(*********************************************************************** 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[ 35837, 1015]*) (*NotebookOutlinePosition[ 36891, 1051]*) (* CellTagsIndexPosition[ 36847, 1047]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[ "Mathematics 161 Laboratory 12 \ November 19, 1992\n\t \nName: _____________________________ \ Lab Partner: ___________________________\n\t \nConsulted with: \ ____________________________________________________________"], "Text", CellMargins->{{17, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{1, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["\[Copyright] Lafayette College, 1994"], "SmallText", Evaluatable->False, TextAlignment->Right, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Integrals, Areas and Average Temperatures", Evaluatable->False, AspectRatioFixed->True, FontSize->21, FontWeight->"Plain"], StyleBox["", Evaluatable->False, AspectRatioFixed->True, FontSize->23, FontWeight->"Plain"] }], "Title", Evaluatable->False, AspectRatioFixed->True, FontSize->23], Cell[TextData[{ StyleBox["In this laboratory, you will use ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " to compute indefinite & definite integrals. You will also practice \ interpreting definite integrals as areas of regions in the plane.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 1: Antiderivatives"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["'s ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Integrate", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " command can be used to find antiderivatives of many continuous functions. \ To compute an antiderivative in the indefinite integral", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.750000 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.750000 scale 1 string 48 36 1 [48 0 0 36 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage 03FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 007FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FC0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFF8FFFFFF1FFFFFFFFFFFFFF FF81FFFFFFF8FC7FC7FC7E3FFFFE001F8FF8 FF81FFFFFFFF1C7FF8FF8E3FFFFE3FE3F1FF FF81FFFFFFFF03FFFF1C7FC7FFFE3FE3FE38 FF81FFFFFFFFE3FFFFFC7FC7FFFFC7FC7FF8 FF81FFFFFFFFE3FFFFE38FC7FFFFC7FC7FC7 FF81FFFFFFFFE07FFFE3FE07FFFFF8FF8FC7 FF81FFFFFFFFE00FFFE3FE07FFFFFF000FC7 FF81FFFFFFFFFC0FFFFFFE3FFFFFFFFFF1FF FF81FFFFFFFFFC703FFFFE3FFFFFFFFFF1FF FF81FFFFFFFFFF8FFFFFF1FFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF03FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFE00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFC0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF pop grestore %% End of Graphics MathPictureEnd %% End of picture \ \>"], "Graphics", Evaluatable->False, AspectRatioFixed->True, ImageSize->{48, 36}, ImageMargins->{{168, Inherited}, {Inherited, Inherited}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgIoc>Ic3>IVC>IIS>II03=Voc=Vc3=V VC=VIS=V"], ImageRangeCache->{{{0, 47}, {35, 0}} -> {-0.00357646, -3.75004*^-6, 0.0214288, 0.0214288}}], Cell[TextData["use the command"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Integrate[f[x], x]", AspectRatioFixed->True], StyleBox["", AspectRatioFixed->True, FontWeight->"Plain"] }], "Input", AspectRatioFixed->True], Cell[TextData["For instance, we all know the indefinite integral"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! 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To check that the answer \ supplied is an antiderivative of the original function, take its \ derivative:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["D[%,x]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Some functions have antiderivatives that seem surprising, such as"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Integrate[1/Sqrt[1 - x^2], x]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["If you don't believe that the derivative of arcsin", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ " could make no mention of any sort of trig function (or even if you do), \ check it for yourself; take its derivative:", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["D[ArcSin[x], x]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Finally, notice that some relatively simple functions have no simple \ expression for their antiderivative. 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Check your answer by interpreting \ the integral as the area of a region whose area you can compute by other \ means. Make an appropriate plot (either by hand or by computer) and shade \ the region whose area you have computed.\n\n\n\n\n\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Use ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Integrate", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " to compute the exact value of the definite integral of the function below \ over the interval (0, 2).", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.222222 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.222222 scale 1 string 72 16 1 [72 0 0 16 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFF8FFFFFF1FFFFFFFFFFFFFF1FFFFFFFFFFFFFFFFFFFFFFFFFF FF8FC7FC7FC7E3FFFFFFFFFFFFF1FFFFFFF8FFFFFFFFFE3FE3FFFF FFF1C7FF8FF8E3FFF8000FFF1F8E3FFFFFF8FFFFFFFFFFC7FC7FFF FFF03FFFF1C7FC7FFFFFFFFFE38E3FFC00001FFFFFFFFFF8E3FFFF FFFE3FFFFFC7FC7FFFFFFFFFFC7E3FFC7FF8FF8007FFFFFFE38000 FFFE3FFFFE38FC7FF8000FFFFFFFC7FF8FF8FFFFFFFFFFFF1C71FF FFFE07FFFE3FE07FFFFFFFFFFFFFC7FFF1F8FFFFFFFFFFFF1FF03F FFFE00FFFE3FE07FFFFFFFFFFFFFC7FFFE38FFFFFFFFFFFF1FF1C7 FFFFC0FFFFFFE3FFFFFFFFFFFFFFF8FFFFC0FFFFFFFFFFFFFFFFF8 FFFFC703FFFFE3FFFFFFFFFFFFFFF8FFFFF8FFFFFFFFFFFFFF8FF8 FFFFF8FFFFFF1FFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFFFFFFFF007 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000000000000000000000000 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF pop grestore %% End of Graphics MathPictureEnd %% End of picture \ \>"], "Graphics", Evaluatable->False, AspectRatioFixed->True, ImageSize->{72, 16}, ImageMargins->{{156, Inherited}, {Inherited, Inherited}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgIoc>Ic3>IVC>IIS>II03=Voc=Vc3=V VC=VIS=V"], ImageRangeCache->{{{0, 71}, {15, 0}} -> {-0.0259307, -1.11112*^-6, 0.0148149, 0.0148149}}], Cell[TextData[ "Explain how you can use the area interpretation of the definite integral to \ verify your answer. (Include a sketch.)\n\n\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Determine the area of the region in the first quadrant which is bounded by \ ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" = ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" \[Dash] sin", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" below, and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" = sin", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ " above. Briefly explain how you determined the area of this region. (Use \ complete sentences.)\n\n\n\n\n\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 3:\tAverage Temperature"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "On a typical day this time of year, the temperature (in degrees Fahrenheit) \ might be given by the function"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["temp[t_] := 29.6 + 2.9 t - 0.1 t^2 - 8 Cos[0.26 t]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["(Here, the time, ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[", is in hours with ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" = 0 representing the beginning of the day and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" = 24 representing the end of the day.)", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "To get a feeling for how the temperature fluctuated over the course of the \ day, plot the graph of temp."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Plot[temp[t], {t, 0, 24}, PlotRange->{0,60}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Print a copy of this plot for use in Part 4. One way to give an \"average \ temperature\" for the day would be to average the high and low temperatures \ which were recorded. This is the method most television reporters seem to \ use to calculate the average temperature. Using your plot, estimate the \ highest and lowest temperatures of the day and compute this rough average of \ the temperature. (You don't need much accuracy here; just eyeball it.)\n\n\t \ High __________ Low __________ Average \ __________"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We could get a more meaningful average temperature by taking several \ readings over the course of the day and averaging those readings. For \ example, we might read the temperature each hour, sum those 24 readings, and \ divide by 24 to obtain such an average. Use the command"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Sum[temp[n], {n, 1, 24}]/24"], "Input", AspectRatioFixed->True], Cell[TextData[ "to compute this average.\n\n The 24-readings average \ temperature is __________ ."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We could do even better, by using more temperature readings in our average. \ Modify the command above to compute an average temperature based on readings \ taken 48 times a day (every half-hour).\n\n The 48-readings \ average temperature is __________ ."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["define a function ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["avetemp", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" to compute an average temperature based on taking ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["n ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ "readings a day. Use your command to compute an average temperature which \ is still more meaningful than the 48-readings average.\n\nHow often did you \ take readings for your average? __________ .\n\n The __ \ -readings average temperature is __________ .", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 4: \tDoes This Involve Integrals?"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Use a definite integral to compute the area of the region bounded by the \ graph of ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["temp[t]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and the ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["-axis for 0 \[LessEqual] ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ " \[LessEqual] 24.\n\n The area of \ the region is __________ .", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Suppose you want to build a rectangle with a base 24 units long and an area \ equal to the area you just computed. What height must that rectangle have?\n\ \n\tThe height of the rectangle must be __________ units."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Draw your rectangle on the plot you printed in Part 3 and shade the region \ inside the rectangle. How does the height of your rectangle compare with the \ average temperatures you computed in Part 3? 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