(* 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[ 144095, 6908] NotebookOptionsPosition[ 139566, 6784] NotebookOutlinePosition[ 140163, 6805] CellTagsIndexPosition[ 140120, 6802] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ StyleBox["Lab 4A - Parametrizing Curves", FontSize->24, FontWeight->"Bold", FontVariations->{"Underline"->True}], "\nMath 2374 - University of Minnesota\nhttp://www.math.umn.edu/math2374\n\ Questions to: rogness@math.umn.edu" }], "Text", CellFrame->True, TextAlignment->Center, TextJustification->0, FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction", FontSize->14]], "Section"], Cell[TextData[{ "Other than a short bit about derivatives at the end of this notebook, Lab 4 \ is almost entirely devoid of calculus. Instead, it's about visualizing and \ describing different curves and surfaces. These skills will be very \ important later in this class,which is why we're spending so much time on \ them now.\n\nThis week we will consider parametric equations of curve. The \ term \"curve\" has not been defined yet, but roughly speaking,a curve (or an \ \"arc\") is a curvy line in space. If you like, you could think of a curve \ as a piece of string that's sitting in two- or three-dimensional space.\n\n\ Examples are a segment of a straight line, a circle, or a section of any \ graph of a function ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", RowBox[{"f", "(", "x", ")"}]}], TraditionalForm]]], ". Sometimes the curve will have two ends; sometimes the two ends of the \ curve will actually be the same point -- i.e.the two ends of the string are \ taped together. In this case we call the curve a ", StyleBox["closed", FontSlant->"Italic"], " curve. The following pictures are all examples of curves. 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There are many reasons, but one \ of the most important is that using parametric equations allows us to graph \ things that we otherwise wouldn't be able to graph. For example,consider the \ unit circle ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], "=1. You know from Lab 1A that one way to graph a function y=f(x) is the \ Mathematica command ", StyleBox["Plot", FontWeight->"Bold"], ". To use this command we need to solve our equation for y, and this is \ where the trouble starts; we get two solutions, ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", RowBox[{"-", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]]}]}], TraditionalForm]]], ".\n\nWe can plot these separately, but they each represent only half of the \ circle:" }], "Text"], Cell[BoxData[{ RowBox[{"upper", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sqrt", "[", RowBox[{"1", "-", RowBox[{"x", "^", "2"}]}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"lower", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"-", RowBox[{"Sqrt", "[", RowBox[{"1", "-", RowBox[{"x", "^", "2"}]}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.412358943113787*^9, 3.412358948457499*^9}}], Cell[TextData[{ "(The ", StyleBox["AspectRatio\[Rule]Automatic", FontWeight->"Bold"], " option forces ", StyleBox["Mathematica", FontSlant->"Italic"], " to scale the graph so it looks like a piece of a circle; try removing the \ option to see what happens.)\n\nIf we wish to see the whole circle, we can \ show the two graphs together:" }], "Text"], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"upper", ",", "lower"}], "]"}]], "Input"], Cell[TextData[{ "So we do in fact get to see the whole circle, but we haven't really ", StyleBox["graphed", FontSlant->"Italic"], " the circle. We've just pasted together two graphs to make it look like \ one circle. Using parametric equations can solve this problem. You've \ probably seen the parametric equations for the unit circle before: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "t", ")"}], "=", RowBox[{"(", RowBox[{ RowBox[{"cos", " ", "t"}], ",", " ", RowBox[{"sin", " ", "t"}]}], ")"}]}], TraditionalForm]]], ", where 0\[LessEqual]t\[LessEqual]2\[Pi]. We can define these equations in \ Mathematica with the following commands:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "t_", "]"}], "=", RowBox[{"Cos", "[", "t", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"y", "[", "t_", "]"}], "=", RowBox[{"Sin", "[", "t", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "t", "]"}], ",", RowBox[{"y", "[", "t", "]"}]}], "}"}]}]}], "Input"], Cell[TextData[{ "(We don't actually have to define x and y separately; we could the whole \ definition in one step by typing ", StyleBox["f[t_]={Cos[t],Sin[t]}", FontFamily->"Courier", FontWeight->"Bold"], ".) \n\n Once we have the parametric equation set up,we can graph it using \ the command ParametricPlot:" }], "Text"], Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]], "Input"], Cell[TextData[{ "Now we have the same picture as above, but we produced it with one graphing \ command, instead of three! Also -- and perhaps more importantly, in the long \ run -- we have a function of one variable which describes the entire circle, \ instead of the implicit function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], "=", "1"}], TraditionalForm]]], ", which has two variables in it and is harder to work with.\n\nWe call f[t] \ a ", StyleBox["parametrization", FontSlant->"Italic"], " of the circle. This is an important term, and you need to learn it. \ Later this semester (and even later in this lab) we will ask you to \"find a \ parametrization for...\" This means you need to find a parametric equation \ whose graph matches a certain picture. If you are asked to find a \ parametrization for the unit circle, you now know that the answer would be:\n\ \nf(t) = (cos t, sin t), 0 \[LessEqual] t \[LessEqual] 2\[Pi].\n" }], "Text"], Cell[TextData[{ "Here are some other common examples of parametric equations:\n\n", StyleBox["Example 1", FontSize->14, FontWeight->"Bold"], StyleBox[". A circle of radius r\n", FontSize->14], "\nWe simply take our parametrization of the unit circle and multiply each \ component by r to get ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "t", ")"}], "=", RowBox[{"(", RowBox[{ RowBox[{"r", "*", "cos", " ", "t"}], ",", " ", RowBox[{"r", "*", "sin", " ", "t"}]}], ")"}]}], TraditionalForm]]], ", 0\[LessEqual]t\[LessEqual]2\[Pi]. Before you go any further, you should \ use ", StyleBox["ParametricPlot", FontWeight->"Bold"], " to graph two different circles where r=3 and r=5. Remember to include the \ ", StyleBox["AspectRatio\[RightArrow]Automatic", FontWeight->"Bold"], " option if you want the result to look like a circle." }], "Text", PageWidth->WindowWidth], Cell[TextData[{ StyleBox["Example 2", FontSize->14, FontWeight->"Bold"], StyleBox[". An ellipse ", FontSize->14], Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["x", "2"], SuperscriptBox["a", "2"]], TraditionalForm]], FontSize->14], StyleBox[" + ", FontSize->14], Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["y", "2"], SuperscriptBox["b", "2"]], TraditionalForm]], FontSize->14], StyleBox[" = 1", FontSize->14], "\n\nThis is an ellipse centered at the origin. To graph it by hand, you \ need to know that an ellipse written in this form goes through the points (\ \[PlusMinus]a, 0) and (0, \[PlusMinus] b). You would plot these four points \ and then try to draw a smooth ellipse through them. Mathematica can draw \ this precisely using parametric equations. (If you tried to use the ", StyleBox["Plot", FontWeight->"Bold"], " command you'd have the same problem as with the circle; when you solve for \ y you get " }], "Text", TextAlignment->Left], Cell[TextData[{ StyleBox["y = ", FontSize->16], Cell[BoxData[ FormBox[ RowBox[{"\[PlusMinus]", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "(", RowBox[{"1", "-", FractionBox[ SuperscriptBox["x", "2"], SuperscriptBox["a", "2"]]}], ")"}]]}], TraditionalForm]], FontSize->15] }], "Text", TextAlignment->Center], Cell[TextData[{ "and you'd have to graph it in two pieces.)\n\nTo parametrize the ellipse, \ we start with the parametrization for the unit circle, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "t", ")"}], "=", RowBox[{"(", RowBox[{ RowBox[{"cos", " ", "t"}], ",", " ", "sint"}], ")"}]}], TraditionalForm]]], ". You can think of the variables a and b as instructions for how far to \ stretch/compress this circle in the x and y directions to make our ellipse. \ In mathematical terms, this means we should multiply the x-component of f(t) \ by a, and the y-component of f(t) by b. So the parametrization of the \ ellipse is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"f", "(", "t", ")"}], "=", RowBox[{"(", RowBox[{ RowBox[{"a", "*", "cos", " ", "t"}], ",", " ", RowBox[{"b", "*", "sin", " ", "t"}]}], ")"}]}], ","}], TraditionalForm]]], " 0\[LessEqual]t\[LessEqual]2\[Pi]. You can see this is similar to the \ parametrization for a circle of radius r, and in fact if a=b then the result \ is a circle!\n\nAs an example, here is the graph of the ellipse ", StyleBox[" ", FontSize->14], Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["x", "2"], SuperscriptBox["4", "2"]], TraditionalForm]], FontSize->14], " + ", Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["y", "2"], SuperscriptBox["3", "2"]], TraditionalForm]], FontSize->14], " = 1" }], "Text", TextAlignment->Left], Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"4", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"3", RowBox[{"Sin", "[", "t", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]], "Input"], Cell[TextData[{ StyleBox["Example 3", FontSize->14, FontWeight->"Bold"], StyleBox[". Circles and Ellipses centered away from the origin.", FontSize->14], StyleBox["\n", FontSize->14], "\nIt's actually surprisingly easy to parametrize a circle or an ellipse \ which is centered at a point other than the origin. For example, consider \ the circle of radius 3 centered at the point (1,-2). If the circle were \ centered at the origin, its parametrization would be g(t) = (3cos t, 3sin t). \ All we need to do is move all of our points 1 unit to the right and 2 units \ down to get the circle we want, so we add these numbers to the components of \ g(t):" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"1", "+", RowBox[{"3", RowBox[{"Cos", "[", "t", "]"}]}]}], ",", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", RowBox[{"Sin", "[", "t", "]"}]}]}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]}], "Input"], Cell["\<\ In general the parametrization for a circle of radius r, centered at (h,k), \ is f(t) = (h + r*cos t, k + r*sin t), where of course 0\[LessEqual]t\ \[LessEqual]2\[Pi]. You won't be surprised to find that this works for an \ ellipse, as well; to shift the ellipse from above so that it's centered at \ (1,-2), simply add 1 and -2 to the x and y components:\ \>", "Text"], Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"1", "+", RowBox[{"4", RowBox[{"Cos", "[", "t", "]"}]}]}], ",", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", RowBox[{"Sin", "[", "t", "]"}]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]], "Input"], Cell[TextData[{ StyleBox["Example 4", FontSize->14, FontWeight->"Bold"], StyleBox[". Spirals", FontSize->14], "\n\nConsider the following function. Try to think about what its graph \ will look like, and why." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"t", "*", RowBox[{"Cos", "[", RowBox[{"2", "Pi", "*", "t"}], "]"}]}], ",", RowBox[{"t", "*", RowBox[{"Sin", "[", RowBox[{"2", "Pi", "*", "t"}], "]"}]}]}], "}"}]}]], "Input"], Cell["\<\ It looks kind of like a circle of radius t, but because t is increasing, it \ means that instead of tracing out a circle, the graph of this function will \ spin out from the origin:\ \>", "Text"], Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "5"}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]], "Input"], Cell[TextData[{ StyleBox["Example 5", FontSize->14, FontWeight->"Bold"], StyleBox[". Straight Line Segment", FontSize->14], StyleBox["\n", FontSize->13], "\nOne important example of a parametric equation is a straight line \ segment. 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The colored arrows \ are all multiples of the vector ", Cell[BoxData[ FormBox[ OverscriptBox["PQ", "\[RightVector]"], TraditionalForm]]], ". (Note that most of each colored arrow is covered up by the previous \ arrows; all of the colored arrows actually start at P.) So once we're at the \ point P, we want to add on everything from 0% of ", Cell[BoxData[ FormBox[ OverscriptBox["PQ", "\[RightVector]"], TraditionalForm]]], " to 100% of ", Cell[BoxData[ FormBox[ OverscriptBox["PQ", "\[RightVector]"], TraditionalForm]]], ". Since percentages are really just numbers between 0 and 1, our \ parametrization for the line segment between the points P and Q is:\n\nf(t) = \ ", Cell[BoxData[ FormBox[ OverscriptBox["OP", "\[RightVector]"], TraditionalForm]]], " + t\[CenterDot]", Cell[BoxData[ FormBox[ OverscriptBox["PQ", "\[RightVector]"], TraditionalForm]]], ", where 0 \[LessEqual] t \[LessEqual] 1.\n\nFor our specific example, we \ have:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "2"}], "}"}], " ", "+", " ", RowBox[{"t", "*", RowBox[{"(", RowBox[{ RowBox[{"{", RowBox[{"2", ",", "1"}], "}"}], "-", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "2"}], "}"}]}], ")"}]}]}]}], "\n", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "1"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.412359012981342*^9, 3.412359046124329*^9}}], Cell[TextData[{ "To make sure you understand this process, you should parametrize the line \ segments from (-1,-1) to (5,1) and from (4,2) to (4,-3). Plot them and \ verify that they are correct.\n\n", StyleBox["Example 6", FontSize->14, FontWeight->"Bold"], StyleBox[". A Line", FontSize->14], "\n\nIf we take the parametrization f(t) = ", Cell[BoxData[ FormBox[ OverscriptBox["OP", "\[RightVector]"], TraditionalForm]]], " + t\[CenterDot]", Cell[BoxData[ FormBox[ OverscriptBox["PQ", "\[RightVector]"], TraditionalForm]]], " and let t range from -\[Infinity] to +\[Infinity], instead of just 0 to 1, \ we get the entire line containing P and Q instead of the line segment between \ them. This is the parametric equation for a line.\n\n", StyleBox["Example 7", FontSize->14, FontWeight->"Bold"], StyleBox[". The Graph of a Function y=f(x)", FontSize->14], "\n\nAs we noted above, a piece of the graph of any function is a curve. \ For example, the following commands all produce examples of curves." }], "Text"], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"x", "^", "3"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Pi"}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"x", "^", "2"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Exp", "[", "x", "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "5"}], ",", "1"}], "}"}]}], "]"}]], "Input"], Cell[TextData[{ "There will be times when we want to represent these curves with parametric \ equations. There's an easy, almost silly, way to do this. If we want to \ parametrize the graph of ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"y", "=", RowBox[{"g", "(", "x", ")"}]}], ")"}], TraditionalForm]]], ", we can let ", StyleBox["x", FontSlant->"Italic"], " be the parameter, and our parametrization is simply ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], " ", "=", " ", RowBox[{"(", RowBox[{"x", ",", " ", RowBox[{"g", "(", "x", ")"}]}], ")"}]}], TraditionalForm]]], ". Sometimes we wish to keep ", StyleBox["t", FontSlant->"Italic"], " as our parameter instead of switching to ", StyleBox["x", FontSlant->"Italic"], ". All we need to do is change the name of our variable, so we have ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"f", "(", "t", ")"}], " ", "=", " ", RowBox[{"(", RowBox[{"t", ",", " ", RowBox[{"g", "(", "t", ")"}]}], ")"}]}], ")"}], TraditionalForm]]], ". 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", StyleBox["Mathematica", FontSlant->"Italic"], " can also graph parametric equations in three dimensional space. To do \ this, we use a variant of ", StyleBox["ParametricPlot", FontWeight->"Bold"], " which is named, not surprisingly, ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ". " }], "Text"], Cell[TextData[{ "Note the capitalization in ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ". If the two P's and the D are not capitalized, you will get an error \ message! " }], "Text", CellFrame->True, CellChangeTimes->{{3.412359083525*^9, 3.412359084033925*^9}}, Background->GrayLevel[0.900008]], Cell[TextData[{ StyleBox["ParametricPlot3D", FontWeight->"Bold"], " expects a parametric equation with three components instead of two. We \ can actually graph any of the equations above by simply letting z=0. For \ example, the following commands will plot the unit circle:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", RowBox[{"Sin", "[", "t", "]"}], ",", "0"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"circle0", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]}], "Input"], Cell[TextData[{ "As you can see, the difference is that now ", StyleBox["Mathematica", FontSlant->"Italic"], " treats the circle as if it lives in three dimensions instead of two. We \ can plot the unit circle for other values of z, as well:" }], "Text"], Cell[BoxData[{ RowBox[{"circle1", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", RowBox[{"Sin", "[", "t", "]"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"circle2", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", RowBox[{"Sin", "[", "t", "]"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]}], "Input"], Cell["\<\ We can even plot all three circles together, so you can really get a feel for \ the added dimension:\ \>", "Text"], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"circle2", ",", "circle1", ",", "circle0"}], "]"}]], "Input", CellChangeTimes->{{3.412359104389729*^9, 3.412359113888544*^9}}], Cell[TextData[{ "Of course, the z component of the function does not have to stay constant, \ and things get a lot more interesting if we let it change. Here's a circle \ of radius 1 which is not in the xy plane. (Which plane ", StyleBox["is", FontSlant->"Italic"], " it in? The output can sometimes be a little deceiving!)" }], "Text"], Cell[BoxData[ RowBox[{"circle", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", "0", ",", RowBox[{"Sin", "[", "t", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.412359142876514*^9, 3.412359143184431*^9}}], Cell[TextData[{ "(To see if you really understand what plane the circle is in, you could try \ adding the option ", StyleBox["ViewPoint\[Rule]{2,0,0}", FontFamily->"Courier", FontWeight->"Bold"], " to the previous command and see if you can explain the output. \ Alternaticvely, you could rotate the picture to figure out what plane it's \ in.)" }], "Text", CellChangeTimes->{{3.412359150878716*^9, 3.412359152707165*^9}}], Cell[TextData[{ "Let's look at a few examples of curves in three-dimensional space.\n\n", StyleBox["Example 8", FontSize->14, FontWeight->"Bold"], StyleBox[". Helix", FontSize->14], "\n\nTo introduce a helix, let's look at the following function first." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", RowBox[{"Sin", "[", "t", "]"}], ",", RowBox[{"t", "/", "2"}]}], "}"}]}]], "Input"], Cell["\<\ As t moves from 0 to 2\[Pi], you can see that the x- and y-coordinates will \ go around the unit circle. But our z-coordinate starts at zero and \ increases, so instead of a circle, we get a helix twisting upwards.\ \>", "Text"], Cell[BoxData[ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input"], Cell["\<\ This helix only makes one revolution; to go around more times we can let t go \ from 0 to 4\[Pi], or 6\[Pi], etc. (We want to choose a multiple of 2\[Pi] \ because that represents a full trip around the circle.)\ \>", "Text"], Cell[BoxData[ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"6", "Pi"}]}], "}"}]}], "]"}]], "Input"], Cell["\<\ Often we write the parametrization for a helix in a slightly different form, \ like this:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", RowBox[{"2", "Pi", "*", "t"}], "]"}], ",", RowBox[{"Sin", "[", RowBox[{"2", "Pi", "*", "t"}], "]"}], ",", "t"}], "}"}]}]], "Input"], Cell["\<\ Now we make a full revolution around the circle when t increases from 0 to 1, \ instead of 2\[Pi]. Similarly, if we let t go from 0 to 2, we would make 2 \ revolutions around the helix. In general, to make n revolutions, we let t go \ from 0 to n. For example, to graph a helix which makes three revolutions, \ using the new parametrization, evaluate the following command:\ \>", "Text"], Cell[BoxData[ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "3"}], "}"}]}], "]"}]], "Input"], Cell[TextData[{ "Note one crucial difference: in this last example, the z values range from \ 0 to 3. In the previous example, the z values ranges from 0 to more than 9. \ Why is this?\n\n", StyleBox["Example 9", FontSize->14, FontWeight->"Bold"], StyleBox[". A Tornado\n", FontSize->14], "\nIf we \"combine\" the spiral from above with the helix we can make an \ interesting effect that looks like a tornado:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"t", " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "\[Pi]", " ", "t"}], "]"}]}], ",", RowBox[{"t", " ", RowBox[{"Sin", "[", RowBox[{"2", " ", "\[Pi]", " ", "t"}], "]"}]}], ",", RowBox[{"2", " ", "t"}]}], "}"}]}], ";"}], "\n", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}], ",", RowBox[{"ViewPoint", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "10"}], ",", "1"}], "}"}]}]}], "]"}]}], "Input"], Cell[TextData[{ StyleBox["Example 10", FontSize->14, FontWeight->"Bold"], StyleBox[". A Randomly Chosen Graph\n", FontSize->14], "\nThis example, unlike many of the others, will probably never be used in \ class. The point is just to show you that all kinds of silly things are \ possible -- and to continue demonstrating that we can make pretty pictures \ using parametric equations.\n\nConsider the graph of this function:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"g", "[", "x_", "]"}], "=", RowBox[{"Sin", "[", RowBox[{"6", "Pi", " ", "x"}], "]"}]}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"g", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "]"}]}], "Input"], Cell[TextData[{ "For our silly example, let's create functions whose x- and y- values will \ follow this graph above, but whose z-values will vary with time t, instead of \ just being 0. If we ", StyleBox["do", FontSlant->"Italic"], " let z=0, the we just get the same graph again, but now \"living\" in \ three-dimensions:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{"t", ",", RowBox[{"Sin", "[", RowBox[{"6", "Pi", " ", "t"}], "]"}], ",", " ", "0"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"PlotPoints", "\[Rule]", "300"}]}], "]"}]}], "Input"], Cell[TextData[{ "Now let's play around with the z values. Try to predict what each graph \ will look like before evaluating the commands. You may want to use the ", StyleBox["ShowLive", FontWeight->"Bold"], " or ", StyleBox["ParametricPlot3DLive", FontWeight->"Bold"], " ", "functions to get a better look at some of these." }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{"t", ",", RowBox[{"Sin", "[", RowBox[{"6", "Pi", " ", "t"}], "]"}], ",", " ", "t"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"PlotPoints", "\[Rule]", "300"}]}], "]"}]}], "Input"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{"t", ",", RowBox[{"Sin", "[", RowBox[{"6", "Pi", " ", "t"}], "]"}], ",", " ", RowBox[{"4", "-", "t"}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"PlotPoints", "\[Rule]", "300"}]}], "]"}]}], "Input"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{"t", ",", RowBox[{"Sin", "[", RowBox[{"6", "Pi", " ", "t"}], "]"}], ",", " ", RowBox[{"4", "-", RowBox[{"Abs", "[", "t", "]"}]}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"PlotPoints", "\[Rule]", "300"}]}], "]"}]}], "Input"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{"t", ",", RowBox[{"Sin", "[", RowBox[{"6", "Pi", " ", "t"}], "]"}], ",", " ", RowBox[{"t", "^", "2"}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"PlotPoints", "\[Rule]", "300"}]}], "]"}]}], "Input"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{"t", ",", RowBox[{"Sin", "[", RowBox[{"6", "Pi", " ", "t"}], "]"}], ",", " ", RowBox[{ RowBox[{"t", "^", "3"}], " ", "/", " ", "2"}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "]"}]}], "Input"], Cell[TextData[{ StyleBox["Exercise 3\n", FontSize->14, FontWeight->"Bold"], "\nParametrize the intersection of the cylinder ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " = 9 with the plane z = 4-x. First you should plot the cylinder with the \ command ", StyleBox["ContourPlot3D", FontWeight->"Bold"], ", which was introduced in Lab 1B. You can plot the plane using ", StyleBox["Plot3D", FontWeight->"Bold"], ". Then use ", StyleBox["Show", FontWeight->"Bold"], " to plot them together so you can see the intersection and get an idea of \ what it should look like. (Suggested ranges of x and y for plotting both the \ cylinder and the plane are {x,-4,4}, {y,-4,4}. For ", StyleBox["ContourPlot3D", FontWeight->"Bold"], " you'll also need to specify a z range, such as {z,0,8}.)\n\nTo parametrize \ the curve which is the intersection, first note that the entire curve lies \ above the circle ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " = 9. You know how to choose x(t) and y(t) so that the x- and y-values are \ on this circle. All that's left is to determine what z(t) is.\n\n", StyleBox["In exercises 4 and 5, follow the directions for exercise three, \ changing the plotting ranges as needed.", FontSlant->"Italic"], "\n\n", StyleBox["Exercise 4\n", FontSize->14, FontWeight->"Bold"], "\nParametrize the intersection of the cylinder ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " = 4 with the surface z = ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " - ", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], ".\n\n", StyleBox["Exercise 5\n", FontSize->14, FontWeight->"Bold"], "\nParametrize the intersection of the cylinder ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " = 16 with the surface z = ", Cell[BoxData[ FormBox[ FractionBox[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm], "4"], TraditionalForm]]], " - ", Cell[BoxData[ FormBox[ FractionBox[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm], "9"], TraditionalForm]]], ".\n\n", StyleBox["Exercise 6", FontSize->14, FontWeight->"Bold"], "\n\nImagine you're in a Geography professor's office and there is a globe \ in front of you. Suppose you put a pen at the North Pole, start spinning the \ globe, and then you slowly move the pen from the North Pole to the South \ Pole. 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You should turn \ in your parametrization, with the corresponding graph, and an explanation of \ how you arrived at your answer. Simply stumbling across the correct answer \ will result in little credit. You may use the following hints:\n\n\ \[FilledSmallSquare] Every point on this curve is on the unit sphere ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+ ", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["z", "2"], TraditionalForm]]], " = 1, so when you think you have your final parametrization, you should \ make sure that if you square x(t), y(t), and z(t), and add them all together \ you'll get 1.\n\n\[FilledSmallSquare] There are many possible \ parametrizations, but in the one originally used to create this picture, \ z(t)=Cos[\[Pi]*t] and t goes from 0 to 1.\n\n\[FilledSmallSquare] Seen from \ the top down, the path makes twenty revolutions around the sphere. So you \ could start by parametrizing a helix which makes 20 revolutions and has the \ z(t) component given above. Then decide how to change x(t) and y(t). (Extra \ hint: you multiply them by the same thing, something which goes from 0 to 1 \ and back to 0 again as t goes from 0 to 1 itself.)\n\n", StyleBox["Exercise 7", FontSize->14, FontWeight->"Bold"], "\n\nParametrize the following curve. 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".\n\n\[FilledSmallSquare] There are many possible parametrizations, but in \ the one originally used to create this picture, z(t)=t, and t goes from -1 \ to 1.\n\n\[FilledSmallSquare] Seen from the top down, the path makes twenty \ revolutions around the z-axis. So you could start by parametrizing a helix \ which makes 20 revolutions and has the z(t) component given above. Then \ decide how to change x(t) and y(t)." }], "Text", CellFrame->True, CellChangeTimes->{{3.412359229027891*^9, 3.412359232352401*^9}}, Background->RGBColor[0.996109, 0.597665, 0.597665]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Derivatives of Parametrizations", FontSize->14]], "Section"], Cell[TextData[{ "Note: depending on your instructor, you may have already seen this material \ as a part of an assigned reading. In that case, you can very quickly skim \ through the material, but you should read about how to use the ", StyleBox["PathAnimate", FontWeight->"Bold"], " and ", StyleBox["PathTangentAnimate", FontWeight->"Bold"], " commands. You'll be asked to use them in later labs.\n\nFor simplicity, \ we're only going to deal with two dimensions here, but the same ideas apply \ in three dimensions as well." }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[TextData[{ "Suppose we have a parametrization ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"f", "(", "t", ")"}], " ", "=", " ", RowBox[{"(", RowBox[{ RowBox[{"x", "(", "t", ")"}], ",", " ", RowBox[{"y", "(", "t", ")"}]}], ")"}]}], ")"}], TraditionalForm]]], ", where ", StyleBox["t", FontSlant->"Italic"], " ranges from 0 to 10. If we say that ", StyleBox["t", FontSlant->"Italic"], " represents time in seconds, then we can say ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "t", ")"}], TraditionalForm]]], " represents the position of an object at time ", StyleBox["t", FontSlant->"Italic"], ". For example, consider the following function:" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", RowBox[{"Sin", "[", "t", "]"}]}], "}"}]}], ";"}]], "Input"], Cell[TextData[{ "You should know by now that this is a parametrization for the unit circle. \ As you learned above you can plot this function using ", StyleBox["ParametricPlot", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]], "Input"], Cell[TextData[{ "This doesn't give you a feel for what I'm telling you, however: that ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "t", ")"}], TraditionalForm]]], " can be viewed as the location function for a particle moving around the \ unit circle, where the particle finishes its trip when ", Cell[BoxData[ FormBox[ RowBox[{"t", "=", RowBox[{"2", "\[Pi]"}]}], TraditionalForm]]], " seconds (or minutes, or hours, or whatever our unit is).\n\nTry looking at \ the output of the following command. It will produce an animation of the \ particle moving around. You can pause/play the animation as desired." }], "Text", CellChangeTimes->{{3.4123908200225773`*^9, 3.412390831122733*^9}}], Cell[BoxData[ RowBox[{"PathAnimate", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.412390702277132*^9, 3.412390702595289*^9}}], Cell[TextData[{ "Now you can see the particle moving around the curve. It looks as if the \ speed of the particle is constant; let's see if we can prove that somehow.\n\n\ Recall from your lecture (or textbook) that if ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "t", ")"}], TraditionalForm]]], " represents the location of a particle at time ", StyleBox["t", FontSlant->"Italic"], ", then ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"v", "(", "t", ")"}], "=", RowBox[{ FormBox[ SuperscriptBox["f", "\[Prime]"], TraditionalForm], "(", "t", ")"}]}], TraditionalForm]]], ", the derivative of ", StyleBox["f", FontSlant->"Italic"], ", represents the ", StyleBox["velocity", FontSlant->"Italic"], " of the particle at time ", StyleBox["t", FontSlant->"Italic"], ". For example, in our case we have:" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"v", "[", "t_", "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", "t"}], "]"}]}]], "Input"], Cell[TextData[{ "Velocity is a little different than speed. Speed is just a number \ representing how fast a particle is moving. Velocity is a vector whose ", StyleBox["length", FontSlant->"Italic"], " represents speed, and whose ", StyleBox["direction", FontSlant->"Italic"], " represents the direction the particle is moving at that specific instance \ of time.\n\nTo see an animation of the particle moving around the circle \ along with its tangent vectors, run this command. (We're increasing the ", StyleBox["PlotRange", FontWeight->"Bold"], " a bit so we can see the whole vector):" }], "Text", CellChangeTimes->{{3.412390743335991*^9, 3.412390760576367*^9}}], Cell[BoxData[ RowBox[{"PathTangentAnimate", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.412390728856162*^9, 3.412390740616765*^9}}], Cell[TextData[{ "It certainly looks as if the length of the velocity vector is constant, \ which would confirm that the speed of the particle is constant. In fact, we \ can show this algebraically. You should find the length of the tangent \ vector v(t) on paper and show that it is constant -- it does not depend on ", StyleBox["t", FontSlant->"Italic"], ".\n\nLet's look at a different parametrization of the unit circle,\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", "t", ")"}], " ", "=", " ", RowBox[{"(", FormBox[ RowBox[{ RowBox[{"Cos", "(", SuperscriptBox["t", "2"], ")"}], ",", " ", RowBox[{"Sin", "(", SuperscriptBox["t", "2"], ")"}], " "}], TraditionalForm], ")"}]}], TraditionalForm]]], ",\t0\[LessEqual]t\[LessEqual]", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"2", "\[Pi]"}]], TraditionalForm]]], ".\n\nThis is a parametrization of the unit circle because as t ranges from \ 0 to ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"2", "\[Pi]"}]], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SuperscriptBox["t", "2"], TraditionalForm]]], " ranges from 0 to 2\[Pi]:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"g", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", RowBox[{"t", "^", "2"}], "]"}], ",", RowBox[{"Sin", "[", RowBox[{"t", "^", "2"}], "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"g", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"Sqrt", "[", RowBox[{"2", "Pi"}], "]"}]}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]}], "Input"], Cell["\<\ Now let's watch the particle whose motion is represented by g[t]:\ \>", "Text"], Cell[BoxData[ RowBox[{"PathTangentAnimate", "[", RowBox[{ RowBox[{"g", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"Sqrt", "[", RowBox[{"2", "Pi"}], "]"}]}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "5"}], ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "5"}], ",", "5"}], "}"}]}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.4123907820180264`*^9, 3.412390792678294*^9}}], Cell["\<\ As you can see, the particle starts out very slowly and picks up speed as \ time goes on. You can see this either by watching the particle itself, or by \ watching the length of the tangent vector grow. (Remember, the length of the \ tangent vector represents the speed of the particle!) Find the velocity of this particle by computing the derivative of g[t]. Can \ you see why the particle is speeding up? You'll be asked to investigate this \ parametrization of the circle a little more in Lab 4A.\ \>", "Text"], Cell[TextData[{ "If you'd like to play around with these two animation commands, their \ syntax is:\n\n", StyleBox["PathAnimate[f[t],{t,tmin,tmax}]\n\ PathTangentAnimate[f[t],{t,tmin,tmax}]", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["\n", FontWeight->"Bold"], "\nAs shown above, you can also use the ", StyleBox["PlotRange", FontWeight->"Bold"], " option with ", StyleBox["PathTangentAnimate", FontFamily->"Courier", FontWeight->"Bold"], " to make sure you see the entire tangent vector; for now, trial and error \ is probably the best way to determine the optimal range. (If nothing else, \ ", StyleBox["PlotRange \[Rule] {{-5, 5}, {-5, 5}}", FontWeight->"Bold"], " should work fine with all of the functions below.)\n\nHere are a couple of \ parametrizations to try; you can copy these and paste them into commands:\n\n\ ", StyleBox["f[t_]={Cos[t],Sin[t]^3}, {t, 0, 2Pi}\nf[t_]={Cos[2t],Sin[3t]}, {t, \ 0, 2Pi}\nf[t_]={Cos[2t],Sin[4t]}, {t, 0, 2Pi}\nf[t_]={Cos[5t],Sin[3t]}, {t, \ 0, 2Pi}\nf[t_]={t,t^2}, {t, -1,1}\nf[t_]={t^3,t^2}, {t, -1,1}", FontFamily->"Courier"] }], "Text", CellFrame->True, CellChangeTimes->{{3.412390855004236*^9, 3.412390942902337*^9}, { 3.412390993424848*^9, 3.412391032424966*^9}}, Background->GrayLevel[0.849989]], Cell[CellGroupData[{ Cell["Credits", "Subsection"], Cell[TextData[{ "This lab was written from scratch in December 2001/January 2002. I made \ some minor updates and added a few exercises in January 2004. The biggest \ change was the inclusion of the Animation commands, which used to live in Lab \ 4A (Arclength). Parametrizations and their derivatives are covered in \ lecture during the second week of class, so this was a more natural location \ for this material to appear in a lab.\n\nFall 2004 Update: just minor changes \ -- instead of Lab 2A this is now Lab 4. Go figure. A few updates in Spring \ 2008 for ", StyleBox["Mathematica", FontSlant->"Italic"], " 6.0 and the new Path*Animate* commands.\n\nI did not think of the curve in \ exercise 7 on my own. I can't remember who first showed it to me, but \ presumably it was Dr. Emil Knapp at Augustana College, who was my advisor and \ my Calc III professor. I've noticed that Harvard's multivariable calculus \ class also used this curve as an example in one of their labs. Their course \ has apparently been revamped, because I tried to find a URL for their lab to \ include here, but it no longer exists.\n\nThis lab is copyright 2002, 2004 by \ Jonathan Rogness (rogness@math.umn.edu) and is protected by the Creative \ Commons Attribution-NonCommercial-ShareAlike License. You can find more \ information on this license at \ http://creativecommons.org/licenses/by-nc-sa/1.0/\n\nAlthough it's not \ specifically required by the license, I'd appreciate it if you let me know if \ you use parts of our labs, just so I can keep track of it. Please send me \ any questions or comments!" }], "Text", CellChangeTimes->{{3.412391044289447*^9, 3.4123910669941072`*^9}}] }, Closed]] }, Closed]] }, WindowSize->{598, 691}, WindowMargins->{{219, Automatic}, {Automatic, 57}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"Magnification"->1, "PaperOrientation"->"Portrait", "PaperSize"->{612, 792}, "PostScriptOutputFile":>FrontEnd`FileName[{"user002", "rogness"}, "Newlab.nb.ps", CharacterEncoding -> "iso8859-1"]}, FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (April 20, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 384, 12, 99, "Text"], Cell[CellGroupData[{ Cell[977, 37, 66, 1, 61, "Section"], Cell[1046, 40, 1224, 22, 299, "Text"], Cell[2273, 64, 7596, 387, 97, 5515, 350, "GraphicsData", "PostScript", \ "Graphics"] }, Closed]], Cell[CellGroupData[{ Cell[9906, 456, 78, 1, 31, "Section"], Cell[9987, 459, 1109, 34, 159, "Text"], Cell[11099, 495, 768, 23, 52, "Input"], Cell[11870, 520, 357, 10, 83, "Text"], Cell[12230, 532, 88, 2, 31, "Input"], Cell[12321, 536, 707, 17, 101, "Text"], Cell[13031, 555, 445, 14, 72, "Input"], Cell[13479, 571, 331, 8, 102, "Text"], Cell[13813, 581, 258, 7, 31, "Input"], Cell[14074, 590, 1039, 21, 245, "Text"], Cell[15116, 613, 927, 27, 158, "Text"], Cell[16046, 642, 1019, 32, 147, "Text"], Cell[17068, 676, 367, 15, 44, "Text"], Cell[17438, 693, 1519, 46, 219, "Text"], Cell[18960, 741, 388, 12, 52, "Input"], Cell[19351, 755, 679, 15, 140, "Text"], Cell[20033, 772, 567, 18, 52, "Input"], Cell[20603, 792, 380, 6, 83, "Text"], Cell[20986, 800, 466, 15, 52, "Input"], Cell[21455, 817, 231, 8, 68, "Text"], Cell[21689, 827, 307, 10, 31, "Input"], Cell[21999, 839, 205, 4, 47, "Text"], Cell[22207, 845, 236, 6, 31, "Input"], Cell[22446, 853, 861, 17, 230, "Text"], Cell[23310, 872, 13385, 596, 261, 9514, 529, "GraphicsData", "PostScript", \ "Graphics"], Cell[36698, 1470, 1207, 32, 207, "Text"], Cell[37908, 1504, 620, 20, 52, "Input"], Cell[38531, 1526, 1050, 27, 255, "Text"], Cell[39584, 1555, 196, 6, 31, "Input"], Cell[39783, 1563, 184, 6, 31, "Input"], Cell[39970, 1571, 196, 6, 31, "Input"], Cell[40169, 1579, 1203, 37, 119, "Text"], Cell[41375, 1618, 365, 11, 52, "Input"], Cell[41743, 1631, 463, 7, 101, "Text"], Cell[42209, 1640, 12126, 604, 685, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[54372, 2249, 80, 1, 31, "Section"], Cell[54455, 2252, 478, 13, 83, "Text"], Cell[54936, 2267, 313, 9, 63, "Text"], Cell[55252, 2278, 293, 6, 65, "Text"], Cell[55548, 2286, 442, 14, 52, "Input"], Cell[55993, 2302, 260, 6, 47, "Text"], Cell[56256, 2310, 673, 20, 52, "Input"], Cell[56932, 2332, 124, 3, 47, "Text"], Cell[57059, 2337, 174, 3, 31, "Input"], Cell[57236, 2342, 344, 7, 65, "Text"], Cell[57583, 2351, 400, 11, 31, "Input"], Cell[57986, 2364, 431, 10, 66, "Text"], Cell[58420, 2376, 276, 8, 104, "Text"], Cell[58699, 2386, 224, 7, 31, "Input"], Cell[58926, 2395, 239, 4, 65, "Text"], Cell[59168, 2401, 202, 6, 31, "Input"], Cell[59373, 2409, 236, 4, 65, "Text"], Cell[59612, 2415, 202, 6, 31, "Input"], Cell[59817, 2423, 113, 3, 29, "Text"], Cell[59933, 2428, 263, 8, 31, "Input"], Cell[60199, 2438, 400, 6, 101, "Text"], Cell[60602, 2446, 180, 5, 31, "Input"], Cell[60785, 2453, 435, 11, 140, "Text"], Cell[61223, 2466, 685, 21, 52, "Input"], Cell[61911, 2489, 448, 10, 140, "Text"], Cell[62362, 2501, 330, 10, 52, "Input"], Cell[62695, 2513, 342, 8, 65, "Text"], Cell[63040, 2523, 463, 14, 52, "Input"], Cell[63506, 2539, 348, 10, 65, "Text"], Cell[63857, 2551, 463, 14, 52, "Input"], Cell[64323, 2567, 484, 14, 52, "Input"], Cell[64810, 2583, 517, 15, 52, "Input"], Cell[65330, 2600, 484, 14, 52, "Input"], Cell[65817, 2616, 473, 15, 52, "Input"], Cell[66293, 2633, 61860, 3815, 1802, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[128190, 6453, 85, 1, 31, "Section"], Cell[128278, 6456, 601, 14, 132, "Text"], Cell[128882, 6472, 735, 25, 56, "Text"], Cell[129620, 6499, 212, 7, 27, "Input"], Cell[129835, 6508, 220, 6, 41, "Text"], Cell[130058, 6516, 258, 7, 27, "Input"], Cell[130319, 6525, 708, 15, 101, "Text"], Cell[131030, 6542, 263, 7, 27, "Input"], Cell[131296, 6551, 865, 29, 103, "Text"], Cell[132164, 6582, 160, 5, 27, "Input"], Cell[132327, 6589, 683, 15, 101, "Text"], Cell[133013, 6606, 527, 16, 27, "Input"], Cell[133543, 6624, 1198, 36, 159, "Text"], Cell[134744, 6662, 565, 18, 43, "Input"], Cell[135312, 6682, 89, 2, 26, "Text"], Cell[135404, 6686, 564, 17, 43, "Input"], Cell[135971, 6705, 528, 9, 101, "Text"], Cell[136502, 6716, 1289, 32, 297, "Text"], Cell[CellGroupData[{ Cell[137816, 6752, 29, 0, 34, "Subsection"], Cell[137848, 6754, 1690, 26, 326, "Text"] }, Closed]] }, Closed]] } ] *) (* End of internal cache information *)