(*********************************************************************** 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). 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[ 58804, 1768]*) (*NotebookOutlinePosition[ 59706, 1797]*) (* CellTagsIndexPosition[ 59662, 1793]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData["Simpson's Rule\[LongDash]III:\nTheorectical Error"], "Title"], Cell["\<\ by Robin Sue Sanders, Spring 1998 copyright 1998\ \>", "Subtitle", TextAlignment->Center], Cell[CellGroupData[{ Cell["Review of Simpson's Rule", "Section"], Cell[TextData[{ "In the past several labs, we've learned several rules that allow us to \ find integrals numerically. This is particularly useful when the Fundamental \ Theorem fails us because there is no easy-to-find antiderivative for the \ integrand. Last week we looked at Simpson's Rule which uses parabolic slices \ to approximate the area trapped under ", Cell[BoxData[ \(TraditionalForm\`y\ = \ f[x]\)]], " and over the interval [", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], "]." }], "Text"], Cell[TextData[{ "Recall that Simpson's Rule uses 2", StyleBox["n", FontSlant->"Italic"], " slices and ", StyleBox["n", FontSlant->"Italic"], " parabolas to approximate the integral of ", StyleBox["f", FontSlant->"Italic"], "[", StyleBox["x", FontSlant->"Italic"], "] from ", StyleBox["a", FontSlant->"Italic"], " to ", StyleBox["b", FontSlant->"Italic"], ". The appropriate formula (which was derived in ", StyleBox["SimpsonsRuleI.nb", FontSlant->"Italic"], ") is given by:" }], "Text"], Cell[BoxData[ \(Clear[a, \ b, \ f, \ n, \ x, SimpsonsRule]; \n SimpsonsRule[n_]\ := \ \n\t\ N[\ \ \(\(b - a\)\/\(6\ n\)\) \((f[a]\ + f[b]\ + \ \[Sum]\+\(i = 1\)\%\(n - 1\)\(( 2\ f[a\ + \ 2\ i\ \ \(b - a\)\/\(2 n\)]\ )\) + \ \[Sum]\+\(i = 0\)\%\(n - 1\)\((\ 4 f[a\ + \ \((2\ i\ + 1)\)\ \(b - a\)\/\(2 n\)])\))\)] \)], "Input", InitializationCell->True], Cell[TextData[{ "Remember to activate the above cell if there is NOT a blue ", StyleBox["In[#]:=", FontColor->RGBColor[0, 0, 1]], " in front of its contents!" }], "Text", FontColor->RGBColor[1, 0, 0]], Cell["The error for Simpson's Rule is given by:", "Text"], Cell[BoxData[ \(SimpsonsError[n_]\ := \ Abs[SimpsonsRule[n]\ - \ trueValue]\)], "Input", InitializationCell->True], Cell[TextData[{ "where trueValue is the actual value of the integral. ", StyleBox["Remember to activate the above cell if there is NOT a blue", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["In[#]:=", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["in front of its contents!", FontColor->RGBColor[1, 0, 0]] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Theorectical error in Simpson's Rule", "Section"], Cell[TextData[{ "We've seen experimental evidence that Simpson's Rule converges to the true \ value of a definite integral pretty fast for \"nice\" functions. Often, \ doubling the number of parabolas will reduce the error by over 90%. (In \ other words, the long term behavior of the ratio ", Cell[BoxData[ \(TraditionalForm\`SimpsonsError[2\ n]\/SimpsonsError[n]\)]], " < 0.10 for many functions.)\nThis tells us that we usually don't need \ very many parabolas to get a good estimate for an integral, but it does NOT \ tell us how to figure out how many parabolas (and slices) we need in order to \ get an estimate with a desired degree of accuracy. " }], "Text"], Cell["We have two new tasks before us:", "Text"], Cell["\<\ 1) How do we determine the accuracy of a given Simpson's Rule \ estimate for an integral when we DON'T know the exact value of the \ integral?\ \>", "Text"], Cell["\<\ 2) How do we determine the number of parabolas (and slices) that \ are needed to insure that the estimate we get using Simpson's Rule is \ sufficiently accurate?\ \>", "Text"], Cell[TextData[{ "These questions are related to each other, and the answers to both are \ based on what is known as the ", StyleBox["error term", FontSlant->"Italic"], " for Simpson's Rule. The theory behind this error term properly belongs \ in a numerical anaylisis course or an advanced calculus course, but the \ result is usually stated as the following theorem: " }], "Text"], Cell[TextData[{ "Theorem: (Error Term for Simpson's Rule) If ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], " has a continuous ", StyleBox["fourth", FontSlant->"Italic"], " derivative on the interval \n[", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], "], then the error in using ", StyleBox["n", FontSlant->"Italic"], " parabolas (and 2", StyleBox["n ", FontSlant->"Italic"], "slices) in Simpson's Rule to approximate ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], " satisfies the inequality:" }], "Text", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm \`SimpsonsError[n]\ \[LessEqual] \ \((b - a)\)\/180\ M\ \ \((width\ of\ one\ slice)\)\^4\)]] }], "Text", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "where ", StyleBox["M", FontSlant->"Italic"], " is an ", StyleBox["upper bound ", FontSlant->"Italic"], " for ", Cell[BoxData[ \(TraditionalForm\`\(|\( f\^\((4)\)\)[x]\ | \)\)]], " on the interval [", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], "]. In other words, for all ", StyleBox["x", FontSlant->"Italic"], " in the interval [", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], "], we know that ", Cell[BoxData[ \(TraditionalForm \`\(\(-M\) \[LessEqual] \(f\^\((4)\)\)[x] \[LessEqual] \ M\ \)\)]], "." }], "Text", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "Now recall that we need 2", StyleBox["n", FontSlant->"Italic"], " slices to construct ", StyleBox["n", FontSlant->"Italic"], " parabolas. So the width of each slice (the \[CapitalDelta]x in the usual \ notation) is just ", Cell[BoxData[ \(TraditionalForm\`\(b - a\)\/\(2 n\)\)]], ". So we can simplify the result in the above theorem to read:" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm \`SimpsonsError[n]\ \[LessEqual] \ \((b - a)\)\/180\ M\ \ \((\ \(b - a\)\/\(2 n\))\)\^4\)]] }], "Text", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "where ", StyleBox["M", FontSlant->"Italic"], " is an upper bound on ", Cell[BoxData[ \(TraditionalForm\`\( | \(f\^\((4)\)\)[x]\ | \)\)]], " on the interval [", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], "]" }], "Text"], Cell[TextData[{ "To make sense of what the theorem says about the error term for Simpson's \ Rule, let's look at what it says for an integral we know: ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%1\( 4\/\(1 + x\^2\)\) \[DifferentialD]x = \ \[Pi]\)]], ". To find an error bound, we need to know something about the SIZE of the \ forth derivative of ", Cell[BoxData[ \(TraditionalForm\`f[x]\ = 4\/\(1 + x\^2\)\)]], " on the interval [0, 1]. Fortunately, we can have ", StyleBox["Mathematica", FontSlant->"Italic"], " both compute the fourth derivative ", StyleBox["and", FontSlant->"Italic"], " plot it so that we can see how big it is. Here's the function and its \ fourth derivative:" }], "Text"], Cell[BoxData[{ \(Clear[f, \ derivative4]\), \(f[x_]\ := \ 4/\((1 + x^2)\); \n derivative4[x_]\ = D[f[x], {x, 4}]\)}], "Input"], Cell["\<\ (The syntax of the derivative command, D[ ] is: D[function, {x, \ number}]. So D[f[x], {x, 2}] is the second derivative, D[f[x], {x, 3}] is \ the third derivative, D[f[x], {x, 4}] is the fourth derivative and so on. \ The first derivative of f[x] can be found using D[f[x], x] or f'[x].)\ \>", "Text"], Cell[TextData[{ "Here's a plot of the fourth derivative of the integrand ", Cell[BoxData[ \(TraditionalForm\`f[x]\ = 4\/\(1 + x\^2\)\)]], " over the interval [0, 1]:" }], "Text"], Cell[BoxData[ \(\(Plot[derivative4[x], {x, 0, 1}]; \)\)], "Input"], Cell[TextData[ "From the graph, it appears that -40 \[LessEqual] deriv4[x] \[LessEqual] 90. \ We need to be more careful, so we can check out our idea with a plot:"], "Text"], Cell[BoxData[ \(\(<< Graphics`Colors`; \)\)], "Input", InitializationCell->True], Cell[TextData[{ "Remember to activate the above cell if there is NOT a blue ", StyleBox["In[#]:=", FontColor->RGBColor[0, 0, 1]], " in front of its contents! If you don't, you won't see the colors." }], "Text", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\(Plot[{\(-40\), derivative4[x], \ 90}, {x, 0, 1}, \n \t\ \ \ \ \ \ \ \ \ \ PlotStyle -> {{Red}, {Black}, {Red}}]; \)\)], "Input"], Cell["\<\ The plot, unfortunately, shows that our guessed bounds are not big \ enough. So we modify our idea. We need to go up to 100 and down to \ -45:\ \>", "Text"], Cell[BoxData[ \(\(Plot[{\(-45\), derivative4[x], \ 100}, {x, 0, 1}, \n \t\ \ \ \ \ \ \ \ \ \ PlotStyle -> {{Red}, {Black}, {Red}}]; \)\)], "Input"], Cell[TextData[{ "Good, we know that ", Cell[BoxData[ \(TraditionalForm \`\(-45\) \[LessEqual] \(f\^\((4)\)\)[x] \[LessEqual] \ 100\)]], ", so we can take ", Cell[BoxData[ \(TraditionalForm\`M = 100\)]], ". So the error in using ", StyleBox["n", FontSlant->"Italic"], " parabolas to estimate ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%1\( 4\/\(1 + x\^2\)\) \[DifferentialD]x\)]], " is less than or equal to ", Cell[BoxData[ \(TraditionalForm \`\((b - a)\)\/180\ M\ \((\ \(b - a\)\/\(2 n\))\)\^4\)]], " = 100 ", Cell[BoxData[ \(TraditionalForm \`\((1 - 0)\)\/180\ \((\ \(1 - 0\)\/\(2 n\))\)\^4 = \ \(100\/180\) \((\ 1\/\(16\ n\^4\))\)\)]], ". Let's check this out with a table:" }], "Text"], Cell[BoxData[ \(f[x_]\ := \ 4/\((x^2\ + \ 1)\); \na\ = \ 0; \nb\ = \ 1; \n trueValue\ = \ Pi; Table[{n, \ SimpsonsError[n], \ N[\ \(100\/180\) \((1\/\(16\ n\^4\))\)]}, {n, 1, \ 10}] // TableForm \)], "Input"], Cell["\<\ In each case, the theorectical error bound is larger than the \ actual error in Simpson's Rule.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " code for the theorectical error and the fourth derivative" }], "Section"], Cell[TextData[{ "In order to easily work with the theorectical error term, we need to write \ some ", StyleBox["Mathematica", FontSlant->"Italic"], " code for it. Recall that the theorectical maximum error for the estimate \ based on ", StyleBox["n", FontSlant->"Italic"], " parabolas is given by: \n\t\t\t\t\t", Cell[BoxData[ \(TraditionalForm\`\(\ \((b - a)\)\/180\ M\ \ \((\ \(b - a\)\/\(2 n\))\)\^4\)\)]], " \nwhere ", StyleBox["M", FontSlant->"Italic"], " is a bound on the size of the fourth derivative. The following cell is \ the appropriate ", StyleBox["Mathematica", FontSlant->"Italic"], " code for the error term:" }], "Text"], Cell[BoxData[ \(Clear[TheorecticalError, \ M]; TheorecticalError[n_]\ := \ N[\((\ \(b - a\)\/180)\)\ M\ \((\ \(b - a\)\/\(2 n\))\)\^4]\)], "Input",\ InitializationCell->True], Cell[TextData[{ "Remember to activate the above cell if there is NOT a blue ", StyleBox["In[#]:=", FontColor->RGBColor[0, 0, 1]], " in front of its contents!" }], "Text", FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ "Before ", StyleBox["using", FontSlant->"Italic"], " the function TheorecticalError[n], we ", StyleBox["must", FontSlant->"Italic"], " tell ", StyleBox["Mathematica", FontSlant->"Italic"], " the values of ", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", and ", StyleBox["M", FontSlant->"Italic"], ". The values of ", StyleBox["a", FontSlant->"Italic"], " and ", StyleBox["b", FontSlant->"Italic"], " come directly off the integral in question, but we have to work for ", StyleBox["M", FontSlant->"Italic"], ", the bound on the fourth derivative. So we'll also need a quick way of \ computing the fourth derivative of a function. The following cell defines a \ function, ", StyleBox["deriv4[x]", FontWeight->"Bold"], ", that computes the fourth derivative of f[x] for us:" }], "Text"], Cell[BoxData[ \(Clear[deriv4]; \nderiv4[x_]\ := \ \(\(\(\(f'\)'\)'\)'\)[x]; \)], "Input", InitializationCell->True], Cell[TextData[{ "Remember to activate the above cell if there is NOT a blue ", StyleBox["In[#]:=", FontColor->RGBColor[0, 0, 1]], " in front of its contents! You must define f[x] before you can use \ deriv4[x]." }], "Text", FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ "One (easy) way to find a good value for ", StyleBox["M", FontSlant->"Italic"], " is to graph the fourth derivative over the interval [", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], "] and use the graph to determine a reasonable bound for the size of the \ fourth derivative. It's also a good idea to check that ", StyleBox["M", FontSlant->"Italic"], " is large enough by making plots of ", Cell[BoxData[ \(TraditionalForm\`y\ = \ f[x]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y\ = \ M\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`y = \(-M\)\)]], " on the same set of axes." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Using the theorectical error bound to find out how accurate a \ particular estimate is.\ \>", "Section"], Cell[TextData[{ "Let ", Cell[BoxData[ \(TraditionalForm\`f[x]\ = \ Sin[x\^2]\)]], ". Here is the picture for the Simpson Rule estimate with 4 parabolas (and \ 8 slices):" }], "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.537323 0.0147151 0.588604 [ [.15814 .00222 -12 -9 ] [.15814 .00222 12 0 ] [.29247 .00222 -9 -9 ] [.29247 .00222 9 0 ] [.4268 .00222 -12 -9 ] [.4268 .00222 12 0 ] [.56113 .00222 -3 -9 ] [.56113 .00222 3 0 ] [.69546 .00222 -12 -9 ] [.69546 .00222 12 0 ] [.82979 .00222 -9 -9 ] [.82979 .00222 9 0 ] [.96413 .00222 -12 -9 ] [.96413 .00222 12 0 ] [.01131 .13244 -18 -4.5 ] [.01131 .13244 0 4.5 ] [.01131 .25016 -18 -4.5 ] [.01131 .25016 0 4.5 ] [.01131 .36788 -18 -4.5 ] [.01131 .36788 0 4.5 ] [.01131 .4856 -18 -4.5 ] [.01131 .4856 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .15814 .01472 m .15814 .02097 L s [(0.25)] .15814 .00222 0 1 Mshowa .29247 .01472 m .29247 .02097 L s [(0.5)] .29247 .00222 0 1 Mshowa .4268 .01472 m .4268 .02097 L s [(0.75)] .4268 .00222 0 1 Mshowa .56113 .01472 m .56113 .02097 L s [(1)] .56113 .00222 0 1 Mshowa .69546 .01472 m .69546 .02097 L s [(1.25)] .69546 .00222 0 1 Mshowa .82979 .01472 m .82979 .02097 L s [(1.5)] .82979 .00222 0 1 Mshowa .96413 .01472 m .96413 .02097 L s [(1.75)] .96413 .00222 0 1 Mshowa .125 Mabswid .05068 .01472 m .05068 .01847 L s .07754 .01472 m .07754 .01847 L s .10441 .01472 m .10441 .01847 L s .13127 .01472 m .13127 .01847 L s .18501 .01472 m .18501 .01847 L s .21187 .01472 m .21187 .01847 L s .23874 .01472 m .23874 .01847 L s .26561 .01472 m .26561 .01847 L s .31934 .01472 m .31934 .01847 L s .3462 .01472 m .3462 .01847 L s .37307 .01472 m .37307 .01847 L s .39994 .01472 m .39994 .01847 L s .45367 .01472 m .45367 .01847 L s .48053 .01472 m .48053 .01847 L s .5074 .01472 m .5074 .01847 L s .53427 .01472 m .53427 .01847 L s .588 .01472 m .588 .01847 L s .61487 .01472 m .61487 .01847 L s .64173 .01472 m .64173 .01847 L s .6686 .01472 m .6686 .01847 L s .72233 .01472 m .72233 .01847 L s .7492 .01472 m .7492 .01847 L s .77606 .01472 m .77606 .01847 L s .80293 .01472 m .80293 .01847 L s .85666 .01472 m .85666 .01847 L s .88353 .01472 m .88353 .01847 L s .91039 .01472 m .91039 .01847 L s .93726 .01472 m .93726 .01847 L s .99099 .01472 m .99099 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .13244 m .03006 .13244 L s [(0.2)] .01131 .13244 1 0 Mshowa .02381 .25016 m .03006 .25016 L s [(0.4)] .01131 .25016 1 0 Mshowa .02381 .36788 m .03006 .36788 L s [(0.6)] .01131 .36788 1 0 Mshowa .02381 .4856 m .03006 .4856 L s [(0.8)] .01131 .4856 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(1)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .04415 m .02756 .04415 L s .02381 .07358 m .02756 .07358 L s .02381 .10301 m .02756 .10301 L s .02381 .16187 m .02756 .16187 L s .02381 .1913 m .02756 .1913 L s .02381 .22073 m .02756 .22073 L s .02381 .27959 m .02756 .27959 L s .02381 .30902 m .02756 .30902 L s .02381 .33845 m .02756 .33845 L s .02381 .39731 m .02756 .39731 L s .02381 .42674 m .02756 .42674 L s .02381 .45617 m .02756 .45617 L s .02381 .51503 m .02756 .51503 L s .02381 .54446 m .02756 .54446 L s .02381 .57389 m .02756 .57389 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .02499 .01472 L .02605 .01473 L .02729 .01474 L .02846 .01476 L .03053 .01481 L .03279 .01488 L .03527 .01498 L .0379 .01512 L .04262 .01544 L .04749 .01586 L .05205 .01634 L .06244 .01776 L .07305 .01966 L .08274 .0218 L .10458 .02801 L .12357 .035 L .14429 .04429 L .18493 .06757 L .22406 .09621 L .26565 .13313 L .30571 .17469 L .34426 .21968 L .38527 .27207 L .42475 .32577 L .46273 .37896 L .50315 .43524 L .54206 .48664 L .58342 .53513 L .60412 .55584 L .62326 .57229 L .64303 .58607 L .66159 .59562 L .67058 .59895 L .6752 .60031 L .68026 .60152 L .68549 .60245 L .68835 .60282 L .68972 .60296 L .69101 .60307 L .6922 .60316 L .6935 .60323 L .69468 .60328 L .69577 .60331 L .69695 .60332 L .69824 .60331 L .69889 .6033 L .6996 .60328 L .70086 .60324 L .70155 .6032 L Mistroke .7022 .60316 L .70341 .60307 L .70612 .60281 L .70846 .6025 L .71099 .60208 L .71675 .60081 L .72204 .59924 L .73182 .59528 L .74214 .58959 L .76181 .57424 L .78016 .55439 L .80055 .52584 L .81912 .49376 L .85902 .40524 L .8974 .29603 L .93824 .15772 L .97619 .01472 L Mfstroke .628 .126 .941 r .02381 .01472 m .02381 .01472 L s .14286 .01472 m .14286 .0436 L s .2619 .01472 m .2619 .12955 L s .38095 .01472 m .38095 .26638 L s .5 .01472 m .5 .43092 L s .61905 .01472 m .61905 .56891 L s .7381 .01472 m .7381 .59201 L s .85714 .01472 m .85714 .41 L s .97619 .01472 m .97619 .01472 L s 0 1 0 r .02381 .01472 m .0241 .01472 L .02437 .01472 L .02497 .01472 L .02553 .01473 L .02605 .01473 L .02723 .01475 L .02851 .01477 L .03109 .01484 L .03347 .01493 L .03858 .0152 L .044 .01559 L .05346 .01657 L .06353 .01801 L .07323 .01978 L .08353 .02207 L .09346 .02469 L .10301 .02757 L .11317 .03105 L .12295 .03479 L .13335 .03919 L .14336 .04384 L .153 .0487 L .16325 .05427 L .17312 .06004 L .18262 .06595 L .19272 .07265 L .20245 .07949 L .21279 .08717 L .22275 .09498 L .23233 .10287 L .24253 .11167 L .25235 .12054 L .26178 .12943 L .2619 .12955 L s .2619 .12955 m .27156 .13961 L .2821 .1508 L .29199 .16151 L .3015 .17198 L .31163 .18333 L .32138 .19444 L .33174 .20645 L .34172 .21823 L .35133 .22974 L .36154 .24218 L .37138 .25435 L .38084 .26623 L .39091 .27908 L .40061 .29163 L .41091 .30517 L .42084 .31842 L .43039 .33134 L .44055 .34528 L .45033 .3589 L .46073 .37357 L .47074 .38791 L .48038 .4019 L .49063 .41697 L .5 .43092 L s .5 .43092 m .50966 .4464 L .52019 .46242 L .53009 .47664 L .5396 .48957 L .54973 .50253 L .55948 .51422 L .56984 .5258 L .57982 .53613 L .58942 .54531 L .59964 .55425 L .60948 .56207 L .61894 .56884 L .62901 .57525 L .6387 .58064 L .64901 .58554 L .65893 .58945 L .66848 .59245 L .67381 .59381 L .67864 .59484 L .68374 .59572 L .68843 .59634 L .69072 .59658 L .69324 .5968 L .69577 .59696 L .69715 .59703 L .69844 .59708 L .69969 .59711 L .7004 .59713 L .70074 .59713 L .70106 .59714 L .70135 .59714 L .70163 .59714 L .70193 .59714 L .70226 .59714 L .70255 .59714 L .70288 .59714 L .70304 .59714 L .70322 .59714 L .70353 .59714 L .70384 .59714 L .70413 .59713 L .70478 .59712 L .70535 .59711 L .70596 .5971 L .70701 .59706 L .70817 .59701 L .71071 .59687 L .71303 .59669 L .71825 .59614 L Mistroke .72301 .59544 L .72819 .59447 L .7381 .59201 L Mfstroke .7381 .59201 m .74775 .58519 L .75829 .57616 L .76818 .56615 L .77769 .55514 L .78782 .54192 L .79757 .52774 L .80793 .5111 L .81791 .49353 L .82752 .47522 L .83773 .45422 L .84757 .43251 L .85703 .41027 L .8671 .3851 L .8768 .35944 L .8871 .33061 L .89703 .30132 L .90658 .27175 L .91674 .23878 L .92652 .20556 L .93692 .16869 L .94693 .13163 L .95657 .09454 L .96682 .05356 L .97619 .01472 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOol7000=0n0C00000e2N0000000M0000 0e2N0000000M00000e2N0000000M00000e2N0000000M00000e2N0000000N00000e2N0000000M0000 0e2N0000000M00000`?P000000040003Ool001Eoo`03001oogoo00Aoo`03001oogoo00=oo`<3h0<0 00=oo`03001oogoo00Aoo`04001oogooD9h3Ool00`00Oomoo`04Ool00`00Oomoo`05Ool00`00Oomo o`04Ool00`00Oomoo`03Ool00e2N001oo`05Ool00`00Oomoo`05Ool00`00Oomoo`04Ool00`00Oomo o`04Ool00`00Oom@WP04Ool00`00Oomoo`05Ool00`00Oomoo`04Ool00`00Oomoo`04Ool00`00Oomo o`02Ool00e2NOol00007Ool00`00Oomoo`04Ool00`00Oomoo`04Ool00`00Oomoo`04Ool00`00D9io o`05Ool00`00Oomoo`04Ool00`00Oomoo`04Ool00`00Oomoo`04Ool00`00Oomoo`02Ool0152NOomo o`001Woo00<007ooOol017oo00<007ooOol017oo00<007ooOol01Goo00=@WWooOol017oo00<007oo Ool017oo00<007ooOol017oo00<007ooOol01Goo00@007ooOol3h0=oo`03001oogoo00=oo`005Goo 00<007ooOol03Goo1@?P00<007ooOol027oo00=@WWooOol07Goo00=@WWooOol07Goo00=@WWooOol0 7Goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol07Goo00<3h7ooOol0 1goo000EOol00`00Oomoo`0BOol30n08Ool00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0M Ool00e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomoo`0LOol00`?POomoo`08 Ool001Eoo`03001oogoo01Eoo`<3h0Eoo`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01eo o`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo01aoo`030n1oogoo00Qo o`005Goo00<007ooOol067oo0P?P0goo00=@WWooOol07Goo00=@WWooOol07Goo00=@WWooOol07Goo 00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol077oo00<3h7ooOol027oo 000EOol00`00Oomoo`0JOol30n000e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0MOol00e2N Oomoo`0MOol00e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomoo`0KOol00`?P001oo`09Ool001Eo o`03001oogoo01aoo`030003h0?P01ioo`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01eo o`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo01]oo`030n1oogoo00Uoo`005Goo0P007Woo 0P0000<3h7ooOol06goo00=@WWooOol07Goo00=@WWooOol07Goo00=@WWooOol07Goo00=@WWooOol0 7Woo00=@WWooOol07Goo00=@WWooOol06goo00<3h7ooOol02Goo000EOol00`00Oomoo`0OOol00`00 0n03h00KOol00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0NOol00e2N Oomoo`0MOol00e2NOomoo`0JOol00`?P001oo`0:Ool001Eoo`03001oogoo025oo`030003h0?P01Uo o`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eo o`03D9ioogoo01Yoo`030n1oogoo00Yoo`005Goo00<007ooOol08goo00<000?POol05goo00=@WWoo Ool07Goo00=@WWooOol07Goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWoo Ool06Woo00<3h7ooOol02Woo000EOol00`00Oomoo`0TOol010000n03h0?P5Goo00=@WWooOol07Goo 00=@WWooOol07Goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol06Woo 00<3h7ooOol02Woo000EOol00`00Oomoo`0VOol200000`?POomoo`0BOol00e2NOomoo`0MOol00e2N Oomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomoo`0IOol00`?P 001oo`0;Ool001Eoo`03001oogoo02Qoo`030003h7oo019oo`03D9ioogoo01eoo`03D9ioogoo01eo o`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo01Uoo`030n1oogoo00]o o`005Goo0P00:goo0P?P4Goo00=@WWooOol07Goo00=@WWooOol07Goo00=@WWooOol07Goo00=@WWoo Ool07Woo00=@WWooOol07Goo00=@WWooOol06Goo00<3h7ooOol02goo000EOol00`00Oomoo`0[Ool0 0`000n1oo`0?Ool00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0NOol0 0e2NOomoo`0MOol00e2NOomoo`0HOol00`?P001oo`0Ool001Eoo`03001oogoo03Qoo`83h0=oo`03D9ioogoo01eo o`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo01Io o`030n1oogoo00ioo`005Goo00<007ooOol0>Goo00D000?POomooe2N01moo`03D9ioogoo01eoo`03 D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo01Ioo`030n1oogoo00ioo`00 5Goo00<007ooOol0>Woo00@000?POom@WQmoo`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo 01ioo`03D9ioogoo01eoo`03D9ioogoo01Ioo`030n1oogoo00ioo`000Woo0P0017oo0P000goo1000 17oo00<007ooOol0>goo00<3h7ooD9h07goo00=@WWooOol07Goo00=@WWooOol07Goo00=@WWooOol0 7Woo00=@WWooOol07Goo00=@WWooOol05Goo00<3h7ooOol03goo00001Goo001oogoo00000goo0P00 17oo00<007ooOol017oo00<007ooOol0?7oo00<3h52NOol07Woo00=@WWooOol07Goo00=@WWooOol0 7Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol05Goo00<3h7ooOol03goo00001Goo001o ogoo00002Woo00<007ooOol00goo0`00?Goo0P?P7Woo00=@WWooOol07Goo00=@WWooOol07Goo00=@ WWooOol07Woo00=@WWooOol07Goo00=@WWooOol05Goo00<3h7ooOol03goo00001Goo001oogoo0000 2goo00<007ooOol00Woo00<007ooOol0?Woo00<000?POol077oo00=@WWooOol07Goo00=@WWooOol0 7Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol05Goo00<3h7ooOol03goo00001Goo001o ogoo000027oo00@007ooOol000Aoo`03001oogoo03moo`030003h7oo01]oo`03D9ioogoo01eoo`03 D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo01Aoo`030003h7oo011oo`00 0Woo0P002Woo0P001Goo00<007ooOol0@7oo00<000?POol06Woo00=@WWooOol07Goo00=@WWooOol0 7Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol057oo00<3h7ooOol047oo000EOol00`00 Oomoo`11Ool00`000n1oo`0IOol00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0NOol00e2N Oomoo`0MOol00e2NOomoo`0DOol00`?POomoo`0@Ool001Eoo`03001oogoo049oo`030003h7oo01Qo o`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo01Ao o`030n1oogoo011oo`005Goo00<007ooOol0@goo00<3h7ooOol05goo00=@WWooOol07Goo00=@WWoo Ool07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol04goo00<3h7ooOol04Goo000EOol0 0`00Oomoo`14Ool00`?POomoo`0FOol00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0NOol0 0e2NOomoo`0MOol00e2NOomoo`0COol00`?POomoo`0AOol001Eoo`8004Ioo`030n1oogoo01Eoo`03 D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo01=oo`03 0n1oogoo015oo`005Goo00<007ooOol0AWoo00<3h7ooOol057oo00=@WWooOol07Goo00=@WWooOol0 7Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol04Woo00<3h000Ool04Woo000EOol00`00 Oomoo`17Ool00`?POomoo`0COol00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0NOol00e2N Oomoo`0MOol00e2NOomoo`0BOol00`?POomoo`0BOol001Eoo`03001oogoo04Qoo`030n1oogoo019o o`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo019o o`030n1oogoo019oo`005Goo00<007ooOol0BGoo00<3h7ooOol04Goo00=@WWooOol07Goo00=@WWoo Ool07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol04Woo00<3h7ooOol04Woo000EOol0 0`00Oomoo`19Ool00`000n1oo`0AOol00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0NOol0 0e2NOomoo`0MOol00e2NOomoo`0AOol00`?POomoo`0COol001Eoo`03001oogoo04Yoo`030003h7oo 011oo`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo 015oo`030n1oogoo01=oo`005Goo00<007ooOol0Bgoo00<3h7ooOol03goo00=@WWooOol07Goo00=@ WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol04Goo00<3h7ooOol04goo000E Ool2001=Ool00`?POomoo`0>Ool00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0NOol00e2N Oomoo`0MOol00e2NOomoo`0@Ool00`?POomoo`0DOol001Eoo`03001oogoo04eoo`030n1oogoo00eo o`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo011o o`030n1oogoo01Aoo`005Goo00<007ooOol0CWoo00<3h7ooOol037oo00=@WWooOol07Goo00=@WWoo Ool07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol047oo00<3h7ooOol057oo000EOol0 0`00Oomoo`1>Ool00`000n1oo`0Ool00`?POomoo`0FOol001Eoo`03001oogoo05Aoo`030n1oogoo 00Ioo`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo 00eoo`030003h7oo01Moo`005Goo00<007ooOol0EGoo00<3h7ooOol01Goo00=@WWooOol07Goo00=@ WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol03Goo00<000?POol05goo000E Ool00`00Oomoo`1FOol00`?POomoo`04Ool00e2NOomoo`0MOol00e2NOomoo`0MOol00e2NOomoo`0N Ool00e2NOomoo`0MOol00e2NOomoo`0=Ool00`?POomoo`0GOol001Eoo`03001oogoo05Moo`030n1o ogoo00=oo`03D9ioogoo01eoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9io ogoo00eoo`030n1oogoo01Moo`000Woo0P0017oo0P0017oo0`0017oo00<007ooOol0Egoo00<000?P Ool00goo00=@WWooOol07Goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWoo Ool037oo00<000?POol067oo00001Goo001oogoo00000goo0P001Goo00<007ooOol00goo00<007oo Ool0F7oo00<000?POol00Woo00=@WWooOol07Goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWoo Ool07Goo00=@WWooOol037oo00<3h7ooOol067oo00001Goo001oogoo000027oo100017oo0`00FGoo 00D000?POomooe2N01moo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo 00aoo`030n1oogoo01Qoo`0000Eoo`00Oomoo`0000Qoo`03001oo`0000Eoo`03001oogoo05Yoo`04 0003h7ooD9hOOol00e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomoo`0;Ool0 0`000n1oo`0IOol00005Ool007ooOol00008Ool00`00Ool00005Ool00`00Oomoo`1JOol010000n1o oe2N7goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol02goo00<3h7oo Ool06Goo0002Ool2000:Ool20005Ool00`00Oomoo`1KOol00`000n1@WP0OOol00e2NOomoo`0MOol0 0e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomoo`0;Ool00`?POomoo`0IOol001Eoo`03001oogoo 05aoo`030003h7oo01ioo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo 00]oo`030n1oogoo01Uoo`005Goo00<007ooOol0GGoo00<3h7ooOol07Goo00=@WWooOol07Goo00=@ WWooOol07Woo00=@WWooOol07Goo00=@WWooOol02Woo00<3h7ooOol06Woo000EOol00`00Oomoo`1N Ool00`?POomoo`0LOol00e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomoo`0: Ool00`?POomoo`0JOol001Eoo`03001oogoo05ioo`030003h7oo01aoo`03D9ioogoo01eoo`03D9io ogoo01ioo`03D9ioogoo01eoo`03D9ioogoo00Yoo`030n1oogoo01Yoo`005Goo0P00H7oo00<000?P Ool06goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol02Goo00<3h7oo Ool06goo000EOol00`00Oomoo`1POol00`000n1oo`0JOol00e2NOomoo`0MOol00e2NOomoo`0NOol0 0e2NOomoo`0MOol00e2NOomoo`09Ool00`?POomoo`0KOol001Eoo`03001oogoo065oo`030003h7oo 01Uoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo00Uoo`030n1oogoo 01]oo`005Goo00<007ooOol0HWoo00<000?POol067oo00=@WWooOol07Goo00=@WWooOol07Woo00=@ WWooOol07Goo00=@WWooOol027oo00<3h7ooOol077oo000EOol00`00Oomoo`1ROol00`000n1oo`0H Ool00e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomoo`08Ool00`?POomoo`0L Ool001Eoo`03001oogoo06=oo`030003h7oo01Moo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9io ogoo01eoo`03D9ioogoo00Qoo`030n1oogoo01aoo`005Goo00<007ooOol0I7oo00<3h7ooOol05Woo 00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol01goo00<3h7ooOol07Goo 000EOol00`00Oomoo`1UOol00`?POomoo`0EOol00e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomo o`0MOol00e2NOomoo`07Ool00`?POomoo`0MOol001Eoo`8006Ioo`030n1oogoo01Eoo`03D9ioogoo 01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo00Moo`030n1oogoo01eoo`005Goo00<0 07ooOol0IWoo00<3h7ooOol057oo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@ WWooOol01Woo00<3h7ooOol07Woo000EOol00`00Oomoo`1WOol00`?POomoo`0COol00e2NOomoo`0M Ool00e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomoo`06Ool00`?POomoo`0NOol001Eoo`03001o ogoo06Qoo`030n1oogoo019oo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9io ogoo00Ioo`030n1oogoo01ioo`005Goo00<007ooOol0JGoo00<3h7ooOol04Goo00=@WWooOol07Goo 00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol01Goo00<000?POol07goo000EOol00`00Oomo o`1YOol00`000n1oo`0AOol00e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomo o`05Ool00`?POomoo`0OOol001Eoo`03001oogoo06Yoo`030003h7oo011oo`03D9ioogoo01eoo`03 D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo00Eoo`030n1oogoo01moo`005Goo00<007ooOol0 Jgoo00<3h7ooOol03goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol0 17oo00<000?POol087oo000EOol2001/Ool00`000n1oo`0?Ool00e2NOomoo`0MOol00e2NOomoo`0N Ool00e2NOomoo`0MOol00e2NOomoo`04Ool00`?POomoo`0POol001Eoo`03001oogoo06aoo`030n1o ogoo00ioo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioogoo00=oo`030003 h7oo025oo`005Goo00<007ooOol0KGoo00<3h7ooOol03Goo00=@WWooOol07Goo00=@WWooOol07Woo 00=@WWooOol07Goo00=@WWooOol00goo00<000?POol08Goo000EOol00`00Oomoo`1]Ool00`000n1o o`0=Ool00e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomoo`03Ool00`?POomo o`0QOol001Eoo`03001oogoo06ioo`030003h7oo00aoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03 D9ioogoo01eoo`03D9ioogoo009oo`030003h7oo029oo`005Goo00<007ooOol0Kgoo00<000?POol0 2goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WWooOol00Woo00<000?POol0 8Woo0002Ool20004Ool20004Ool20005Ool00`00Oomoo`1_Ool00`000n1oo`0;Ool00e2NOomoo`0M Ool00e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOomoo`02Ool00`?POomoo`0ROol00005Ool007oo Ool00003Ool20003Ool01000Oomoo`0017oo00<007ooOol0L7oo00<000?POol02Woo00=@WWooOol0 7Goo00=@WWooOol07Woo00=@WWooOol07Goo00I@WWooOomoo`000n0TOol00005Ool007ooOol00008 Ool01000Oomoo`0017oo0`00LGoo00<000?POol02Goo00=@WWooOol07Goo00=@WWooOol07Woo00=@ WWooOol07Goo00E@WWooOomoo`?P02Eoo`0000Eoo`00Oomoo`0000Qoo`<000Eoo`03001oogoo075o o`03001oo`?P00Uoo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`05D9ioogoo0003 h00UOol00005Ool007ooOol00008Ool00`00Oomoo`05Ool00`00Oomoo`1bOol00`000n1oo`08Ool0 0e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomoo`0MOol0152NOomoo`?P9Woo0002Ool2000:Ool3 0004Ool00`00Oomoo`1cOol00`000n1oo`07Ool00e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomo o`0MOol0152NOomoo`?P9Woo000EOol00`00Oomoo`1cOol00`000n1oo`07Ool00e2NOomoo`0MOol0 0e2NOomoo`0NOol00e2NOomoo`0MOol00e2NOol3h00WOol001Eoo`03001oogoo07Aoo`030003h7oo 00Ioo`03D9ioogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9ioo`?P02Moo`005Goo00<0 07ooOol0MGoo00<3h7ooOol01Goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@ WWoo0n009goo000EOol00`00Oomoo`1fOol00`?POomoo`04Ool00e2NOomoo`0MOol00e2NOomoo`0N Ool00e2NOomoo`0MOol00e2N0n1oo`0WOol001Eoo`8007Moo`030003h7oo00Aoo`03D9ioogoo01eo o`03D9ioogoo01ioo`03D9ioogoo01eoo`03D9h3h7oo02Moo`005Goo00<007ooOol0Mgoo00<000?P Ool00goo00=@WWooOol07Goo00=@WWooOol07Woo00=@WWooOol07Goo00=@WP?POol09goo000EOol0 0`00Oomoo`1hOol00`?POomoo`02Ool00e2NOomoo`0MOol00e2NOomoo`0NOol00e2NOomoo`0MOol0 0`?POomoo`0WOol001Eoo`03001oogoo07Uoo`050n1oogooOom@WP0OOol00e2NOomoo`0NOol00e2N Oomoo`0MOol00`?POomoo`0WOol001Eoo`03001oogoo07Uoo`050003h7ooOom@WP0OOol00e2NOomo o`0NOol00e2NOomoo`0LOol00`000n1oo`0XOol001Eoo`03001oogoo07Yoo`040003h7ooD9hOOol0 0e2NOomoo`0NOol00e2NOomoo`0LOol00`?POomoo`0XOol001Eoo`03001oogoo07]oo`030n1ooe2N 01moo`03D9ioogoo01ioo`03D9ioogoo01]oo`030003h7oo02Uoo`005Goo00<007ooOol0Ngoo00<0 00?PD9h07goo00=@WWooOol07Woo00=@WWooOol06goo00<3h7ooOol0:Goo000EOol2001mOol00`?P D9ioo`0NOol00e2NOomoo`0NOol00e2NOomoo`0JOol00`000n1oo`0ZOol001Eoo`03001oogoo07eo o`030n1oogoo01eoo`03D9ioogoo01ioo`03D9ioogoo01Yoo`030n1oogoo02Yoo`005Goo00<007oo Ool0OGoo00<3h000Ool07Goo00=@WWooOol07Woo00=@WWooOol06Woo00<3h7ooOol0:Woo000EOol0 0`00Oomoo`1nOol00`?POomoo`0LOol00e2NOomoo`0NOol00e2NOomoo`0IOol00`?POomoo`0[Ool0 01Eoo`03001oogoo07moo`030n1oogoo01]oo`03D9ioogoo01ioo`03D9ioogoo01Uoo`030n1oogoo 02]oo`005Goo00<007ooOol0P7oo00<3h7ooOol06Woo00=@WWooOol07Woo00=@WWooOol067oo00<3 h7ooOol0;7oo000EOol00`00Oomoo`20Ool00`?P001oo`0JOol00e2NOomoo`0NOol00e2NOomoo`0H Ool00`?POomoo`0/Ool001Eoo`03001oogoo085oo`030n0007oo01Uoo`03D9ioogoo01ioo`03D9io ogoo01Moo`030n1oogoo02eoo`005Goo0P00Pgoo00<3h7ooOol067oo00=@WWooOol07Woo00=@WWoo Ool05goo00<3h7ooOol0;Goo000EOol00`00Oomoo`23Ool00`?POomoo`0GOol00e2NOomoo`0NOol0 0e2NOomoo`0FOol00`?POomoo`0^Ool001Eoo`03001oogoo08=oo`030n0007oo01Moo`03D9ioogoo 01ioo`03D9ioogoo01Ioo`030n1oogoo02ioo`005Goo00<007ooOol0Q7oo00<3h000Ool05Woo00=@ WWooOol07Woo00=@WWooOol05Goo00<000?POol0;goo000EOol00`00Oomoo`24Ool00`?P001oo`0F Ool00e2NOomoo`0NOol00e2NOomoo`0EOol00`?POomoo`0_Ool001Eoo`03001oogoo08Eoo`030n00 07oo01Eoo`03D9ioogoo01ioo`03D9ioogoo01Eoo`030n1oogoo02moo`000Woo0P0017oo0P0017oo 0P001Goo00<007ooOol0QGoo00<3h7oo00005Goo00=@WWooOol07Woo00=@WWooOol057oo00<000?P Ool0<7oo00001Goo001oogoo00000goo0P000goo00@007ooOol000Aoo`03001oogoo08Ioo`030n1o o`0001Aoo`03D9ioogoo01ioo`03D9ioogoo01Aoo`030n1oogoo031oo`0000Eoo`00Oomoo`0000Qo o`04001oogoo0004Ool30027Ool00`?POol0000COol00e2NOomoo`0NOol00e2NOomoo`0COol00`?P Oomoo`0aOol00005Ool007ooOol00009Ool20005Ool00`00Oomoo`28Ool00`?P001oo`0BOol00e2N Oomoo`0NOol00e2NOomoo`0COol00`?POomoo`0aOol00005Ool007ooOol00008Ool01000Oomoo`00 17oo00<007ooOol0RGoo00<3h000Ool04Goo00=@WWooOol07Woo00=@WWooOol04Woo00<3h7ooOol0 Ool0 0e2NOomoo`0NOol00e2NOomoo`0?Ool00`?P001oo`0eOol001Eoo`03001oogoo08eoo`030n1oo`00 00eoo`03D9ioogoo01ioo`03D9ioogoo00moo`030n1oogoo03Eoo`005Goo0P00Sgoo00<3h000Ool0 37oo00=@WWooOol07Woo00=@WWooOol03Woo00<3h000Ool0=Woo000EOol00`00Oomoo`2?Ool00`?P 001oo`0;Ool00e2NOomoo`0NOol00e2NOomoo`0>Ool00`?POomoo`0fOol001Eoo`03001oogoo091o o`030n0007oo00Yoo`03D9ioogoo01ioo`03D9ioogoo00eoo`030003h7oo03Moo`005Goo00<007oo Ool0TGoo00<3h000Ool02Goo00=@WWooOol07Woo00=@WWooOol03Goo00<3h7ooOol0=goo000EOol0 0`00Oomoo`2BOol00`?P001oo`08Ool00e2NOomoo`0NOol00e2NOomoo`0Goo000EOol2002EOol00`?P001oo`06Ool00e2NOomoo`0NOol00e2NOomoo`0:Ool00`?P Oomoo`0jOol001Eoo`03001oogoo09Eoo`83h0Ioo`03D9ioogoo01ioo`03D9ioogoo00Uoo`030n00 07oo03]oo`005Goo00<007ooOol0Ugoo00<3h7ooOol00goo00=@WWooOol07Woo00=@WWooOol027oo 00<3h000Ool0?7oo000EOol00`00Oomoo`2HOol20n03Ool00e2NOomoo`0NOol00e2NOomoo`07Ool0 0`?P001oo`0mOol001Eoo`03001oogoo09Uoo`050003h7ooOom@WP0POol00e2NOomoo`07Ool00`?P 001oo`0mOol001Eoo`03001oogoo09]oo`030n1ooe2N021oo`03D9ioogoo00Ioo`030n0007oo03io o`005Goo00<007ooOol0W7oo00<3h52NOol07goo00=@WWooOol01Goo00<3h000Ool0?goo000EOol0 0`00Oomoo`2MOol20n0OOol00e2NOomoo`04Ool00`?P001oo`10Ool001Eoo`800:1oo`030n0007oo 01aoo`03D9ioogoo00=oo`030n0007oo045oo`005Goo00<007ooOol0X7oo0`?P6goo00A@WWooOomo o`83h003001oogoo045oo`005Goo00<007ooOol0XWoo00<000?P0n006Goo00E@WWooOol3h00004Eo o`005Goo00<007ooOol0Xgoo00<007oo0n0067oo00A@WWoo0n0004Ioo`005Goo00<007ooOol0Y7oo 0P000P?P5Woo00=@WP?P0000Agoo000EOol00`00Oomoo`2VOol200040n0AOol20n000`00Oomoo`16 Ool000eoo`<000Eoo`03001oogoo0:Uoo`800003Ool3h0?P00<3h0Moo`D3h003001oogoo04Qoo`00 3Woo00<007ooOol017oo00<007ooOol0Zgoo0P0017oo1`?P00=oo`000000CGoo000>Ool00`00Oomo o`04Ool3002]Ool<001?Ool000ioo`03001oogoo00Aoo`03001oogoo0?moo`Uoo`003Woo00<007oo Ool017oo00<007ooOol0ogoo2Goo000=Ool20006Ool00`00Oomoo`3oOol9Ool001Eoo`03001oogoo 0?moo`Uoo`00ogoo8Goo0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.147471, -0.0820341, 0.00689996, 0.00629882}}], Cell[TextData[{ "This estimate looks good, but how accurate is it? In other words, when we \ ask ", StyleBox["Mathematica", FontSlant->"Italic"], " for:" }], "Text"], Cell[BoxData[ \(f[x_]\ := \ Sin[x^2]; \na\ = \ 0; \nb\ = \ Sqrt[Pi]; \n SimpsonsRule[4]\)], "Input"], Cell["How many decimal places of accuracy can we guarentee?", "Text"], Cell["\<\ We start by finding a bound for the fourth derivative. Here's the \ trigonometric expression for the fourth derivative:\ \>", "Text"], Cell[BoxData[ \(f[x_]\ := \ Sin[x^2]; \nderiv4[x]\)], "Input"], Cell[TextData[{ "To find the bound, let's plot over the interval [0, ", Cell[BoxData[ \(TraditionalForm\`\@\[Pi]\)]], "]:" }], "Text"], Cell[BoxData[ \(\(Plot[deriv4[x], \ {x, \ 0, \ Sqrt[Pi]}]; \)\)], "Input"], Cell["\<\ Looks like the absolute value of deriv4[x] is always less than \ about 170. Here's the check:\ \>", "Text"], Cell[BoxData[ \(\(Plot[{\(-170\), \ deriv4[x], \ 170}, \ {x, \ 0, \ Sqrt[Pi]}, \n \ \ \ \ \ \ \ \ \ \ \ \ PlotStyle -> {{Red}, {Black}, {Red}}]; \)\)], "Input"], Cell[TextData[{ "The plot shows 170 works\[LongDash]i.e. we can take ", StyleBox["M", FontSlant->"Italic"], " in the theorectical error bound to be 170. The theorecical error in \ using 4 parabolas is given by TheorecticalError[4]:" }], "Text"], Cell[BoxData[ \(f[x_]\ := \ Sin[x^2]; \na\ = \ 0; \nb\ = \ Sqrt[Pi]; \nM\ = \ 170; \nTheorecticalError[4]\)], "Input"], Cell["\<\ Since the theorectical error is less than 0.005, we can guarentee \ that SimpsonsRule[4] is accurate to two decimal places.\ \>", "Text"], Cell[TextData[{ "Exercise 1: Using the same integral as in the example, ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%\(\@\[Pi]\)Sin[x\^2] \[DifferentialD]x\)]], ", determine the accuracy of \nSimpson's Rule if we use 8 parabolas \ instead of 4." }], "Text", FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ "Exercise 2: Consider the integral ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_\(\(-1\)/2\)\%1 Sec[x] \[DifferentialD]x\)]], ". Determine the accuracy of the Simpson's Rule estimate if we use 5 \ parabolas. What's the accuracy if we use 20 parabolas? (Hint: You may need \ to add the option PlotRange->All to the plot command used to make the plot of \ the fourth derivative.)" }], "Text", FontColor->RGBColor[1, 0, 0]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Using the theorectical error bound to figure out how many parabolas \ (and slices) are needed to insure that the estimate is sufficiently accurate.\ \ \>", "Section"], Cell[TextData[{ "We can also use the theorectical error to figure out ", StyleBox["how many parabolas ", FontSlant->"Italic"], " are needed to insure a suffiecently accurate estimate. For example, \ suppose we need an estimate of ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_1\%4\(\@\( x\^2 + \ 1\)\%3\) \[DifferentialD]x\)]], " that is \naccurate to 6 decimal places. How many parabolas (and slices) \ do we need to make sure that the Simpson's Rule estimate is accurate enough?" }], "Text"], Cell[TextData[{ "First, we need to realize that \"accurate to 6 decimal places\" means that \ the size of the error must be less than ", Cell[BoxData[ \(TraditionalForm\`5\[Cross]10\^\(-7\)\)]], ". We also know that:\n\t\t\t|true error using ", StyleBox["n", FontSlant->"Italic"], " parabolas| \[LessEqual] TheorecticalError[", StyleBox["n", FontSlant->"Italic"], "]\nSo if we find an ", StyleBox["n", FontSlant->"Italic"], " that makes TheorecticalError[", StyleBox["n", FontSlant->"Italic"], "] \[LessEqual] ", Cell[BoxData[ \(TraditionalForm\`5\[Cross]10\^\(-7\)\)]], ", we'll know how many parabolas (and slices) to use." }], "Text"], Cell[TextData[{ "Step 1: We must find a bound on the fourth derivative of ", Cell[BoxData[ \(TraditionalForm\`\@\(x\^2 + \ 1\)\%3\)]], " over the interval [1, 4]. Here's the fourth derivative:" }], "Text"], Cell[BoxData[ \(f[x_]\ := \ \((x^2\ + \ 1)\)^\((1/3)\); \nderiv4[x]\)], "Input"], Cell[TextData[ "What a mess. Here's the plot\[LongDash]note the PlotRange->All makes sure \ we see all the y-values:"], "Text"], Cell[BoxData[ \(\(Plot[deriv4[x], {x, 1, 4}, \ PlotRange -> All]; \)\)], "Input"], Cell[TextData[{ "These y-values are all less than 1. So we can take ", StyleBox["M", FontSlant->"Italic"], " = 1 in the theorectical error formula. " }], "Text"], Cell[TextData[{ "Step 2) We need to find an ", StyleBox["n", FontSlant->"Italic"], " that makes TheorecticalError small enough (i.e. smaller than ", Cell[BoxData[ \(TraditionalForm\`5\[Cross]10\^\(-7\)\)]], "). One way to do this is to make a table of values of the form {n, \ TheorecticalError[", StyleBox["n", FontSlant->"Italic"], "]}:" }], "Text"], Cell[BoxData[ \(f[x_]\ := \ \((x^2\ + \ 1)\)^\((1/3)\); \na\ = \ 1; \nb\ = \ 4; \n M\ = \ 1; \nTable[{n, \ TheorecticalError[n]}, {n, \ 1, \ 25}]\)], "Input"], Cell["\<\ If we use 21 parabolas (and 42 slices) the estimate will be \ accurate to 7 decimal places. So we can say that the estimate\ \>", "Text"], Cell[BoxData[ \(f[x_]\ := \ \((x^2\ + \ 1)\)^\((1/3)\); \na\ = \ 1; \nb\ = \ 4; \n N[SimpsonsRule[21], \ 8]\)], "Input"], Cell["\<\ is accurate to 7 decimal places (the number of places displayed \ here.)\ \>", "Text"], Cell[TextData[{ "Exericise 3. Find an estimate of ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_1\%3\( 1\/x\) \[DifferentialD]x\)]], " that is accurate to 5 decimal places. (Hint: You will first need to \ determine the number of parabolas (and slices) needed to insure that the \ Simpson's Rule estimate for ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_1\%3\( 1\/x\) \[DifferentialD]x\)]], " is accurate to 5 decimal places. Then find the estimate.)" }], "Text", FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ "Exericise 4. Find an estimate of ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%1\ Tan[x]\ \[DifferentialD]x\)]], " that is accurate to 5 decimal places. (Hint: You will first need to \ determine the number of parabolas (and slices) needed to insure that the \ Simpson's Rule estimate for ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_1\%3\( 1\/x\) \[DifferentialD]x\)]], " is accurate to 5 decimal places. Then find the estimate.)" }], "Text", FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ "Exericise 5. Find an estimate of ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%1\ e\^\(x\^2\)\ \[DifferentialD]x\)]], " that is accurate to 5 decimal places. (Hints: You will first need to \ determine the number of parabolas (and slices) needed to insure that the \ Simpson's Rule estimate for ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_1\%3\( 1\/x\) \[DifferentialD]x\)]], " is accurate to 5 decimal places. Then find the estimate. Also, recall \ that ", StyleBox["Mathematica", FontSlant->"Italic"], "'s name for the number ", StyleBox["e", FontSlant->"Italic"], " is E or \[ExponentialE], which can be found on the pallete.)" }], "Text", FontColor->RGBColor[1, 0, 0]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[ "Slow convergence of Simpson's Rule\[LongDash]one possible cause"], "Section"], Cell[TextData[{ "In Exercise 7 of ", StyleBox["Simpson's Rule\[LongDash]II: Behavior of Error", FontSlant->"Italic"], ", you looked at what happens to the error when the number of slices is \ doubled for the integral ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%1\(\@\( 1 - x\^2\)\) \[DifferentialD]x = \[Pi]\)]], " for the various rules. As you may recall, the ratio for Simpson's Rule \ was not really any better than either Trapezoids or MiddleRectangles. In all \ three rules, the new error was about 35% the size of the old error.) This \ means is that the estimates from Simpson's Rule are not converging any faster \ than the estimates from MiddleRectangles or Trapezoids. What might cause \ this kind of behavior? That's what you will investigate in this section." }], "Text"], Cell["Exercise 6. ", "Text"], Cell["\<\ A) (Identify the problem) The input cell below builds a table that \ shows the ratio of error size when we double the number of parabolas in \ Simpson's rule.\ \>", "Text"], Cell[BoxData[ \(f[x_]\ := \ Sqrt[1 - x^2]; \na\ = \ 0; \nb\ = \ 1; \n trueValue\ = \ Pi/4; \n Table[{\ n, SimpsonsError[2\ n]/SimpsonsError[n]}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ {n, \ 1, \ 30}]\)], "Input"], Cell["\<\ Does this ratio seem rather large compared to the other \ SimpsonsError ratios? Do these ratios say that we'll need more parabolas \ than usual to get a good estimate for the integral? Why?\ \>", "Text"], Cell[TextData[{ "B) (Identify where the problem occurs.) Below are tables that build the \ same kind of tables for the ratio of error sizes for integrating the same \ function, ", Cell[BoxData[ \(TraditionalForm\`f[x]\ = \ \@\(1 - x\^2\)\)]], ", but over different intervals. " }], "Text"], Cell[TextData[{ "First, a table for ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%\(1/2\)\(\@\(1 - x\^2\)\) \[DifferentialD]x\)]], ":" }], "Text"], Cell[BoxData[ \(f[x_]\ := \ Sqrt[1 - x^2]; \na\ = \ 0; \nb\ = \ 1/2; \n trueValue\ = \ 0.478305738745259212; \n Table[{2\ n, SimpsonsError[2\ n]/SimpsonsError[n]}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ {n, \ 1, \ 30}]\)], "Input"], Cell[TextData[{ "Are the error ratios for using Simpson's Rule to compute ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%\(1/2\)\(\@\(1 - x\^2\)\) \[DifferentialD]x\)]], " more typical of what we've seen in other problems? Does the problem with \ convergence for ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%1\(\@\( 1 - x\^2\)\) \[DifferentialD]x\)]], " appear to lie with what is going on over the interval [0, 1/2]? Why or \ why not?" }], "Text"], Cell[TextData[{ "First, a table for ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%\(1/2\)\(\@\(1 - x\^2\)\) \[DifferentialD]x\)]], ":" }], "Text"], Cell[TextData[{ "Next a table for ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_\(1/2\)\%1\(\@\( 1 - x\^2\)\) \[DifferentialD]x\)]] }], "Text"], Cell[BoxData[ \(f[x_]\ := \ Sqrt[1 - x^2]; \na\ = \ 1/2; \nb\ = \ 1; \n trueValue\ = \ 0.307092424652649187; \n Table[{2\ n, SimpsonsError[2\ n]/SimpsonsError[n]}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ {n, \ 1, \ 30}]\)], "Input"], Cell[TextData[{ "Are the error ratios for using Simpson's Rule to compute ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_\(1/2\)\%1\(\@\( 1 - x\^2\)\) \[DifferentialD]x\)]], " more typical of what we've seen in other problems? Does the problem with \ convergence for ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%1\(\@\( 1 - x\^2\)\) \[DifferentialD]x\)]], " appear to lie with what is going on over the interval [1/2, 1]? Why or \ why not?" }], "Text"], Cell[TextData[{ "C) Repeat part B) for the integrals ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_\(3/4\)\%1\(\@\( 1 - x\^2\)\) \[DifferentialD]x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(9/10\)\%1\@\( 1 - x\^2\)\)]], "\[DifferentialD]x", ". Do these integrals have large SimpsonError ratios (indicating slow \ convergence)? " }], "Text"], Cell[TextData[{ "Hint: Use trueValue = 0.113327938494565 for ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_\(3/4\)\%1\(\@\( 1 - x\^2\)\) \[DifferentialD]x\)]], ".\n " }], "Text"], Cell[TextData[{ "and use trueValue = 0.02936295343884203 for ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(9/10\)\%1\@\( 1 - x\^2\)\)]], "\[DifferentialD]x." }], "Text"], Cell[TextData[{ "D) (Using the theorectical error bound) We now have compeling evidence \ that something that happens near ", Cell[BoxData[ \(TraditionalForm\`x = 1\)]], " is what must be causing the slow rate of convergence in Simpson's Rule. \ Recall that the theorectical error bound in Simpson's Rule is given by:\n\t\t\ \t\t\t", Cell[BoxData[ \(TraditionalForm\`\(\ \((b - a)\)\/180\ M\ \ \((\ \(b - a\)\/\(2 n\))\)\^4\)\)]], " \nwhere M is a bound on the size of the fourth derivative of the \ integrand. Let's look at the fourth derivative of ", Cell[BoxData[ \(TraditionalForm\`f[x]\ = \ \@\(1 - x\^2\)\)]], " overt the interval [", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], "]:" }], "Text"], Cell[BoxData[ \(f[x_]\ := \ Sqrt[1\ - \ x^2]; \nSimplify[deriv4[x]]\)], "Input"], Cell[TextData[{ "Is ", Cell[BoxData[ \(TraditionalForm\`deriv4[1]\)]], " defined? Why or why not? Plot deriv4[x] over some intervals of the type \ [", StyleBox["a", FontSlant->"Italic"], ", 1] where you let ", StyleBox["a", FontSlant->"Italic"], " approach 1. What happens to ", Cell[BoxData[ \(TraditionalForm\`\( | deriv4[x]\ | \)\)]], " as ", Cell[BoxData[ \(TraditionalForm\`x \[RightArrow] 1\)]], "? Does this explain why it takes Simpson's Rule so long to converge for ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_0\%1\(\@\( 1 - x\^2\)\) \[DifferentialD]x\)]], "? Why?" }], "Text"], Cell[TextData[{ "A final note: Your plots should show that as ", Cell[BoxData[ \(TraditionalForm\`x \[RightArrow] 1\)]], ", deriv4[x] \[RightArrow] -\[Infinity] . In other words, we can't can't \ find an ", StyleBox["M", FontSlant->"Italic"], " such that |deriv4[x]| \[LessEqual]", StyleBox[" ", FontSlant->"Italic"], " ", StyleBox["M", FontSlant->"Italic"], " on the interval [0, 1]. Thus, we cannot ", StyleBox["theorectically", FontSlant->"Italic"], " force the error to ", StyleBox["quickly", FontSlant->"Italic"], " drop to 0 by simply making ", StyleBox["n", FontSlant->"Italic"], " large enough. " }], "Text"] }, Closed]] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 800}, {0, 580}}, AutoGeneratedPackage->None, WindowSize->{671, 540}, WindowMargins->{{33, Automatic}, {Automatic, 5}}, StyleDefinitions -> "ACStyles.nb", MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@0000000000000006P801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** 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[CellGroupData[{ Cell[1731, 51, 76, 0, 152, "Title"], Cell[1810, 53, 101, 4, 87, "Subtitle"], Cell[CellGroupData[{ Cell[1936, 61, 43, 0, 55, "Section"], Cell[1982, 63, 568, 15, 90, "Text"], Cell[2553, 80, 545, 23, 52, "Text"], Cell[3101, 105, 463, 10, 153, "Input", InitializationCell->True], Cell[3567, 117, 212, 6, 33, "Text"], Cell[3782, 125, 57, 0, 33, "Text"], Cell[3842, 127, 125, 3, 28, "Input", InitializationCell->True], Cell[3970, 132, 340, 10, 52, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[4347, 147, 55, 0, 35, "Section"], Cell[4405, 149, 682, 11, 132, "Text"], Cell[5090, 162, 48, 0, 33, "Text"], Cell[5141, 164, 166, 4, 52, "Text"], Cell[5310, 170, 185, 4, 52, "Text"], Cell[5498, 176, 393, 8, 71, "Text"], Cell[5894, 186, 667, 24, 73, "Text"], Cell[6564, 212, 216, 7, 35, "Text"], Cell[6783, 221, 715, 31, 52, "Text"], Cell[7501, 254, 403, 12, 55, "Text"], Cell[7907, 268, 216, 7, 36, "Text"], Cell[8126, 277, 307, 14, 33, "Text"], Cell[8436, 293, 762, 19, 115, "Text"], Cell[9201, 314, 141, 3, 64, "Input"], Cell[9345, 319, 316, 6, 71, "Text"], Cell[9664, 327, 191, 5, 36, "Text"], Cell[9858, 334, 70, 1, 28, "Input"], Cell[9931, 337, 180, 3, 52, "Text"], Cell[10114, 342, 86, 2, 28, "Input", InitializationCell->True], Cell[10203, 346, 253, 6, 52, "Text"], Cell[10459, 354, 159, 3, 46, "Input"], Cell[10621, 359, 169, 4, 52, "Text"], Cell[10793, 365, 160, 3, 46, "Input"], Cell[10956, 370, 795, 25, 75, "Text"], Cell[11754, 397, 243, 5, 138, "Input"], Cell[12000, 404, 119, 3, 33, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[12156, 412, 145, 4, 35, "Section"], Cell[12304, 418, 692, 21, 112, "Text"], Cell[12999, 441, 195, 5, 67, "Input", InitializationCell->True], Cell[13197, 448, 212, 6, 33, "Text"], Cell[13412, 456, 919, 34, 91, "Text"], Cell[14334, 492, 125, 3, 46, "Input", InitializationCell->True], Cell[14462, 497, 266, 7, 52, "Text"], Cell[14731, 506, 709, 24, 90, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[15477, 535, 114, 3, 58, "Section"], Cell[15594, 540, 196, 6, 33, "Text"], Cell[15793, 548, 28660, 765, 186, 7166, 495, "GraphicsData", "PostScript", "Graphics"], Cell[44456, 1315, 176, 6, 33, "Text"], Cell[44635, 1323, 111, 2, 82, "Input"], Cell[44749, 1327, 69, 0, 33, "Text"], Cell[44821, 1329, 144, 3, 52, "Text"], Cell[44968, 1334, 67, 1, 46, "Input"], Cell[45038, 1337, 146, 5, 33, "Text"], Cell[45187, 1344, 78, 1, 28, "Input"], Cell[45268, 1347, 118, 3, 33, "Text"], Cell[45389, 1352, 173, 3, 46, "Input"], Cell[45565, 1357, 255, 6, 52, "Text"], Cell[45823, 1365, 131, 2, 100, "Input"], Cell[45957, 1369, 147, 3, 52, "Text"], Cell[46107, 1374, 317, 8, 60, "Text"], Cell[46427, 1384, 466, 10, 75, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[46930, 1399, 176, 4, 58, "Section"], Cell[47109, 1405, 525, 11, 105, "Text"], Cell[47637, 1418, 692, 21, 109, "Text"], Cell[48332, 1441, 219, 5, 58, "Text"], Cell[48554, 1448, 86, 1, 46, "Input"], Cell[48643, 1451, 128, 2, 33, "Text"], Cell[48774, 1455, 85, 1, 28, "Input"], Cell[48862, 1458, 173, 5, 33, "Text"], Cell[49038, 1465, 381, 12, 52, "Text"], Cell[49422, 1479, 174, 3, 100, "Input"], Cell[49599, 1484, 148, 3, 52, "Text"], Cell[49750, 1489, 132, 2, 82, "Input"], Cell[49885, 1493, 96, 3, 33, "Text"], Cell[49984, 1498, 528, 11, 78, "Text"], Cell[50515, 1511, 528, 11, 75, "Text"], Cell[51046, 1524, 752, 19, 96, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[51835, 1548, 93, 1, 35, "Section"], Cell[51931, 1551, 824, 15, 143, "Text"], Cell[52758, 1568, 29, 0, 33, "Text"], Cell[52790, 1570, 183, 4, 52, "Text"], Cell[52976, 1576, 226, 4, 118, "Input"], Cell[53205, 1582, 215, 4, 52, "Text"], Cell[53423, 1588, 305, 7, 58, "Text"], Cell[53731, 1597, 171, 6, 48, "Text"], Cell[53905, 1605, 245, 4, 118, "Input"], Cell[54153, 1611, 498, 12, 101, "Text"], Cell[54654, 1625, 171, 6, 48, "Text"], Cell[54828, 1633, 163, 5, 50, "Text"], Cell[54994, 1640, 245, 4, 118, "Input"], Cell[55242, 1646, 499, 12, 103, "Text"], Cell[55744, 1660, 396, 11, 69, "Text"], Cell[56143, 1673, 203, 6, 69, "Text"], Cell[56349, 1681, 187, 5, 50, "Text"], Cell[56539, 1688, 796, 22, 137, "Text"], Cell[57338, 1712, 86, 1, 46, "Input"], Cell[57427, 1715, 665, 23, 86, "Text"], Cell[58095, 1740, 681, 24, 71, "Text"] }, Closed]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)