(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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[ 38344, 1230]*) (*NotebookOutlinePosition[ 39043, 1254]*) (* CellTagsIndexPosition[ 38999, 1250]*) (*WindowFrame->Normal*) Notebook[{ Cell["Functions, Derivatives, and Plots", "Title"], Cell["\<\ Tuesday, September 12, 2006 Created by Jason Sherman\ \>", "Subsubtitle"], Cell["Functions (a quick review)", "Subsubsection"], Cell[TextData[{ "Remember when defining your own functions, use an ", StyleBox["underscore ", FontSlant->"Italic"], "after each variable on the left side of the equals sign, not on the \ right." }], "Text"], Cell[BoxData[ \(f[x_] = E^\(-x\)*Sin[x]\)], "Input"], Cell[TextData[{ "When using the function, ", StyleBox["do not", FontSlant->"Italic"], " use an underscore. The underscore is ", StyleBox["only", FontSlant->"Italic"], " used in the function definition." }], "Text"], Cell[BoxData[ \(f[t]\)], "Input"], Cell[BoxData[ \(f[0]\)], "Input"], Cell[BoxData[ \(f[Pi/2]\)], "Input"], Cell["Vectors and vector functions", "Subsubsection"], Cell[TextData[{ "Remember, to enter a vector like ", StyleBox["v", FontWeight->"Bold", FontSlant->"Italic"], " = 2", StyleBox[" i ", FontWeight->"Bold", FontSlant->"Italic"], "- 3 ", StyleBox["j", FontWeight->"Bold", FontSlant->"Italic"], " + 7 ", StyleBox["k", FontWeight->"Bold", FontSlant->"Italic"], ", we use curly brackets {}" }], "Text"], Cell[BoxData[ \(v = {2, \(-3\), 7}\)], "Input"], Cell[TextData[{ "To access the i-th component, use ", StyleBox["double square brackets", FontSlant->"Italic"] }], "Text"], Cell[BoxData[ \(v[\([1]\)]\)], "Input"], Cell[BoxData[ \(v[\([2]\)]\)], "Input"], Cell[BoxData[ \(v[\([3]\)]\)], "Input"], Cell["\<\ You can devine vector functions in pretty much the same way you define \ vectors\ \>", "Text"], Cell["For example,r(t)=Sin(t)i+Cos(t)j+tkcould be entered by", "Text"], Cell[BoxData[ \(r[t_] = {\ Sin[t]\ , \ Cos[t]\ , \ t\ }\)], "Input"], Cell[TextData[{ "How could we get the second component (the ", StyleBox["j", FontWeight->"Bold", FontSlant->"Italic"], " component) of ", StyleBox["r ", FontWeight->"Bold", FontSlant->"Italic"], "?" }], "Text"], Cell[BoxData[ \( (*\ Enter\ the\ command\ here\ *) \)], "Input"], Cell["Derivatives", "Subsubsection"], Cell["\<\ The command to take a derivative is D. Let's get some help for D.\ \>", "Text"], Cell[BoxData[ \(\(?D\)\)], "Input"], Cell["Here are some examples:", "Text"], Cell[BoxData[ \(f[x]\)], "Input"], Cell[TextData[{ "The derivative of f( x ). Note, we have to tell ", StyleBox["Mathematica", FontSlant->"Italic"], " what our variable is!" }], "Text"], Cell[BoxData[ \(D[f[x], x]\)], "Input"], Cell["We can use % (previous output) to take a second derivative.", "Text"], Cell[BoxData[ \(D[%, x]\)], "Input"], Cell["But, there is an easier way!", "Text"], Cell[BoxData[ \(D[f[x], {x, 2}]\)], "Input"], Cell[BoxData[ \(D[ArcSin[x], {x, 1}]\)], "Input"], Cell["------ ! ! ! Important ! ! ! ------", "Subtitle"], Cell["\<\ We can create a function that is the derivative of another function! Just \ use square brackets and an underscore! Let's define g( x ) as the derivative of f( x ), and h( x ) as the second \ derivative of f( x )\ \>", "Text"], Cell[BoxData[ \(g[x_] = D[f[x], x]\)], "Input"], Cell[BoxData[ \(h[x_] = D[f[x], {x, 2}]\)], "Input"], Cell[BoxData[ \(g[Pi/6]\)], "Input"], Cell[BoxData[ \(h[Pi/6]\)], "Input"], Cell["D also works for vector functions!", "Text"], Cell[BoxData[ \(D[r[t], t]\)], "Input"], Cell[BoxData[ \(D[r[t], {t, 2}]\)], "Input"], Cell["2-D Plots", "Subsubsection"], Cell[BoxData[ \(\(?Plot\)\)], "Input"], Cell[BoxData[ \(Plot[x^2, {x, \(-3\), 3}]\)], "Input"], Cell["\<\ If we have a function defined, we can use it inside the Plot command!\ \>", "Text"], Cell[BoxData[ \(Plot[f[x], {x, 0, 5}]\)], "Input"], Cell[TextData[{ "If we want to change the *RANGE* (the ", StyleBox["y", FontSlant->"Italic"], " values displayed) of the plot, use the PlotRange option: PlotRange - > { \ y_min , y_max}" }], "Text"], Cell[BoxData[ \(Plot[x^2, {x, \(-3\), 3}, PlotRange -> {\(-4\), 4}]\)], "Input"], Cell["\<\ To see *all* the y-values, and cut off unnecessary y-values, use: PlotRange - \ > All\ \>", "Text"], Cell[BoxData[ \(Plot[x^2, {x, \(-3\), 3}, PlotRange \[Rule] All]\)], "Input"], Cell[TextData[{ "We can change the color using the PlotStyle - > RBGColor option. \ RGBColor[ ", StyleBox["red, green, blue ", FontSlant->"Italic"], "] will display a plot in color. ", StyleBox["Red, Green, and Blue", FontSlant->"Italic"], " should all be numbers between 0 and 1." }], "Text"], Cell[BoxData[ \(Plot[x^2, {x, \(-3\), 3}, PlotStyle \[Rule] RGBColor[1, 0, 1]]\)], "Input"], Cell[BoxData[ \(Plot[x^2, {x, \(-3\), 3}, PlotStyle \[Rule] RGBColor[0, 0, 1]]\)], "Input"], Cell[TextData[{ "Another way of changing the color is the PlotStyle -> Hue option. Hue[ ", StyleBox["n", FontSlant->"Italic"], " ] set the color. ", StyleBox["n", FontSlant->"Italic"], " should be between 0 and 1. A hue of 0 corresponds to red, increasing \ from 0 to 1 goes through the colors of the rainbow!" }], "Text"], Cell[BoxData[ \(Plot[x^2, {x, \(-3\), 3}, PlotStyle \[Rule] Hue[0]]\)], "Input"], Cell[TextData[{ "Label your axes with the AxesLabel - > { \" ", StyleBox["horizontal-axis-label", FontSlant->"Italic"], " \" , \" ", StyleBox["vertical-axis-label", FontSlant->"Italic"], " \" } option." }], "Text"], Cell[BoxData[ \(Plot[x^2, {x, \(-3\), 3}, AxesLabel \[Rule] {"\", "\"}]\)], "Input"], Cell["All of these options can be used together!", "Text"], Cell[BoxData[ \(Plot[x^2, {x, \(-2\), 2}, PlotRange -> {\(-2\), 4}, PlotStyle \[Rule] RGBColor[0.1, 0.6, 0.7], AxesLabel \[Rule] {"\", "\"}]\)], "Input"], Cell[BoxData[ \(Plot[x^2, {x, \(-2\), 2}, PlotRange -> {\(-2\), 4}, PlotStyle \[Rule] RGBColor[0.1, 0.6, 0.7], AxesLabel \[Rule] {"\", "\"}, Background \[Rule] RGBColor[0.1, 0.9, 0.9]]\)], "Input"], Cell[TextData[{ "There are 2 ways to display more than one plot on a single graph. \nThe \ first is to enter several functions in curly brackets in the Plot command. \n\ The second is to store the different ", StyleBox["plots", FontSlant->"Italic"], " as variables, and then use the command Show[ ] to display them all. 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