
 RowBox[{"Clear", "[", 
  RowBox[{"x", ",", "y", ",", "z"}], "]"}], "\[IndentingNewLine]", 
 RowBox[{"y", "=", 
  RowBox[{"x", "^", "2"}]}]}], "Input",
 CellChangeTimes->{{3.4605745298201103`*^9, 3.46057453272511*^9}, {
  3.460574606635293*^9, 3.4605746070309134`*^9}, {3.460658121984784*^9, 
  3.4606581228071117`*^9}}],

Cell[TextData[{
 "The values of these variable can be checked using ",
 StyleBox["?", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.460574549617649*^9, 3.460574551831312*^9}, {
  3.460574613754168*^9, 3.460574629896989*^9}}],

Cell[BoxData[{
 RowBox[{"?", "x"}], "\[IndentingNewLine]", 
 RowBox[{"?", "y"}], "\[IndentingNewLine]", 
 RowBox[{"?", "z"}]}], "Input",
 CellChangeTimes->{{3.460574638866939*^9, 3.4605746413674726`*^9}, {
  3.460658276455662*^9, 3.460658278222292*^9}}],

Cell[TextData[{
 "(Ignore \"Global`x\" for now.) What happens to ",
 StyleBox["y", "Input"],
 " if we change the value of ",
 StyleBox["x", "Input"],
 "?"
}], "Text",
 CellChangeTimes->{{3.460657836331396*^9, 3.460657909914488*^9}}],

Cell[BoxData[{
 RowBox[{"x", "=", "4"}], "\[IndentingNewLine]", "y"}], "Input",
 CellChangeTimes->{{3.460657913884939*^9, 3.4606579193011503`*^9}}],

Cell[BoxData[{
 RowBox[{"x", "=", 
  RowBox[{
   RowBox[{"z", "^", "2"}], "+", 
   "1"}]}], "\[IndentingNewLine]", "y"}], "Input",
 CellChangeTimes->{{3.460657930575305*^9, 3.460657948684677*^9}, 
   3.4606581303784723`*^9}],

Cell[BoxData[{
 RowBox[{"z", "=", "3"}], "\[IndentingNewLine]", "y"}], "Input",
 CellChangeTimes->{{3.4606581710294123`*^9, 3.460658175108438*^9}}],

Cell[TextData[{
 "Note that the definition of ",
 StyleBox["y", "Input"],
 " has not changed - unlike the definitions of ",
 StyleBox["x", "Input"],
 " and ",
 StyleBox["z", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.460657956991048*^9, 3.460657977908616*^9}, {
   3.460658040107978*^9, 3.460658065036619*^9}, {3.460658200355597*^9, 
   3.46065821034381*^9}, 3.4606620637143173`*^9}],

Cell[BoxData[{
 RowBox[{"?", "x"}], "\[IndentingNewLine]", 
 RowBox[{"?", "y"}], "\[IndentingNewLine]", 
 RowBox[{"?", "z"}]}], "Input",
 CellChangeTimes->{{3.46065798209019*^9, 3.460657982751233*^9}, {
  3.4606580342745028`*^9, 3.4606580351699753`*^9}, {3.460658286528647*^9, 
  3.460658288701947*^9}}],

Cell[TextData[{
 " There is another way of evaluating ",
 StyleBox["y", "Input"],
 " when ",
 StyleBox["x", "Input"],
 " is ",
 StyleBox["4", "Input"],
 ". Let's start again:"
}], "Text",
 CellChangeTimes->{{3.460658472689242*^9, 3.4606585001921864`*^9}, {
  3.460658673552548*^9, 3.4606586945379066`*^9}, {3.460658795570878*^9, 
  3.460658814596095*^9}}],

Cell[BoxData[{
 RowBox[{"Clear", "[", 
  RowBox[{"x", ",", "y", ",", "z"}], "]"}], "\[IndentingNewLine]", 
 RowBox[{"y", "=", 
  RowBox[{"x", "^", "2"}]}]}], "Input",
 CellChangeTimes->{{3.4605745298201103`*^9, 3.46057453272511*^9}, {
  3.460574606635293*^9, 3.4605746070309134`*^9}, {3.460658121984784*^9, 
  3.4606581228071117`*^9}}],

Cell[BoxData[
 RowBox[{"y", "/.", 
  RowBox[{"{", 
   RowBox[{"x", "\[Rule]", "4"}], "}"}]}]], "Input",
 CellChangeTimes->{{3.460658698040574*^9, 3.460658712572945*^9}, {
  3.460830179157827*^9, 3.460830186574677*^9}}],

Cell[TextData[{
 "The ",
 StyleBox["/.", "Input",
  FontWeight->"Bold"],
 " can be thought of as \"",
 StyleBox["given that",
  FontWeight->"Bold"],
 "\", and the ",
 StyleBox["\[RightArrow]", "Input"],
 " can be thought of as \"",
 StyleBox["takes the value",
  FontWeight->"Bold"],
 "\".  Thus ",
 StyleBox["y/.{x\[Rule]4}", "Input"],
 " computes the value of ",
 StyleBox["y", "Input"],
 " given that ",
 StyleBox["x", "Input"],
 " takes the value ",
 StyleBox["4", "Input"],
 ", without changing ",
 StyleBox["y", "Input"],
 " or assigning a value to ",
 StyleBox["x", "Input"],
 ". The symbol ",
 StyleBox["\[Rule]", "Input"],
 " appears automatically when ",
 StyleBox["-", "Input"],
 " is followed by ",
 StyleBox[">", "Input"],
 ". The braces are not actually needed in this simple example."
}], "Text",
 CellChangeTimes->{{3.460824399618456*^9, 3.460824465132139*^9}, 
   3.460824509390689*^9, {3.4608245629375973`*^9, 3.460824602065321*^9}, {
   3.46082465939287*^9, 3.460824680293343*^9}, {3.4608248047443123`*^9, 
   3.4608248063128357`*^9}, {3.460830094187327*^9, 3.460830151072847*^9}, {
   3.460830198433505*^9, 3.4608302398041687`*^9}}],

Cell[BoxData[{
 RowBox[{"?", "x"}], "\[IndentingNewLine]", 
 RowBox[{"?", "y"}]}], "Input",
 CellChangeTimes->{{3.4606590081069593`*^9, 3.460659012074669*^9}}],

Cell["This is frequently a useful trick. For example, ", "Text",
 CellChangeTimes->{{3.460659057196014*^9, 3.460659076193166*^9}, {
  3.460659117906651*^9, 3.4606591242234077`*^9}}],

Cell[BoxData[{
 RowBox[{"Clear", "[", 
  RowBox[{"x", ",", "p"}], "]"}], "\[IndentingNewLine]", 
 RowBox[{"p", "=", 
  RowBox[{
   RowBox[{"x", "^", "3"}], "+", "x", "+", "3"}]}]}], "Input",
 CellChangeTimes->{{3.460659129816271*^9, 3.460659134122402*^9}, {
  3.460659225118689*^9, 3.460659230458329*^9}}],

Cell[TextData[{
 " defines a polynomial. If we set ",
 StyleBox["x=5", "Input"],
 ", then, ",
 StyleBox["p", "Input"],
 " acts like a number and ",
 StyleBox["Expand", "Input"],
 ", ",
 StyleBox["Coefficient", "Input"],
 " don't work the way you might hope."
}], "Text",
 CellChangeTimes->{{3.46065913848811*^9, 3.460659209810796*^9}, {
  3.4606592918363934`*^9, 3.460659381903323*^9}}],

Cell[BoxData[{
 RowBox[{"x", "=", "5"}], "\[IndentingNewLine]", "p", "\[IndentingNewLine]", 
 RowBox[{"Expand", "[", 
  RowBox[{"p", "^", "2"}], "]"}]}], "Input",
 CellChangeTimes->{{3.4606592127602043`*^9, 3.460659215747075*^9}, {
  3.46065941231457*^9, 3.460659412759223*^9}}],

Cell[TextData[{
 "The definition of ",
 StyleBox["p", "Input"],
 " has not changed, but when ",
 StyleBox["p", "Input"],
 " is evaluated, ",
 StyleBox["x", "Input"],
 " is replaced by ",
 StyleBox["5", "Input"],
 ". To make ",
 StyleBox["p", "Input"],
 " behave like a polynomial again we have to clear the value of ",
 StyleBox["x", "Input"],
 ".  "
}], "Text",
 CellChangeTimes->{{3.460659432740755*^9, 3.460659489812293*^9}, {
  3.460659533508642*^9, 3.460659539022378*^9}, {3.4606595795395*^9, 
  3.460659609598673*^9}, {3.460661473970449*^9, 3.460661535669918*^9}}],

Cell[BoxData[{
 RowBox[{"Clear", "[", "x", 
  "]"}], "\[IndentingNewLine]", "p", "\[IndentingNewLine]", 
 RowBox[{"Expand", "[", 
  RowBox[{"p", "^", "2"}], "]"}]}], "Input",
 CellChangeTimes->{{3.460659262952186*^9, 3.460659267031612*^9}, {
  3.4606595012790956`*^9, 3.460659507078506*^9}}],

Cell[TextData[{
 "All this can be avoided if ",
 StyleBox["p", "Input"],
 " is evaluated this way:"
}], "Text",
 CellChangeTimes->{{3.4606615675410643`*^9, 3.460661596628367*^9}}],

Cell[BoxData[{
 RowBox[{"p", "/.", 
  RowBox[{"{", 
   RowBox[{"x", "\[Rule]", "5"}], "}"}]}], "\[IndentingNewLine]", 
 RowBox[{"?", "p"}], "\[IndentingNewLine]", 
 RowBox[{"?", "x"}]}], "Input",
 CellChangeTimes->{{3.460661608080041*^9, 3.4606616502127237`*^9}, {
  3.4608302524336557`*^9, 3.4608302545555773`*^9}}],

Cell["\<\
Here are some other variations on this theme using a polynomial in two \
variables:\
\>", "Text",
 CellChangeTimes->{{3.46066166456262*^9, 3.4606616766629467`*^9}, {
  3.460823881760119*^9, 3.460823897394781*^9}}],

Cell[BoxData[{
 RowBox[{"Clear", "[", 
  RowBox[{"x", ",", "y", ",", "z", ",", "p"}], "]"}], "\[IndentingNewLine]", 
 RowBox[{"p", "=", 
  RowBox[{
   RowBox[{"x", "^", "2"}], "+", " ", 
   RowBox[{"x", " ", "y"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"p", "/.", 
  RowBox[{"{", 
   RowBox[{"x", "\[Rule]", "3"}], "}"}]}]}], "Input",
 CellChangeTimes->{{3.460661684777446*^9, 3.460661768903345*^9}}],

Cell[TextData[{
 "Two rules applied to ",
 StyleBox["p", "Input"],
 ":"
}], "Text",
 CellChangeTimes->{{3.46066199268139*^9, 3.460662003594427*^9}}],

Cell[BoxData[
 RowBox[{"p", "/.", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"x", "\[Rule]", "2"}], ",", 
    RowBox[{"y", "\[Rule]", "3"}]}], "}"}]}]], "Input",
 CellChangeTimes->{{3.4606617717760067`*^9, 3.460661808334502*^9}}],

Cell[TextData[{
 "Evaluation at ",
 StyleBox["x=1", "Input"],
 " and ",
 StyleBox["x=2", "Input"],
 " produces a list of values."
}], "Text",
 CellChangeTimes->{{3.460661903164719*^9, 3.460661980061503*^9}}],

Cell[BoxData[
 RowBox[{"p", "/.", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"x", "\[Rule]", "1"}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", "\[Rule]", "2"}], "}"}]}], "}"}]}]], "Input",
 CellChangeTimes->{{3.460661827740888*^9, 3.460661842104578*^9}}],

Cell["Three different evaluations:", "Text",
 CellChangeTimes->{{3.4606619470420237`*^9, 3.460661961266885*^9}}],

Cell[BoxData[
 RowBox[{"p", "/.", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{
      RowBox[{"x", "\[Rule]", "3"}], ",", 
      RowBox[{"y", "\[Rule]", "2"}]}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", "\[Rule]", "1"}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", "\[Rule]", "z"}], "}"}]}], "}"}]}]], "Input",
 CellChangeTimes->{{3.4606618507498703`*^9, 3.460661876102852*^9}}],

Cell["Three different evaluations of a list:", "Text",
 CellChangeTimes->{{3.4608252136872807`*^9, 3.460825250259807*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"{", 
   RowBox[{"x", ",", "y"}], "}"}], "/.", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{
      RowBox[{"x", "\[Rule]", "3"}], ",", 
      RowBox[{"y", "\[Rule]", "2"}]}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", "\[Rule]", "1"}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", "\[Rule]", "z"}], "}"}]}], "}"}]}]], "Input",
 CellChangeTimes->{{3.4608251488653393`*^9, 3.4608252036135073`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Practice", "Subsection",
 CellFrame->{{0, 0}, {0, 2}},
 CellChangeTimes->{{3.46082398906501*^9, 3.46082399092842*^9}}],

Cell["\<\
1) Before evaluating each of the cells below, predict its output.\
\>", "Text",
 CellChangeTimes->{{3.460823996336668*^9, 3.460824031793532*^9}, {
  3.460824130724292*^9, 3.460824131415477*^9}}],

Cell[BoxData[{
 RowBox[{"Clear", "[", 
  RowBox[{"a", ",", "b", ",", "c", ",", "x", ",", "y", ",", "f"}], 
  "]"}], "\[IndentingNewLine]", 
 RowBox[{"f", "=", 
  RowBox[{
   RowBox[{"a", " ", 
    RowBox[{"x", "^", "2"}]}], "+", " ", 
   RowBox[{"b", " ", "x", " ", "y"}], " ", "+", 
   RowBox[{"c", " ", 
    RowBox[{"y", "^", "2"}]}]}]}]}], "Input",
 CellChangeTimes->{{3.460824035705024*^9, 3.460824117366995*^9}}],

Cell[BoxData[
 RowBox[{"f", "/.", 
  RowBox[{"{", 
   RowBox[{"x", "\[Rule]", "a"}], "}"}]}]], "Input",
 CellChangeTimes->{{3.460824195860013*^9, 3.460824202826638*^9}, {
  3.4608302976534033`*^9, 3.4608303000904512`*^9}}],

Cell[BoxData[
 RowBox[{"f", "/.", 
  RowBox[{"{", 
   RowBox[{"x", "\[Rule]", "y"}], "}"}]}]], "Input",
 CellChangeTimes->{{3.460824250453766*^9, 3.46082425862192*^9}, {
  3.460830302370515*^9, 3.460830304429378*^9}}],

Cell[BoxData[
 RowBox[{"f", "/.", 
  RowBox[{"{", 
   RowBox[{"f", "\[Rule]", "x"}], "}"}]}]], "Input",
 CellChangeTimes->{{3.460824277677479*^9, 3.4608242823718348`*^9}, {
  3.460830307349177*^9, 3.460830309707438*^9}}],

Cell[BoxData[
 RowBox[{"f", "/.", 
  RowBox[{"{", 
   RowBox[{"x", "\[Rule]", "f"}], "}"}]}]], "Input",
 CellChangeTimes->{{3.460824293979866*^9, 3.4608242967358418`*^9}, {
  3.460830312127274*^9, 3.460830314126326*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Solving equations", "Subsection",
 CellFrame->{{0, 0}, {0, 2}},
 CellChangeTimes->{{3.460825346454295*^9, 3.4608253543648663`*^9}, {
  3.4608284647677803`*^9, 3.460828470322959*^9}}],

Cell[BoxData[
 RowBox[{"Clear", "[", 
  RowBox[{"x", ",", "a"}], "]"}]], "Input",
 CellChangeTimes->{{3.460829117982581*^9, 3.4608291218394527`*^9}}],

Cell[TextData[{
 "The ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " function for solving equations is ",
 StyleBox["Solve", "Input"],
 ".  Here is a simple example:"
}], "Text",
 CellChangeTimes->{{3.460828527197851*^9, 3.460828557566955*^9}, {
  3.460828588384185*^9, 3.460828594445023*^9}}],

Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"3", "x"}], "+", "5"}], "\[Equal]", "7"}], ",", "x"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.460828564535461*^9, 3.4608285842330103`*^9}}],

Cell[TextData[{
 "The equation to be solved is given as an argument, with equality indicated \
by ",
 StyleBox["==", "Input"],
 " (otherwise it would be interpreted as an assignment statement). ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " also needs to know which variable to solve for - ",
 StyleBox["x", "Input"],
 " in this example. If ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " can solve the equations, the solution is given as a replacement rule. \
Here's an example that shows why the variable to solve for has to be \
specified."
}], "Text",
 CellChangeTimes->{{3.4608285971586933`*^9, 3.460828741189436*^9}, 
   3.460829151556521*^9, {3.4608292007480288`*^9, 3.460829239426734*^9}, {
   3.460829276597537*^9, 3.4608292875476427`*^9}}],

Cell[BoxData[{
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"a", " ", "x"}], "+", "5"}], "\[Equal]", "7"}], ",", "x"}], 
  "]"}], "\[IndentingNewLine]", 
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"a", " ", "x"}], " ", "+", "5"}], "\[Equal]", "7"}], ",", "a"}], 
  "]"}]}], "Input",
 CellChangeTimes->{{3.4608290696483183`*^9, 3.460829100550577*^9}}],

Cell[TextData[{
 "If there is more than one solution, ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " produces a list of replacement rules:"
}], "Text",
 CellChangeTimes->{{3.4608292510767193`*^9, 3.460829314651599*^9}}],

Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"x", "^", "2"}], "\[Equal]", "37"}], ",", "x"}], "]"}]], "Input",
 CellChangeTimes->{{3.460829318387183*^9, 3.460829342137301*^9}}],

Cell[TextData[{
 "The replacement rules are useful as input to other expressions. For \
example, to find the three roots of the polynomial ",
 StyleBox[Cell[BoxData[
  FormBox[
   SuperscriptBox["x", "3"], TraditionalForm]], "Input"], "Input"],
 StyleBox["-3", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   SuperscriptBox["x", "2"], TraditionalForm]], "Input"], "Input"],
 StyleBox["-17x+51", "Input"],
 ", we solve the equation ",
 StyleBox[Cell[BoxData[
  FormBox[
   SuperscriptBox["x", "3"], TraditionalForm]], "Input"], "Input"],
 StyleBox["-3", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   SuperscriptBox["x", "2"], TraditionalForm]], "Input"], "Input"],
 StyleBox["-17x+51=0", "Input"],
 ":"
}], "Text",
 CellChangeTimes->{{3.4608294304192953`*^9, 3.460829491535911*^9}, {
  3.460829564654366*^9, 3.460829610090016*^9}, {3.4608296612451763`*^9, 
  3.460829680786603*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"poly", "=", 
   RowBox[{
    SuperscriptBox["x", "3"], "-", 
    RowBox[{"3", 
     FormBox[
      SuperscriptBox["x", "2"],
      TraditionalForm]}], "-", 
    RowBox[{"17", "x"}], " ", "+", " ", "51"}]}], ";"}], "\n", 
 RowBox[{"sol", "=", 
  RowBox[{"Solve", "[", 
   RowBox[{
    RowBox[{"poly", "==", "0"}], ",", "x"}], "]"}]}]}], "Input"],

Cell[TextData[{
 "By assigning the name ",
 StyleBox["sol", "Input",
  FontWeight->"Bold"],
 StyleBox[" ",
  FontWeight->"Bold"],
 "to the list of solutions, we can do further calculations."
}], "Text",
 CellChangeTimes->{
  3.460829708736269*^9, {3.460829746353956*^9, 3.460829781549777*^9}}],

Cell[BoxData[
 RowBox[{"sol", "[", 
  RowBox[{"[", "1", "]"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"poly", "/.", 
  RowBox[{"sol", "[", 
   RowBox[{"[", "1", "]"}], "]"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"poly", "/.", "sol"}]], "Input",
 CellChangeTimes->{{3.4608248736283417`*^9, 3.460824876423699*^9}}],

Cell["Here is a list of the squares of the solutions.", "Text",
 CellChangeTimes->{{3.460825404390551*^9, 3.460825418162589*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"x", "^", "2"}], "/.", "sol"}]], "Input",
 CellChangeTimes->{{3.460825382568708*^9, 3.4608253910181437`*^9}, {
  3.460830360752453*^9, 3.4608303883225813`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Practice", "Subsection",
 CellFrame->{{0, 0}, {0, 2}},
 CellChangeTimes->{{3.4608249832625513`*^9, 3.460824984987585*^9}}],

Cell["\<\
1) How do we get the solutions to the above equation as a list of numbers \
rather than a list of replacement rules? \
\>", "Text",
 CellChangeTimes->{{3.460824990118333*^9, 3.460825082911685*^9}, {
  3.460830701572014*^9, 3.460830732971895*^9}}],

Cell[TextData[{
 "2 What happens if you ask ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " to solve",
 StyleBox[" ", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SuperscriptBox["ax", "2"], "+", "bx", "+", "c"}], "=", "0"}], 
   TraditionalForm]], "Input"], "Input"],
 " for ",
 StyleBox["x", "Input"],
 "?"
}], "Text",
 CellChangeTimes->{{3.4479725431062717`*^9, 3.447972622493424*^9}, {
  3.447972746211347*^9, 3.4479727481428823`*^9}, {3.4479728489525557`*^9, 
  3.447972949340362*^9}, {3.448022620783722*^9, 3.448022656946389*^9}, {
  3.448022726172607*^9, 3.448022850963038*^9}, {3.4480229041078587`*^9, 
  3.448022975356772*^9}, {3.4480230730887947`*^9, 3.4480230737275677`*^9}, {
  3.460830557942738*^9, 3.4608305905109777`*^9}, {3.460836191323539*^9, 
  3.460836250007669*^9}, {3.460836287499507*^9, 3.4608362963184443`*^9}}],

Cell[TextData[{
 "3) What happens if you ask ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " to solve",
 StyleBox[" ", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SuperscriptBox["ax", "2"], "+", "bx", "+", "c"}], "=", "0"}], 
   TraditionalForm]], "Input"], "Input"],
 " for ",
 StyleBox["a", "Input"],
 "?"
}], "Text",
 CellChangeTimes->{{3.4479725431062717`*^9, 3.447972622493424*^9}, {
  3.447972746211347*^9, 3.4479727481428823`*^9}, {3.4479728489525557`*^9, 
  3.447972949340362*^9}, {3.448022620783722*^9, 3.448022656946389*^9}, {
  3.448022726172607*^9, 3.448022850963038*^9}, {3.4480229041078587`*^9, 
  3.448022975356772*^9}, {3.4480230730887947`*^9, 3.4480230737275677`*^9}, {
  3.460830557942738*^9, 3.4608305905109777`*^9}, {3.460836191323539*^9, 
  3.460836250007669*^9}}],

Cell[TextData[{
 "4) What happens if you ask ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " to solve",
 StyleBox[" ", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SuperscriptBox["ax", "3"], "+", 
     SuperscriptBox["bx", "2"], "+", "cx", "+", "d"}], "=", "0"}], 
   TraditionalForm]], "Input"], "Input"],
 " for ",
 StyleBox["x", "Input"],
 "?"
}], "Text",
 CellChangeTimes->{{3.4479725431062717`*^9, 3.447972622493424*^9}, {
  3.447972746211347*^9, 3.4479727481428823`*^9}, {3.4479728489525557`*^9, 
  3.447972949340362*^9}, {3.448022620783722*^9, 3.448022656946389*^9}, {
  3.448022726172607*^9, 3.448022850963038*^9}, {3.4480229041078587`*^9, 
  3.448022975356772*^9}, {3.4480230730887947`*^9, 3.4480230737275677`*^9}, {
  3.460830565918425*^9, 3.4608305927206593`*^9}, {3.460836100294495*^9, 
  3.4608361445350027`*^9}, {3.46083631003036*^9, 3.460836312569867*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Simultaneous Equations", "Subsection",
 CellFrame->{{0, 0}, {0, 2}},
 CellChangeTimes->{{3.460830606436481*^9, 3.460830613388199*^9}}],

Cell[TextData[{
 "If we have several equations that need to be solved simultaneously, we \
create a list containing all the equations and a list of all the variables. \
Here is an example of two simultaneous equations in two variables. Remember \
to use ",
 StyleBox["==", "Input"],
 ".\n\tx = 2x - 3 x y\n\ty = -y + x y"
}], "Text",
 CellChangeTimes->{{3.460830470232234*^9, 3.46083047910315*^9}}],

Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"x", "==", 
      RowBox[{
       RowBox[{"2", "x"}], "-", " ", 
       RowBox[{"3", " ", "x", " ", "y"}]}]}], ",", 
     RowBox[{"y", " ", "==", " ", 
      RowBox[{
       RowBox[{"-", "y"}], " ", "+", " ", 
       RowBox[{"x", " ", "y"}]}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "y"}], "}"}]}], "]"}]], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Practice", "Subsection",
 CellFrame->{{0, 0}, {0, 2}},
 CellChangeTimes->{3.460830640641706*^9}],

Cell[TextData[{
 "1) Solve",
 StyleBox[" ", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   StyleBox[
    RowBox[{
     RowBox[{"xy", "+", 
      SuperscriptBox["x", "2"], "+", 
      SuperscriptBox["y", "2"]}], "=", "7"}], "Input"], TraditionalForm]], 
  "Input"], "Input"],
 " and",
 StyleBox[" ", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   StyleBox[
    RowBox[{
     RowBox[{"x", "+", "y"}], "=", "3"}], "Input"], TraditionalForm]], 
  "Input"], "Input"],
 " simultaneously."
}], "Text",
 CellChangeTimes->{{3.4479725431062717`*^9, 3.447972622493424*^9}, {
   3.447972746211347*^9, 3.4479727481428823`*^9}, {3.4479728489525557`*^9, 
   3.447972949340362*^9}, {3.448022620783722*^9, 3.448022656946389*^9}, {
   3.448022726172607*^9, 3.448022850963038*^9}, {3.4480229041078587`*^9, 
   3.448022975356772*^9}, {3.4480230730887947`*^9, 3.4480230737275677`*^9}, {
   3.460830626748472*^9, 3.4608306458640947`*^9}, {3.460836348181381*^9, 
   3.460836428261436*^9}, 3.46083647842857*^9, {3.461000785488744*^9, 
   3.461000793471814*^9}}],

Cell[TextData[{
 "2) Evaluate the expression ",
 StyleBox[Cell[BoxData[
  FormBox[
   RowBox[{
    SuperscriptBox["x", "3"], "+", 
    SuperscriptBox["y", "3"]}], TraditionalForm]], "Input"], "Input"],
 " at the solutions you found in 1)."
}], "Text",
 CellChangeTimes->{{3.460830820525009*^9, 3.4608308881117163`*^9}, {
  3.460835967621711*^9, 3.4608359880671177`*^9}, {3.460836031504468*^9, 
  3.4608360344037333`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Numerical Solutions", "Subsection",
 CellFrame->{{0, 0}, {0, 2}},
 CellChangeTimes->{{3.460830652176094*^9, 3.4608306571247473`*^9}}],

Cell[TextData[{
 "Solving equations is inherently very difficult. ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " knows many tricks but will fail to find the solutions in many cases. Here \
are two examples, "
}], "Text",
 CellChangeTimes->{{3.4608309734786053`*^9, 3.460831037443781*^9}, {
  3.460831075400146*^9, 3.460831132640622*^9}, {3.460831278527247*^9, 
  3.460831284861524*^9}}],

Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"x", "^", "5"}], "+", "x"}], "\[Equal]", "7"}], ",", "x"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.46083129001016*^9, 3.460831303378284*^9}, {
  3.460836747032374*^9, 3.460836748381865*^9}, {3.460837259721458*^9, 
  3.460837260337262*^9}}],

Cell[TextData[{
 "There is no general \"formula\" for the solution of quintic equations. Even \
so, ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " has a way of representing these solutions using the ",
 StyleBox["Root", "Input"],
 " function. This is actually more useful than it seems, but you won't be \
tested on this!"
}], "Text",
 CellChangeTimes->{{3.4608365142106733`*^9, 3.460836640638777*^9}, {
  3.460836670968334*^9, 3.460836800449481*^9}, {3.460836958631496*^9, 
  3.460836975730124*^9}, {3.460837332095039*^9, 3.460837333708888*^9}}],

Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"Cos", "[", "x", "]"}], "==", "x"}], ",", "x"}], "]"}]], "Input",
 CellChangeTimes->{{3.4608311396261044`*^9, 3.460831220925503*^9}}],

Cell[TextData[{
 "Here ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " produces an error message claiming essentially that it does not know how \
to solve this equation."
}], "Text",
 CellChangeTimes->{{3.4608370061704473`*^9, 3.4608370592073402`*^9}}],

Cell[TextData[{
 "Despite these problems, ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " has several ways of finding approximate solutions of these and many other \
equations. For polynomial equations, the ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " function ",
 StyleBox["NSolve", "Input"],
 " works:"
}], "Text",
 CellChangeTimes->{{3.460837152386136*^9, 3.460837244888719*^9}}],

Cell[BoxData[
 RowBox[{"NSolve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"x", "^", "5"}], "+", "x"}], "\[Equal]", "7"}], ",", "x"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.460837252963901*^9, 3.460837269011022*^9}}],

Cell[BoxData[{
 RowBox[{"poly", "=", 
  RowBox[{
   SuperscriptBox["x", "3"], "-", 
   RowBox[{"3", 
    FormBox[
     SuperscriptBox["x", "2"],
     TraditionalForm]}], "-", 
   RowBox[{"17", "x"}], " ", "+", " ", "51"}]}], "\[IndentingNewLine]", 
 RowBox[{"NSolve", "[", 
  RowBox[{
   RowBox[{"poly", "==", "0"}], ",", "x"}], "]"}]}], "Input",
 CellChangeTimes->{
  3.464524579181231*^9, {3.464524696378772*^9, 3.464524701585318*^9}}],

Cell[TextData[{
 "From above, we know that the exact solutions of the second equation are ",
 StyleBox["3", "Input"],
 ",",
 StyleBox[" ", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   StyleBox[
    SqrtBox["17"], "Input"], TraditionalForm]], "Input"], "Input"],
 "and",
 StyleBox[" -", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   SqrtBox["17"], TraditionalForm]], "Input"], "Input"],
 ". Note that the exact solution ",
 StyleBox["3", "Input"],
 " is now displayed as a decimal approximation (as indicated by the decimal \
point). "
}], "Text",
 CellChangeTimes->{{3.4608374522514153`*^9, 3.460837534863826*^9}, {
  3.460837575088632*^9, 3.4608376448081703`*^9}, {3.46083768987691*^9, 
  3.460837696820842*^9}, {3.4610008166263723`*^9, 3.461000827075253*^9}, {
  3.4645245975137577`*^9, 3.4645246007038393`*^9}}],

Cell[TextData[{
 "Let us try ",
 StyleBox["NSolve", "Input"],
 " on our second problem:"
}], "Text",
 CellChangeTimes->{{3.460837717432247*^9, 3.460837734113006*^9}}],

Cell[BoxData[
 RowBox[{"NSolve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"Cos", "[", "x", "]"}], "==", "x"}], ",", "x"}], "]"}]], "Input",
 CellChangeTimes->{{3.4608377399176188`*^9, 3.460837765084448*^9}}],

Cell[TextData[{
 "Once again, ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " has no method for finding numerical solutions of this equation. This leads \
us to the remaining ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " function we need...\n\nYou may be familiar with Newton's Method for \
solving equations and know that it is an iterative procedure that takes, in \
each step, a given approximation to a solution and calculates a better \
approximation. The ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " function ",
 StyleBox["FindRoot", "Input"],
 " works essentially that way. It finds only one solution at a time and may \
not find any if the initial starting value is too far off. The advantage of \
",
 StyleBox["FindRoot", "Input",
  FontWeight->"Bold"],
 StyleBox[" ",
  FontWeight->"Bold"],
 "is, however, that it can be used for any type of equation, not just \
polynomial ones."
}], "Text",
 CellChangeTimes->{{3.4608377737442093`*^9, 3.460837801589291*^9}, {
   3.4608378730848007`*^9, 3.460838022775131*^9}, 3.460838100855571*^9, {
   3.460838160619227*^9, 3.460838167320558*^9}, {3.461000844997129*^9, 
   3.4610008918953543`*^9}}],

Cell[TextData[{
 "Here we use ",
 StyleBox["FindRoot", "Input"],
 " to find roots of the the polynomial ",
 StyleBox["poly", "Input"],
 " from above. Notice that we need to provide a starting approximation, ",
 StyleBox["5", "Input"],
 ", to the solution (root) we are seeking."
}], "Text",
 CellChangeTimes->{{3.460838209989727*^9, 3.460838329267313*^9}, {
  3.4608391200608683`*^9, 3.4608391257488413`*^9}, {3.464524683904564*^9, 
  3.464524685688448*^9}}],

Cell[BoxData[{
 RowBox[{"poly", "=", 
  RowBox[{
   SuperscriptBox["x", "3"], "-", 
   RowBox[{"3", 
    FormBox[
     SuperscriptBox["x", "2"],
     TraditionalForm]}], "-", 
   RowBox[{"17", "x"}], " ", "+", " ", "51"}]}], "\[IndentingNewLine]", 
 RowBox[{"FindRoot", "[", 
  RowBox[{
   RowBox[{"poly", "==", "0"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "5"}], "}"}]}], "]"}]}], "Input",
 CellChangeTimes->{
  3.460838201780809*^9, {3.464524672532971*^9, 3.4645246737817497`*^9}}],

Cell[TextData[{
 "Now we can finally solve ",
 StyleBox["Cos[x] = x", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.460839059439143*^9, 3.4608390905796337`*^9}}],

Cell[BoxData[
 RowBox[{"sol", "=", 
  RowBox[{"FindRoot", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"Cos", "[", "x", "]"}], "==", "x"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", ",", "1"}], "}"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.4608391050423203`*^9, 3.460839108757949*^9}, {
   3.460839159839329*^9, 3.460839165229437*^9}, {3.460839268198739*^9, 
   3.460839269063587*^9}, {3.461000951623464*^9, 3.461000956414864*^9}, 
   3.464524734403472*^9}],

Cell[BoxData[{
 RowBox[{"x", "/.", "sol"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Cos", "[", "x", "]"}], "/.", "sol"}]}], "Input",
 CellChangeTimes->{{3.460839273816495*^9, 3.460839284622645*^9}}],

Cell[TextData[{
 "How did I know to choose ",
 StyleBox["1", "Input"],
 " as the starting approximation? Here a good deal of experience with the \
graphs of these functions is useful. Of course, ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " can also help with graphs. The ",
 StyleBox["Plot", "Input"],
 " function can generate a graph with ease (We will have a lot more to say \
about ",
 StyleBox["Plot", "Input"],
 " starting in Lesson 9.):"
}], "Text",
 CellChangeTimes->{
  3.448023247730098*^9, {3.4608255573148727`*^9, 3.460825578506846*^9}, {
   3.460839173472966*^9, 3.460839253236741*^9}, {3.4608393307753*^9, 
   3.460839419293964*^9}, {3.464524759540012*^9, 3.464524799577138*^9}}],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"Cos", "[", "x", "]"}], "-", "x"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "0", ",", "2"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.460839143029388*^9, 3.460839150164009*^9}}],

Cell[TextData[{
 "From the graph we can read off that ",
 StyleBox["x=0.7", "Input"],
 " should be a good initial guess."
}], "Text",
 CellChangeTimes->{{3.448023316376433*^9, 3.448023317664606*^9}}],

Cell[TextData[{
 "Here is another example. We solve ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    SuperscriptBox["e", "x"], "=", "2"}], TraditionalForm]]],
 ". Of course, we expect that the solution should be ",
 StyleBox["Log[2]", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.46452487424881*^9, 3.464524958539019*^9}}],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"E", "^", "x"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "0", ",", "2"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.464524962197839*^9, 3.4645249746122503`*^9}}],

Cell[BoxData[
 RowBox[{"FindRoot", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"E", "^", "x"}], "\[Equal]", "2"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "0.5"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.448023322380143*^9, 3.448023344926011*^9}, 
   3.464524994869528*^9}],

Cell[BoxData[
 RowBox[{"N", "[", 
  RowBox[{"Log", "[", "2", "]"}], "]"}]], "Input",
 CellChangeTimes->{{3.464525000242282*^9, 3.464525004284161*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Practice", "Subsection",
 CellFrame->{{0, 0}, {0, 2}}],

Cell[TextData[{
 "1) Find all solutions (exact and approximate) of the equation ",
 Cell[BoxData[
  FormBox[
   StyleBox[
    RowBox[{
     RowBox[{
      SuperscriptBox["x", "5"], "+", "x"}], "=", "1"}], "Input"], 
   TraditionalForm]]],
 "."
}], "Text",
 CellChangeTimes->{{3.4479725431062717`*^9, 3.447972622493424*^9}, {
   3.447972746211347*^9, 3.4479727481428823`*^9}, {3.4479728489525557`*^9, 
   3.447972949340362*^9}, {3.448022620783722*^9, 3.448022656946389*^9}, {
   3.448022726172607*^9, 3.448022850963038*^9}, {3.4480229041078587`*^9, 
   3.448022975356772*^9}, {3.448023091943857*^9, 3.44802312672012*^9}, {
   3.448023165250265*^9, 3.4480232009839*^9}, {3.4480234867873917`*^9, 
   3.448023487522623*^9}, {3.46100097037778*^9, 3.461000978394785*^9}, {
   3.4610010344956408`*^9, 3.4610010451778183`*^9}, 3.461001083310503*^9, {
   3.461001120807241*^9, 3.461001127153165*^9}, 3.46452485412387*^9}],

Cell[TextData[{
 "2) Find all solutions of",
 StyleBox[" ", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   RowBox[{
    SuperscriptBox["e", "x"], "=", 
    RowBox[{"2", "+", "x"}]}], TraditionalForm]], "Input"], "Input"],
 ". There are two."
}], "Text",
 CellChangeTimes->{{3.4479725431062717`*^9, 3.447972622493424*^9}, {
   3.447972746211347*^9, 3.4479727481428823`*^9}, {3.4479728489525557`*^9, 
   3.447972949340362*^9}, {3.448022620783722*^9, 3.448022656946389*^9}, {
   3.448022726172607*^9, 3.448022850963038*^9}, {3.4480229041078587`*^9, 
   3.448022975356772*^9}, {3.448023091943857*^9, 3.44802312672012*^9}, {
   3.448023165250265*^9, 3.4480232009839*^9}, {3.4480234867873917`*^9, 
   3.448023487522623*^9}, {3.46100097037778*^9, 3.461000978394785*^9}, {
   3.4610010344956408`*^9, 3.4610010451778183`*^9}, 3.461001083310503*^9, {
   3.461001120807241*^9, 3.461001127153165*^9}, 3.46452485837018*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Homework", "Subsection",
 CellFrame->{{0, 0}, {0, 2}},
 CellChangeTimes->{{3.460825722876402*^9, 3.460825725610227*^9}}],

Cell[TextData[{
 "1)  Have ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " find the",
 StyleBox[" exact",
  FontWeight->"Bold"],
 " solutions to the following equations:"
}], "Text",
 CellChangeTimes->{{3.416500591551766*^9, 3.416500609472273*^9}, {
   3.416500667925123*^9, 3.4165007194843884`*^9}, 3.41770915185122*^9, {
   3.417709224971787*^9, 3.417709230745022*^9}, {3.460825864733408*^9, 
   3.460825868959683*^9}, {3.460825922441498*^9, 3.4608259242015963`*^9}, 
   3.4610012507895308`*^9},
 FontWeight->"Plain"],

Cell[TextData[{
 "(a) ",
 Cell[BoxData[
  FormBox[
   StyleBox[
    RowBox[{
     RowBox[{
      SuperscriptBox["x", "2"], "-", 
      RowBox[{"5", " ", "x"}], " ", "+", "4"}], "=", "0"}], "Input"], 
   TraditionalForm]]]
}], "Text",
 CellChangeTimes->{{3.460826364871724*^9, 3.4608263874571133`*^9}}],

Cell[TextData[{
 "(b) ",
 StyleBox["x+y=5", "Input"],
 " and ",
 StyleBox["2x+6y=23", "Input"],
 " simultaneously."
}], "Text",
 CellChangeTimes->{{3.460825978143834*^9, 3.460826006771989*^9}, 
   3.460826044152461*^9}],

Cell[TextData[{
 "2) Can ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " find the exact solutions of ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SuperscriptBox["x", "5"], "+", 
     RowBox[{"3", 
      SuperscriptBox["x", "4"]}], "-", 
     RowBox[{"3", 
      SuperscriptBox["x", "2"]}], "+", 
     RowBox[{"5", "x"}], "-", "5"}], "=", "0"}], TraditionalForm]]],
 "? If not, why not?"
}], "Text",
 CellChangeTimes->{{3.460826122383803*^9, 3.460826173387498*^9}, {
  3.4608265073860493`*^9, 3.460826567317264*^9}, {3.4608279366397*^9, 
  3.460827939216076*^9}, {3.4608281646247683`*^9, 3.460828166234248*^9}, {
  3.461001259182382*^9, 3.461001279083243*^9}}],

Cell[TextData[{
 "3) Can ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " find the exact solutions of ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{
     SuperscriptBox["x", "5"], "+", 
     RowBox[{"3", 
      SuperscriptBox["x", "4"]}], "-", 
     RowBox[{"3", 
      SuperscriptBox["x", "2"]}], "+", 
     RowBox[{"5", "x"}], "+", "6"}], "=", "0"}], TraditionalForm]]],
 "? How do you reconcile your observations of your answers to 2) and 3)?"
}], "Text",
 CellChangeTimes->{{3.460826642283739*^9, 3.460826685374817*^9}, {
  3.460827947668837*^9, 3.460827949500822*^9}, {3.460828171032873*^9, 
  3.46082818050056*^9}, {3.4610012641105433`*^9, 3.461001272451923*^9}, {
  3.461001323840065*^9, 3.461001332691093*^9}}],

Cell[TextData[{
 "4)  We want to find the solutions to the equation   ",
 StyleBox["sin x = 1-x/4", "Input"],
 ". To get started on the problem, use the following input command for ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " to get a graph of the two functions together. Solutions to the equation \
above are exactly the ",
 StyleBox["x", "Input"],
 "-values of the points of intersection. "
}], "Text",
 CellChangeTimes->{{3.460828194998019*^9, 3.460828209714355*^9}, 
   3.461001343108363*^9}],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"Sin", "[", "x", "]"}], ",", 
     RowBox[{"1", "-", 
      RowBox[{"x", "/", "4"}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "1"}], ",", "7"}], "}"}]}], "]"}]], "Input"],

Cell[TextData[{
 "Evaluate the following input command which tries to find a solution by \
using ",
 StyleBox["Solve", "Input"],
 "."
}], "Text",
 CellChangeTimes->{3.460828265577881*^9}],

Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"Sin", "[", "x", "]"}], "\[Equal]", 
    RowBox[{"1", "-", 
     RowBox[{"x", "/", "4"}]}]}], ",", "x"}], "]"}]], "Input"],

Cell[TextData[{
 StyleBox["Basically, here ",
  FontColor->GrayLevel[0]],
 StyleBox["Mathematica",
  FontSlant->"Italic",
  FontColor->GrayLevel[0]],
 StyleBox[" tells you that the function to be solved is not a polynomial, so \
it cannot do it. ",
  FontColor->GrayLevel[0]]
}], "Text",
 CellChangeTimes->{{3.417709459315682*^9, 3.4177095034865093`*^9}, 
   3.4610015254814043`*^9},
 FontColor->RGBColor[0, 0, 1]],

Cell[TextData[{
 "Now use ",
 StyleBox["FindRoot", "Input"],
 "  to get a numerical solution. Remember that ",
 StyleBox["FindRoot", "Input"],
 " needs a starting point, i.e., an ",
 StyleBox["x", "Input"],
 "-value that is close to the solution. Use the graph above to read off a \
starting value and compute all three solutions."
}], "Text",
 CellChangeTimes->{{3.4608283429783993`*^9, 3.4608283454760847`*^9}, 
   3.461001879891197*^9}]
}, Open  ]]
}, Open  ]]
},
ScreenStyleEnvironment->"Working",
WindowSize->{850, 641},
WindowMargins->{{122, Automatic}, {Automatic, 30}},
PrintingCopies->1,
PrintingPageRange->{Automatic, Automatic},
ShowSelection->True,
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, 70, 0, 67, "Section"],
Cell[CellGroupData[{
Cell[685, 27, 156, 2, 34, "Subsection"],
Cell[844, 31, 463, 14, 26, "Text"],
Cell[1310, 47, 335, 7, 43, "Input"],
Cell[1648, 56, 227, 6, 26, "Text"],
Cell[1878, 64, 253, 5, 58, "Input"],
Cell[2134, 71, 232, 7, 26, "Text"],
Cell[2369, 80, 147, 2, 43, "Input"],
Cell[2519, 84, 224, 6, 43, "Input"],
Cell[2746, 92, 147, 2, 43, "Input"],
Cell[2896, 96, 389, 11, 26, "Text"],
Cell[3288, 109, 303, 6, 58, "Input"],
Cell[3594, 117, 355, 11, 26, "Text"],
Cell[3952, 130, 335, 7, 43, "Input"],
Cell[4290, 139, 218, 5, 27, "Input"],
Cell[4511, 146, 1152, 36, 56, "Text"],
Cell[5666, 184, 159, 3, 43, "Input"],
Cell[5828, 189, 181, 2, 26, "Text"],
Cell[6012, 193, 305, 7, 43, "Input"],
Cell[6320, 202, 386, 12, 26, "Text"],
Cell[6709, 216, 278, 5, 58, "Input"],
Cell[6990, 223, 570, 17, 41, "Text"],
Cell[7563, 242, 291, 6, 58, "Input"],
Cell[7857, 250, 179, 5, 26, "Text"],
Cell[8039, 257, 316, 7, 58, "Input"],
Cell[8358, 266, 223, 5, 26, "Text"],
Cell[8584, 273, 399, 10, 58, "Input"],
Cell[8986, 285, 148, 5, 26, "Text"],
Cell[9137, 292, 227, 6, 27, "Input"],
Cell[9367, 300, 207, 7, 26, "Text"],
Cell[9577, 309, 277, 8, 27, "Input"],
Cell[9857, 319, 112, 1, 26, "Text"],
Cell[9972, 322, 406, 12, 27, "Input"],
Cell[10381, 336, 122, 1, 26, "Text"],
Cell[10506, 339, 455, 14, 27, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[10998, 358, 124, 2, 44, "Subsection"],
Cell[11125, 362, 204, 4, 26, "Text"],
Cell[11332, 368, 417, 11, 43, "Input"],
Cell[11752, 381, 222, 5, 27, "Input"],
Cell[11977, 388, 217, 5, 27, "Input"],
Cell[12197, 395, 220, 5, 27, "Input"],
Cell[12420, 402, 220, 5, 27, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[12677, 412, 188, 3, 44, "Subsection"],
Cell[12868, 417, 149, 3, 27, "Input"],
Cell[13020, 422, 301, 9, 26, "Text"],
Cell[13324, 433, 229, 7, 27, "Input"],
Cell[13556, 442, 764, 18, 56, "Text"],
Cell[14323, 462, 408, 13, 43, "Input"],
Cell[14734, 477, 227, 6, 26, "Text"],
Cell[14964, 485, 204, 5, 27, "Input"],
Cell[15171, 492, 883, 24, 47, "Text"],
Cell[16057, 518, 380, 13, 47, "Input"],
Cell[16440, 533, 293, 9, 26, "Text"],
Cell[16736, 544, 79, 2, 27, "Input"],
Cell[16818, 548, 107, 3, 27, "Input"],
Cell[16928, 553, 123, 2, 27, "Input"],
Cell[17054, 557, 129, 1, 26, "Text"],
Cell[17186, 560, 194, 4, 27, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[17417, 569, 128, 2, 44, "Subsection"],
Cell[17548, 573, 256, 5, 26, "Text"],
Cell[17807, 580, 870, 22, 29, "Text"],
Cell[18680, 604, 823, 22, 29, "Text"],
Cell[19506, 628, 908, 23, 29, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[20451, 656, 140, 2, 44, "Subsection"],
Cell[20594, 660, 398, 8, 71, "Text"],
Cell[20995, 670, 424, 14, 27, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[21456, 689, 102, 2, 44, "Subsection"],
Cell[21561, 693, 1039, 28, 29, "Text"],
Cell[22603, 723, 421, 11, 29, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[23061, 739, 139, 2, 44, "Subsection"],
Cell[23203, 743, 394, 9, 26, "Text"],
Cell[23600, 754, 329, 9, 27, "Input"],
Cell[23932, 765, 554, 12, 41, "Text"],
Cell[24489, 779, 206, 5, 27, "Input"],
Cell[24698, 786, 259, 7, 26, "Text"],
Cell[24960, 795, 399, 12, 41, "Text"],
Cell[25362, 809, 233, 7, 27, "Input"],
Cell[25598, 818, 437, 13, 47, "Input"],
Cell[26038, 833, 819, 22, 46, "Text"],
Cell[26860, 857, 166, 5, 26, "Text"],
Cell[27029, 864, 207, 5, 27, "Input"],
Cell[27239, 871, 1172, 29, 116, "Text"],
Cell[28414, 902, 458, 11, 41, "Text"],
Cell[28875, 915, 490, 15, 47, "Input"],
Cell[29368, 932, 164, 5, 26, "Text"],
Cell[29535, 939, 460, 11, 27, "Input"],
Cell[29998, 952, 203, 4, 43, "Input"],
Cell[30204, 958, 702, 17, 56, "Text"],
Cell[30909, 977, 261, 7, 27, "Input"],
Cell[31173, 986, 199, 5, 26, "Text"],
Cell[31375, 993, 322, 10, 26, "Text"],
Cell[31700, 1005, 231, 6, 27, "Input"],
Cell[31934, 1013, 283, 8, 27, "Input"],
Cell[32220, 1023, 150, 3, 27, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[32407, 1031, 60, 1, 44, "Subsection"],
Cell[32470, 1034, 912, 19, 30, "Text"],
Cell[33385, 1055, 911, 18, 28, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[34333, 1078, 126, 2, 44, "Subsection"],
Cell[34462, 1082, 526, 14, 26, "Text"],
Cell[34991, 1098, 301, 11, 30, "Text"],
Cell[35295, 1111, 219, 8, 26, "Text"],
Cell[35517, 1121, 680, 20, 30, "Text"],
Cell[36200, 1143, 731, 20, 30, "Text"],
Cell[36934, 1165, 506, 12, 41, "Text"],
Cell[37443, 1179, 295, 10, 27, "Input"],
Cell[37741, 1191, 187, 6, 26, "Text"],
Cell[37931, 1199, 195, 6, 27, "Input"],
Cell[38129, 1207, 414, 12, 26, "Text"],
Cell[38546, 1221, 439, 11, 41, "Text"]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)