(* 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[ 57200, 1944] NotebookOptionsPosition[ 54587, 1863] NotebookOutlinePosition[ 54944, 1879] CellTagsIndexPosition[ 54901, 1876] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell["\<\ Comp 110 Midterm Exam Spring 2009: Due at Noon Thursday, March 12. Send \ Your Notebook by E-mail to: rng@rice.edu.\ \>", "Title", CellChangeTimes->{{3.411305277024578*^9, 3.411305305499592*^9}, { 3.411755411668397*^9, 3.411755451779368*^9}, {3.4125166948165903`*^9, 3.412516707492036*^9}, {3.444416381859082*^9, 3.4444164239535923`*^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 D", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["ocumentation 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 D", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["ocumentation 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 D", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["ocumentation 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.", FontSize->18, FontVariations->{"CompatibilityType"->0}] }], "Subsubtitle", CellChangeTimes->{{3.4113053236785517`*^9, 3.411305398158415*^9}, 3.411311480516227*^9}], Cell[TextData[StyleBox["There is a 3.5 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.", FontSize->18]], "Subsubtitle"], Cell[CellGroupData[{ Cell["Debugging (10 Points)", "Section", CellChangeTimes->{{3.411310841332945*^9, 3.411310847532806*^9}}, FontColor->GrayLevel[0]], Cell["\<\ 1. Explain why it is good programming practice to clear the name of a new \ function (above the definition of the new function but in the same cell), \ even before you define the new function.\ \>", "Subsection", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["\<\ 2. Explain why it is not sufficient simply to delete the old function \ definition when you are redefining a function to remove a bug.\ \>", "Subsection", FontWeight->"Plain"], Cell[CellGroupData[{ Cell["\<\ 3. Debug the following program for computing the sum of the squares of the \ numbers in a list. Give examples to show that your debugged program works for any list of \ numbers.\ \>", "Subsection", CellChangeTimes->{3.411306699549301*^9}, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"sumSquares", "[", "mylist", "]"}], "=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"sum_", "=", "0"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{"sum", "=", SuperscriptBox[ RowBox[{"[", RowBox[{"[", "i", "]"}], "]"}], "2"]}], ",", RowBox[{"{", RowBox[{"i", ",", "n"}], "}"}]}], "]"}], ";", "\[IndentingNewLine]", "sum"}]}], "]"}]}], ";"}]], "Input", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }, Open ]], Cell["\<\ 4. In Problem 3, which variables are local and which variables are global?\ \>", "Subsection", CellChangeTimes->{{3.411306885694175*^9, 3.4113068875215*^9}}, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }, Open ]], Cell[CellGroupData[{ Cell["Functions and Expressions (10 points)", "Section", CellChangeTimes->{{3.411306313699053*^9, 3.411306320982111*^9}, { 3.411310859366459*^9, 3.411310860943942*^9}}, FontFamily->"Times", FontVariations->{"CompatibilityType"->0}, FontColor->GrayLevel[0]], Cell[TextData[{ "1. Explain in your own words the difference in ", StyleBox["Mathematica", FontSlant->"Italic"], " between a function and an expression.\n" }], "Subsection", CellChangeTimes->{{3.392651954872747*^9, 3.392652027218834*^9}, { 3.3926520601791067`*^9, 3.392652063131028*^9}}, FontWeight->"Plain", FontColor->GrayLevel[0]], Cell["\<\ 2. Give 3 examples to illustrate the difference between a function and an \ expression. \ \>", "Subsection", CellChangeTimes->{{3.392651954872747*^9, 3.392652027218834*^9}, 3.392652064344593*^9}, FontWeight->"Plain", FontColor->GrayLevel[0]], Cell[TextData[{ "3. Give 3 examples of procedures in ", StyleBox["Mathematica", FontSlant->"Italic"], " that will work only if you pass functions, but will not work correctly if \ you pass expressions. (Hint: Use the Document Center.)" }], "Subsection", CellChangeTimes->{{3.392651954872747*^9, 3.392652027218834*^9}, { 3.392652074033449*^9, 3.392652131205205*^9}, {3.392652241776976*^9, 3.3926522494410467`*^9}, 3.392740416965523*^9}, FontWeight->"Plain", FontColor->GrayLevel[0]] }, Open ]], Cell[CellGroupData[{ Cell["Prime Numbers (16 Points)", "Section", FontColor->GrayLevel[0]], Cell[TextData[{ "1. Let \[Pi][", StyleBox["n", FontSlant->"Italic"], "] denote the number of prime numbers less than or equal to ", StyleBox["n", FontSlant->"Italic"], ". Explain the difference between the function \[Pi][", StyleBox["n", FontSlant->"Italic"], "] and the ", StyleBox["Mathematica", FontSlant->"Italic"], " function Prime[", StyleBox["n", FontSlant->"Italic"], "]. Give examples to illustrate the difference. " }], "Text"], Cell[TextData[{ "2. Use the Documentation Center to find a built in ", StyleBox["Mathematica", FontSlant->"Italic"], " function that computes \[Pi][", StyleBox["n", FontSlant->"Italic"], "]. What is the name of this built in ", StyleBox["Mathematica", FontSlant->"Italic"], " function?" }], "Text", CellChangeTimes->{{3.4113054964505577`*^9, 3.411305497450097*^9}, { 3.4113069284818907`*^9, 3.4113069367370453`*^9}}], Cell[TextData[{ "3. Plot the ratio ", Cell[BoxData[ FormBox[ FractionBox["p", RowBox[{"\[Pi]", "[", StyleBox["p", FontSlant->"Italic"], "]"}]], TraditionalForm]]], " between the ", StyleBox["p", FontSlant->"Italic"], " and the number of primes less than or equal to ", StyleBox["p,", FontSlant->"Italic"], " for all integers ", StyleBox["p", FontSlant->"Italic"], " between 2 and ", StyleBox["n. ", FontSlant->"Italic"], StyleBox["Consider bot", FontVariations->{"CompatibilityType"->0}], "h small ", StyleBox["(<100)", FontSlant->"Italic"], " and large ", StyleBox["(>10,000)", FontSlant->"Italic"], " values of ", StyleBox["n", FontSlant->"Italic"], ". Join the points on your plot with straight lines. What do you observe?" }], "Text"], Cell[TextData[{ "4. Plot the logarithm of ", StyleBox["p", FontSlant->"Italic"], " for all integers ", StyleBox["p", FontSlant->"Italic"], " between 2 and ", StyleBox["n", FontSlant->"Italic"], StyleBox[".", FontVariations->{"CompatibilityType"->0}], " ", StyleBox["Consider bot", FontVariations->{"CompatibilityType"->0}], "h small and large values of ", StyleBox["n", FontSlant->"Italic"], ". Join the points on your plot with straight lines." }], "Text"], Cell["\<\ 5. Plot the functions in parts 3 and 4 in the same graph. What do you \ observe?\ \>", "Text"], Cell[TextData[{ "6. The built in ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["LogIntegral", FontSlant->"Italic"], " computes the integral of ", Cell[BoxData[ FormBox[ FractionBox["1", RowBox[{"Log", "[", StyleBox["t", FontSlant->"Italic"], "]"}]], TraditionalForm]]], " between 0 and ", StyleBox["x", FontSlant->"Italic"], ". Using the built in function ", StyleBox["LogIntegral, ", FontSlant->"Italic"], "plot the ratio ", Cell[BoxData[ FormBox[ FractionBox["p", RowBox[{ SubsuperscriptBox["\[Integral]", "0", "p"], " ", RowBox[{ RowBox[{"1", "/", RowBox[{"Log", "[", "t", "]"}]}], " ", RowBox[{"\[DifferentialD]", "t"}]}]}]], TraditionalForm]]], " between ", StyleBox["p", FontSlant->"Italic"], " and the integral of ", Cell[BoxData[ FormBox[ FractionBox["1", RowBox[{"Log", "[", StyleBox["t", FontSlant->"Italic"], "]"}]], TraditionalForm]]], " from 0 and ", StyleBox["p,", FontSlant->"Italic"], " for all integers ", StyleBox["p", FontSlant->"Italic"], " between 2 and ", StyleBox["n", FontSlant->"Italic"], StyleBox[".", FontVariations->{"CompatibilityType"->0}], " ", StyleBox["Consider bot", FontVariations->{"CompatibilityType"->0}], "h small and large values of ", StyleBox["n", FontSlant->"Italic"], ". Join the points on your plot with straight lines. What do you observe?" }], "Text"], Cell[TextData[{ "7. Plot the functions in parts 3,4,6 in the same graph. ", StyleBox["Consider bot", FontVariations->{"CompatibilityType"->0}], "h small and large values of ", StyleBox["n", FontSlant->"Italic"], ". What do you observe?" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Calculus (16 Points)", "Section", FontColor->GrayLevel[0]], Cell[TextData[{ "1. The derivative of a function ", StyleBox["f(x) ", FontSlant->"Italic"], "at ", StyleBox["x=t", FontSlant->"Italic"], " is, by definition, the slope of the tangent to the curve ", StyleBox["y=f(x)", FontSlant->"Italic"], " at the parameter ", StyleBox["x=t", FontSlant->"Italic"], ". To compute this slope, we approximate the slope of the tangent at ", StyleBox["x=t", FontSlant->"Italic"], " by the slope of the chords to the curve from the point of interest to \ nearby points and then take the limit of these slopes as the nearby points \ approach the given point. " }], "Text", CellChangeTimes->{{3.444570864317618*^9, 3.44457127739795*^9}, 3.4445713174950933`*^9, {3.444571361925699*^9, 3.444571362441904*^9}}], Cell[TextData[{ "Problem: Using the definition of the derivative in terms of a limit, write \ a ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["diff", FontSlant->"Italic"], " that takes as input a function ", StyleBox["f", FontSlant->"Italic"], " and a parameter ", StyleBox["t", FontSlant->"Italic"], " and returns the derivative of the function ", StyleBox["f", FontSlant->"Italic"], " evaluated at the parameter ", StyleBox["t", FontSlant->"Italic"], ". In your definition of ", StyleBox["diff", FontSlant->"Italic"], ", you are NOT permitted to use the ", StyleBox["Mathematica", FontSlant->"Italic"], " functions ", StyleBox["Derivative,", FontSlant->"Italic"], " or ", StyleBox["D, ", FontSlant->"Italic"], "or ", Cell[BoxData[ SubscriptBox["\[PartialD]", "x"]]], StyleBox[",", FontSlant->"Italic"], " nor can you use the notation ", StyleBox["f '.", FontSlant->"Italic"] }], "Text", CellChangeTimes->{{3.444570864317618*^9, 3.44457127739795*^9}, 3.4445713174950933`*^9}], Cell[TextData[{ "2. Using the function ", StyleBox["diff", FontSlant->"Italic"], " you defined in part 1, compute the following:\n\ta. the derivative of ", StyleBox["sin(x)", FontSlant->"Italic"], " with respect to ", StyleBox["x.", FontSlant->"Italic"], "\n\tb. the derivative of the function ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " at ", StyleBox["x=", FontSlant->"Italic"], "1", StyleBox[".\n\t ", FontSlant->"Italic"], "c. the derivative of ", StyleBox["ln(x)", FontSlant->"Italic"], " at ", StyleBox["x=0.\n\t ", FontSlant->"Italic"], "d. the derivative of ", Cell[BoxData[ FormBox[ SuperscriptBox["e", "x"], TraditionalForm]]], " at ", StyleBox["x=t.\n", FontSlant->"Italic"], "In each case verify that your answer is correct by using the built in ", StyleBox["Mathematica", FontSlant->"Italic"], " function for differentiation." }], "Text"], Cell[TextData[{ "3. The definite integral from ", StyleBox["x=a", FontSlant->"Italic"], " to ", StyleBox["x=b", FontSlant->"Italic"], " of a function ", StyleBox["f(x)", FontSlant->"Italic"], " is, by definition, the area between the ", StyleBox["x", FontSlant->"Italic"], "-axis and the curve ", StyleBox["y=f(x)", FontSlant->"Italic"], " for", StyleBox[" a\[LessEqual]x\[LessEqual]b.", FontSlant->"Italic"], " To compute this integral, we approximate the area under the curve by a \ collection of rectangles, and then take the limit of the sum of the area of \ these rectangles as the number of rectangles approaches infinity. " }], "Text", CellChangeTimes->{{3.4117554950838842`*^9, 3.411755549437751*^9}, { 3.411755643748188*^9, 3.4117556770244417`*^9}, {3.444570718013629*^9, 3.444570729131076*^9}}], Cell[TextData[{ "Problem: Write a function ", StyleBox["integral ", FontSlant->"Italic"], "that takes as input a function ", StyleBox["f", FontSlant->"Italic"], " and computes the limit of the sum of the area of the rectangles that \ approximate the area between the ", StyleBox["x", FontSlant->"Italic"], "-axis and the curve ", StyleBox["y=f(x)", FontSlant->"Italic"], " for 0\[LessEqual]x\[LessEqual]1. In your definition of ", StyleBox["integral", FontSlant->"Italic"], ", you are NOT permitted to use the ", StyleBox["Mathematica", FontSlant->"Italic"], " function Integrate or the symbol \[Integral]", StyleBox[". ", FontSlant->"Italic"], " You must compute the limit of a sum. (Hint: Look up Sum in the \ Documentation Center.)" }], "Text", CellChangeTimes->{{3.4117554950838842`*^9, 3.411755549437751*^9}, { 3.411755643748188*^9, 3.4117556770244417`*^9}, {3.444570718013629*^9, 3.444570735387311*^9}}], Cell[TextData[{ "4. Using the function ", StyleBox["integral", FontSlant->"Italic"], " you defined in part 3, compute the following:\n\ta. the area between the \ ", StyleBox["x", FontSlant->"Italic"], "-axis and the curve ", StyleBox["y=", FontSlant->"Italic"], Cell[BoxData[ FormBox["1", TraditionalForm]], FontSlant->"Italic"], " for ", StyleBox["0"Italic"], StyleBox[".\n\t ", FontSlant->"Italic"], "b. the area between the ", StyleBox["x", FontSlant->"Italic"], "-axis and the curve ", StyleBox["y=", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox["x", "3"], TraditionalForm]], FontSlant->"Italic"], " for ", StyleBox["0"Italic"], StyleBox[".\n\t ", FontSlant->"Italic"], "c. the area between the ", StyleBox["x", FontSlant->"Italic"], "-axis and the curve ", StyleBox["y=", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox["e", "x"], TraditionalForm]], FontSlant->"Italic"], " for ", StyleBox["0"Italic"], StyleBox[".\n\t ", FontSlant->"Italic"], "d. the area between the ", StyleBox["x", FontSlant->"Italic"], "-axis and the curve ", StyleBox["y=", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{"sin", "(", RowBox[{"\[Pi]", " ", "x"}], ")"}], TraditionalForm]], FontSlant->"Italic"], " for ", StyleBox["0"Italic"], StyleBox[".\n", FontSlant->"Italic"], "In each case verify that your answer is correct by using the built in ", StyleBox["Mathematica", FontSlant->"Italic"], " function for integration. 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Cell[BoxData[ FormBox[ SuperscriptBox["a", "2"], TraditionalForm]], FontSlant->"Italic"], StyleBox["+ ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox["b", "2"], TraditionalForm]], FontSlant->"Italic"], StyleBox["\[Dash] 2abCos(C)", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ "Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to derive the Law of Cosines. 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