(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 333294, 8494]*) (*NotebookOutlinePosition[ 334233, 8526]*) (* CellTagsIndexPosition[ 334189, 8522]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Math 126A-01Calculus II Introduction to Using Mathematica\ \>", "Subtitle"], Cell[CellGroupData[{ Cell["Using the Help Browser", "Section"], Cell[TextData[{ "The Help Browser has all the information you need to run ", StyleBox["Mathematica", FontSlant->"Italic"], ", including the complete contents of the ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["Book", FontSlant->"Italic"], ". 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The \ ones for integration and differentiation look like:" }], "Text"], Cell[BoxData[ RowBox[{ StyleBox["\[Integral]", ScriptLevel->0], RowBox[{"\[Placeholder]", StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]\[Placeholder]\), ScriptLevel->0], " ", \(\[PartialD]\_\[Placeholder]\ \[Placeholder]\)}]}]], "Input"], Cell["You can then fill in the blanks. For exmaple:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox["\[Integral]", ScriptLevel->0], RowBox[{ RowBox[{ StyleBox["(", ScriptLevel->0], \(E^\((\(-x\)/3\ )\) Sin[2 x]\), ")"}], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]x\), ScriptLevel->0]}]}]], "Input"], Cell[BoxData[ \(\(-\(3\/37\)\)\ E\^\(\(-x\)/3\)\ \((6\ Cos[2\ x] + Sin[2\ x])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x\((\ E^\((\(-x\)/3\ )\) Sin[2 x])\)\)], "Input"], Cell[BoxData[ \(2\ E\^\(\(-x\)/3\)\ Cos[2\ x] - 1\/3\ E\^\(\(-x\)/3\)\ Sin[2\ x]\)], "Output"] }, Open ]], Cell["\<\ These command can also be entered by writing out their full \ names:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[E^\((\(-x\)/3\ )\) Sin[2 x], \ x]\)], "Input"], Cell[BoxData[ \(\(-\(3\/37\)\)\ E\^\(\(-x\)/3\)\ \((6\ Cos[2\ x] + Sin[2\ x])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[E^\((\(-x\)/3\ )\) Sin[2 x], \ x]\)], "Input"], Cell[BoxData[ \(2\ E\^\(\(-x\)/3\)\ Cos[2\ x] - 1\/3\ E\^\(\(-x\)/3\)\ Sin[2\ x]\)], "Output"] }, Open ]], Cell[TextData[{ "Notice that functions use square brackets: ", StyleBox["[ ]", "Input"], ". Parentheses are used only for grouping terms in a mathematical \ expression." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Variables and Assignments", "Section"], Cell["\<\ Any sequence of characters you enter will be interpreted by the \ Kernel as the name of a variable. For example,\ \>", "Text"], Cell[BoxData[ \(x\ = \ 123; \ y\ = \ 118; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(delta\ = \ N[x\ - \ y]/x\)], "Input"], Cell[BoxData[ \(0.0406504065040650353`\)], "Output"] }, Open ]], Cell[TextData[{ "A single equal sign ", StyleBox["=", "Input"], " denotes ", StyleBox["assignment", FontSlant->"Italic"], ": the value on the right-hand side of the ", StyleBox["=", "Input"], " is stored", " in the variable named on the left-hand side. The function ", StyleBox["Clear[]", "Input"], " removes an assignment." }], "Text"], Cell[BoxData[ \(\(Clear[delta]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(2*delta\ - \ x\)], "Input"], Cell[BoxData[ \(\(-123\) + 2\ delta\)], "Output"] }, Open ]], Cell["\<\ Be careful to put spaces where you intend to have multiplication. \ If you enter\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(xSin[y]\)], "Input"], Cell[BoxData[ \(xSin[118]\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " thinks ", StyleBox["xSin", "Input"], " is the name of a variable. Putting a space between ", StyleBox["x", "Input"], " and ", StyleBox["Sin[]", "Input"], " gives" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x\ Sin[y]\)], "Input"], Cell[BoxData[ \(123\ Sin[118]\)], "Output"] }, Open ]], Cell[TextData[{ "Another form of assignment is the ", StyleBox["delayed assignment", FontSlant->"Italic"], " which is denoted by ", StyleBox[":=", "Input"], ". In this kind of assignment, the right hand side is not evaluated until \ the left-hand side is used. This distinction often does not make a practical \ difference, but it can be useful. 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A list is any sequence of \ objects separated by commas and enclosed in a set of braces: ", StyleBox["{ }", "Input"], ". The following are simple examples:" }], "Text"], Cell[BoxData[{ \({a, b, c}\), \({{1, 2}, {3, 4}}\), \({x^2\ - \ y\ == \ 1, \ x\ + 2\ y\ == \ 4}\)}], "Input"], Cell[TextData[{ "An equation in ", StyleBox["Mathematica", FontSlant->"Italic"], " is denoted by a double equal sign ", StyleBox["==", "Input"], " to distinguish it from an assignment ", StyleBox["=", "Input"], ". Equations are most commonly used with the ", StyleBox["Solve[]", "Input"], " command. For example," }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{x^2\ - \ y\ == \ 1, \ x\ + 2\ y\ == \ 4}]\)], "Input"], Cell[BoxData[ \({{y \[Rule] 5\/4, x \[Rule] 3\/2}, {y \[Rule] 3, x \[Rule] \(-2\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "Notice that both the input and output is a list. The arrow notation is \ used for substitutions. The syntax for substituting a value for a variable \ is: ", StyleBox["expression", FontSlant->"Italic"], " /. {", StyleBox["variable", FontSlant->"Italic"], " -> ", StyleBox["value", FontSlant->"Italic"], "}. For example," }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x^2\ - \ y\ /. \ {y -> 5\/4, x -> 3\/2}\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x\ + 2\ y\ /. \ {y -> 5\/4, x -> 3\/2}\)], "Input"], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell["\<\ As another example, say we want to calculate the critical points of \ the function\ \>", "Text"], Cell[BoxData[ \(f[x_]\ := \ 9 x^4\ - 38 x^3\ + 6 x^2 + 18 x\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\(f'\)[x]\ == \ 0]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(-\(1\/3\)\)}, {x \[Rule] 1\/2}, {x \[Rule] 3}}\)], "Output"] }, Open ]], Cell[TextData[{ "The value of ", StyleBox["f[x]", "Input"], " at the critical points can be calculated easly by substituting the \ previous result into ", StyleBox["f[x]", "Input"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[x]\ /. \ %\)], "Input"], Cell[BoxData[ \({\(-\(103\/27\)\), 101\/16, \(-189\)}\)], "Output"] }, Open ]], Cell[TextData[{ "Many functions in ", StyleBox["Mathematica", FontSlant->"Italic"], " have a numerical form that replaces symbolic computations with numerical \ algorithms. 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