(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 16859, 609]*) (*NotebookOutlinePosition[ 17875, 643]*) (* CellTagsIndexPosition[ 17831, 639]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[StyleBox["Math 161, Fall 2004, Barnier\t\nProject 1 - Tangent \ Lines to a Cubic. ", FontColor->RGBColor[1, 0, 0]]], "Section", CellFrame->True, Evaluatable->False, AspectRatioFixed->False, Background->GrayLevel[0.849989]], Cell[TextData[{ " This project is to be completed in teams of two. After removing this \ from the server or my webpage and placing it on the desktop, use ", StyleBox["Save As . . .", FontWeight->"Bold"], " to change the name of the notebook to include your last name at the \ beginning. It should be written as: ", StyleBox["161,F04-Proj1,yourlastnames.nb", FontWeight->"Bold"], " \n\nThe project is due ", StyleBox["as hard copy to me ", FontWeight->"Plain"], "and", StyleBox[" as a digital file in my drop box by ", FontWeight->"Plain"], "noon, ", StyleBox["Tuesday, October 19 at 4:00 PM", FontWeight->"Bold"], ". Please consult with me if you have any questions or difficulties. \ Bring a 100 M Zip Disk or USB memory device to my office with the notebook on \ it, so that I can see what is going on. ", StyleBox["It is your responsibility to assure that the project is complete \ and in polished form. You will be graded on correctness, clarity, evidence \ for your assertions, and good English.", FontWeight->"Plain"], " " }], "Subsubtitle", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", Background->GrayLevel[0.833326], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ " Let f[x] = ", Cell[BoxData[ \(x\^3\ - \ 3\ s\ x\^2\ - \ 3 \((20\ - \ \ 3 s)\)\ x\ + \ 496\)]], ", where \"s\" is the right-most nonzero digit of your Student \ Identification Number. To first calculate the values 3s and 3(10 + s) place \ your nonzero digit in the red box (", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 0]], ") and then just ", StyleBox["input the cell below by pressing the key. ", FontFamily->"Bookman", FontWeight->"Plain"] }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ RowBox[{ RowBox[{"s", "=", " ", StyleBox["\[Placeholder]", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]]}], ";", \({3\ s, 3 \((20\ - \ 3 s)\)}\)}]], "Input"], Cell[TextData[{ " ", StyleBox["Use the Basic Input palette, found under ", FontFamily->"Bookman", FontWeight->"Plain"], StyleBox["File", FontFamily->"Bookman"], StyleBox[" in the toolbar above, to define the function ", FontFamily->"Bookman", FontWeight->"Plain"], "f[x] = ", Cell[BoxData[ \(x\^3\ - \ 18\ x\^2\ - \ 3 \((20\ - \ \ 3 s)\)\ x\ + \ 496\)]], StyleBox[" in the input cell below. Be sure to input the function by \ pressing the key.", FontFamily->"Bookman", FontWeight->"Plain"] }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ RowBox[{\(Clear[f]\), ";", " ", RowBox[{\(f[x_]\), " ", ":=", " ", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 0]]}]}]], "Input"], Cell[TextData[{ " ", StyleBox["Input the function above by pressing the key every time \ you work on this notebook. ", FontFamily->"Bookman", FontWeight->"Plain"], StyleBox["Remember that any function that is needed in a calculation must \ be entered before the calculation", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], ".", StyleBox[" ", FontFamily->"Bookman", FontWeight->"Plain"] }], "Subsubsection", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", Background->GrayLevel[0.833326], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ Plot the function y = f[x] by placing the cursor in the cell below \ and then pressing the key. The domain chosen is -7 \[LessEqual] x \ \[LessEqual] 21. You may wish to alter the domain to see more or less of the \ plot. \ \>", "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ RowBox[{ RowBox[{"Plot", "[", RowBox[{\(f[x]\), ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", StyleBox[\(-7\), FontColor->RGBColor[1, 0, 0]], ",", " ", StyleBox["21", FontColor->RGBColor[1, 0, 0]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[TextData[{ "The first derivative of f at x is just f'[x] with a single quotation \ mark. The second derivative is f''[x] with two single quotation marks. \ Put your cursor in the input cell below and press to have ", StyleBox["Mathematica", FontSlant->"Italic"], " compute both f'[x] and f''[x]. " }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \({\(f'\)[x], \ \(\(f'\)'\)[x]}\)], "Input"], Cell["\<\ To find critical points, you may use Solve[f'[x] == 0, x]. \ Similarly for inflection points. \ \>", "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["Solve[f'[x] == 0, x]", "Input", AspectRatioFixed->True], Cell["Solve[f''[x] == 0, x]", "Input", AspectRatioFixed->True], Cell[TextData[{ " ", StyleBox["Exercise 1", FontColor->RGBColor[1, 0, 0]], ") Specify the critical points and the inflection point. Does this make \ sense considering the plot above? How are the x-values for the critical \ points and the inflection point related?" }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", Background->GrayLevel[0.666667], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ " ", StyleBox["Explanation:", FontColor->RGBColor[1, 0, 0]], " \n\n\n\t" }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "For a given function ", StyleBox[" f", FontWeight->"Bold"], " define the function ", StyleBox["TangLine", FontWeight->"Bold"], " (as a function of both ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["c", FontSlant->"Italic"], ") to be the point-slope form of the tangent line to the curve y = f[x] \ at x = c. 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Now replace \"c\" in the second cell by a value of your choice and \ then input in the usual way. \ \>", "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "Clear[c]; c = ", StyleBox["13", FontColor->RGBColor[1, 0, 0]], "; Plot[{f[x], TangLine[x, ", StyleBox["c", FontColor->RGBColor[1, 0, 0]], "]}, {x, ", StyleBox["-7", FontColor->RGBColor[1, 0, 0]], ", ", StyleBox["21", FontColor->RGBColor[1, 0, 0]], "}];" }], "Input", AspectRatioFixed->True], Cell[TextData[{ "Clear[c]; c = ", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 0]], "; ", "Plot[{f[x], TangLine[x, ", StyleBox["c", FontColor->RGBColor[1, 0, 0]], "]}, {x, ", StyleBox["-7", FontColor->RGBColor[1, 0, 0]], ", ", StyleBox["21", FontColor->RGBColor[1, 0, 0]], "}];" }], "Input", AspectRatioFixed->True], Cell[TextData[{ "For a given value c, ", StyleBox["Solve", FontWeight->"Bold"], " will find the values of x at which the curve and the tangent line \ cross. Replace \"c\" with a value of your choice and place the cursor in the \ cell below and input Solve by pressing the key." }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "Clear[c]; c = ", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 0]], "; ", "Solve[f[x] == TangLine[x, ", StyleBox["c", FontColor->RGBColor[1, 0, 0]], "], x]" }], "Input", AspectRatioFixed->True], Cell[TextData[{ " ", StyleBox["Exercise 2", FontColor->RGBColor[1, 0, 0]], ") As you can see, the tangent line at x = 13 intersects f(x) at another \ point. Find the x-value of that point. Then find the y-value by plugging \ into f[ ] in the next cell and pressing the key." }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", Background->GrayLevel[0.666667], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["Solve[f[x] == TangLine[x, 13], x]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(f[\ ]\)], "Input"], Cell[TextData[{ " ", StyleBox["Explanation:", FontColor->RGBColor[1, 0, 0]], " \n\n\n" }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ " ", StyleBox["Exercise 3", FontColor->RGBColor[1, 0, 0]], ") Find a tangent line to y = f(x) that intersects the curve only at the \ point of tangency. Plot f(x) and this tangent line on the same set of axes. \ [Hint: You may experiment with different tangent lines in the input cell \ below by changing the c-value in the Plot function.] Specify this c-value." }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", Background->GrayLevel[0.666667], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "Clear[c]; c = ", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 0]], "; ", "Plot[{f[x], TangLine[x, ", StyleBox["c", FontColor->RGBColor[1, 0, 0]], "]}, {x, ", StyleBox["-7", FontColor->RGBColor[1, 0, 0]], ", ", StyleBox["21", FontColor->RGBColor[1, 0, 0]], "}];" }], "Input", AspectRatioFixed->True], Cell[TextData[{ " ", StyleBox["Explanation:", FontColor->RGBColor[1, 0, 0]], " \n\n\n " }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Bookman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ " ", StyleBox["Exercise 4", FontColor->RGBColor[1, 0, 0]], ") Are there any tangent lines other than the one found in Exercise 3) \ that intersect y = f(x) only at the point of tangency? ", StyleBox["Explain", FontWeight->"Bold"], ". Use evidence based on graphs (using Plot) and algebra (using Solve) in \ your clear and well-written explanation. Type your explanation in the cell \ below. 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