(* 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[ 33342, 1104] NotebookOptionsPosition[ 26779, 915] NotebookOutlinePosition[ 29639, 987] CellTagsIndexPosition[ 29596, 984] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "Solutions for a Sample Calculus Exam using ", StyleBox["Mathematica", FontSlant->"Italic"] }], "Title"], Cell[CellGroupData[{ Cell[TextData[{ "Evaluate ", Cell[BoxData[ FormBox[ RowBox[{"\[Integral]", RowBox[{ FractionBox["1", SqrtBox[ RowBox[{ RowBox[{"25", SuperscriptBox["x", "2"]}], "+", "36"}]]], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]]], "." }], "ProblemItem"], Cell[CellGroupData[{ Cell["Solution", "Solution"], Cell["Calculate the indefinite integral:", "Text"], Cell[BoxData[ FormBox[ RowBox[{"int1", " ", "=", " ", RowBox[{"\[Integral]", RowBox[{ FractionBox["1", SqrtBox[ RowBox[{ RowBox[{"25", " ", SuperscriptBox["x", "2"]}], "+", "36"}]]], RowBox[{"\[DifferentialD]", "x"}]}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.397916987268715*^9, 3.397917004956442*^9}, { 3.397917492384556*^9, 3.3979175045878367`*^9}, {3.3980759162258973`*^9, 3.3980759181790347`*^9}}, FontSize->14], Cell[TextData[{ "Verify the answer by differentiation, using ", ButtonBox["D:", BaseStyle->"Link", ButtonData->"paclet:ref/D"] }], "Text"], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"D", "[", RowBox[{"int1", ",", " ", "x"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.3980759197102947`*^9, 3.3980759423823147`*^9}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Evaluate ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "0"}]], "\[ThinSpace]", FractionBox[ RowBox[{ RowBox[{"x", " ", RowBox[{"cos", "(", RowBox[{"3", " ", "x"}], ")"}]}], "-", RowBox[{"x", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["x", "2"]}]]}]}], RowBox[{ SuperscriptBox["sin", "3"], "(", "x", ")"}]]}], TraditionalForm]]], "." }], "ProblemItem"], Cell[CellGroupData[{ Cell["Solution:", "Solution", CellChangeTimes->{{3.398076840091185*^9, 3.3980768421536984`*^9}, 3.3980771532181892`*^9}], Cell["Find the required limit:", "Text"], Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "0"}]], "\[ThinSpace]", FractionBox[ RowBox[{ RowBox[{"x", " ", RowBox[{"cos", "(", RowBox[{"3", " ", "x"}], ")"}]}], "-", RowBox[{"x", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["x", "2"]}]]}]}], RowBox[{ SuperscriptBox["sin", "3"], "(", "x", ")"}]]}], TraditionalForm]], "Input", CellChangeTimes->{{3.397917086785614*^9, 3.3979171162859917`*^9}, { 3.3980750359858885`*^9, 3.3980750693767276`*^9}}, FontSize->14], Cell[TextData[{ "This problem can also be done using ", ButtonBox["Series", BaseStyle->"Link", ButtonData->"paclet:ref/Series"], ":" }], "Text", CellChangeTimes->{{3.3981817247723722`*^9, 3.398181731959918*^9}, { 3.398343869747138*^9, 3.398343877419062*^9}, {3.3983444469695826`*^9, 3.3983444480945897`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"Normal", "[", RowBox[{"Series", "[", RowBox[{ FractionBox[ RowBox[{ RowBox[{"x", " ", RowBox[{"cos", "(", RowBox[{"3", " ", "x"}], ")"}]}], "-", RowBox[{"x", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["x", "2"]}]]}]}], RowBox[{ SuperscriptBox["sin", "3"], "(", "x", ")"}]], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "0"}], "}"}]}], "]"}], "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.398010028063816*^9, 3.398010034329481*^9}, { 3.398075073001751*^9, 3.3980750947362647`*^9}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Find all the roots of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "4"], "+", RowBox[{"64", " ", "\[ImaginaryI]", " ", "x"}]}], "\[LongEqual]", "0"}], TraditionalForm]]], " in Cartesian form." }], "ProblemItem"], Cell[CellGroupData[{ Cell["Solution:", "Solution", CellChangeTimes->{{3.398076840091185*^9, 3.3980768421536984`*^9}, 3.3980771532181892`*^9}], Cell[TextData[{ "Use ", ButtonBox["Roots ", BaseStyle->"Link", ButtonData->"paclet:ref/Roots"], "to compute the roots of the equation:" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{"r", "=", RowBox[{"Roots", "[", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "4"], "+", RowBox[{"64", " ", "\[ImaginaryI]", " ", "x"}]}], "\[LongEqual]", "0"}], ",", "x"}], "]"}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.398181789835288*^9, 3.398181799444725*^9}}, FontSize->14], Cell[TextData[{ "Convert to Cartesian form using ", ButtonBox["ComplexExpand", BaseStyle->"Link", ButtonData->"paclet:ref/ComplexExpand"], ":" }], "Text", CellChangeTimes->{{3.3981817530069275`*^9, 3.398181768132025*^9}, { 3.398181811444802*^9, 3.3981818252886405`*^9}, {3.3983444800947943`*^9, 3.3983444800947943`*^9}, {3.398344563970331*^9, 3.398344570485998*^9}}], Cell[BoxData[ FormBox[ RowBox[{"ComplexExpand", "[", "r", "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.3983445283919783`*^9, 3.398344544907709*^9}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Evaluate ", Cell[BoxData[ FormBox[ RowBox[{"\[Integral]", RowBox[{ RowBox[{"cos", "(", RowBox[{"log", "(", RowBox[{"3", " ", "t"}], ")"}], ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}], TraditionalForm]]], "." }], "ProblemItem"], Cell[CellGroupData[{ Cell["Solution:", "Solution", CellChangeTimes->{{3.398076840091185*^9, 3.3980768421536984`*^9}, 3.3980771532181892`*^9}], Cell["Calculate the indefinite integral:", "Text"], Cell[BoxData[ FormBox[ RowBox[{"int2", "=", " ", RowBox[{"\[Integral]", RowBox[{ RowBox[{"cos", "(", RowBox[{"log", "(", RowBox[{"3", " ", "t"}], ")"}], ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.39807588302256*^9, 3.398075886522582*^9}}, FontSize->14], Cell["Verify the answer by differentiation:", "Text", CellChangeTimes->{{3.398181664037608*^9, 3.3981816765064383`*^9}, { 3.398344020513728*^9, 3.398344025685636*^9}}], Cell[BoxData[ RowBox[{"D", "[", RowBox[{"int2", ",", " ", "t"}], "]"}]], "Input", CellChangeTimes->{{3.3980759051320763`*^9, 3.3980759079914694`*^9}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Is the integral ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ FractionBox[ RowBox[{"cos", "(", "x", ")"}], SuperscriptBox["x", "3"]], RowBox[{"\[DifferentialD]", "x", " "}]}]}], TraditionalForm]]], "convergent or divergent?" }], "ProblemItem"], Cell[CellGroupData[{ Cell["Solution:", "Solution", CellChangeTimes->{{3.398076840091185*^9, 3.3980768421536984`*^9}, 3.3980771532181892`*^9}], Cell[TextData[{ "The message from ", Cell[BoxData[ ButtonBox["Integrate", BaseStyle->"Link", ButtonData->"paclet:ref/Integrate"]], "InlineFormula"], " indicates that the integral is divergent on {0, \[Infinity]}." }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ FractionBox[ RowBox[{"cos", "(", "x", ")"}], SuperscriptBox["x", "3"]], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.3979170111283956`*^9, 3.3979170247848206`*^9}, { 3.3979175135410767`*^9, 3.397917514384837*^9}, {3.3980097463901377`*^9, 3.3980097656715117`*^9}}, FontSize->14], Cell["\<\ As seen below, the integral is divergent on {0, 1} but it converges on {1, \ Infinity}.\ \>", "Text", CellChangeTimes->{{3.3980762482748976`*^9, 3.398076276571954*^9}}], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[{ FractionBox[ RowBox[{"cos", "(", "x", ")"}], SuperscriptBox["x", "3"]], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.398009774624694*^9, 3.39800977482782*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"int3", "=", RowBox[{ SubsuperscriptBox["\[Integral]", "1", "\[Infinity]"], RowBox[{ FractionBox[ RowBox[{"cos", "(", "x", ")"}], SuperscriptBox["x", "3"]], RowBox[{"\[DifferentialD]", "x"}]}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.3980097824372435`*^9, 3.39800978414038*^9}, { 3.398430167927616*^9, 3.398430171505764*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"N", "[", "int3", "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.398075398347583*^9, 3.3980754000663443`*^9}, { 3.3984301753182883`*^9, 3.3984301769589243`*^9}}, FontSize->14], Cell[TextData[{ "Use ", ButtonBox["NIntegrate ", BaseStyle->"Link", ButtonData->"paclet:ref/NIntegrate"], "to verify the answer numerically:" }], "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.398182020180513*^9}, {3.3983440942173247`*^9, 3.39834411148306*^9}, {3.3983441679209213`*^9, 3.3983441893585587`*^9}, { 3.398344223374401*^9, 3.398344230655698*^9}}], Cell[BoxData[ FormBox[ RowBox[{"NIntegrate", "[", RowBox[{ FractionBox[ RowBox[{"cos", "(", "x", ")"}], SuperscriptBox["x", "3"]], ",", RowBox[{"{", RowBox[{"x", ",", "1", ",", "\[Infinity]"}], "}"}]}], "]"}], TraditionalForm]], "Input", CellChangeTimes->{3.398075407128889*^9}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Find the arclength of the curve given by ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{ SuperscriptBox["cos", RowBox[{"-", "1"}]], "(", "t", ")"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", RowBox[{"log", "(", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["t", "2"]}]], ")"}]}], TraditionalForm]]], " between ", Cell[BoxData[ FormBox[ RowBox[{"t", "=", "0"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"t", "=", RowBox[{"3", "/", "4"}]}], TraditionalForm]]], ":" }], "ProblemItem"], Cell[CellGroupData[{ Cell["Solution:", "Solution", CellChangeTimes->{{3.398076840091185*^9, 3.3980768421536984`*^9}, 3.3980771532181892`*^9}], Cell["Set up the parametric equations for the curve:", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"x", "(", "t_", ")"}], ":=", RowBox[{ SuperscriptBox["cos", RowBox[{"-", "1"}]], "(", "t", ")"}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.398008607132847*^9, 3.398008616039154*^9}, 3.3980755480829163`*^9, 3.398075595692596*^9, {3.3980774629545465`*^9, 3.3980774636889257`*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"y", "(", "t_", ")"}], ":=", RowBox[{"log", "(", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["t", "2"]}]], ")"}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.3980086186954207`*^9, 3.398008635945531*^9}, { 3.398075520629616*^9, 3.398075599130118*^9}}, FontSize->14], Cell["Compute the arc length:", "Text"], Cell[BoxData[ FormBox[ RowBox[{"int4", "=", " ", RowBox[{ SubsuperscriptBox["\[Integral]", "0", FractionBox["3", "4"]], RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "2"], "+", SuperscriptBox[ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "2"]}]], RowBox[{"\[DifferentialD]", "t"}]}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.398008718914812*^9, 3.398008757743186*^9}, { 3.398075610239564*^9, 3.3980756109739437`*^9}, {3.3984301820058312`*^9, 3.398430184708974*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"N", "[", "int4", "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.3980087624307156`*^9, 3.398008763711974*^9}, { 3.3984305175079784`*^9, 3.39843051922674*^9}}, FontSize->14], Cell["Verify the answer using numerical integration:", "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.398182020180513*^9}, {3.398344053623315*^9, 3.3983440639202557`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"NIntegrate", "[", RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "2"], "+", SuperscriptBox[ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "2"]}]], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", FractionBox["3", "4"]}], "}"}]}], "]"}], TraditionalForm]], "Input", CellChangeTimes->{ 3.3980087808058333`*^9, {3.3980756163958535`*^9, 3.3980756189114943`*^9}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Does the infinite series ", Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "\[Infinity]"], FractionBox[ RowBox[{ RowBox[{"sin", "(", "n", ")"}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "n"}]]}], SuperscriptBox["n", RowBox[{"13", "/", "12"}]]]}], TraditionalForm]]], " converge or diverge?" }], "ProblemItem"], Cell[CellGroupData[{ Cell["Solution:", "Solution", CellChangeTimes->{{3.398076840091185*^9, 3.3980768421536984`*^9}, 3.3980771532181892`*^9}], Cell["\<\ An answer is returned without a message indicating that the series is \ convergent:\ \>", "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.3981820493525743`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"sum1", "=", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "\[Infinity]"], FractionBox[ RowBox[{ RowBox[{"sin", "(", "n", ")"}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "n"}]]}], SuperscriptBox["n", RowBox[{"5", "/", "4"}]]]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.397917284538145*^9, 3.397917308819706*^9}, { 3.3979173426795144`*^9, 3.397917353726531*^9}, {3.398009648889514*^9, 3.398009678030326*^9}, {3.3980776939403996`*^9, 3.3980777098623767`*^9}, { 3.3984281106644497`*^9, 3.398428135773986*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"Chop", "[", RowBox[{"N", "[", "sum1", "]"}], "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.397917313288513*^9, 3.3979173146010303`*^9}, { 3.398428139227133*^9, 3.3984281403990154`*^9}, {3.3984281750086117`*^9, 3.3984281786023846`*^9}}, FontSize->14], Cell["\<\ Sum the first 10000 terms of the series to verify the answer:\ \>", "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.3981820753371153`*^9}, {3.398182544558869*^9, 3.3981825447463703`*^9}}], Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1."}], "10000"], FractionBox[ RowBox[{ RowBox[{"sin", "(", "n", ")"}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "n"}]]}], SuperscriptBox["n", RowBox[{"5", "/", "4"}]]]}], TraditionalForm]], "Input", CellChangeTimes->{{3.3979173226480083`*^9, 3.397917363523531*^9}, { 3.3980777150655346`*^9, 3.398077754300161*^9}, {3.3981824708708973`*^9, 3.398182481230338*^9}, {3.398182514027423*^9, 3.3981825384963303`*^9}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Find the general solution of ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{"x", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "+", RowBox[{"3", RowBox[{"y", "(", "x", ")"}]}]}], "\[LongEqual]", RowBox[{"cos", "(", "x", ")"}]}]}], TraditionalForm]]], ":" }], "ProblemItem"], Cell[CellGroupData[{ Cell["Solution:", "Solution", CellChangeTimes->{{3.398076840091185*^9, 3.3980768421536984`*^9}, 3.3980771532181892`*^9}], Cell[TextData[{ "Use ", ButtonBox["DSolve", BaseStyle->"Link", ButtonData->"paclet:ref/DSolve"], " to solve the ODE:" }], "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.3981820493525743`*^9}, {3.3981821049935555`*^9, 3.3981821140873632`*^9}, {3.3983442728747177`*^9, 3.3983442728747177`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"Clear", "[", RowBox[{"x", ",", "y"}], "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.3981824131049023`*^9, 3.398182415839295*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"ode1", "=", RowBox[{ RowBox[{ RowBox[{"x", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], " ", "+", " ", RowBox[{"3", " ", RowBox[{"y", "(", "x", ")"}]}]}], "\[LongEqual]", RowBox[{"cos", "(", "x", ")"}]}]}], ";"}], TraditionalForm]], "Input", CellChangeTimes->{{3.398076700402791*^9, 3.398076716777896*^9}, { 3.3981655923722506`*^9, 3.3981655995285463`*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"sol1", "=", RowBox[{"DSolve", "[", RowBox[{"ode1", ",", "y", ",", "x"}], "]"}]}], TraditionalForm]], "Input",\ CellChangeTimes->{{3.397917201630834*^9, 3.397917220537326*^9}, { 3.398076469307562*^9, 3.3980764800888815`*^9}, {3.398076695980888*^9, 3.3980767179654036`*^9}, 3.3980781666309247`*^9}, FontSize->14], Cell["Verify the answer by substitution:", "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.3981820493525743`*^9}, {3.3981821049935555`*^9, 3.3981821343531184`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"Simplify", "[", RowBox[{"ode1", "/.", "\[InvisibleSpace]", RowBox[{"sol1", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.398076713559125*^9, 3.3980767349186373`*^9}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Find the general solution of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]\[Prime]", MultilineFunction->None], "(", "x", ")"}], "-", RowBox[{"4", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "+", RowBox[{"3", " ", RowBox[{"y", "(", "x", ")"}]}]}], "\[LongEqual]", SuperscriptBox["x", "3"]}], TraditionalForm]]], "." }], "ProblemItem"], Cell[CellGroupData[{ Cell["Solution", "Solution"], Cell["Solve the ODE:", "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.3981820493525743`*^9}, {3.3981821049935555`*^9, 3.3981821140873632`*^9}, {3.3983442502026978`*^9, 3.3983442507183266`*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"ode2", "=", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]\[Prime]", MultilineFunction->None], "(", "x", ")"}], "-", RowBox[{"4", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "+", RowBox[{"3", " ", RowBox[{"y", "(", "x", ")"}]}]}], "\[LongEqual]", SuperscriptBox["x", "3"]}]}], ";"}], TraditionalForm]], "Input", CellChangeTimes->{{3.3980767444655733`*^9, 3.398076751153116*^9}, { 3.3981656086223545`*^9, 3.3981656172317843`*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"sol2", "=", RowBox[{"DSolve", "[", RowBox[{"ode2", ",", "y", ",", "x"}], "]"}]}], TraditionalForm]], "Input",\ CellChangeTimes->{{3.3979172417719727`*^9, 3.3979172564909115`*^9}, { 3.3980766596212797`*^9, 3.398076687840211*^9}, {3.3980767407936745`*^9, 3.3980767623094373`*^9}}, FontSize->14], Cell["Verify the answer by substitution:", "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.3981820493525743`*^9}, {3.3981821049935555`*^9, 3.3981821343531184`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"Simplify", "[", RowBox[{"ode2", "/.", "\[InvisibleSpace]", RowBox[{"sol2", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.398076713559125*^9, 3.3980767349186373`*^9}, { 3.398076771184494*^9, 3.398076773481384*^9}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Approximate the integral ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", FractionBox["1", "100"]], RowBox[{ RowBox[{"cos", "(", SqrtBox["t"], ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}], TraditionalForm]]], " with an error of less than ", Cell[BoxData[ FormBox[ FractionBox["1", SuperscriptBox["10", "5"]], TraditionalForm]]], "." }], "ProblemItem"], Cell[CellGroupData[{ Cell["Solution:", "Solution", CellChangeTimes->{{3.398076840091185*^9, 3.3980768421536984`*^9}, 3.3980771532181892`*^9}], Cell["Use direct numerical integration:", "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.3981820493525743`*^9}, {3.3981821049935555`*^9, 3.3981821140873632`*^9}, {3.3981822324943714`*^9, 3.398182245822582*^9}, { 3.3983443406407766`*^9, 3.3983443510314684`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"NIntegrate", "[", RowBox[{ RowBox[{"cos", "(", SqrtBox["t"], ")"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", FractionBox["1", "100"]}], "}"}]}], "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.398095572023569*^9, 3.398095604008148*^9}}, FontSize->14], Cell["\<\ Alternatively, use the first three terms in the series for the integrand to \ calculate the required approximation.\ \>", "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.3981820493525743`*^9}, {3.3981821049935555`*^9, 3.3981821140873632`*^9}, {3.3981822324943714`*^9, 3.398182281775937*^9}, { 3.3983443603284025`*^9, 3.3983443674221983`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"ser", " ", "=", RowBox[{"Normal", "[", RowBox[{"Series", "[", RowBox[{ RowBox[{"Cos", "(", SqrtBox["t"], ")"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", "2"}], "}"}]}], "]"}], "]"}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.3980783739760017`*^9, 3.398078389757353*^9}, { 3.398078456039027*^9, 3.3980784576484127`*^9}, {3.398078700759343*^9, 3.3980787018687253`*^9}, {3.3980822279069166`*^9, 3.398082228078793*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"Table", "[", RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "0", FractionBox["1", "100"]], RowBox[{ RowBox[{"ser", "\[LeftDoubleBracket]", "i", "\[RightDoubleBracket]"}], RowBox[{"\[DifferentialD]", "t"}]}]}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"Length", "[", "ser", "]"}]}], "}"}]}], "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.39807852994575*^9, 3.3980785484302435`*^9}, { 3.3980786286338816`*^9, 3.398078663384104*^9}, {3.398078705040621*^9, 3.3980787057906256`*^9}, 3.3980787952755733`*^9, {3.3980788506040525`*^9, 3.3980788516196837`*^9}, {3.398428370337987*^9, 3.3984283883849773`*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"int5", "=", RowBox[{ SubsuperscriptBox["\[Integral]", "0", FractionBox["1", "100"]], RowBox[{"ser", RowBox[{"\[DifferentialD]", "t"}]}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.3980787150250597`*^9, 3.3980787157906895`*^9}, 3.398078864901019*^9, {3.3984301995059433`*^9, 3.3984302015372066`*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"N", "[", "int5", "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.3980787184157066`*^9, 3.398078720134467*^9}, { 3.3984302053497305`*^9, 3.398430206974741*^9}}, FontSize->14], Cell["\<\ This example can also be done using exact, symbolic integration.\ \>", "Text", CellChangeTimes->{{3.398181950711318*^9, 3.398181982086519*^9}, { 3.3981820128367157`*^9, 3.3981820493525743`*^9}, {3.3981821049935555`*^9, 3.3981821140873632`*^9}, {3.3981822324943714`*^9, 3.398182281775937*^9}, { 3.398182317151163*^9, 3.39818233693254*^9}, {3.3983443854066887`*^9, 3.398344412719363*^9}}], Cell[BoxData[ FormBox[ RowBox[{"int6", " ", "=", " ", RowBox[{ SubsuperscriptBox["\[Integral]", "0", FractionBox["1", "100"]], RowBox[{ RowBox[{"cos", "(", SqrtBox["t"], ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.3984283433065634`*^9, 3.3984283463690834`*^9}, { 3.398430210427888*^9, 3.3984302107560153`*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"N", "[", "int6", "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.398078733243926*^9, 3.398078752462799*^9}, { 3.3984283498691053`*^9, 3.398428356916026*^9}, {3.398430213052905*^9, 3.398430213365407*^9}}, FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"Clear", "[", RowBox[{ "x", ",", "y", ",", "t", ",", "ser", ",", " ", "sum1", ",", " ", "int1", ",", " ", "int2", ",", " ", "int3", ",", " ", "int4", ",", " ", "int5", ",", " ", "int6", ",", " ", "r"}], "]"}], TraditionalForm]], "Input", CellChangeTimes->{{3.398182340245061*^9, 3.3981823518388853`*^9}, { 3.39842824510281*^9, 3.3984282467434454`*^9}, {3.398430267428253*^9, 3.3984302804908366`*^9}, {3.4009271729062705`*^9, 3.4009271732187724`*^9}}, FontSize->14] }, Closed]] }, Closed]] }, WindowSize->{1020, 665}, WindowMargins->{{Automatic, 0}, {Automatic, 0}}, Magnification->1.25, FrontEndVersion->"6.0 for Microsoft Windows (32-bit) (June 19, 2007)", StyleDefinitions->Notebook[{ Cell[ CellGroupData[{ Cell[ StyleData["Title"], CellMargins -> {{27, Inherited}, {60, 30}}, MenuPosition -> 10000, FontFamily -> "Times New Roman", FontSize -> 36, FontWeight -> "Bold", FontSlant -> "Plain", FontVariations -> {"StrikeThrough" -> False, "Underline" -> False}], Cell[ StyleData[ StyleDefinitions -> FrontEnd`FileName[{"Book"}, "Textbook.nb", CharacterEncoding -> "WindowsANSI"]]]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["ProblemItem"], ShowCellBracket -> True, ShowGroupOpener -> True, CellMargins -> {{30, 0}, {12, 12}}, CellGroupingRules -> {"SectionGrouping", 30}, CellFrameLabels -> {{ Cell[ TextData[{"Problem ", CounterBox["ExerciseNumber"], ":"}], "Text", CellBaseline -> Baseline, FontWeight -> "Bold", FontColor -> RGBColor[0.380392, 0.431373, 0.705882], CellSize -> {78, Inherited}], None}, {None, None}}, TextAlignment -> Left, CounterIncrements -> "ExerciseNumber", StyleMenuListing -> None, CounterStyleMenuListing -> None, FontFamily -> "Times", FontSize -> 14], Cell[ StyleData["ProblemItem", "Presentation"], FontSize -> 22], Cell[ StyleData["ProblemItem", "SlideShow"], FontSize -> 18], Cell[ StyleData["ProblemItem", "Printout"], CellMargins -> {{Inherited, Inherited}, {Inherited, 0}}, Hyphenation -> True, FontSize -> 10]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Solution"], ShowCellBracket -> True, ShowGroupOpener -> True, CellMargins -> {{48, 0}, {12, 12}}, CellGroupingRules -> {"SectionGrouping", 40}, TextAlignment -> Left, FontFamily -> "Times New Roman", FontSize -> 14, FontColor -> RGBColor[0.576471, 0.584314, 0.596078]], Cell[ StyleData["Solution", "Presentation"], FontSize -> 22], Cell[ StyleData["Solution", "SlideShow"], FontSize -> 18], Cell[ StyleData["Solution", "Printout"], CellMargins -> {{Inherited, Inherited}, {Inherited, 0}}, Hyphenation -> True, FontSize -> 10]}, Open]], Cell[ StyleData["Text"], CellMargins -> {{48, 10}, {3, 5}}, MenuPosition -> 10000, FontSize -> 14]}, Visible -> False, FrontEndVersion -> "6.0 for Microsoft Windows (32-bit) (June 19, 2007)", StyleDefinitions -> "PrivateStylesheetFormatting.nb"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 124, 4, 218, "Title"], Cell[CellGroupData[{ Cell[717, 29, 313, 13, 75, "ProblemItem"], Cell[CellGroupData[{ Cell[1055, 46, 28, 0, 50, "Solution"], Cell[1086, 48, 50, 0, 30, "Text"], Cell[1139, 50, 491, 14, 89, "Input"], Cell[1633, 66, 144, 5, 30, "Text"], Cell[1780, 73, 206, 5, 39, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[2035, 84, 524, 19, 58, "ProblemItem"], Cell[CellGroupData[{ Cell[2584, 107, 125, 2, 50, "Solution"], Cell[2712, 111, 40, 0, 30, "Text"], Cell[2755, 113, 601, 19, 78, "Input"], Cell[3359, 134, 321, 9, 30, "Text"], Cell[3683, 145, 682, 21, 78, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[4414, 172, 274, 10, 39, "ProblemItem"], Cell[CellGroupData[{ Cell[4713, 186, 125, 2, 50, "Solution"], Cell[4841, 190, 154, 6, 30, "Text"], Cell[4998, 198, 368, 11, 44, "Input"], Cell[5369, 211, 379, 9, 30, "Text"], Cell[5751, 222, 178, 4, 38, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[5978, 232, 283, 11, 43, "ProblemItem"], Cell[CellGroupData[{ Cell[6286, 247, 125, 2, 50, "Solution"], Cell[6414, 251, 50, 0, 30, "Text"], Cell[6467, 253, 346, 10, 60, "Input"], Cell[6816, 265, 170, 2, 30, "Text"], Cell[6989, 269, 170, 4, 39, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[7208, 279, 353, 12, 48, "ProblemItem"], Cell[CellGroupData[{ Cell[7586, 295, 125, 2, 50, "Solution"], Cell[7714, 299, 236, 7, 31, "Text"], Cell[7953, 308, 452, 12, 67, "Input"], Cell[8408, 322, 179, 4, 30, "Text"], Cell[8590, 328, 333, 10, 67, "Input"], Cell[8926, 340, 427, 12, 67, "Input"], Cell[9356, 354, 222, 5, 38, "Input"], Cell[9581, 361, 426, 10, 30, "Text"], Cell[10010, 373, 330, 11, 67, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[10389, 390, 624, 26, 60, "ProblemItem"], Cell[CellGroupData[{ Cell[11038, 420, 125, 2, 50, "Solution"], Cell[11166, 424, 62, 0, 30, "Text"], Cell[11231, 426, 372, 10, 40, "Input"], Cell[11606, 438, 340, 10, 63, "Input"], Cell[11949, 450, 39, 0, 30, "Text"], Cell[11991, 452, 726, 21, 69, "Input"], Cell[12720, 475, 219, 5, 38, "Input"], Cell[12942, 482, 230, 3, 30, "Text"], Cell[13175, 487, 618, 19, 65, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[13842, 512, 439, 15, 49, "ProblemItem"], Cell[CellGroupData[{ Cell[14306, 531, 125, 2, 50, "Solution"], Cell[14434, 535, 226, 5, 30, "Text"], Cell[14663, 542, 660, 17, 73, "Input"], Cell[15326, 561, 305, 7, 38, "Input"], Cell[15634, 570, 255, 5, 30, "Text"], Cell[15892, 577, 570, 15, 77, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[16511, 598, 423, 15, 39, "ProblemItem"], Cell[CellGroupData[{ Cell[16959, 617, 125, 2, 50, "Solution"], Cell[17087, 621, 358, 9, 30, "Text"], Cell[17448, 632, 194, 5, 38, "Input"], Cell[17645, 639, 530, 15, 38, "Input"], Cell[18178, 656, 364, 9, 38, "Input"], Cell[18545, 667, 222, 3, 30, "Text"], Cell[18770, 672, 298, 7, 38, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[19117, 685, 510, 17, 39, "ProblemItem"], Cell[CellGroupData[{ Cell[19652, 706, 28, 0, 50, "Solution"], Cell[19683, 708, 252, 3, 30, "Text"], Cell[19938, 713, 635, 18, 40, "Input"], Cell[20576, 733, 344, 9, 38, "Input"], Cell[20923, 744, 222, 3, 30, "Text"], Cell[21148, 749, 347, 8, 38, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[21544, 763, 446, 17, 58, "ProblemItem"], Cell[CellGroupData[{ Cell[22015, 784, 125, 2, 50, "Solution"], Cell[22143, 788, 322, 4, 30, "Text"], Cell[22468, 794, 332, 10, 65, "Input"], Cell[22803, 806, 412, 7, 54, "Text"], Cell[23218, 815, 547, 14, 51, "Input"], Cell[23768, 831, 739, 18, 69, "Input"], Cell[24510, 851, 389, 10, 69, "Input"], Cell[24902, 863, 220, 5, 38, "Input"], Cell[25125, 870, 407, 7, 30, "Text"], Cell[25535, 879, 421, 12, 69, "Input"], Cell[25959, 893, 267, 6, 38, "Input"], Cell[26229, 901, 522, 10, 38, "Input"] }, Closed]] }, Closed]] } ] *) (* End of internal cache information *)