(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 18204, 497]*) (*NotebookOutlinePosition[ 18882, 520]*) (* CellTagsIndexPosition[ 18838, 516]*) (*WindowFrame->Normal*) Notebook[{ Cell["Section 3.3 - The Shape of a Graph", "Title"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ StyleBox["In this lab you will first compare the graphs of a function, its \ first derivative, and its second derivative. Each derivative tells important \ information about the original function - your job is to decipher the graphs \ to determine what information is conveyed by each derivative. ", FormatType->StandardForm, FontFamily->"Arial", Background->RGBColor[0.898451, 0.863294, 0.808606]], "Next, you will explore some functions whose behavior may not be obvious \ from the first graph you view. You will see how first and second derivative \ information can be used to verify that your calculator or computer is indeed \ displaying the entire graph with all features visible." }], "Text"], Cell[TextData[{ "As usual, questions that you need to provide answers to are in a ", StyleBox["different color", FontColor->RGBColor[1, 0, 0]], ". Answers to those questions should be typed in complete sentences. When \ explanations are called for, they should be as thorough as possible. It is \ not necessary to print out this ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook, although some questions may require that you print specific \ graphs. ", StyleBox["This lab is due on Tuesday, October 21.", FontColor->RGBColor[1, 0, 0]] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["A Function and Its First Two Derivatives", "Section"], Cell["First we will define a function.", "Text"], Cell[BoxData[ \(f[x_] = 3 + 3.5\ x - 6\ x^2 - 1.5\ x^3 + x^4\)], "Input"], Cell[TextData[{ StyleBox["Now we graph that function for the ", FormatType->StandardForm, FontFamily->"Arial", Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["x", FormatType->StandardForm, FontFamily->"Arial", FontSlant->"Italic", Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox[" values -5 to 5.", FormatType->StandardForm, FontFamily->"Arial", Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["\:f35f", FormatType->StandardForm, Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["", FormatType->StandardForm] }], "Text"], Cell[BoxData[ \(Plot[f[x], {x, \(-5\), 5}]\)], "Input"], Cell[TextData[{ StyleBox["Use the graph above to answer the following questions. \ Approximate the ", FormatType->StandardForm, FontFamily->"Arial", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["x", FormatType->StandardForm, FontFamily->"Arial", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox[" values as well as you can.\n1. On what intervals is the function \ increasing? \n2. On what intervals is the function decreasing?\n3. Where are \ the local maxima of the function?\n4. Where are the local minima of the \ function?\n5. Where is the function concave up (concave up means that the \ curve of the function opens upward - like a bowl)?\n6. Where is the function \ concave down (concave down mens that the curve of the function opens \ downward)?\n7. Where are the inflection points (an inflection point is a \ point where the concavity of the function changes)?", FormatType->StandardForm, FontFamily->"Arial", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["\n", FormatType->StandardForm, Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["Now we calculate and then plot the derivative of the function.", FormatType->StandardForm, FontFamily->"Arial", Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["\:f35f", FormatType->StandardForm, Background->RGBColor[0.898451, 0.863294, 0.808606]] }], "Text"], Cell[BoxData[ \(\(\(Deriv[x_] = D[f[x], x]\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(Plot[Deriv[x], {x, \(-5\), 5}]\)], "Input"], Cell[TextData[{ StyleBox["Use the graph of the derivative (above) to answer the following \ questions. Again, approximate the ", FormatType->StandardForm, FontFamily->"Arial", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["x ", FormatType->StandardForm, FontFamily->"Arial", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["values.", FormatType->StandardForm, FontFamily->"Arial", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["\n", FormatType->StandardForm, FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["8. Where is the derivative positive?\n9. Where is the derivative \ negative?\n10. Where is the derivative zero?\n11. Where is the derivative \ increasing?\n12. Where is the derivative decreasing?\n13. Where does the \ derivative have maxima?\n14. Where does the derivative have minima?", FormatType->StandardForm, FontFamily->"Arial", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["\n", FormatType->StandardForm, FontColor->RGBColor[0.996109, 0, 0.996109], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["Now we calculate and plot the second derivative of the function.", FormatType->StandardForm, FontFamily->"Arial", Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["\:f35f", FormatType->StandardForm, Background->RGBColor[0.898451, 0.863294, 0.808606]] }], "Text"], Cell[BoxData[ \(Deriv2[x_] = D[Deriv[x], x]\)], "Input"], Cell[BoxData[ \(Plot[Deriv2[x], {x, \(-5\), 5}]\)], "Input"], Cell[TextData[{ StyleBox["Use the graph of the second derivative (above) to answer the \ following questions. Again, approximate the ", FormatType->StandardForm, FontFamily->"Arial", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["x", FormatType->StandardForm, FontFamily->"Arial", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox[" values.\n15. Where is the second derivative positive?\n16. \ Where is the second derivative negative?\n17. Where is the second derivative \ zero?\n\nNow write a paragraph or two tying all of this information together. \ What information does the first derivative give you about the function? What \ information does the second derivative give you about the function? How does \ the infomation given by the second derivative tie in with the first \ derivative? Explain in as much detail as possible. The graph below might be \ of use as it shows all three functions on the same axes - the red graph is \ the original function, the green graph is the first derivative, and the blue \ graph is the second derivative. ", FormatType->StandardForm, FontFamily->"Arial", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]], StyleBox["\:f35f", FormatType->StandardForm, FontColor->RGBColor[0.996109, 0, 0.996109], Background->RGBColor[0.898451, 0.863294, 0.808606]] }], "Text"], Cell[BoxData[ \(Plot[{f[x], Deriv[x], Deriv2[x]}, {x, \(-5\), 5}, PlotStyle \[Rule] {{RGBColor[1, 0, 0]}, {RGBColor[0, 1, 0]}, {RGBColor[ 0, 0, 1]}}]\)], "Input"], Cell[TextData[StyleBox["If you haven't already covered this point in your \ explanations from above,explain how to use the second derivative to determine \ whether a critical point is a maximum or a minimum of the function.", FormatType->StandardForm, FontFamily->"Arial", FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.898451, 0.863294, 0.808606]]], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Better Graphing Through Derivatives", "Section"], Cell[TextData[{ "Now you will see how to use everything you just learned about \ relationships between a function and its first two derivatives to determine \ whether you are seeing a complete graph of a function.\nConsider the function \ ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(f(x)\), Cell[""]}], "=", \(\(x\^2\) e\^\(\(-2\) x\)\)}], TraditionalForm]]], ". Let's try graphing this function on a standard interval of -10 to 10." }], "Text"], Cell[BoxData[{ \(f[x_] = \(x\^2\) \[ExponentialE]\^\(\(-2\) x\)\), \ "\[IndentingNewLine]", \(Plot[f[x], {x, \(-10\), 10}]\)}], "Input"], Cell[TextData[{ "From this graph, it appears that the function decreases rapidly to a \ horizontal asymptote of y = 0. Are we seeing all of the features of the \ function in this graph? Can we be sure that it is always decreasing? One way \ to check this out would be to keep plotting the function for different ranges \ of x values, but how would we know when we had tried enough sets of values? A \ more definitive way of checking is to look at the derivative of the function. \ ", StyleBox["How can the derivative help us? ", FontColor->RGBColor[1, 0, 0]], "Let's have ", StyleBox["Mathematica", FontSlant->"Italic"], " compute that derivative for us." }], "Text"], Cell[BoxData[ \(Deriv[x_] = D[f[x], x]\)], "Input"], Cell[TextData[{ "If the function really is always decreasing, as we suspect from our graph, \ then the derivative should always be negative (why?). ", StyleBox["Just from looking at the expression for the derivative, explain \ one way you can tell that it is not always negative.", FontColor->RGBColor[1, 0, 0]], " One way to determine if the derivative is always negative is by looking \ at a graph. Let's try that." }], "Text"], Cell[BoxData[ \(Plot[Deriv[x], {x, \(-10\), 10}]\)], "Input"], Cell["\<\ Well, what do you think? This graph has as many difficulties as the graph of \ our original function. It does appear likely that the derivative is always \ negative. On the other hand, we have some evidence that the derivative is not \ always negative. We'll turn to alebraic methods to resolve this definitively. \ The derivative is continuous, so if it is not always negative, then at some \ point (or points) it crosses the x-axis. We can find those points by setting \ the derivative equal to zero and solving the resulting equation.\ \>", "Text"], Cell[BoxData[ \(Solve[Deriv[x] \[Equal] 0, x]\)], "Input"], Cell["\<\ Hmmm...now we know that the derivative is zero at x = 0 and at x = 1. From \ the graph it is evident that the derivative is negative to the left of x = 0, \ so unless the graph just bumps the x-axis at x = 0, it is positive on the \ interval (0,1). We can check this algebraically by evaluating the derivative \ at a point between 0 and 1, say at x = .5.\ \>", "Text"], Cell[BoxData[ \(Deriv[ .5]\)], "Input"], Cell[TextData[{ "Whaddya know??! The derivative is postive on that interval! That means we \ have found an interval where our function is ", StyleBox["increasing", FontSlant->"Italic"], "! This also means that our function has some local extreme values. So what \ about the interval (1,\[Infinity])? The derivative is probably negative on \ this interval, unless the graph bumps the axis at x = 1. Let's check." }], "Text"], Cell[BoxData[ \(Deriv[2]\)], "Input"], Cell[TextData[{ "Yup! It's negative, alright. So...given this information we can determine \ where our original function is increasing and decreasing and where it has \ local maxima and minima. ", StyleBox["Do that now. Include a brief justification of why you know that \ there aren't any other maxima or minima of your function.", FontColor->RGBColor[1, 0, 0]] }], "Text"], Cell["\<\ I would also like to know something about concavity and inflection points of \ that function. The second derivative can give me that information, through a \ process much like the one we used with the first derivative. Let's compute \ the second derivative.\ \>", "Text"], Cell[BoxData[ \(Deriv2[x_] = D[Deriv[x], x]\)], "Input"], Cell["As before, we begin with a graph of the second derivative. ", "Text"], Cell[BoxData[ \(Plot[Deriv2[x], {x, \(-10\), 10}]\)], "Input"], Cell["\<\ Once again, we probably aren't seeing everything we should. This graph would \ lead us to believe that the second derivative is always postive, which would \ mean that the function is always concave up. But by now we know better than \ to trust our eyes, so we quickly turn to algebraic methods and look for zeros \ of the second derivative.\ \>", "Text"], Cell[BoxData[ \(Solve[Deriv2[x] \[Equal] 0, x]\)], "Input"], Cell["\<\ Aha! It is as we suspected. The second derivative is not always positive. To \ do some testing on the intervals, it might be helpful to have a decimal \ approximation of these zeros (although in reporting about the function, the \ exact values should be used).\ \>", "Text"], Cell[BoxData[ \(N[%]\)], "Input"], Cell[TextData[{ "So, on the interval (-\[Infinity], ", Cell[BoxData[ \(1\/2\ \((2 - \@2)\)\)]], "), the second derivative is postive. How about on the interval between the \ two zeros? Let's test it at ", StyleBox["x", FontSlant->"Italic"], " = 1." }], "Text"], Cell[BoxData[ \(Deriv2[1]\)], "Input"], Cell["Negative. Okay, how about to the right of the second zero?", "Text"], Cell[BoxData[ \(Deriv2[2]\)], "Input"], Cell[TextData[{ "Postive. Okay, now we know all about the concavity of our function and the \ location of inflection points. ", StyleBox["Record that information now. ", FontColor->RGBColor[1, 0, 0]] }], "Text"], Cell[TextData[{ "Armed now with all of the pertinent information about the shape of the \ graph of our function, let's see if we can find a range of ", StyleBox["x", FontSlant->"Italic"], " values for the graph that would allow us to see all of those features. We \ know that all important features occur between ", StyleBox["x", FontSlant->"Italic"], " = -1 and ", StyleBox["x", FontSlant->"Italic"], " = 2, so let's try graphing on that interval." }], "Text"], Cell[BoxData[ \(Plot[f[x], {x, \(-1\), 2}]\)], "Input"], Cell[TextData[{ "Not bad! We can make the graph look even a little bit better by extending \ the interval to ", StyleBox["x", FontSlant->"Italic"], " = 4 or so." }], "Text"], Cell[BoxData[ \(Plot[f[x], {x, \(-1\), 4}]\)], "Input"], Cell[TextData[{ "Finally! We have a good graph of our function. What's more, we are \ confident that we are seeing all of the features. We might have hit on this \ particular window had we just played with the values after the first graph we \ generated, but we still would not have known for sure that there was not \ another interval to the right of ", StyleBox["x", FontSlant->"Italic"], " = 4 on which the function would again increase. Now we know for sure." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["You Try It!", "Section"], Cell[TextData[StyleBox["For each of the functions below, determine the values \ (to 4 decimal places) of any maxima, minima, and inflection points, and give \ the intervals on which the function is increasing, decreasing, concave up, \ and concave down. Once you have located all of the significant features of \ the graph, produce a graph that clearly displays all of those features. You \ should print and turn in those graphs. (Or copy and paste them into your \ report at the appropriate places.) Be sure to label the graphs so that it is \ clear which function each represents.", FontColor->RGBColor[1, 0, 0]]], "Text"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " tips: You can use all of the commands from above. You'll need to change \ the orginal function definition and the points you evaluate the derivatives \ at. To type in a new function definition, reduce the size of this window or \ narrow it, so that you can see the palette on the right to help with \ exponents and other mathematics." }], "Text"], Cell[TextData[{ "1. ", Cell[BoxData[ \(TraditionalForm\`g(x) = 3\ x\^5 + x\^4 - 2 x + x\^2\)]], "\n2. ", Cell[BoxData[ \(TraditionalForm\`h( x) = .002 x\^5 + .06 x\^4 - .001 x\^3 - .2 x + 15\)]] }], "Text"] }, Closed]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 677}}, WindowSize->{901, 644}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, StyleDefinitions -> "DemoText.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1705, 50, 52, 0, 70, "Title"], Cell[CellGroupData[{ Cell[1782, 54, 31, 0, 54, "Section"], Cell[1816, 56, 731, 12, 86, "Text"], Cell[2550, 70, 581, 13, 67, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[3168, 88, 59, 0, 34, "Section"], Cell[3230, 90, 48, 0, 29, "Text"], Cell[3281, 92, 77, 1, 40, "Input"], Cell[3361, 95, 638, 19, 29, "Text"], Cell[4002, 116, 59, 1, 40, "Input"], Cell[4064, 119, 1582, 35, 277, "Text"], Cell[5649, 156, 86, 1, 60, "Input"], Cell[5738, 159, 63, 1, 40, "Input"], Cell[5804, 162, 1685, 42, 277, "Text"], Cell[7492, 206, 60, 1, 40, "Input"], Cell[7555, 209, 64, 1, 40, "Input"], Cell[7622, 212, 1515, 31, 241, "Text"], Cell[9140, 245, 187, 3, 60, "Input"], Cell[9330, 250, 372, 6, 48, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[9739, 261, 54, 0, 34, "Section"], Cell[9796, 263, 486, 11, 79, "Text"], Cell[10285, 276, 145, 3, 60, "Input"], Cell[10433, 281, 686, 14, 86, "Text"], Cell[11122, 297, 55, 1, 40, "Input"], Cell[11180, 300, 439, 8, 67, "Text"], Cell[11622, 310, 65, 1, 40, "Input"], Cell[11690, 313, 561, 8, 86, "Text"], Cell[12254, 323, 62, 1, 40, "Input"], Cell[12319, 326, 378, 6, 67, "Text"], Cell[12700, 334, 43, 1, 40, "Input"], Cell[12746, 337, 433, 8, 67, "Text"], Cell[13182, 347, 41, 1, 40, "Input"], Cell[13226, 350, 382, 7, 48, "Text"], Cell[13611, 359, 281, 5, 48, "Text"], Cell[13895, 366, 60, 1, 40, "Input"], Cell[13958, 369, 75, 0, 29, "Text"], Cell[14036, 371, 66, 1, 40, "Input"], Cell[14105, 374, 365, 6, 67, "Text"], Cell[14473, 382, 63, 1, 40, "Input"], Cell[14539, 385, 284, 5, 48, "Text"], Cell[14826, 392, 37, 1, 40, "Input"], Cell[14866, 395, 280, 9, 32, "Text"], Cell[15149, 406, 42, 1, 40, "Input"], Cell[15194, 409, 74, 0, 29, "Text"], Cell[15271, 411, 42, 1, 40, "Input"], Cell[15316, 414, 219, 5, 29, "Text"], Cell[15538, 421, 487, 13, 48, "Text"], Cell[16028, 436, 59, 1, 40, "Input"], Cell[16090, 439, 183, 6, 29, "Text"], Cell[16276, 447, 59, 1, 40, "Input"], Cell[16338, 450, 485, 9, 67, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[16860, 464, 30, 0, 34, "Section"], Cell[16893, 466, 626, 8, 86, "Text"], Cell[17522, 476, 418, 8, 67, "Text"], Cell[17943, 486, 245, 8, 60, "Text"] }, Closed]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)