(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 119308, 4268] NotebookOptionsPosition[ 115965, 4164] NotebookOutlinePosition[ 116324, 4180] CellTagsIndexPosition[ 116281, 4177] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "Comp 110 Final Exam: Spring 2009. Due at Noon on Wednesday, April 29. \ Send Your ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook by E-mail to: rng@rice.edu" }], "Title", CellChangeTimes->{{3.4158870635005703`*^9, 3.415887068309574*^9}, { 3.415973357036235*^9, 3.415973444121129*^9}, {3.415974641715996*^9, 3.415974643062335*^9}, 3.4486283658265247`*^9, {3.448634212564517*^9, 3.4486342647712383`*^9}, {3.448634400920775*^9, 3.448634405125901*^9}}], Cell[TextData[{ StyleBox["All work must be entirely your own. You are not allowed to work \ with a partner or to consult on these problems with anyone except the \ instructor or the labbie.\n\nThis exam is closed book and closed notes. You \ are not permitted to look at your work on previous modules or previous \ homeworks while taking this exam, but you are encouraged to make extensive \ use of the Documentation Center. ", FontSize->18], StyleBox["\n\n", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["Caution: ", FontSize->18], StyleBox["Although we expect you to make extensive use of the ", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["Documentation Center", FontSize->18], StyleBox[" to assist you with these problems, when you are asked to explain \ concepts or notation, you should not copy your answers verbatim from the ", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["Documentation Center", FontSize->18], StyleBox[". Present the answers in your own words. Avoid jargon and \ special notation; use standard English and classical Mathematical notation. \ When you are asked to present examples, provide your own examples; do not \ copy examples from the ", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["Documentation Center", FontSize->18], StyleBox[".", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["\n", FontSize->12, FontVariations->{"CompatibilityType"->0}], StyleBox["\nAll proofs and computations must be done using ", FontSize->18], StyleBox["Mathematica", FontSize->18, FontSlant->"Italic"], StyleBox["; no other proofs or computations will be accepted. \n\nPlease \ format your solutions appropriately. Use text format, not input format, when \ you are typing text. Write coherent sentences and paragraphs; part of your \ grade will depend on how clearly you present your ideas.\n\n", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["There is a 4 hour time limit for this exam. If you get stuck on a \ problem, go on to the next problem and come back later to the problem that is \ giving you trouble. Do not waste time.\n\nThe exam is due at noon on May 5. \ For graduating seniors the exam is due at noon on May 2.", FontSize->18] }], "Subsubtitle", CellChangeTimes->{ 3.415885553887953*^9, {3.415973282536766*^9, 3.4159733046608467`*^9}, { 3.4159737376219883`*^9, 3.415973737790701*^9}, 3.4166591120581284`*^9}], Cell[TextData[StyleBox["", FontSize->18]], "Subsubtitle", CellChangeTimes->{3.4158855693048*^9}], Cell[CellGroupData[{ Cell["Geometry: Volume of a Cone (20 Points)", "Section", FontColor->GrayLevel[0]], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .70288 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics3D %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 5.55112e-17 1 -0.10878 1 [ [ 0 0 0 0 ] [ 1 .70288 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 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p F P 0 g s .784 .998 .822 r .20958 .44896 m .19946 .48422 L .25623 .43741 L p F P 0 g s .795 .992 .754 r .26827 .40075 m .25623 .43741 L p .28998 .38217 L F P 0 g s 0 0 0 r .26827 .40075 m .29096 .38471 L p .23995 .42498 L F P 0 g s 0 0 0 r .20958 .44896 m .26827 .40075 L p .23995 .42498 L F P 0 g s .795 .992 .754 r .29482 .37426 m .26827 .40075 L p .28998 .38217 L F P 0 g s 0 .3 .796 r .84944 .53168 m .80054 .48422 L .81184 .46394 L p F P 0 g s .063 .42 .866 r .81184 .46394 m .80054 .48422 L .79042 .44896 L closepath p F P 0 g s .767 .995 .866 r .18816 .46394 m .19946 .48422 L .20958 .44896 L closepath p F P 0 g s .832 .999 .796 r .15056 .53168 m .19946 .48422 L .18816 .46394 L p F P 0 g s .699 .266 .217 r .5 .26009 m .5 .28913 L .60186 .30293 L p F P 0 g s .699 .266 .217 r .60186 .30293 m .60142 .29703 L .5 .26009 L p F P 0 g s .482 .088 .217 r .39858 .29703 m .5 .26009 L .5 .28913 L p F P 0 g s .036 .363 .834 r .90627 .5798 m .84944 .53168 L .84696 .49786 L p F P 0 g s 0 .3 .796 r .81184 .46394 m .84696 .49786 L .84944 .53168 L p F P 0 g s .816 .999 .834 r .15056 .53168 m .15304 .49786 L .09419 .54747 L p F P 0 g s .816 .999 .834 r .09419 .54747 m .09373 .5798 L .15056 .53168 L p F P 0 g s .832 .999 .796 r .18816 .46394 m .15304 .49786 L .15056 .53168 L p F P 0 g s .838 .397 .18 r .61003 .30633 m .60142 .29703 L .60186 .30293 L closepath p F P 0 g s .482 .088 .217 r .5 .28913 m .39814 .30293 L .39858 .29703 L p F P 0 g s .224 0 .18 r .39814 .30293 m .38997 .30633 L .39858 .29703 L closepath p F P 0 g s .098 0 0 r .38997 .30633 m .39814 .30293 L .39529 .32636 L p F P 0 g s .036 .363 .834 r .84696 .49786 m .90581 .54747 L .90627 .5798 L p F P 0 g s .742 .224 0 r .67046 .35321 m .60471 .32636 L .60186 .30293 L p F P 0 g s .597 .075 0 r .5 .28913 m .60186 .30293 L .60471 .32636 L p F P 0 g s .365 0 0 r .39814 .30293 m .5 .28913 L .5 .31245 L p F P 0 g s .365 0 0 r .5 .31245 m .39529 .32636 L .39814 .30293 L p F P 0 g s .605 .035 0 r .67046 .35321 m .60471 .32636 L p .70904 .38471 L F P 0 g s .605 .035 0 r .70904 .38471 m .70518 .37426 L .67046 .35321 L p F P 0 g s 0 0 0 r .32954 .35321 m .39529 .32636 L p .34878 .36143 L F P 0 g s 0 0 0 r .29482 .37426 m .32954 .35321 L p .34878 .36143 L F P 0 g s .671 .146 0 r .73173 .40075 m .70518 .37426 L .70904 .38471 L closepath p F P 0 g s 0 0 0 r .34878 .36143 m .29096 .38471 L .29482 .37426 L p F P 0 g s 0 0 0 r .29096 .38471 m .26827 .40075 L .29482 .37426 L closepath p F P 0 g s .597 .075 0 r .60471 .32636 m .5 .31245 L .5 .28913 L p F P 0 g s .532 0 0 r .73655 .41112 m .79042 .44896 L p .721 .39778 L F P 0 g s .532 0 0 r .71448 .39296 m .73655 .41112 L p .721 .39778 L F P 0 g s .532 0 0 r .70904 .38471 m .71448 .39296 L p .79042 .44896 L F P 0 g s 0 .145 .583 r .79042 .44896 m .73655 .41112 L p .82198 .47087 L F P 0 g s 0 .145 .583 r .82198 .47087 m .81184 .46394 L .79042 .44896 L p F P 0 g s 0 .145 .583 r .80871 .46155 m .82198 .47087 L p .77931 .44102 L F P 0 g s 0 .145 .583 r .73655 .41112 m .80871 .46155 L p .77931 .44102 L F P 0 g s 0 .052 .557 r .84696 .49786 m .81184 .46394 L .82198 .47087 L closepath p F P 0 g s 0 .153 .644 r .90581 .54747 m .84696 .49786 L .82198 .47087 L p F P 0 g s 0 .153 .644 r .82198 .47087 m .82544 .47172 L p .90581 .54747 L F P 0 g s 0 .251 .677 r .82544 .47172 m .82198 .47087 L .80871 .46155 L closepath p F P 0 g s .607 .037 0 r .83485 .49061 m .82659 .47402 L p .80871 .46155 L F P 0 g s .607 .037 0 r .82659 .47402 m .82544 .47172 L .80871 .46155 L p F P 0 g s .687 .142 0 r .73655 .41112 m .80871 .46155 L p .75756 .42914 L F P 0 g s .687 .142 0 r .71664 .40321 m .73655 .41112 L p .75756 .42914 L F P 0 g s 0 0 0 r .26345 .41112 m .20958 .44896 L p .279 .39778 L F P 0 g s 0 0 0 r .28552 .39296 m .26345 .41112 L p .279 .39778 L F P 0 g s .037 0 0 r .27634 .4285 m .21575 .45204 L p .26345 .41112 L F P 0 g s 0 0 0 r .29096 .38471 m .28552 .39296 L p .20958 .44896 L F P 0 g s .408 .807 .583 r .20958 .44896 m .26345 .41112 L .19129 .46155 L p F P 0 g s .408 .807 .583 r .18816 .46394 m .20958 .44896 L p .19129 .46155 L F P 0 g s .037 0 0 r .21575 .45204 m .19129 .46155 L .26345 .41112 L p F P 0 g s .408 .807 .583 r .19129 .46155 m .17802 .47087 L .18816 .46394 L p F P 0 g s .546 .857 .557 r .17802 .47087 m .15304 .49786 L .18816 .46394 L closepath p F P 0 g s .574 .899 .644 r .09419 .54747 m .15304 .49786 L .17802 .47087 L p F P 0 g s .453 0 0 r .71448 .39296 m .70904 .38471 L .65122 .36143 L closepath p F P 0 g s .605 .035 0 r .60631 .33413 m .65122 .36143 L .70904 .38471 L p F P 0 g s .605 .035 0 r .60471 .32636 m .60631 .33413 L p .70904 .38471 L F P 0 g s 0 0 0 r .28552 .39296 m .29096 .38471 L .34878 .36143 L closepath p F P 0 g s .454 0 0 r .60631 .33413 m .60471 .32636 L .5 .31245 L closepath p F P 0 g s .223 0 0 r .39369 .33413 m .39529 .32636 L .5 .31245 L closepath p F P 0 g s 0 0 0 r .39529 .32636 m .39369 .33413 L .34878 .36143 L p F P 0 g s 0 .153 .644 r .82544 .47172 m .87414 .51272 L .90581 .54747 L p F P 0 g s .574 .899 .644 r .17456 .47172 m .12586 .51272 L .09419 .54747 L p F P 0 g s .574 .899 .644 r .17802 .47087 m .17456 .47172 L p .09419 .54747 L F P 0 g s .642 .193 .169 r .5 .31245 m .5 .34278 L .60772 .35114 L p F P 0 g s .642 .193 .169 r .60772 .35114 m .60631 .33413 L .5 .31245 L p F P 0 g s .508 .083 .169 r .39369 .33413 m .5 .31245 L .5 .34278 L p F P 0 g s .747 .285 .15 r .65122 .36143 m .60631 .33413 L .60772 .35114 L closepath p F P 0 g s .508 .083 .169 r .5 .34278 m .39228 .35114 L .39369 .33413 L p F P 0 g s .351 0 .15 r .39228 .35114 m .34878 .36143 L .39369 .33413 L closepath p F P 0 g s .689 .177 .013 r .65122 .36143 m .60772 .35114 L p .71664 .40321 L F P 0 g s .689 .177 .013 r .71664 .40321 m .71448 .39296 L .65122 .36143 L p F P 0 g s .282 0 .013 r .34878 .36143 m .39228 .35114 L .38923 .37878 L p F P 0 g s .282 0 .013 r .28552 .39296 m .34878 .36143 L p .38923 .37878 L F P 0 g s .76 .25 0 r .73655 .41112 m .71448 .39296 L .71664 .40321 L closepath p F P 0 g s .282 0 .013 r .38923 .37878 m .28336 .40321 L .28552 .39296 L p F P 0 g s .109 0 0 r .28336 .40321 m .26345 .41112 L .28552 .39296 L closepath p F P 0 g s .45 .855 .677 r .17456 .47172 m .17802 .47087 L .19129 .46155 L closepath p F P 0 g s .037 0 0 r .26345 .41112 m .28336 .40321 L .27634 .4285 L p F P 0 g s .689 .177 .013 r .60772 .35114 m .61077 .37878 L .71664 .40321 L p F P 0 g s .581 .081 .029 r .5 .34278 m .60772 .35114 L .61077 .37878 L p F P 0 g s .443 0 .029 r .39228 .35114 m .5 .34278 L .5 .37039 L p F P 0 g s .443 0 .029 r .5 .37039 m .38923 .37878 L .39228 .35114 L p F P 0 g s .687 .142 0 r .80871 .46155 m .72366 .4285 L .71664 .40321 L p F P 0 g s .616 .064 0 r .61077 .37878 m .71664 .40321 L .72366 .4285 L p F P 0 g s .208 0 0 r .28336 .40321 m .38923 .37878 L p .30317 .42244 L F P 0 g s .208 0 0 r .30317 .42244 m .27634 .4285 L .28336 .40321 L p F P 0 g s .607 .037 0 r .80871 .46155 m .72366 .4285 L p .83485 .49061 L F P 0 g s 0 0 0 r .19129 .46155 m .27634 .4285 L p .20531 .4753 L F 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Display the right circular cone of height 1, whose vertex is at the \ origin, whose axis is the ", StyleBox["z", FontSlant->"Italic"], "-axis, and whose cross section with the plane ", StyleBox["z", FontSlant->"Italic"], "=1 is a unit circle." }], "Text", CellChangeTimes->{{3.415974144919763*^9, 3.415974148975062*^9}}, FontColor->RGBColor[0.5, 0, 0.5]], Cell[TextData[{ "2. Write a function ptin", StyleBox["Cone", FontSlant->"Italic"], " that given a point in 3-space determines whether the point lies inside the \ cone in Problem 1. The function ptin", StyleBox["Cone", FontSlant->"Italic"], " should return 1 if the point lies inside the cone and 0 if the point lies \ outside the cone." }], "Text", FontColor->RGBColor[0.5, 0, 0.5]], Cell[TextData[{ "3. Write a function ", StyleBox["randomPt", FontSlant->"Italic"], " that generates a random point inside a rectangular box that contains the \ cone in Problem 1. " }], "Text", FontColor->RGBColor[0.5, 0, 0.5]], Cell[TextData[{ "4. Write a function ", StyleBox["dartlist", FontSlant->"Italic"], " that generates a list of ", StyleBox["n", FontSlant->"Italic"], " random points inside the rectangular box that contains the cone in Problem \ 1. (Hint: Use the helper function in Problem 3.)" }], "Text", FontColor->RGBColor[0.5, 0, 0.5]], Cell["\<\ 5. Use Monte Carlo methods to determine the ratio of the volume of the cone \ in Problem 1 to the volume of the box that contains the cone by applying the \ functions in Problems 2 and 4.\ \>", "Text", FontColor->RGBColor[0.5, 0, 0.5]], Cell["\<\ 6. Using the results of Problems 5, determine numerically the volume of the \ cone in Problem 1.\ \>", "Text", FontColor->RGBColor[0.5, 0, 0.5]], Cell["\<\ 7. Using the result of Problems 6 and the known formula for the volume of a \ right circular cone, determine numerically the value of \[Pi].\ \>", "Text", CellChangeTimes->{{3.415973646663774*^9, 3.415973647556342*^9}}, FontColor->RGBColor[0.5, 0, 0.5]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Quaternion Algebra (20 Points)", "Section", FontColor->GrayLevel[0]], Cell["\<\ In this problem you will investigate identities for quaternions.\t\t\ \>", "Text", CellChangeTimes->{{3.4159741239021606`*^9, 3.415974124311331*^9}, { 3.415974307113163*^9, 3.415974319821858*^9}}], Cell[TextData[{ "A ", StyleBox["quaternion", FontSlant->"Italic"], " is a pair (", StyleBox["c,v", FontSlant->"Italic"], "), where ", StyleBox["c", FontSlant->"Italic"], " is a scalar (a real number) and ", StyleBox["v", FontSlant->"Italic"], " is a vector in 3-dimensions. Quaternion addition and quaternion \ multiplication are defined in the following fashion: Let ", StyleBox["p", FontSlant->"Italic"], "=(", StyleBox["c,v", FontSlant->"Italic"], ") and ", StyleBox["q", FontSlant->"Italic"], "=(", StyleBox["d,w", FontSlant->"Italic"], ") be quaternions.\n", StyleBox["Quaternion Addition: ", FontSlant->"Italic"], "\t\n\t", StyleBox["p+q", FontSlant->"Italic"], " = (", StyleBox["c+d, v+w", FontSlant->"Italic"], ")\n", StyleBox["Quaternion Multiplication:", FontSlant->"Italic"], "\n\t", StyleBox["pq = ", FontSlant->"Italic"], "(", StyleBox["cd - v\[CenterDot]w, cw + dv + v\[Times]w", FontSlant->"Italic"], ")\nwhere \[Times] denotes cross product, \[CenterDot] denotes dot product, \ and juxtaposition (i.e. ", StyleBox["cw or dv) ", FontSlant->"Italic"], "denotes the product of a scalar with a vector." }], "Text", CellChangeTimes->{{3.415965900697665*^9, 3.415965902120125*^9}, { 3.415972648213141*^9, 3.415972657259891*^9}, {3.4486286164879913`*^9, 3.4486286730329857`*^9}, {3.4486332595766478`*^9, 3.448633295927723*^9}}], Cell[CellGroupData[{ Cell["Problems", "Exercise", FontColor->RGBColor[0.570001, 0.200015, 0.570001]], Cell[TextData[{ StyleBox["Using ", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox[", either prove or provide a counterexample for each of the \ following four identities:\ni.\t", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["p+q = q+p", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["\n", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["ii.\tpq = qp", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["\niii.\t(", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["pq", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox[")", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["r = p", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["(", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["qr", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox[")", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["\n", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["iv.\t", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["p", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["(", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox["q+r", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox[")", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox[" = pq + pr", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]] }], "Text", CellChangeTimes->{{3.4159660408898*^9, 3.4159661478067636`*^9}, { 3.415966195904956*^9, 3.41596620112953*^9}, 3.41596627477781*^9, { 3.415966324624482*^9, 3.41596634926339*^9}, 3.415972689873921*^9}], Cell[TextData[{ "Note that numerical examples are fine as ", StyleBox["counterexamples", FontSlant->"Italic"], ". But a numerical example is NOT a proof. NO CREDIT will be given if you \ present a numerical example in place of a proof." }], "Text", CellChangeTimes->{{3.4113119857752733`*^9, 3.411311987384079*^9}}, FontColor->RGBColor[0.5, 0, 0.5]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Polygons, Circles, Wheels, and Rosettes (20 Points)", "Section", FontColor->GrayLevel[0]], Cell[TextData[{ "The vertices of a regular polygon lie on a circle. If the circle is \ centered at the origin and the polygon has ", StyleBox["n", FontSlant->"Italic"], " sides, then the vertices can be placed at the points:" }], "Text", FontWeight->"Bold"], Cell[TextData[{ "\t", Cell[BoxData[ FormBox[ SubscriptBox["P", "k"], TraditionalForm]]], "=(", StyleBox["r cos(2\[Pi]k/n), r sin(2\[Pi]k/n)", FontSlant->"Italic"], ") \t\t", StyleBox["k=0,...,n", FontSlant->"Italic"] }], "Text", CellChangeTimes->{{3.4488074316931763`*^9, 3.448807436472687*^9}}, FontWeight->"Bold"], Cell[TextData[{ "where ", StyleBox["r", FontSlant->"Italic"], " is the radius of the circle." }], "Text", FontWeight->"Bold"] }, Open ]], Cell[CellGroupData[{ Cell["Problems", "Section", FontColor->RGBColor[0.570001, 0.200015, 0.570001]], Cell[TextData[{ "In the following problems you must write your own functions. You may NOT \ use the built in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions for circles and polygons." }], "Text", CellChangeTimes->{{3.448633364119233*^9, 3.448633365676276*^9}}, FontWeight->"Bold"], Cell[TextData[{ "1. Write a function ", StyleBox["polyverts", FontSlant->"Italic"], " that takes as input a positive integer ", StyleBox["n", FontSlant->"Italic"], " and a positive real number ", StyleBox["r,", FontSlant->"Italic"], " and generates a list of vertices for a polygon with ", StyleBox["n", FontSlant->"Italic"], " sides lying on a circle of radius ", StyleBox["r", FontSlant->"Italic"], ". (You may want to use the function N to speed up the calculations of \ sines and cosines.)" }], "Text", CellChangeTimes->{{3.448629025637081*^9, 3.4486290287470617`*^9}}, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.200015, 0.570001]], Cell[TextData[{ "2. Using the function ", StyleBox["polyverts", FontSlant->"Italic"], " in Problem 1, write a function ", StyleBox["displaypoly", FontSlant->"Italic"], " that takes as input a positive integer ", StyleBox["n", FontSlant->"Italic"], " and a positive real number ", StyleBox["r", FontSlant->"Italic"], ", and displays the corresponding polygon with ", StyleBox["n", FontSlant->"Italic"], " sides. Turn off the axes so that only the polygon is displayed. Generate \ some examples to illustrate that your program works." }], "Text", FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.200015, 0.570001]], Cell[TextData[{ "3. Write a function ", StyleBox["circle", FontSlant->"Italic"], " that takes as input a positive integer ", StyleBox["r", FontSlant->"Italic"], " and displays the corresponding circle of radius ", StyleBox["r", FontSlant->"Italic"], ". Turn off the axes so that only the circle is displayed. (Hint: A \ circle is just a polygon with lots of sides.) Generate some examples to \ illustrate that your program works. Use your ", StyleBox["circle", FontSlant->"Italic"], " program together with your ", StyleBox["displaypoly ", FontSlant->"Italic"], "program to display polygons with ", StyleBox["n", FontSlant->"Italic"], " sides inscribed in circles of radius ", StyleBox["r", FontSlant->"Italic"], "." }], "Text", FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.200015, 0.570001]], Cell[TextData[{ "4. A wheel is a polygon with all its vertices joined to its center (see \ below). Using the function ", StyleBox["polyverts", FontSlant->"Italic"], " in Problem 1, write a function ", StyleBox["wheel", FontSlant->"Italic"], " that takes as input a positive real number ", StyleBox["r, ", FontSlant->"Italic"], "and displays a wheel inscribed in a circle of radius ", StyleBox["r", FontSlant->"Italic"], ". Use a slider to control the number of sides of the wheel. Turn off the \ axes so that only the wheel is displayed. Generate some examples to \ illustrate that your program works. 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A rosette is a polygon with all its diagonals connected by straight \ lines (see below). Using the function ", StyleBox["polyverts", FontSlant->"Italic"], " in Problem 1, write a function ", StyleBox["rosette", FontSlant->"Italic"], " that takes as input a positive real number ", StyleBox["r", FontSlant->"Italic"], ", and displays a rosette inscribed in a circle of radius ", StyleBox["r", FontSlant->"Italic"], ". Use a slider to control the number of sides of the rosette. Turn off \ the axes so that only the rosette is displayed. Generate some examples to \ illustrate that your program works. 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In this problem you will use ", StyleBox["Mathematica", FontSlant->"Italic"], " to derive Heron's formula." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Problems", "Section", FontColor->RGBColor[0.570001, 0.200015, 0.570001]], Cell[TextData[{ "1. Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to derive a formula for the area of a triangle in terms the lengths of its \ sides, based upon simple geometry and trigonometry. [Hint: Use the standard \ formula for the area of a triangle: Area = 1/2 Base \[Star] Height, and \ apply the Pythagorean Theorem to obtain two additional equations involving ", StyleBox["h", FontSlant->"Italic"], " and ", StyleBox["x", FontSlant->"Italic"], ". Then invoke ", StyleBox["Mathematica", FontSlant->"Italic"], " to ", "eliminate", " the variables ", StyleBox["h", FontSlant->"Italic"], " and ", StyleBox["x", FontSlant->"Italic"], ".]" }], "Text", FontColor->RGBColor[0.5, 0, 0.5], Background->RGBColor[1, 0.9, 1]], Cell[TextData[{ "2. Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to derive a symmetric version of this area formula in terms of the \ semiperimeter ", Cell[BoxData[ FormBox[ RowBox[{"s", "=", RowBox[{ FractionBox["1", "2"], RowBox[{"(", RowBox[{"a", "+", "b", "+", "c"}], ")"}]}]}], TraditionalForm]]], ", where ", StyleBox["a, b, c", FontSlant->"Italic"], " are the lengths of the sides of the triangle. [", "Hint: Factor the expression for the area you generated in part 1, and then \ use replacment rules.", "]" }], "Text", FontColor->RGBColor[0.5, 0, 0.5], Background->RGBColor[1, 0.9, 1]] }, Open ]], Cell[CellGroupData[{ Cell["Fractal Tennis (20 Points)", "Section", FontColor->GrayLevel[0]], Cell[TextData[{ "The fractals you generated in class were constructed by a", StyleBox[" deterministic algorithm", FontSlant->"Italic"], ". Here you will generate fractals using a non-deterministic or ", StyleBox["random algorithm", FontSlant->"Italic"], ". In this random algorithm, you will start with an arbitrary set of \ transformation W={", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["w", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["w", "p"]}], TraditionalForm]]], "}. These transformations are usually represented by 3x3 matrices, and \ points in the plane are typically represented using 3 coordinates, where the \ first two coordinates are the standard rectangular coordinates and the third \ coordinate is always 1. If you select any point ", Cell[BoxData[ FormBox[ SubscriptBox["P", "0"], TraditionalForm]]], "=(", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], ",", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox["y", "0"]}], TraditionalForm]]], ",1) in the fractal and compute\n\t ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["P", "1"], "=", " ", RowBox[{ SubscriptBox["P", "0"], "*", SubscriptBox["w", SubscriptBox["i", "1"]]}]}], TraditionalForm]]], "\n\t \[VerticalEllipsis]\n\t ", Cell[BoxData[ FormBox[ SubscriptBox["P", "n"], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["P", RowBox[{"n", "-", "1"}]], "*", SubscriptBox["w", SubscriptBox["i", "n"]]}], TraditionalForm]]], "\nwhere the transformation matrices ", Cell[BoxData[ FormBox[ SubscriptBox["w", SubscriptBox["i", "1"]], TraditionalForm]]], ",\[Ellipsis],", Cell[BoxData[ FormBox[ SubscriptBox["w", SubscriptBox["i", "n"]], TraditionalForm]]], " are chosen at random from the set ", StyleBox["W", FontSlant->"Italic"], ", then it turns out that the points ", Cell[BoxData[ FormBox[ SubscriptBox["P", "0"], TraditionalForm]]], ",\[Ellipsis],", Cell[BoxData[ FormBox[ SubscriptBox["P", "n"], TraditionalForm]]], ",\[Ellipsis] form a dense subset of the fractal determined by the the \ functions ", StyleBox["W", FontSlant->"Italic"], ", so if you display enough of these points you will get a good idea of the \ shape of the fractal. Your code will be written so that you will first \ display ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["P", "0"], ",", RowBox[{"then", " ", SubscriptBox["P", "0"]}], ",", SubscriptBox["P", "1"], ",", RowBox[{"then", " ", SubscriptBox["P", "0"]}], ",", SubscriptBox["P", "1"], ",", SubscriptBox["P", "2"], ",", RowBox[{"and", " ", "so", " ", "on"}]}], TraditionalForm]]], ". You will stop when the number of points is sufficiently large that you \ get a good idea of the shape of the fractal. Note that when we apply a \ transformation matrix to a point, we multiply by placing the point on the \ left and the matrix on the right." }], "Text", CellChangeTimes->{{3.448633647572942*^9, 3.448633668071055*^9}, { 3.448808094218994*^9, 3.448808106666692*^9}, {3.448808227866069*^9, 3.4488082955778646`*^9}}], Cell[TextData[{ "The functions below build transformation matrices. In particular\n\ttrans[", StyleBox["v", FontSlant->"Italic"], "] generates the 3x3 matrix that translate points by the vector ", StyleBox["v", FontSlant->"Italic"], ";\n\trotQ(\[Theta], ", StyleBox["Q", FontSlant->"Italic"], ") generates the 3x3 matrix that rotates points about the point ", StyleBox["Q", FontSlant->"Italic"], " by the angle \[Theta];\n\tscl(", StyleBox["s,Q", FontSlant->"Italic"], ") generates the 3x3 matrix that scales the distance of arbitrary points to \ the fixed point ", StyleBox["Q", FontSlant->"Italic"], " by the scale factor ", StyleBox["s", FontSlant->"Italic"], ".\nThe transformations you need to build fractals will be given to you, so \ you will not need to define these functions. Nevertheless you must enable \ these functions so that you can use them later in the problem." }], "Text", CellChangeTimes->{{3.4486328502070503`*^9, 3.448632850821385*^9}, 3.4486337317525673`*^9}, FontWeight->"Bold"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"rot", "[", "angle_", "]"}], ":=", RowBox[{"N", "[", RowBox[{"(", GridBox[{ { RowBox[{"Cos", "[", RowBox[{"angle", " ", "Degree"}], "]"}], RowBox[{"Sin", "[", RowBox[{"angle", " ", "Degree"}], "]"}], "0"}, { RowBox[{"-", RowBox[{"Sin", "[", RowBox[{"angle", " ", "Degree"}], "]"}]}], RowBox[{"Cos", "[", RowBox[{"angle", " ", "Degree"}], "]"}], "0"}, {"0", "0", "1"} }], ")"}], "]"}]}], ";"}], "\n"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"scl", "[", "scalefactor_", "]"}], ":=", RowBox[{"(", GridBox[{ {"scalefactor", "0", "0"}, {"0", "scalefactor", "0"}, {"0", "0", "1"} }], ")"}]}], ";"}], "\n"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"trans", "[", "vector_", "]"}], ":=", RowBox[{"(", GridBox[{ {"1", "0", "0"}, {"0", "1", "0"}, { RowBox[{"vector", "[", RowBox[{"[", "1", "]"}], "]"}], RowBox[{"vector", "[", RowBox[{"[", "2", "]"}], "]"}], "1"} }], ")"}]}], "\n", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rotQ", "[", RowBox[{"angle_", ",", "Q_"}], "]"}], ":=", RowBox[{ RowBox[{"rot", "[", "angle", "]"}], ".", RowBox[{"trans", "[", RowBox[{"Q", "-", RowBox[{"Q", ".", RowBox[{"rot", "[", "angle", "]"}]}]}], "]"}]}]}], "\n"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"sclQ", "[", RowBox[{"scalefactor_", ",", "Q_"}], "]"}], ":=", RowBox[{ RowBox[{"scl", "[", "scalefactor", "]"}], ".", RowBox[{"trans", "[", RowBox[{"Q", "-", RowBox[{"Q", ".", RowBox[{"scl", "[", "scalefactor", "]"}]}]}], "]"}]}]}]}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problems", "Section", FontColor->GrayLevel[0]], Cell[TextData[{ "In the following problems you will generate code that builds a fractal \ point by point from a set of matrices by repeatedly multiplying a point in \ the fractal by a matrix chosen at random from a fixed set of matrices. When \ performing the matrix multiplication, the point should be on the left of the \ multiplication sign and the matrix on the right. The points will be stored \ in a global variable called ", StyleBox["ptlist. ", FontSlant->"Italic"], "I will give you the matrices; you will write the code to generate points \ on the corresponding fractal. " }], "Text", FontColor->GrayLevel[0]], Cell[TextData[{ "1. Write a function ", StyleBox["buildnewpts", FontSlant->"Italic"], " that takes as input a list of 3x3 matrices, and appends one new point to \ the global variable ", StyleBox["ptlist", FontSlant->"Italic"], " by selecting at random a matrix from the list of matrices and applying \ this matrix to the last point in ", StyleBox["ptlist", FontSlant->"Italic"], ". (Hint: Remember to represent your points with 3 coordinates, where the \ third coordinate is always 1.)" }], "Text", FontWeight->"Bold", FontColor->RGBColor[0.5, 0, 0.5]], Cell[TextData[{ "2. Modify your function ", StyleBox["buildnewpts", FontSlant->"Italic"], " so that it takes as input a list of 3x3 matrices and an integer ", StyleBox["n", FontSlant->"Italic"], ", and appends ", StyleBox["n", FontSlant->"Italic"], " new points to ", StyleBox["ptlist", FontSlant->"Italic"], " by repeating the function ", StyleBox["buildnewpts", FontSlant->"Italic"], " in Problem 1 a total of ", StyleBox["n", FontSlant->"Italic"], " times." }], "Text", FontWeight->"Bold", FontColor->RGBColor[0.5, 0, 0.5]], Cell[TextData[{ "3. To get started, you need to find a single point on the fractal. To \ find a point on the fractal, you can select a random point in the plane and \ apply many times (10 is usually enough) a random a matrix from the list of \ matrices. Write a function ", StyleBox["startpt", FontSlant->"Italic"], " that takes as input a list of matrices and produces as output a single \ point on the associated fractal." }], "Text", FontWeight->"Bold", FontColor->RGBColor[0.5, 0, 0.5]], Cell[TextData[{ "4. Modify your function ", StyleBox["startpt", FontSlant->"Italic"], " so that it takes as input a list of matrices and initializes ", StyleBox["ptlist", FontSlant->"Italic"], " to a list consisting of a single point on the fractal. (Hint: Remember \ that ", StyleBox["ptlist", FontSlant->"Italic"], " must contain a list of points, and that a point is a list of numbers, so a \ point is not a list of points.)" }], "Text", FontWeight->"Bold", FontColor->RGBColor[0.5, 0, 0.5]], Cell[TextData[{ "5. Write a function ", StyleBox["displayptlist", FontSlant->"Italic"], " that displays the points in the global variable ", StyleBox["ptlist", FontSlant->"Italic"], ". Remember that the points in ", StyleBox["ptlist", FontSlant->"Italic"], " are represented by three coordinates, where the third coordinate is always \ one. You may want to get rid of this third coordinate before you try to \ display your points in the plane. Turn off the axes so that only the points \ are displayed. " }], "Text", FontWeight->"Bold", FontColor->RGBColor[0.5, 0, 0.5]], Cell[TextData[{ "6. Using your functions ", StyleBox["buildnewpts, startpt, and displayptlist", FontSlant->"Italic"], ", display the four fractals generated by the following four sets of \ transformation matrices ", StyleBox["W", FontSlant->"Italic"], ". 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