(* 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, NotebookDataLength[ 27601, NotebookOptionsPosition[ 25480, NotebookOutlinePosition[ 26084, CellTagsIndexPosition[ 26041, WindowFrame->Normal*) 7] 789] 722] 744] 741] (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "Inverse, Natural Log, and Exponential ", Cell[BoxData[ FormBox["Functions", TraditionalForm]], "None"] }], "Title", Editable->False], Cell[CellGroupData[{ Cell["Exercises", "Section"], Cell[CellGroupData[{ Cell["\\<\\ Problem #1: An animation of the exponential function and its tangent line. \\ \\>", "Subsection"], Cell[TextData[{ "The objective of this assignment is for you to produce an animation of the \\ graph of the exponential function ", Cell[BoxData[ FormBox[ RowBox[{"y", " ", "=", " ", SuperscriptBox["e", "x"]}], TraditionalForm]]], " together with a tangent line to the graph through a point on the graph \\ which will be allowed to vary. In particular here is what your output should \\ look like: " }], "Text"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 4., Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\\"untitled\\"", Typeset`specs$$ = {{ Hold[$CellContext`a$$], -2, 4}}, Typeset`size$$ = {540., {175., 182.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$10466$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = -2}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$10466$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Show[ Plot[ Exp[$CellContext`x], {$CellContext`x, -2, 4}, PlotRange -> {-1, 60}, PlotStyle -> { Thickness[0.005]}], Graphics[{Green, PointSize[0.02], Point[{$CellContext`a$$, E^$CellContext`a$$}]}], Plot[ E^$CellContext`a$$ ($CellContext`x - $CellContext`a$$) + E^$CellContext`a$$, {$CellContext`x, $CellContext`a$$ 1, $CellContext`a$$ + 1}, PlotStyle -> { Thickness[0.005], Red}]], "Specifications" :> {{$CellContext`a$$, -2, 4}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{604., {233., 240.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output"], Cell[TextData[{ "Here is a brief description of the ", StyleBox["Mathematica", FontSlant->"Italic"], " commands that were used to make the above animation and some advice to \\ help you to reproduce it. More details about these functions can (and should) \\ be looked up using the help menu. " }], "Text"], Cell[TextData[{ "The ", StyleBox["Plot", FontFamily->"Courier"], " command; you can use the ", StyleBox["PlotStyle", FontFamily->"Courier"], " option to specify the color of the graph and the ", StyleBox["Thickness", FontFamily->"Courier"], " command to make the curve more easily seen. You may also specify the ", Cell[BoxData[ FormBox["y", TraditionalForm]]], "-coordinates of the graph shown with the ", StyleBox["PlotRange", FontFamily->"Courier"], " option. For example" }], "Item"], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ SuperscriptBox["x", "3"], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", " ", "3"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"Thickness", "[", ".003", "]"}], ",", " ", "Red"}], "}"}]}], ",", " ", RowBox[{"PlotRange", "\\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "20"}], ",", " ", "40"}], "}"}]}]}], "]"}]], "Input"], Cell[TextData[{ "The ", StyleBox["Show", FontFamily->"Courier"], " command; with this command you can combine two or more graphics objects \\ into one. Care must be given to the order in which the objects are listed in \\ the command. For example: " }], "Item"], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ SuperscriptBox["x", "3"], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", " ", "3"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"Thickness", "[", ".003", "]"}], ",", " ", "Red"}], "}"}]}], ",", " ", RowBox[{"PlotRange", "\\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "20"}], ",", " ", "40"}], "}"}]}]}], "]"}], ",", "\\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"10", SqrtBox["x"]}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "0", ",", " ", "4"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\\[Rule]", " ", "Blue"}]}], "]"}]}], " ", "]"}]], "Input"], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"10", SqrtBox["x"]}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "0", ",", " ", "4"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\\[Rule]", " ", "Blue"}]}], "]"}], " ", ",", " ", RowBox[{"Plot", "[", RowBox[{ SuperscriptBox["x", "3"], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", " ", "3"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"Thickness", "[", ".003", "]"}], ",", " ", "Red"}], "}"}]}], ",", " ", RowBox[{"PlotRange", "\\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "20"}], ",", " ", "40"}], "}"}]}]}], "]"}]}], "]"}]], "Input"], Cell[TextData[{ "The ", StyleBox["Graphics", FontFamily->"Courier"], " command; this command is convenient to produce some basic graphics objects \\ (like points, circles, or lines). For example, " }], "Item"], Cell[BoxData[ RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"Green", ",", RowBox[{"PointSize", "[", ".05", "]"}], ",", " ", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"{", RowBox[{"1", ",", "1"}], "}"}], "}"}], "]"}]}], "}"}], "]"}]], "Input"], Cell["\\<\\ The Manipulate command allows one to fairly easily produce animations which \\ the end-user will be able to interact with through the use of slider buttons. \\ The output may be of a graphical nature or otherwise. For example\\ \\>", "Item"], Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Show", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ SuperscriptBox["x", "2"], ",", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "5"}], ",", " ", "5"}], "}"}]}], "]"}], ",", " ", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"Red", ",", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"{", RowBox[{"a", ",", SuperscriptBox["a", "2"]}], "}"}], "}"}], "]"}]}], "}"}], "]"}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"a", ",", " ", RowBox[{"-", "4"}], ",", " ", "3"}], "}"}]}], "]"}]], "Input"], Cell[TextData[{ StyleBox["Advice:", FontFamily->"Times New Roman", FontWeight->"Bold", FontVariations->{"Underline"->True}], " ", StyleBox["When trying to do something complicated with Mathematica, build up your code \\"from the inside out\\". In other words, first get one basic aspect of your desired output working. Then embed that within some outer function \\ (like Show or Manipulate) and make any necessary additions or changes to your working code to get the more complicated expression to also work. Add \\ additional details as you go, all the while making sure things work with each addition or change. ", FontFamily->"Times New Roman", FontSlant->"Italic"] }], "Text"], \\ \\ \\ \\ Cell[TextData[{ "In this problem you will have to do a little bit of mathematical thought to \\ come up for an expression for the tangent line to the exponential curve \\ through a variable point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", SuperscriptBox["\\[ExponentialE]", "a"]}]}], TraditionalForm]]], ") --- you will then Manipulate over some range of values for ", Cell[BoxData[ FormBox["a", TraditionalForm]]], ". Explain how you get this expression in your lab solutions. " }], "Text"], Cell[BoxData[""], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem #2: Seeing a property of logs", "Subsection"], Cell[TextData[{ "Consider the following graph of ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", FractionBox["1", "x"]}], TraditionalForm]]], "on the interval ", Cell[BoxData[ FormBox[ RowBox[{"[", RowBox[{"1", ",", "26"}], "]"}], TraditionalForm]]], ". The region under the curve (and above the ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "-axis) between ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "2"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "6"}], TraditionalForm]]], " has been shaded red, while that between ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", " ", "8"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"x", " ", "=", "24"}], TraditionalForm]]], " has been shaded blue. 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A good explanation \\ should involve discussing or using some property of logarithms. " }], "Item"], Cell[TextData[{ "Give the endpoints of a third region under the graph of ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", FractionBox["1", "x"]}], TraditionalForm]]], " (not overlapping the given two regions) which would also have the same \\ area. " }], "Item"], Cell[TextData[{ "Give a ", StyleBox["Mathematica", FontSlant->"Italic"], " command that draws the above graph. You can form this command by combining \\ several ", StyleBox["Plot", FontFamily->"Courier"], " commands using the ", StyleBox["Show", FontFamily->"Courier"], " command. Within the ", StyleBox["Plot", FontFamily->"Courier"], " commands you may want to use the folllowing options: ", StyleBox["AxesOrigin, Ticks, PlotStyle, Thickness, PlotRange, FillingStyle, \\ Filling", FontFamily->"Courier"], ". See the help documentation for more information. " }], "Item"], Cell[BoxData[""], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem #3: A most beautiful equation", "Subsection"], Cell[TextData[{ "In your Calculus class lecture, we were careful to point out that the \\ inverse function relationship ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["e", RowBox[{"ln", "(", "x", ")"}]], "=", " ", "x"}], TraditionalForm]]], " is only true for ", Cell[BoxData[ FormBox[ RowBox[{"x", ">", "0"}], TraditionalForm]]], ", since the domain of the natural logarithm function is ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"0", ",", "\\[Infinity]"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], ". However, ", StyleBox["Mathematica", FontSlant->"Italic"], "'s function Log[x] is more general than ", Cell[BoxData[ FormBox[ RowBox[{"ln", "(", "x", ")"}], TraditionalForm]]], ". In other words ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Log", "[", "x", "]"}], " ", "=", " ", RowBox[{"ln", "(", "x", ")"}]}], TraditionalForm]]], " for all ", Cell[BoxData[ FormBox[ RowBox[{"x", " ", ">", "0"}], TraditionalForm]]], ", but ", Cell[BoxData[ FormBox[ RowBox[{"Log", "[", "x", "]"}], TraditionalForm]]], " is defined for all complex numbers ", Cell[BoxData[ FormBox["x", TraditionalForm]]], ". " }], "Text"], Cell[TextData[{ " Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to verify that ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["e", RowBox[{"Log", "[", "x", "]"}]], "=", " ", "x"}], TraditionalForm]]], " for all non-zero real numbers ", Cell[BoxData[ FormBox["x", TraditionalForm]]], ". You can do this very easily in ", StyleBox["Mathematica", FontSlant->"Italic"], ". Look up the command ", StyleBox["Simplify", FontFamily->"Courier"], " in the help menu. " }], "Item"], Cell[TextData[{ "Examine the equation ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["e", RowBox[{"Log", "[", "x", "]"}]], "=", " ", "x"}], TraditionalForm]], FormatType->"TraditionalForm"], " for the value of ", Cell[BoxData[ FormBox[ RowBox[{"x", " ", "=", RowBox[{"-", "1"}]}], TraditionalForm]]], ". In particular get ", StyleBox["Mathematica", FontSlant->"Italic"], " to compute ", Cell[BoxData[ FormBox[ RowBox[{"Log", "[", RowBox[{"-", "1"}], "]"}], TraditionalForm]]], ", substitute that result into the equation and re-write the equation by \\ bringing all terms to one side of the equals sign. " }], "Item"], Cell["\\<\\ Explain why the equation you wrote in the previous part is viewed by many \\ mathematicians as the most beautiful in all of Mathematics. If you are \\ unsure, you can google the equation to get some help. \\ \\>", "Item"], Cell[BoxData[""], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem #4: Determining an inverse function. ", "Subsection"], Cell[TextData[{ "Consider the function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], " ", "=", " ", RowBox[{ SuperscriptBox["x", "5"], "+", RowBox[{"4", SuperscriptBox["x", "3"]}], "+", RowBox[{"2", "x"}], " ", "+", "7"}]}], TraditionalForm]]], ". " }], "Text"], Cell["Explain why this function is invertible", "Item"], Cell[TextData[{ "Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to create a function ", Cell[BoxData[ FormBox["g", TraditionalForm]], FormatType->"TraditionalForm"], " which is the inverse of ", Cell[BoxData[ FormBox["f", TraditionalForm]], FormatType->"TraditionalForm"], " . Note in this case ", StyleBox["Mathematica", FontSlant->"Italic"], " won't be able to explicitly solve the equation ", Cell[BoxData[ FormBox[ RowBox[{"y", " ", "=", " ", RowBox[{"f", "(", "x", ")"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " for ", Cell[BoxData[ FormBox["x", TraditionalForm]], FormatType->"TraditionalForm"], " in terms of ", Cell[BoxData[ FormBox["y", TraditionalForm]], FormatType->"TraditionalForm"], ", but it will give a five solutions in terms of a built-in function called ", StyleBox["Root", FontFamily->"Courier"], ". Use the first of the five solutions to create the definition of ", Cell[BoxData[ FormBox["g", TraditionalForm]], FormatType->"TraditionalForm"], " by doing something similar to what was done in the discussion file (in \\ this problem you will also need to use the numerical approximation command N \\ in front of the Solve command). " }], "Item"], Cell[TextData[{ "Verify that ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["g", TraditionalForm]]], " satisfy ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", RowBox[{"g", "(", "x", ")"}], ")"}], "=", "x"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", RowBox[{"f", "(", "x", ")"}], ")"}], "=", "x"}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Item"], Cell["\\<\\ Show a graph of both functions on a single coordinate system that illustrates \\ the symmetric property of the graphs of inverse functions. 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