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Notebook[{

Cell[CellGroupData[{
Cell["The Intermediate Value Theorem", "Subtitle"],

Cell["\<\
I. Cnop
Vrije Universiteit Brussel
icnop@vub.ac.be\
\>", "Subsubtitle"],

Cell[CellGroupData[{

Cell["Continuous functions", "Section"],

Cell["\<\
In  the age of technology, a continuous dependance (function) means \
that we can obtain its values with any (finite) precision if we know the \
value of the independant variable with sufficient (finite) precision.This is \
exactly what Augustin Cauchy meant when he gave his first definition od \
continuity, and since computers work with finite precision this definition is \
the one we will use. If a dependence is continuous, we can obtain values with \
arbitrary precision.
The same is true for the definition of a limit.
We will now show how we can approximate in the same way the zeroes of a \
continuous function when we know its graph crosses the axis of the \
independant variables. The existence of a zero in this case is shown by a \
construction, on which the classical proof is based. It is an example of a \
constructive proof. Many constructive proofs use repeated computations called \
algorithms, and they also are the basis of algorithms themselves.
We are going to locate the zero with any precision by the following \
algorithmic construction.\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["The bisection method", "Section"],

Cell[CellGroupData[{

Cell["Set f the name of any function on an interval", "Subsection"],

Cell[BoxData[
    \(\(\(\ \)\(f[x_] := x\^3 + \@\(x + 2\) - 4\)\)\)], "Input"],

Cell["\<\
Here we have chosen a function for which there is no algebraic way \
to locate a zero. We will first check that it has negative and positive \
values on a certain interval by taking a product of values:\
\>", "Text"],

Cell[CellGroupData[{

Cell["Control", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(f[0. ]\ \ f[2]\)\(//\)\(N\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    \(\(-15.51471862576143`\)\)], "Output"]
}, Closed]],

Cell[TextData[{
  "This will be our initial interval where we look for a zero. Recall that in \
 ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "  , an interval  [a,b]  is denoted by the listing of both endpoints  \
{a,b}"
}], "Text"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Divide the interval", "Subsection"],

Cell["\<\
If the value of the function in the middle of an interval  [a,b]  \
has a different sign of the value in the left endpoint we keep the left half \
of the interval, otherwise we keep the right half of the interval.\
\>", \
"Text"],

Cell[BoxData[
    \(newHalf[{a_, b_}] := 
      If[f[a]\ f[\((a + b)\)/2] <= 0, {a, \((a + b)\)/2}, {\((a + b)\)/2, 
          b}]\)], "Input"],

Cell[CellGroupData[{

Cell["Repeat this division:", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(intervals = NestList[newHalf, {0. , 2. }, 10]\)], "Input"],

Cell[BoxData[
    \({{0.`, 2.`}, {1.`, 2.`}, {1.`, 1.5`}, {1.25`, 1.5`}, {1.25`, 
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        1.3125`}, {1.296875`, 1.3046875`}, {1.296875`, 
        1.30078125`}, {1.296875`, 1.298828125`}}\)], "Output"]
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        ColumnSpacings->3,
        RowAlignments->Baseline,
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}, Closed]],

Cell["More steps will give more precision", "Text"],

Cell[CellGroupData[{

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          {"1.2974472045898438`", "1.297454833984375`"},
          {"1.2974510192871094`", "1.297454833984375`"},
          {"1.2974510192871094`", "1.2974529266357422`"},
          {"1.2974519729614258`", "1.2974529266357422`"},
          {"1.297452449798584`", "1.2974529266357422`"},
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Cell["\<\
The value of f in the endpoints of the last interval obtained is \
indeed close to zero\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Closed]]
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Cell[CellGroupData[{

Cell["Plotting our results", "Subsubsection"],

Cell["Left endpoints", "Text"],

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Cell[BoxData[
    \(ListPlot[
      Transpose[{First[Transpose[intervals]], 
          Range[Length[intervals]]}]]\)], "Input"],

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