(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 169538, 4921]*) (*NotebookOutlinePosition[ 170691, 4957]*) (* CellTagsIndexPosition[ 170647, 4953]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["Math 225: Calculus III Assignment 1"], "Subsection", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Name:"], "Subsection", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Section:"], "Subsection", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "\nI affirm that the solutions presented in this assignment are entirely my \ own work."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Signature:"], "Subsection", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Instructions"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This notebook is an introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " and some of its basic functions which we shall be using throughout the \ course. You should look through the notebooks in the Notebook folder (most of \ these are reprinted in the back of the Lecture Notes). ", StyleBox["Mathematica", FontSlant->"Italic"], " reference books are available in the labs. Be sure to check out the ", StyleBox["Help", FontWeight->"Bold"], " menu.\n\nBelow you will find some simple examples of how to execute ", StyleBox["Mathematica", FontSlant->"Italic"], " commands. Each example is followed by a problem for which you must type \ in a solution, mimicking the preceding example. Before you move on to the \ next problem, you should \"clear\" the variables you used in the current \ probem, so ", StyleBox["Mathematica", FontSlant->"Italic"], " won't remember any special values or expression you may have assigned to \ them. To do this use the command,\n", StyleBox[" Clear[x,y,...]\n", FontFamily->"Courier", FontWeight->"Bold"], "listing any variable names you used. If you get unexpected expressions \ back from ", StyleBox["Mathematica", FontSlant->"Italic"], ", try clearing the variable names and starting over. You may need to use \ the command ", StyleBox["Remove[x,y,...]", FontFamily->"Courier", FontWeight->"Bold"], " (a more \"powerful' version of ", StyleBox["Clear[]", FontFamily->"Courier", FontWeight->"Bold"], ") or even save, quit, and restart ", StyleBox["Mathematica ", FontSlant->"Italic"], "to get back to normal.\n\nIf a section of this notebook seems to \ \"disappear\" it may be because a \"cell\" has been \"closed\". A cell is one \ or more lines in a notebook grouped together. The cell is indicated by a bar \ on the right hand side of the cell. To select a cell, click on this bar. To \ open or close a cell double-click on this bar.\n\nBefore you print out the \ assignment, make sure ", StyleBox["Show In/Out Names", FontWeight->"Bold"], " is not checked in the ", StyleBox["Kernel", FontWeight->"Bold"], " menu. It would also be a good idea to close this ", StyleBox["Intructions", FontWeight->"Bold"], " cell and all of the ", StyleBox["Example", FontWeight->"Bold"], " cells (double-click on the bar on the far right) before printing.\n\nThe \ following ", StyleBox["Initialization", FontWeight->"Bold"], " cell should always be executed before you type in any other statements. \ You were asked when you opened the notebook if you wanted to do this. You \ should normally click OK to such a question. If you did not click OK, select \ the cell below by clicking on the bar on the right and then hit the ENTER \ key." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Initialization"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["<{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, InitializationCell->True, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Example 1"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Find the value of", StyleBox[" (2 + 5*Sqrt[3])^3/2 ", FontFamily->"Courier", FontWeight->"Bold"], "to 10 significant digits." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Arithmetic operations are the usual : ", StyleBox["+ - * / ", FontFamily->"Courier", FontWeight->"Bold"], "and parentheses are used to group expressions. Exponents are indicated by \ ", StyleBox["^", FontFamily->"Courier", FontWeight->"Bold"], ". ", StyleBox["Sqrt[x]", FontFamily->"Courier", FontWeight->"Bold"], " stands for the square root, ", StyleBox["x^(1/2)", FontFamily->"Courier", FontWeight->"Bold"], ".\nMultiplication is normally indicated by ", StyleBox["*", FontFamily->"Courier", FontWeight->"Bold"], ", but you can use a SPACE instead, for example,\n", StyleBox["2 x", FontFamily->"Courier", FontWeight->"Bold"], " is the same as ", StyleBox["2*x", FontFamily->"Courier", FontWeight->"Bold"], ". However, be careful not to enter something like ", StyleBox["x2 ", FontFamily->"Courier", FontWeight->"Bold"], "when you mean", StyleBox[" x*2 ", FontFamily->"Courier", FontWeight->"Bold"], "since ", StyleBox["Mathematica", FontSlant->"Italic"], " will read ", StyleBox["x2", FontFamily->"Courier", FontWeight->"Bold"], " as a variable with the name \"x2\". Although ", StyleBox["Mathematica", FontSlant->"Italic"], " is smart enough to recognize ", StyleBox["2x", FontFamily->"Courier", FontWeight->"Bold"], " as ", StyleBox["2*x", FontFamily->"Courier", FontWeight->"Bold"], ", you should get in the habit of always using a space or ", StyleBox["*", FontFamily->"Courier", FontWeight->"Bold"], " for multiplication.\n\nTo solve the problem, first type ", StyleBox["(2 + 5*Sqrt[3])^3/2 ", FontFamily->"Courier", FontWeight->"Bold"], "and the hit the ENTER key (not the RETURN key which will just take you to \ the next line). A substitute for ENTER is SHIFT-RETURN." }], "Text", CellMargins->{{Inherited, 18}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["(2 + 5*Sqrt[3])^3/2", "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\/2\ \((2 + 5\ \@3)\)\^3\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has left the expression in its original form since it tries to keep all \ expressions in their most exact form. To get a decimal approximation we must \ apply the function ", StyleBox["N[]", FontFamily->"Courier", FontWeight->"Bold"], " that converts an expression to a decimal." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["N[(2 + 5*Sqrt[3])^3/2]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(605.721050646230629`\)], "Output"] }, Open ]], Cell[TextData[{ "To get 10 significant digits, we must add the optional second argument to \ ", StyleBox["N[]", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["N[(2 + 5*Sqrt[3])^3/2, 10]", "Input", AspectRatioFixed->True], Cell[BoxData[ StyleBox["605.721050646230629`", StyleBoxAutoDelete->True, PrintPrecision->10]], "Output"] }, Open ]], Cell[TextData[{ "To find out more information about a function you can enter ", StyleBox["?", FontFamily->"Courier", FontWeight->"Bold"], " followed by the function name. For example," }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["?N", "Input", AspectRatioFixed->True], Cell[BoxData[ \("N[expr] gives the numerical value of expr. N[expr, n] attempts to give \ a result with n\[Hyphen]digit precision."\)], "Print"] }, Open ]], Cell[TextData[{ "You can also get information in the \"", StyleBox["Help", FontWeight->"Bold"], "\" menu." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 1"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Find the value of ", StyleBox["(2*Pi + Sqrt[3])^(5/2)", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["20", FontFamily->"Courier", FontWeight->"Bold"], " significant digits. (", StyleBox["Pi", FontFamily->"Courier", FontWeight->"Bold"], " is a constant built into ", StyleBox["Mathematica", FontSlant->"Italic"], " that represents the exact mathematical value of \[Pi]. Other important \ constants to be aware of like this are", StyleBox[" E == 2.71828...", FontFamily->"Courier", FontWeight->"Bold"], " for the base of the natural logarithms and ", StyleBox["I == Sqrt[-1]", FontFamily->"Courier", FontWeight->"Bold"], "). Click in the line BELOW the word \"", StyleBox["Solution", FontWeight->"Bold"], "\" to enter the expression. Then hit ENTER (not RETURN)." }], "Text", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[" ", "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Example 2"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Find the first and second derivatives of", StyleBox[" x^2 Sin[Cos[x]]", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "The function to use here is ", StyleBox["D[f,x]", FontFamily->"Courier", FontWeight->"Bold"], " which takes the derivative of ", StyleBox["f", FontFamily->"Courier", FontWeight->"Bold"], " with respect to ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["D[ x^2 Sin[Cos[x]] , x ]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-x\^2\)\ Cos[Cos[x]]\ Sin[x] + 2\ x\ Sin[Cos[x]]\)], "Output"] }, Open ]], Cell[TextData[ "The second derivative is the derivative of the previous output:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["D[ %, x ]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-x\^2\)\ Cos[x]\ Cos[Cos[x]] - 4\ x\ Cos[Cos[x]]\ Sin[x] + 2\ Sin[Cos[x]] - x\^2\ Sin[x]\^2\ Sin[Cos[x]]\)], "Output"] }, Open ]], Cell[TextData[{ "The symbol ", StyleBox["%", FontFamily->"Courier", FontWeight->"Bold"], " stands for the previous output. Be careful when you use this that you \ know what statement is being substituted for ", StyleBox["%", FontFamily->"Courier", FontWeight->"Bold"], ". Each output also has a number (check \"", StyleBox["Show In/Out names", FontWeight->"Bold"], "\" in the \"", StyleBox["Kernel", FontWeight->"Bold"], "\" menu to see them). To substitute the 17th output into another \ expression, use ", StyleBox["%17", FontFamily->"Courier", FontWeight->"Bold"], ". Another way to take the second derivative is to enter", StyleBox[" D[f,x,x]", FontFamily->"Courier", FontWeight->"Bold"], " or\n", StyleBox["D[f,{x,2}]", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["D[ x^2 Sin[Cos[x]] , x, x ]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-x\^2\)\ Cos[x]\ Cos[Cos[x]] - 4\ x\ Cos[Cos[x]]\ Sin[x] + 2\ Sin[Cos[x]] - x\^2\ Sin[x]\^2\ Sin[Cos[x]]\)], "Output"] }, Open ]], Cell[TextData[{ "You can also set up a function as follows (the underscore ", StyleBox["_", FontFamily->"Courier", FontWeight->"Bold"], " is VERY important):" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["f[x_] = x^2 Sin[Cos[x]]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(x\^2\ Sin[Cos[x]]\)], "Output"] }, Open ]], Cell[TextData[{ "After entering this, you can take the derivatives of ", StyleBox["f", FontFamily->"Courier", FontWeight->"Bold"], " by entering ", StyleBox["f'[x] ", FontFamily->"Courier", FontWeight->"Bold"], "and", StyleBox[" f''[x]", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["f'[x]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-x\^2\)\ Cos[Cos[x]]\ Sin[x] + 2\ x\ Sin[Cos[x]]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["f''[x]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-x\^2\)\ Cos[x]\ Cos[Cos[x]] - 4\ x\ Cos[Cos[x]]\ Sin[x] + 2\ Sin[Cos[x]] - x\^2\ Sin[x]\^2\ Sin[Cos[x]]\)], "Output"] }, Open ]], Cell[TextData[{ "Notice how all built in functions in ", StyleBox["Mathematica", FontSlant->"Italic"], " , like ", StyleBox["Sin[]", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["Cos[]", FontFamily->"Courier", FontWeight->"Bold"], ", are capitalized and that functions always use SQUARE braces, NOT \ parentheses." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 2"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Find the first and second derivatives of the function ", StyleBox["Sin[Log[Sqrt[1-x^2]]] ", FontFamily->"Courier", FontWeight->"Bold"], "using the function ", StyleBox["D[]", FontFamily->"Courier", FontWeight->"Bold"], " and using the prime notation ", StyleBox["f'[x]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["f''[x]", FontFamily->"Courier", FontWeight->"Bold"], " as above." }], "Text", CellMargins->{{18, 39}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["", "Input", CellMargins->{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Example 3"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Find an anti-derivative of ", StyleBox["1/(1-x^2)", FontFamily->"Courier", FontWeight->"Bold"], ". Integrate ", StyleBox["1/(1-x^2)", FontFamily->"Courier", FontWeight->"Bold"], " from ", StyleBox["0", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["1/2", FontFamily->"Courier", FontWeight->"Bold"], ". Find an approximation to the integral of ", StyleBox["1/Sqrt[1+x^3]", FontFamily->"Courier", FontWeight->"Bold"], " from ", StyleBox["0", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["1", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "To find an antiderivative, use the function ", StyleBox["Integrate[f,x]", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "Integrate[ ", StyleBox[" ", FontFamily->"Times", FontWeight->"Plain"], "1/(1-x^2), x ]" }], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-\(1\/2\)\)\ Log[\(-1\) + x] + 1\/2\ Log[1 + x]\)], "Output"] }, Open ]], Cell[TextData[{ "The definite integral is found by grouping the limits of integration with \ the variable:\n", StyleBox["{x,0,1/2}", FontFamily->"Courier", FontWeight->"Bold"], ". (By the way, curly braces, ", StyleBox["{", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["}", FontFamily->"Courier", FontWeight->"Bold"], ", are always used to lists things in ", StyleBox["Mathematica", FontSlant->"Italic"], " and it is the usual way arguments are grouped together in functions.)" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "Integrate[ ", StyleBox[" ", FontFamily->"Times", FontWeight->"Plain"], "1/(1-x^2), {x, 0, 1/2} ] //Simplify" }], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Log[3]\/2\)], "Output"] }, Open ]], Cell[TextData["A numerical approximation is given by"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["N[%]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.549306144334054913`\)], "Output"] }, Open ]], Cell[TextData[{ "If you only want a numerical approximation to begin with, then use the \ function ", StyleBox["NIntegrate[]", FontFamily->"Courier", FontWeight->"Bold"], " (limits MUST be provided for this function)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "NIntegrate[ ", StyleBox[" ", FontFamily->"Times", FontWeight->"Plain"], "1/(1-x^2), {x, 0, 1/2} ]" }], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 0.5493061443340557\ \>", "\<\ 0.549306\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData["To do the other integral:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "Integrate[ ", StyleBox[" ", FontFamily->"Times", FontWeight->"Plain"], "1/Sqrt[1+x^3], x ]" }], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(1\/\(3\^\(1/4\)\ \@\(1 + x\^3\)\)\(( 2\ \((\(-1\))\)\^\(1/6\)\ \@\(\((\(-1\))\)\^\(5/6\)\ \((\(-1\) + \((\(-1\))\)\^\(1/3\)\ x)\)\)\ \@\(1 + \((\(-1\))\)\^\(1/3\)\ x + \((\(-1\))\)\^\(2/3\)\ x\^2\)\ EllipticF[ ArcSin[\@\(\(-\((\(-1\))\)\^\(5/6\)\) - \((\(-1\))\)\^\(5/6\)\ x\)\/3\^\(1/4\)], \((\(-1\))\)\^\(1/3\)])\)\)\)], "Output"] }, Open ]], Cell["To find the value numerically:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "NIntegrate[ ", StyleBox[" ", FontFamily->"Times", FontWeight->"Plain"], "1/Sqrt[1+x^3], {x,0,1} ]" }], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.909604242638405935`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 3"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Find an anti-derivative of ", StyleBox["2/(1-x^2)", FontFamily->"Courier", FontWeight->"Bold"], ". Integrate ", StyleBox["2/(1-x^2)", FontFamily->"Courier", FontWeight->"Bold"], " from ", StyleBox["2", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["3", FontFamily->"Courier", FontWeight->"Bold"], ". Find an approximation to the integral of ", StyleBox["1/(2 + E^Sqrt[x])", FontFamily->"Courier", FontWeight->"Bold"], " from ", StyleBox["0", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["1", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["", "Input", CellMargins->{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Example 4"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Solve the system of equations ", StyleBox["2 x^2 - 2 x y - x^2 y + x y^2 == 0,\nx^2 + y^2 == 1", FontFamily->"Courier", FontWeight->"Bold"], ".", StyleBox["", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Note that EQUATIONS in ", StyleBox["Mathematica", FontSlant->"Italic"], " are represented by a DOUBLE equal sign, ", StyleBox["==", FontFamily->"Courier", FontWeight->"Bold"], ". A single equal sign, ", StyleBox["=", FontFamily->"Courier", FontWeight->"Bold"], ", means you want to assign what is on the right hand side to the variable \ on the left hand side. This is a very common mistake, so beware. To solve the \ system \"by hand\", we factor the first equation ", StyleBox["x(y-x)(y-2) == 0", FontFamily->"Courier", FontWeight->"Bold"], ", so that ", StyleBox["x == 0", FontFamily->"Courier", FontWeight->"Bold"], ", or", StyleBox[" y == x", FontFamily->"Courier", FontWeight->"Bold"], ", or", StyleBox[" y == 2. ", FontFamily->"Courier", FontWeight->"Bold"], "Plugging these possibilities into the second equation gives ", StyleBox["0^2 + y^2 == 1", FontFamily->"Courier", FontWeight->"Bold"], ", or \n", StyleBox["x^2 + x^2 == 1", FontFamily->"Courier", FontWeight->"Bold"], ", or ", StyleBox["x^2 + 2^2 == 1", FontFamily->"Courier", FontWeight->"Bold"], ", respectively. The soultions are then ", StyleBox["{0,\[PlusMinus]1}", FontFamily->"Courier", FontWeight->"Bold"], ", or ", StyleBox["\[PlusMinus]{1/Sqrt[2],1/Sqrt[2]}", FontFamily->"Courier", FontWeight->"Bold"], ". The ", StyleBox["Solve[]", FontFamily->"Courier", FontWeight->"Bold"], " function does this in one step. The equations must be grouped in curly \ braces ", StyleBox["{", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["}", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ Solve[{2 x^2 - 2 x y - x^2 y + x y^2 == 0, x^2 + y^2 == 1}]\ \>", "Input", CellMargins->{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \({{x \[Rule] \(-\(1\/\@2\)\), y \[Rule] \(-\(1\/\@2\)\)}, { x \[Rule] 1\/\@2, y \[Rule] 1\/\@2}, {x \[Rule] \(-I\)\ \@3, y \[Rule] 2}, {x \[Rule] I\ \@3, y \[Rule] 2}, {y \[Rule] \(-1\), x \[Rule] 0}, {y \[Rule] 1, x \[Rule] 0}}\)], "Output"] }, Open ]], Cell[TextData["To convert to decimal approximations:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["N[%]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x \[Rule] \(-0.707106781186547461`\), y \[Rule] \(-0.707106781186547461`\)}, { x \[Rule] 0.707106781186547461`, y \[Rule] 0.707106781186547461`}, { x \[Rule] \(-1.73205080756887692`\)\ I, y \[Rule] 2.`}, { x \[Rule] 1.73205080756887692`\ I, y \[Rule] 2.`}, { y \[Rule] \(-1.`\), x \[Rule] 0}, {y \[Rule] 1.`, x \[Rule] 0}}\)], "Output"] }, Open ]], Cell[TextData[{ "Note that some of these answers contain ", StyleBox["I", FontFamily->"Courier", FontWeight->"Bold"], " which stands for ", StyleBox["Sqrt[-1]", FontFamily->"Courier", FontWeight->"Bold"], ". You should ignore\nsolutions that have ", StyleBox["I", FontFamily->"Courier", FontWeight->"Bold"], " in them." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "You can also use ", StyleBox["NSolve[]", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ NSolve[{2 x^2 - 2 x y - x^2 y + x y^2 ==0, x^2 + y^2 ==1}]\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x \[Rule] \(-0.70710678118654755`\), y \[Rule] \(-0.70710678118654755`\)}, { x \[Rule] \(0.`\[InvisibleSpace]\) - 1.73205080756887692`\ I, y \[Rule] 2.`}, { x \[Rule] \(0.`\[InvisibleSpace]\) + 1.73205080756887692`\ I, y \[Rule] 2.`}, {x \[Rule] 0.70710678118654755`, y \[Rule] 0.70710678118654755`}, {y \[Rule] \(-1.`\), x \[Rule] 0.`}, {y \[Rule] 1.`, x \[Rule] 0.`}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 4 "], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Solve the system of equations ", StyleBox["x^2 y - x y + x y^2 == 0,\nx^2 + y - y^2 == 10.", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["", "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Example 5"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Plot the graph of the function ", StyleBox["x + Sin[x]", FontFamily->"Courier", FontWeight->"Bold"], ".\nPlot the graph of the function ", StyleBox["x y + Sin[x y]", FontFamily->"Courier", FontWeight->"Bold"], ".\nPlot the graph of the curve ", StyleBox["{t^2, t^3}", FontFamily->"Courier", FontWeight->"Bold"], ".\nPlot the graph of the curve ", StyleBox["{t, t^2, t^3}", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "To plot a function of 1 variable use ", StyleBox["Plot[]", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Plot[x + Sin[x], {x,0,4Pi}]", "Input", 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