(* 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[ 22482, 833] NotebookOptionsPosition[ 20058, 746] NotebookOutlinePosition[ 20472, 764] CellTagsIndexPosition[ 20429, 761] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Lesson 6 Answers - Differentiation and Integration", "Section", CellChangeTimes->{{3.461019705701769*^9, 3.461019711482246*^9}, { 3.4610853024185762`*^9, 3.461085309692246*^9}, {3.461268916597555*^9, 3.461268918280714*^9}, {3.4651303528136873`*^9, 3.465130353730649*^9}}], Cell[TextData[{ "1) ", "Find the following derivatives.\n(a) the second derivative of ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SuperscriptBox["sin", "2"], "(", "x", ")"}], "Input"], TraditionalForm]], "Input"], "." }], "Text", CellChangeTimes->{{3.461084570791834*^9, 3.461084578883746*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "^", "2"}], ",", RowBox[{"{", RowBox[{"x", ",", "2"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.418227903639594*^9, 3.418227912786407*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{ RowBox[{"2", " ", SuperscriptBox[ RowBox[{"Cos", "[", "x", "]"}], "2"]}], "-", RowBox[{"2", " ", SuperscriptBox[ RowBox[{"Sin", "[", "x", "]"}], "2"]}]}]], "Output", CellChangeTimes->{3.418227914724745*^9, 3.4610846654633408`*^9}, FontSlant->"Italic"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"(", "b", ")"}], " ", StyleBox[Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["d", "2"], RowBox[{ StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]], StyleBox[" ", FontSlant->"Italic"], "dy"}]], TraditionalForm]], "Input"], "Input"], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"2", "x"}]], RowBox[{"sin", "(", "xy", ")"}]}], TraditionalForm]], "Input"], "Input"]}]], "Text", FontColor->GrayLevel[0]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"E", "^", RowBox[{"(", RowBox[{"2", "x"}], ")"}]}], RowBox[{"Sin", "[", RowBox[{"x", "*", "y"}], "]"}]}], ",", RowBox[{"{", "x", "}"}], ",", RowBox[{"{", "y", "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.418227919883328*^9, 3.418227941771563*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "x"}]], " ", RowBox[{"Cos", "[", RowBox[{"x", " ", "y"}], "]"}]}], "+", RowBox[{"2", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "x"}]], " ", "x", " ", RowBox[{"Cos", "[", RowBox[{"x", " ", "y"}], "]"}]}], "-", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "x"}]], " ", "x", " ", "y", " ", RowBox[{"Sin", "[", RowBox[{"x", " ", "y"}], "]"}]}]}]], "Output", CellChangeTimes->{3.418227942341765*^9, 3.461084667973654*^9}, FontSlant->"Italic"] }, Open ]], Cell[TextData[{ "2) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to compute the ", StyleBox["exact", FontWeight->"Bold"], " values for the integral of the following expressions. Be aware that ", StyleBox["Mathematica", FontSlant->"Italic"], " can find closed form solutions for many, not all integrals.\n(a) ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{" ", RowBox[{"1", "/", SqrtBox[ RowBox[{ RowBox[{"2", "a", " ", "x"}], "+", SuperscriptBox["x", "2"]}]]}]}], "Input"], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.448195852226054*^9, 3.44819585222609*^9}, { 3.448195893203958*^9, 3.448195911037004*^9}, {3.448195966891161*^9, 3.448195982599103*^9}, {3.46108521681492*^9, 3.461085217411747*^9}}, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"2", " ", "a", " ", "x"}], "+", RowBox[{"x", "^", "2"}]}], "]"}]}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.418228103498261*^9, 3.418228135661632*^9}, 3.448196055348044*^9}, FontSlant->"Italic"], Cell[BoxData[ FractionBox[ RowBox[{"2", " ", SqrtBox["x"], " ", SqrtBox[ RowBox[{ RowBox[{"2", " ", "a"}], "+", "x"}]], " ", RowBox[{"Log", "[", RowBox[{ SqrtBox["x"], "+", SqrtBox[ RowBox[{ RowBox[{"2", " ", "a"}], "+", "x"}]]}], "]"}]}], SqrtBox[ RowBox[{"x", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "a"}], "+", "x"}], ")"}]}]]]], "Output", CellChangeTimes->{3.418228136524729*^9, 3.448196057475375*^9}, FontSlant->"Italic"] }, Open ]], Cell[TextData[{ "(b)", Cell[BoxData[ FormBox[ RowBox[{" ", StyleBox[ RowBox[{"1", "/", SqrtBox[ RowBox[{ RowBox[{"2", "a", " ", "x"}], "-", SuperscriptBox["x", "2"]}]]}], "Input"]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.448195852226054*^9, 3.44819585222609*^9}, { 3.448195893203958*^9, 3.448195911037004*^9}, {3.448195986602634*^9, 3.448195989005582*^9}, {3.448196036341547*^9, 3.448196063067308*^9}}, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"2", " ", "a", " ", "x"}], "-", RowBox[{"x", "^", "2"}]}], "]"}]}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.418228103498261*^9, 3.418228135661632*^9}}, FontSlant->"Italic"], Cell[BoxData[ FractionBox[ RowBox[{"2", " ", SqrtBox[ RowBox[{ RowBox[{"2", " ", "a"}], "-", "x"}]], " ", SqrtBox["x"], " ", RowBox[{"ArcTan", "[", FractionBox[ SqrtBox["x"], SqrtBox[ RowBox[{ RowBox[{"2", " ", "a"}], "-", "x"}]]], "]"}]}], SqrtBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", " ", "a"}], "-", "x"}], ")"}], " ", "x"}]]]], "Output", CellChangeTimes->{3.418228136524729*^9}, FontSlant->"Italic"] }, Open ]], Cell[TextData[{ "(c) ", Cell[BoxData[ FormBox[ RowBox[{"1", "/", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["sin", "2"], "(", RowBox[{"a", " ", "x"}], ")"}], RowBox[{ SuperscriptBox["cos", "2"], "(", RowBox[{"a", " ", "x"}], ")"}]}], ")"}]}], TraditionalForm]], "Input"] }], "Text", CellChangeTimes->{{3.418228089414297*^9, 3.418228098509897*^9}, 3.461084965833276*^9}, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Sin", "[", RowBox[{"a", "*", "x"}], "]"}], "^", "2"}], RowBox[{ RowBox[{"Cos", "[", RowBox[{"a", "*", "x"}], "]"}], "^", "2"}]}], ")"}]}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.4182283948606653`*^9, 3.418228421300301*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{"-", FractionBox[ RowBox[{"2", " ", RowBox[{"Cot", "[", RowBox[{"2", " ", "a", " ", "x"}], "]"}]}], "a"]}]], "Output", CellChangeTimes->{3.418228421953095*^9}, FontSlant->"Italic"] }, Open ]], Cell[TextData[{ "Here are some mistakes to avoid: \n1) Forgetting the space between ", StyleBox["a", "Input"], " and ", StyleBox["x", "Input"], ", in which case the functions become much simpler. \n2) Putting exponents \ for ", StyleBox["sin", "Input"], " and ", StyleBox["cos", "Input"], " functions in the incorrect place\n3) In part c), forgetting to put \ parentheses into the denominator. \n\nHere are examples of the incorrect \ answers. 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", FontVariations->{"CompatibilityType"->0}], "Be careful to use the proper ", StyleBox["Mathematica", FontSlant->"Italic"], " functions for ln and ", Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["tan", RowBox[{"-", "1"}]]}], TraditionalForm]]], "(look them up!)." }], "Text", CellChangeTimes->{{3.461084702208061*^9, 3.4610847138696012`*^9}, { 3.4610848374603148`*^9, 3.461084845260159*^9}, 3.46108522919835*^9}], Cell[TextData[{ "(a) ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ RowBox[{"sin", "(", "x", ")"}], " ", SuperscriptBox["e", RowBox[{"-", SuperscriptBox["x", "2"]}]], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]], "Input"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NIntegrate", "[", RowBox[{ RowBox[{ RowBox[{"Sin", "[", "x", "]"}], RowBox[{"Exp", "[", RowBox[{"-", RowBox[{"x", "^", "2"}]}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.418227993486293*^9, 3.418228016626802*^9}}], Cell[BoxData["0.4244363835023353`"], "Output", CellChangeTimes->{3.418228017926881*^9}] }, Open ]], Cell[TextData[{ "(b) ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "1", RowBox[{"\[Pi]", " "}]], RowBox[{ RowBox[{"ln", "(", "x", ")"}], " ", RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", RowBox[{"2", SqrtBox["x"]}], ")"}], RowBox[{"\[DifferentialD]", "x"}]}]}], "Input"], TraditionalForm]]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NIntegrate", "[", RowBox[{ RowBox[{ RowBox[{"Log", "[", "x", "]"}], RowBox[{"ArcTan", "[", RowBox[{"2", RowBox[{"Sqrt", "[", "x", "]"}]}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "1", ",", "Pi"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.418228034979554*^9, 3.4182280798116817`*^9}}], Cell[BoxData["1.8184537239363852`"], "Output", CellChangeTimes->{{3.418228058582076*^9, 3.418228080391178*^9}}] }, Open ]], Cell[TextData[{ "4) Integrate the quantity ", StyleBox[Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"1", "/", RowBox[{"(", RowBox[{ SuperscriptBox["x", "3"], "+", "1"}], ")"}]}], "Input"], TraditionalForm]], "Input"], "Input"], " with respect to ", Cell[BoxData[ FormBox[ StyleBox["x", "Input"], TraditionalForm]]], ". Use the ", StyleBox["D", "Input", FontWeight->"Bold"], " function to differentiate the result you obtained. Show that the result \ you obtain when differentiating is equal to ", StyleBox[Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"1", "/", RowBox[{"(", RowBox[{ SuperscriptBox["x", "3"], "+", "1"}], ")"}]}], "Input"], TraditionalForm]], "Input"], "Input"], " by using ", StyleBox["Simplify", "Input", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], "or a similar function to modify the result of the differentiation." }], "Text", CellChangeTimes->{{3.448195852226054*^9, 3.44819585222609*^9}, { 3.448195893203958*^9, 3.448195911037004*^9}, {3.448195986602634*^9, 3.448195989005582*^9}, {3.448196124335108*^9, 3.448196124818327*^9}, { 3.448196497266223*^9, 3.448196500960291*^9}, {3.461085133062455*^9, 3.461085177827814*^9}, 3.461085232106866*^9, 3.461085490843808*^9, 3.464525213792528*^9}, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{ RowBox[{"x", "^", "3"}], "+", "1"}], ")"}]}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.418228437184041*^9, 3.418228449026379*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"ArcTan", "[", FractionBox[ RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", "x"}]}], SqrtBox["3"]], "]"}], SqrtBox["3"]], "+", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"Log", "[", RowBox[{"1", "+", "x"}], "]"}]}], "-", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"Log", "[", RowBox[{"1", "-", "x", "+", SuperscriptBox["x", "2"]}], "]"}]}]}]], "Output", CellChangeTimes->{3.418228457285659*^9}, FontSlant->"Italic"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{ FractionBox[ RowBox[{"ArcTan", "[", FractionBox[ RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", "x"}]}], SqrtBox["3"]], "]"}], SqrtBox["3"]], "+", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"Log", "[", RowBox[{"1", "+", "x"}], "]"}]}], "-", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"Log", "[", RowBox[{"1", "-", "x", "+", SuperscriptBox["x", "2"]}], "]"}]}]}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.418228459430743*^9, 3.4182284664318037`*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{ FractionBox["1", RowBox[{"3", " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}]], "-", FractionBox[ RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", "x"}]}], RowBox[{"6", " ", RowBox[{"(", RowBox[{"1", "-", "x", "+", SuperscriptBox["x", "2"]}], ")"}]}]], "+", FractionBox["2", RowBox[{"3", " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{ FractionBox["1", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", "x"}]}], ")"}], "2"]}]}], ")"}]}]]}]], "Output", CellChangeTimes->{3.4182284673007812`*^9}, FontSlant->"Italic"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ FractionBox["1", RowBox[{"3", " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}]], "-", FractionBox[ RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", "x"}]}], RowBox[{"6", " ", RowBox[{"(", RowBox[{"1", "-", "x", "+", SuperscriptBox["x", "2"]}], ")"}]}]], "+", FractionBox["2", RowBox[{"3", " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{ FractionBox["1", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", "x"}]}], ")"}], "2"]}]}], ")"}]}]]}], "]"}]], "Input", CellChangeTimes->{{3.4182284692668953`*^9, 3.418228477152275*^9}}, FontSlant->"Italic"], Cell[BoxData[ FractionBox["1", RowBox[{"1", "+", SuperscriptBox["x", "3"]}]]], "Output", CellChangeTimes->{3.418228477723166*^9}, FontSlant->"Italic"] }, Open ]], Cell["\<\ So after the simplification, we are back to the function we started with.\ \>", "Text", CellChangeTimes->{{3.418228481883294*^9, 3.4182284939680433`*^9}}, FontSlant->"Italic"], Cell[TextData[{ "5) Create a table containing the values of", StyleBox[" ", "Input"], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ SuperscriptBox["x", "n"], SuperscriptBox["e", RowBox[{"-", "x"}]], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]], "Input"], "Input"], " for ", StyleBox["n=1,2,...,10", "Input"], ". 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