(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 16, "Times"; ; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = leftheader, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = leftfooter, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] ArcLength Problems :[font = subtitle; inactive; preserveAspect] From the article ``Arclength: An Example from the New Calculus'' in Mathematica in Education Vol.3 No.2 Spring 1994 (c) TELOS/Springer-Verlag ;[s] 2:0,0;92,1;143,-1; 2:1,17,13,Times,3,18,0,0,0;1,16,12,Times,1,18,0,0,0; :[font = subsubtitle; inactive; preserveAspect] by Jack K. Cohen Center for Wave Phenomenon Colorado School of Mines Golden, CO 80401-1887 jkc@keller.mines.colorado.edu :[font = section; inactive; Cclosed; preserveAspect; startGroup] Introduction :[font = text; inactive; preserveAspect] Texts do arclength problems that seem artificial and avoid more interesting cases such as the arclength of an arch under the sine or the arclength of an ellipse because they don't come out in terms of functions with elementary antiderivatives. But this is a good opportunity to contrast using the Existence Theorem and the Fundamental Theorem. ;[s] 3:0,0;279,1;287,2;346,-1; 3:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Laying the groundwork :[font = text; inactive; preserveAspect] Interactively develop with the class an arclength approximator based on computing the sum of the chord lengths of adjacent points. There are many ideas, here's one possible development: :[font = input; initialization; preserveAspect] *) Pythagorean[{x1_, y1_}, {x2_, y2_}] := Sqrt[(x1 - x2)^2 + (y1 - y2)^2] (* :[font = input; initialization; preserveAspect] *) ChordLength[f_, x_, dx_] := Pythagorean[{x, f[x]}, {x + dx, f[x + dx]}] (* :[font = input; initialization; preserveAspect] *) ArcLength[f_, a_, b_, n_] := Module[ {h = (b - a)/n, k}, N@Sum[ChordLength[f, a + k h, h], {k, 0, n-1}] ] (* :[font = text; inactive; preserveAspect] For convenience, also define the formula for differential arclength: :[font = input; initialization; preserveAspect; endGroup; endGroup] *) ds[f_, x_] := Sqrt[1 + f'[x]^2] (* :[font = section; inactive; Cclosed; preserveAspect; startGroup] Some Elementary Examples :[font = text; inactive; preserveAspect] For certain special functions, antiderivatives exist for the differential arclength and we can then use the FTC to compute the arclength. For example: :[font = input; preserveAspect] f[x_] := x^2 :[font = input; Cclosed; preserveAspect; startGroup] dsParabola = ds[f, x] :[font = output; output; inactive; preserveAspect; endGroup] (1 + 4*x^2)^(1/2) ;[o] 2 Sqrt[1 + 4 x ] :[font = input; Cclosed; preserveAspect; startGroup] sParabola[x_] = Integrate[dsParabola, x] :[font = output; output; inactive; preserveAspect; endGroup] (x*(1 + 4*x^2)^(1/2))/2 + ArcSinh[2*x]/4 ;[o] 2 x Sqrt[1 + 4 x ] ArcSinh[2 x] ---------------- + ------------ 2 4 :[font = text; inactive; preserveAspect] So, by the FTC, the arclength for 0 to 1 is :[font = input; Cclosed; preserveAspect; startGroup] sParabola[1] - sParabola[0] :[font = output; output; inactive; preserveAspect; endGroup] 5^(1/2)/2 + ArcSinh[2]/4 ;[o] Sqrt[5] ArcSinh[2] ------- + ---------- 2 4 :[font = text; inactive; preserveAspect] or, in numerical terms: :[font = input; Cclosed; preserveAspect; startGroup] sParabola[1] - sParabola[0] //N :[font = output; output; inactive; preserveAspect; endGroup] 1.478942857544597 ;[o] 1.47894 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Note to the instructor :[font = text; inactive; preserveAspect] The students may not have seen the trigonometric substitution technique routinely used for the integrand above. Moreover, when they do, they will express the answer in terms of the logarithm, not the arcsinh function. They should be reminded that all antiderivatives of a given function differ only by a constant, no matter how different they may seem. ;[s] 3:0,0;349,1;353,2;354,-1; 3:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = text; inactive; preserveAspect] But, for most common functions f(x) the evaluation portion of the FTC cannot be used because an antiderivative in terms of elementary functions doesn't exist. For example, just replace x^2 by x^3 : :[font = input; preserveAspect] f[x_] := x^3 :[font = input; Cclosed; preserveAspect; startGroup] Integrate[ds[f,x], x] :[font = output; output; inactive; preserveAspect; endGroup] (x*(1 + 9*x^4)^(1/2))/3 + (2*Integrate[(1 + 9*x^4)^(-1/2), x])/3 ;[o] 1 2 Integrate[--------------, x] 4 4 x Sqrt[1 + 9 x ] Sqrt[1 + 9 x ] ---------------- + ------------------------------ 3 3 :[font = text; inactive; preserveAspect] Notice that the ``answer'' involves an integral that Mathematica could not ``do''. Indeed, integrands involving square roots of quartics usually involve the non-elementary functions known as elliptic functions. ;[s] 7:0,0;39,1;47,2;53,3;64,4;192,5;210,6;212,-1; 7:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = text; inactive; preserveAspect] PROBLEM: Verify that the functions sqrt(1 + x^n) with n = 2, 1, 0, -1 and -2 have elementary antiderivatives. Then determine which functions of the form y = x^p have differential arclengths with elementary antiderivatives. Two of the powers p correspond to lines; which two? Look at the problems in your calculus book and identify arclength exercises involving the non-trivial powers you have found. ;[s] 4:0,0;222,1;225,2;401,3;403,-1; 4:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0; :[font = text; inactive; preserveAspect] This being the case, textbooks concoct rather exotic functions with the property that 1 + y'^2 becomes a perfect square of powers. For example, y = 1/6 x^3 + 1/2 1/x y = 1/8 x^4 + 1/4 1/x^2 y = 1/3 x^3 + 1/4 1/x . :[font = text; inactive; preserveAspect] PROBLEM: Verify that the above three examples miraculously lead to a perfect square and an arclength integral without a square root. :[font = text; inactive; preserveAspect] A reasonable person just has to be wondering: How in the world do they come up with these dreadful looking things? Well, see the next problem, where you are given a chance to answer that question for yourself. ;[s] 3:0,0;46,1;113,2;214,-1; 3:1,14,11,Times,0,16,0,0,0;1,14,11,Times,1,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = text; inactive; preserveAspect; endGroup; endGroup] PROBLEM: Using the experience gained in solving the last problem, find out which functions of the form, y = x^p + c x^q, have differential arclengths with elementary antiderivatives? Here, c, p, and q are constants. Put the above three examples in the form considered in this problem and verify that everything is consistent. Find other problems in your text that exploit the result you have rediscovered. :[font = section; inactive; Cclosed; preserveAspect; startGroup] Arch of Sine Curve :[font = text; inactive; preserveAspect] Find the arclength of one arch of the sine curve---that is, the arclength from 0 to p. ;[s] 3:0,0;84,1;85,2;86,-1; 3:1,14,11,Times,0,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; preserveAspect] f[x_] := Sin[x] :[font = input; Cclosed; preserveAspect; startGroup] ds[f, x] :[font = output; output; inactive; preserveAspect; endGroup] (1 + Cos[x]^2)^(1/2) ;[o] 2 Sqrt[1 + Cos[x] ] :[font = text; inactive; preserveAspect] This differential arclength does not have an elementary antiderivative: ;[s] 3:0,0;33,1;36,2;71,-1; 3:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup] Integrate[ds[f, x], x] :[font = output; output; inactive; preserveAspect; endGroup] Integrate[(1 + Cos[x]^2)^(1/2), x] ;[o] 2 Integrate[Sqrt[1 + Cos[x] ], x] :[font = text; inactive; preserveAspect] However, since ds[f,x] is continuous (verify this!), the Existence Theorem guarantees that any standard sequence of Riemann sums will converge to the numerical answer. Before using such a standard sequence, let's get an approximation by using our idea of summing chord lengths: :[font = input; Cclosed; preserveAspect; startGroup] ArcLength[f, 0, Pi, 20] :[font = output; output; inactive; preserveAspect; endGroup] 3.818966756502719 ;[o] 3.81897 :[font = input; Cclosed; preserveAspect; startGroup] TableForm[ Table[ {n, ArcLength[f, 0, Pi, n]}, {n, 10, 90, 20}], TableHeadings -> {None, {"N", "Arclength"}} ] :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup] TableForm[{{10, 3.815282726088384}, {30, 3.819650476802025}, {50, 3.820000722216176}, {70, 3.820097239898859}, {90, 3.820136961807257}}, TableHeadings -> {None, {"N", "Arclength"}}] ;[o] N Arclength 10 3.81528 30 3.81965 50 3.82 70 3.8201 90 3.82014 :[font = text; inactive; preserveAspect] Question: Is the above arclength approximator a Riemann sum? :[font = text; inactive; preserveAspect] Now turn to some examples of standard Riemann sum limits. Here is the result of using the midpoint rule on the differential arclength: :[font = input; Cclosed; preserveAspect; startGroup] TableForm[ Table[ {n, N@MidpointRule[ds[f, x], {x, 0, Pi, n}]}, {n, 1, 10}], TableHeadings -> {None, {"N", "Arclength"}} ] :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup] TableForm[{{1, 3.141592653589793}, {2, 3.847649490485592}, {3, 3.817821845218625}, {4, 3.820451787590731}, {5, 3.820167296077245}, {6, 3.820201710511331}, {7, 3.82019726059101}, {8, 3.820197862670103}, {9, 3.820197778501158}, {10, 3.820197790562557}}, TableHeadings -> {None, {"N", "Arclength"}}] ;[o] N Arclength 1 3.14159 2 3.84765 3 3.81782 4 3.82045 5 3.82017 6 3.8202 7 3.8202 8 3.8202 9 3.8202 10 3.8202 :[font = text; inactive; preserveAspect] This converges much quicker than our chord approximator! How about the lowly Rectangle rule? :[font = input; Cclosed; preserveAspect; startGroup] TableForm[ Table[ {n, N@LeftEndpointRule[ds[f, x], {x, 0, Pi, n}]}, {n, 1, 10}], TableHeadings -> {None, {"N", "Arclength"}} ] :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup] TableForm[{{1, 4.442882938158366}, {2, 3.79223779587408}, {3, 3.82256588973303}, {4, 3.819943643179835}, {5, 3.820228278908489}, {6, 3.820193867475827}, {7, 3.820198317462826}, {8, 3.820197715385283}, {9, 3.820197799554266}, {10, 3.820197787492867}}, TableHeadings -> {None, {"N", "Arclength"}}] ;[o] N Arclength 1 4.44288 2 3.79224 3 3.82257 4 3.81994 5 3.82023 6 3.82019 7 3.8202 8 3.8202 9 3.8202 10 3.8202 :[font = text; inactive; preserveAspect] Remark:NIntegrate gives Mathematica's best numerical integrator: ;[s] 3:0,0;24,1;35,2;64,-1; 3:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[ds[f, x], {x, 0, Pi}] :[font = output; output; inactive; preserveAspect; endGroup] 3.820197789023307 ;[o] 3.8202 :[font = text; inactive; preserveAspect] Remark: LeftEndpointRule and MidpointRule are not part of Mathematica, but rather are supplied as part of the CSM calculus environment. ;[s] 5:0,0;46,1;49,2;58,3;69,4;136,-1; 5:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = text; inactive; preserveAspect] PROBLEM: Why is our chordlength approximator so slowly convergent compared with standard numerical techniques (including even the lowly rectangle rule)? :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Remarks for the instructor :[font = text; inactive; preserveAspect; endGroup; endGroup] The integration techniques convert the problem into a related area problem that uses the exact differential arclength. The intuitive formula only crudely approximates that arclength. We might think of improving the approximation by computing the arclength of the parabola through 3 adjacent points---this would not be the same as Simpson's rule applied to the integrand ds[f,x] (this could be an additional exercise). ;[s] 9:0,0;62,1;66,2;89,3;94,4;248,5;257,6;313,7;316,8;419,-1; 9:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = section; inactive; Cclosed; preserveAspect; startGroup] Ellipse -- Rectangular Form :[font = text; inactive; preserveAspect] The standard form for an ellipse is (x/a)^2 + (y/b)^2 = 1. Solving for y and introducing the eccentricity e (no connection to the base of the natural logarithms!) leads to the following expression for y: ;[s] 11:0,0;36,1;57,2;72,3;73,4;107,5;108,6;110,7;162,8;202,9;203,10;211,-1; 11:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,1,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; preserveAspect] f[x_] := Sqrt[(1 - e^2) (a^2 - x^2)] :[font = text; inactive; preserveAspect] PROBLEM: Explain in full detail, why this formula gives the upper arch of an ellipse with semi-major axis a and eccentricity e. If necessary, use the index of your text to get the definition of e and other facts about the ellipse. Note especially that e is between 0 and 1 for an ellipse. ;[s] 9:0,0;106,1;107,2;125,3;126,4;195,5;196,6;254,7;255,8;290,-1; 9:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup] ds[f, x] :[font = output; output; inactive; preserveAspect; endGroup] (1 + ((1 - e^2)*x^2)/(a^2 - x^2))^(1/2) ;[o] 2 2 (1 - e ) x Sqrt[1 + -----------] 2 2 a - x :[font = text; inactive; preserveAspect] The parameter `a' just scales the whole result proportionately, so take a = 1 henceforth (we have to take some particular value to apply numerical methods--here, we are lucky that we don't lose generality by doing so): ;[s] 7:0,0;15,1;16,2;72,3;77,4;137,5;146,6;220,-1; 7:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; preserveAspect] a = 1; :[font = text; inactive; preserveAspect] PROBLEM: Justify the claim in the previous paragraph and explain, in clear english, how to get arclengths for arbitrary values of a from the values for a = 1. ;[s] 7:0,0;110,1;119,2;130,3;131,4;152,5;157,6;160,-1; 7:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[4 ds[f,x] /. {e -> .5}, {x, 0, a}] :[font = output; output; inactive; preserveAspect; endGroup] 5.869848837397313 ;[o] 5.86985 :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[4 ds[f,x] /. {e -> 0}, {x, 0, a}] :[font = output; output; inactive; preserveAspect; endGroup] 6.283185307225316 ;[o] 6.28319 :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[4 ds[f,x] /. {e -> 1}, {x, 0, a}] :[font = output; output; inactive; preserveAspect; endGroup] 4. ;[o] 4. :[font = text; inactive; preserveAspect] PROBLEM: Explain the results obtained for e = 0 and for e = 1. Generalize these two results to arbitrary a. ;[s] 7:0,0;42,1;47,2;56,3;61,4;106,5;107,6;108,-1; 7:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[4 ds[f,x] /. {e -> .5}, {x, 0, a}] //Timing :[font = output; output; inactive; preserveAspect; endGroup] {81.1333333333333*Second, 5.869848837397313} ;[o] {81.1333 Second, 5.86985} :[font = text; inactive; preserveAspect] PROBLEM: Make a graph of arclength vs. eccentricity. Note: Since the NIntegrates take a long time, consider making a table and using ListPlot instead of using Plot. We will cure the timing problem in the next section. ;[s] 3:0,0;89,1;93,2;219,-1; 3:1,14,11,Times,0,16,0,0,0;1,14,11,Times,1,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Solution for the instructor :[font = input; Cclosed; preserveAspect; startGroup] aTable = Table[{e, NIntegrate[4 ds[f,x], {x, 0, a}]}, {e, 0.0, 1.0, .10}] :[font = output; output; inactive; preserveAspect; endGroup] {{0., 6.283185307225316}, {0.1, 6.267447768132176}, {0.2, 6.219874185014924}, {0.3, 6.139333859736621}, {0.4, 6.023766449482074}, {0.5, 5.869848837397312}, {0.6, 5.672333577831482}, {0.7, 5.42264454232048}, {0.8, 5.105399772707064}, {0.9, 4.68678821114639}, {1., 4.000000000000002}} ;[o] {{0., 6.28319}, {0.1, 6.26745}, {0.2, 6.21987}, {0.3, 6.13933}, {0.4, 6.02377}, {0.5, 5.86985}, {0.6, 5.67233}, {0.7, 5.42264}, {0.8, 5.1054}, {0.9, 4.68679}, {1., 4.}} :[font = input; Cclosed; preserveAspect; startGroup] ListPlot[aTable, PlotJoined -> True] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 -1.016482 0.257799 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa [(4.5)] 0.01131 0.14361 1 0 Msboxa [(5)] 0.01131 0.27251 1 0 Msboxa [(5.5)] 0.01131 0.40141 1 0 Msboxa [(6)] 0.01131 0.53031 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.01472 moveto 0.21429 0.02097 lineto stroke grestore [(0.2)] 0.21429 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.01472 moveto 0.40476 0.02097 lineto stroke grestore [(0.4)] 0.40476 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.01472 moveto 0.59524 0.02097 lineto stroke grestore [(0.6)] 0.59524 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.01472 moveto 0.78571 0.02097 lineto stroke grestore [(0.8)] 0.78571 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(1)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.01472 moveto 0.0619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.01472 moveto 0.1 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.01472 moveto 0.1381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.01472 moveto 0.17619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.01472 moveto 0.25238 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.01472 moveto 0.29048 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.01472 moveto 0.32857 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.01472 moveto 0.36667 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.01472 moveto 0.44286 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.01472 moveto 0.48095 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.01472 moveto 0.51905 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.01472 moveto 0.55714 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.01472 moveto 0.63333 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.01472 moveto 0.67143 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.01472 moveto 0.70952 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.01472 moveto 0.74762 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.01472 moveto 0.82381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.01472 moveto 0.8619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.01472 moveto 0.9 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.01472 moveto 0.9381 0.01847 lineto stroke grestore gsave 0.002 setlinewidth 0 0.01472 moveto 1 0.01472 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.14361 moveto 0.03006 0.14361 lineto stroke grestore [(4.5)] 0.01131 0.14361 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.27251 moveto 0.03006 0.27251 lineto stroke grestore [(5)] 0.01131 0.27251 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.40141 moveto 0.03006 0.40141 lineto stroke grestore [(5.5)] 0.01131 0.40141 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.53031 moveto 0.03006 0.53031 lineto stroke grestore [(6)] 0.01131 0.53031 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.0405 moveto 0.02756 0.0405 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.06627 moveto 0.02756 0.06627 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.09205 moveto 0.02756 0.09205 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.11783 moveto 0.02756 0.11783 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.16939 moveto 0.02756 0.16939 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.19517 moveto 0.02756 0.19517 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.22095 moveto 0.02756 0.22095 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.24673 moveto 0.02756 0.24673 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.29829 moveto 0.02756 0.29829 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.32407 moveto 0.02756 0.32407 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.34985 moveto 0.02756 0.34985 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.37563 moveto 0.02756 0.37563 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.42719 moveto 0.02756 0.42719 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.45297 moveto 0.02756 0.45297 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.47875 moveto 0.02756 0.47875 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.50453 moveto 0.02756 0.50453 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.55609 moveto 0.02756 0.55609 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58187 moveto 0.02756 0.58187 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.60765 moveto 0.02756 0.60765 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath 0.004 setlinewidth 0.02381 0.60332 moveto 0.11905 0.59926 lineto 0.21429 0.587 lineto 0.30952 0.56623 lineto 0.40476 0.53644 lineto 0.5 0.49676 lineto 0.59524 0.44584 lineto 0.69048 0.38147 lineto 0.78571 0.29969 lineto 0.88095 0.19177 lineto 0.97619 0.01472 lineto stroke % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] ListPlot[aTable, PlotRange -> {0, 6.5}, PlotJoined -> True] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 1.040834e-17 0.095082 [ [(0)] 0.02381 0 0 2 Msboxa [(0.2)] 0.21429 0 0 2 Msboxa [(0.4)] 0.40476 0 0 2 Msboxa [(0.6)] 0.59524 0 0 2 Msboxa [(0.8)] 0.78571 0 0 2 Msboxa [(1)] 0.97619 0 0 2 Msboxa [(1)] 0.01131 0.09508 1 0 Msboxa [(2)] 0.01131 0.19016 1 0 Msboxa [(3)] 0.01131 0.28525 1 0 Msboxa [(4)] 0.01131 0.38033 1 0 Msboxa [(5)] 0.01131 0.47541 1 0 Msboxa [(6)] 0.01131 0.57049 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.00625 lineto stroke grestore [(0)] 0.02381 0 0 2 Mshowa gsave 0.002 setlinewidth 0.21429 0 moveto 0.21429 0.00625 lineto stroke grestore [(0.2)] 0.21429 0 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0 moveto 0.40476 0.00625 lineto stroke grestore [(0.4)] 0.40476 0 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0 moveto 0.59524 0.00625 lineto stroke grestore [(0.6)] 0.59524 0 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0 moveto 0.78571 0.00625 lineto stroke grestore [(0.8)] 0.78571 0 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0 moveto 0.97619 0.00625 lineto stroke grestore [(1)] 0.97619 0 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0 moveto 0.0619 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0 moveto 0.1 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0 moveto 0.1381 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0 moveto 0.17619 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0 moveto 0.25238 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0 moveto 0.29048 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0 moveto 0.32857 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0 moveto 0.36667 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0 moveto 0.44286 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0 moveto 0.48095 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0 moveto 0.51905 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0 moveto 0.55714 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0 moveto 0.63333 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0 moveto 0.67143 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0 moveto 0.70952 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0 moveto 0.74762 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0 moveto 0.82381 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0 moveto 0.8619 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0 moveto 0.9 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0 moveto 0.9381 0.00375 lineto stroke grestore gsave 0.002 setlinewidth 0 0 moveto 1 0 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.09508 moveto 0.03006 0.09508 lineto stroke grestore [(1)] 0.01131 0.09508 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.19016 moveto 0.03006 0.19016 lineto stroke grestore [(2)] 0.01131 0.19016 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.28525 moveto 0.03006 0.28525 lineto stroke grestore [(3)] 0.01131 0.28525 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.38033 moveto 0.03006 0.38033 lineto stroke grestore [(4)] 0.01131 0.38033 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.47541 moveto 0.03006 0.47541 lineto stroke grestore [(5)] 0.01131 0.47541 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.57049 moveto 0.03006 0.57049 lineto stroke grestore [(6)] 0.01131 0.57049 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.01902 moveto 0.02756 0.01902 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.03803 moveto 0.02756 0.03803 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.05705 moveto 0.02756 0.05705 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.07607 moveto 0.02756 0.07607 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.1141 moveto 0.02756 0.1141 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.13312 moveto 0.02756 0.13312 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.15213 moveto 0.02756 0.15213 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.17115 moveto 0.02756 0.17115 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.20918 moveto 0.02756 0.20918 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.2282 moveto 0.02756 0.2282 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.24721 moveto 0.02756 0.24721 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.26623 moveto 0.02756 0.26623 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.30426 moveto 0.02756 0.30426 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.32328 moveto 0.02756 0.32328 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.3423 moveto 0.02756 0.3423 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.36131 moveto 0.02756 0.36131 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.39935 moveto 0.02756 0.39935 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.41836 moveto 0.02756 0.41836 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.43738 moveto 0.02756 0.43738 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.45639 moveto 0.02756 0.45639 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.49443 moveto 0.02756 0.49443 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.51344 moveto 0.02756 0.51344 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.53246 moveto 0.02756 0.53246 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.55148 moveto 0.02756 0.55148 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58951 moveto 0.02756 0.58951 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.60853 moveto 0.02756 0.60853 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath 0.004 setlinewidth 0.02381 0.59742 moveto 0.11905 0.59592 lineto 0.21429 0.5914 lineto 0.30952 0.58374 lineto 0.40476 0.57275 lineto 0.5 0.55812 lineto 0.59524 0.53934 lineto 0.69048 0.5156 lineto 0.78571 0.48543 lineto 0.88095 0.44563 lineto 0.97619 0.38033 lineto stroke % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = section; inactive; Cclosed; preserveAspect; startGroup] Ellipse -- Polar Form :[font = text; inactive; preserveAspect] PROBLEM: In the rectangular form of the ellipse, make the substitution x = a cos(q) and show that the equation reduces to y = b sin(q). Then show that ds = sqrt(dx^2 + dy^2) reduces to ds = a sqrt(1 - e^2 cos^2(q)) dq ;[s] 19:0,0;71,1;81,2;82,3;83,4;133,5;143,6;144,7;148,8;165,9;170,10;174,11;187,12;229,13;236,14;240,15;255,16;256,17;260,18;261,-1; 19:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,0,0,Symbol,0,16,0,0,0; :[font = input; preserveAspect] Clear[a]; :[font = input; Cclosed; preserveAspect; startGroup] ds = a Sqrt[1 - e^2 Cos[theta]^2] :[font = output; output; inactive; preserveAspect; endGroup] a*(1 - e^2*Cos[theta]^2)^(1/2) ;[o] 2 2 a Sqrt[1 - e Cos[theta] ] :[font = text; inactive; preserveAspect] Again, the parameter `a' just scales the whole result proportionately, so take a = 1 henceforth: ;[s] 5:0,0;22,1;23,2;79,3;84,4;97,-1; 5:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; preserveAspect] a = 1; :[font = text; inactive; preserveAspect] PROBLEM: Justify this claim and explain how to get arclengths for arbitrary values of a. Explain why integrating from q = 0 to q = p/2 gives 1/4 of the arclength of the ellipse ;[s] 11:0,0;86,1;87,2;119,3;120,4;124,5;128,6;129,7;132,8;134,9;135,10;178,-1; 11:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[4 ds /. {e -> .5}, {theta, 0, Pi/2}] :[font = output; output; inactive; preserveAspect; endGroup] 5.869848837357673 ;[o] 5.86985 :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[4 ds /. {e -> 0}, {theta, 0, Pi/2}] :[font = output; output; inactive; preserveAspect; endGroup] 6.283185307179587 ;[o] 6.28319 :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[4 ds /. {e -> 1}, {theta, 0, Pi/2}] :[font = output; output; inactive; preserveAspect; endGroup] 4. ;[o] 4. :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[4 ds /. {e -> .5}, {theta, 0, Pi/2}] //Timing :[font = output; output; inactive; preserveAspect; endGroup] {0.6166666666666667*Second, 5.869848837357673} ;[o] {0.616667 Second, 5.86985} :[font = text; inactive; preserveAspect] PROBLEM: Explain the results obtained for e = 0 and for e = 1. Generalize these two results to arbitrary a. ;[s] 7:0,0;42,1;47,2;56,3;61,4;106,5;107,6;108,-1; 7:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = text; inactive; preserveAspect] PROBLEM: Make a graph of arclength vs. eccentricity. Notice that now, NIntegrate goes like the wind! So you can use a Plot command like this: :[font = input; Cclosed; preserveAspect; startGroup] Plot[NIntegrate[4 ds, {theta, 0, Pi/2}], {e, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 -1.016482 0.257799 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa [(4.5)] 0.01131 0.14361 1 0 Msboxa [(5)] 0.01131 0.27251 1 0 Msboxa [(5.5)] 0.01131 0.40141 1 0 Msboxa [(6)] 0.01131 0.53031 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.01472 moveto 0.21429 0.02097 lineto stroke grestore [(0.2)] 0.21429 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.01472 moveto 0.40476 0.02097 lineto stroke grestore [(0.4)] 0.40476 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.01472 moveto 0.59524 0.02097 lineto stroke grestore [(0.6)] 0.59524 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.01472 moveto 0.78571 0.02097 lineto stroke grestore [(0.8)] 0.78571 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(1)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.01472 moveto 0.0619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.01472 moveto 0.1 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.01472 moveto 0.1381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.01472 moveto 0.17619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.01472 moveto 0.25238 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.01472 moveto 0.29048 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.01472 moveto 0.32857 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.01472 moveto 0.36667 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.01472 moveto 0.44286 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.01472 moveto 0.48095 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.01472 moveto 0.51905 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.01472 moveto 0.55714 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.01472 moveto 0.63333 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.01472 moveto 0.67143 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.01472 moveto 0.70952 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.01472 moveto 0.74762 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.01472 moveto 0.82381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.01472 moveto 0.8619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.01472 moveto 0.9 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.01472 moveto 0.9381 0.01847 lineto stroke grestore gsave 0.002 setlinewidth 0 0.01472 moveto 1 0.01472 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.14361 moveto 0.03006 0.14361 lineto stroke grestore [(4.5)] 0.01131 0.14361 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.27251 moveto 0.03006 0.27251 lineto stroke grestore [(5)] 0.01131 0.27251 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.40141 moveto 0.03006 0.40141 lineto stroke grestore [(5.5)] 0.01131 0.40141 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.53031 moveto 0.03006 0.53031 lineto stroke grestore [(6)] 0.01131 0.53031 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.0405 moveto 0.02756 0.0405 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.06627 moveto 0.02756 0.06627 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.09205 moveto 0.02756 0.09205 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.11783 moveto 0.02756 0.11783 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.16939 moveto 0.02756 0.16939 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.19517 moveto 0.02756 0.19517 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.22095 moveto 0.02756 0.22095 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.24673 moveto 0.02756 0.24673 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.29829 moveto 0.02756 0.29829 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.32407 moveto 0.02756 0.32407 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.34985 moveto 0.02756 0.34985 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.37563 moveto 0.02756 0.37563 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.42719 moveto 0.02756 0.42719 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.45297 moveto 0.02756 0.45297 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.47875 moveto 0.02756 0.47875 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.50453 moveto 0.02756 0.50453 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.55609 moveto 0.02756 0.55609 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58187 moveto 0.02756 0.58187 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.60765 moveto 0.02756 0.60765 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave 0.004 setlinewidth 0.02381 0.60332 moveto 0.02505 0.60332 lineto 0.02629 0.60332 lineto 0.02753 0.60331 lineto 0.02877 0.60331 lineto 0.03001 0.6033 lineto 0.03125 0.60329 lineto 0.03373 0.60327 lineto 0.03621 0.60325 lineto 0.03869 0.60322 lineto 0.04365 0.60314 lineto 0.04861 0.60304 lineto 0.05357 0.60292 lineto 0.06349 0.60262 lineto 0.07341 0.60222 lineto 0.08333 0.60174 lineto 0.10317 0.6005 lineto 0.12302 0.59892 lineto 0.14286 0.59697 lineto 0.18254 0.59201 lineto 0.22222 0.5856 lineto 0.2619 0.5777 lineto 0.30159 0.5683 lineto 0.34127 0.55734 lineto 0.38095 0.54478 lineto 0.42063 0.53054 lineto 0.46032 0.51457 lineto 0.5 0.49676 lineto 0.53968 0.47701 lineto 0.57937 0.45519 lineto 0.61905 0.43112 lineto 0.65873 0.40461 lineto 0.69841 0.3754 lineto 0.7381 0.34312 lineto 0.77778 0.30732 lineto 0.81746 0.26733 lineto 0.85714 0.2221 lineto 0.89683 0.16986 lineto 0.93651 0.10682 lineto 0.95635 0.06809 lineto 0.97619 0.01472 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = text; inactive; preserveAspect] PROBLEM: Why is Mathematica able to obtain integrals so much faster in the polar representation? ;[s] 3:0,0;16,1;27,2;97,-1; 3:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Investigation problem :[font = text; inactive; preserveAspect; endGroup] Let d be a number between 0 and 1. Which is greater: the arclength from c < x < a on the upper branch of the ellipse (x/a)^2 + (y/b)^2 = 1 with eccentricity e = 1 - d or the arclength from a < x < c on the upper right branch of the hyperbola (x/a)^2 - (y/b)^2 = 1 with eccentricity e = 1 + d ? Hint: Use a representation in terms of hyperbolic functions for the hyperbola! ;[s] 18:0,0;4,1;5,2;73,3;82,4;118,5;140,6;158,7;167,8;190,9;199,10;243,11;264,12;295,13;305,14;346,15;356,16;375,17;393,-1; 18:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Solution for the instructor :[font = text; inactive; preserveAspect] For the elliptical case, the x limits are defined by x = a <--> cos(q) = 1 ==> q = 0 and x = c = a e <--> cos(q) = e. For the hyperbolic case, use x = a cosh(u), y = b sinh(u) and the relations cosh^2 u - sinh^2 u = 1, b^2 = c^2 - a^2 = a^2 (e^2 - 1) to derive ds = a sqrt(e^2 cosh^2 u - 1) du. ;[s] 32:0,0;29,1;30,2;86,3;97,4;100,5;101,6;102,7;112,8;113,9;115,10;117,11;123,12;140,13;143,14;144,15;145,16;150,17;181,18;188,19;192,20;203,21;207,22;210,23;233,24;240,25;244,26;285,27;296,28;303,29;307,30;312,31;344,-1; 32:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,0,0,Symbol,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; :[font = input; preserveAspect] Clear[a]; :[font = input; Cclosed; preserveAspect; startGroup] dsE = a Sqrt[1 - (1-d)^2 Cos[theta]^2] :[font = output; output; inactive; preserveAspect; endGroup] a*(1 - (1 - d)^2*Cos[theta]^2)^(1/2) ;[o] 2 2 a Sqrt[1 - (1 - d) Cos[theta] ] :[font = input; Cclosed; preserveAspect; startGroup] dsH = a Sqrt[(1+d)^2 Cosh[u]^2 - 1] :[font = output; output; inactive; preserveAspect; endGroup] a*(-1 + (1 + d)^2*Cosh[u]^2)^(1/2) ;[o] 2 2 a Sqrt[-1 + (1 + d) Cosh[u] ] :[font = input; preserveAspect] a = 1; :[font = input; Cclosed; preserveAspect; startGroup] pEllipse = Plot[NIntegrate[dsE, {theta, 0, ArcCos[1-d]}], {d, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.014715 0.374717 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa [(0.25)] 0.01131 0.10839 1 0 Msboxa [(0.5)] 0.01131 0.20207 1 0 Msboxa [(0.75)] 0.01131 0.29575 1 0 Msboxa [(1)] 0.01131 0.38943 1 0 Msboxa [(1.25)] 0.01131 0.48311 1 0 Msboxa [(1.5)] 0.01131 0.57679 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.01472 moveto 0.21429 0.02097 lineto stroke grestore [(0.2)] 0.21429 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.01472 moveto 0.40476 0.02097 lineto stroke grestore [(0.4)] 0.40476 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.01472 moveto 0.59524 0.02097 lineto stroke grestore [(0.6)] 0.59524 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.01472 moveto 0.78571 0.02097 lineto stroke grestore [(0.8)] 0.78571 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(1)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.01472 moveto 0.0619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.01472 moveto 0.1 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.01472 moveto 0.1381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.01472 moveto 0.17619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.01472 moveto 0.25238 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.01472 moveto 0.29048 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.01472 moveto 0.32857 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.01472 moveto 0.36667 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.01472 moveto 0.44286 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.01472 moveto 0.48095 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.01472 moveto 0.51905 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.01472 moveto 0.55714 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.01472 moveto 0.63333 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.01472 moveto 0.67143 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.01472 moveto 0.70952 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.01472 moveto 0.74762 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.01472 moveto 0.82381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.01472 moveto 0.8619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.01472 moveto 0.9 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.01472 moveto 0.9381 0.01847 lineto stroke grestore gsave 0.002 setlinewidth 0 0.01472 moveto 1 0.01472 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.10839 moveto 0.03006 0.10839 lineto stroke grestore [(0.25)] 0.01131 0.10839 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.20207 moveto 0.03006 0.20207 lineto stroke grestore [(0.5)] 0.01131 0.20207 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.29575 moveto 0.03006 0.29575 lineto stroke grestore [(0.75)] 0.01131 0.29575 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.38943 moveto 0.03006 0.38943 lineto stroke grestore [(1)] 0.01131 0.38943 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.48311 moveto 0.03006 0.48311 lineto stroke grestore [(1.25)] 0.01131 0.48311 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.57679 moveto 0.03006 0.57679 lineto stroke grestore [(1.5)] 0.01131 0.57679 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.03345 moveto 0.02756 0.03345 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.05219 moveto 0.02756 0.05219 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.07092 moveto 0.02756 0.07092 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.08966 moveto 0.02756 0.08966 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.12713 moveto 0.02756 0.12713 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.14587 moveto 0.02756 0.14587 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.1646 moveto 0.02756 0.1646 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.18334 moveto 0.02756 0.18334 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.22081 moveto 0.02756 0.22081 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.23955 moveto 0.02756 0.23955 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.25828 moveto 0.02756 0.25828 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.27702 moveto 0.02756 0.27702 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.31449 moveto 0.02756 0.31449 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.33322 moveto 0.02756 0.33322 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.35196 moveto 0.02756 0.35196 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.3707 moveto 0.02756 0.3707 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.40817 moveto 0.02756 0.40817 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.4269 moveto 0.02756 0.4269 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.44564 moveto 0.02756 0.44564 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.46438 moveto 0.02756 0.46438 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.50185 moveto 0.02756 0.50185 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.52058 moveto 0.02756 0.52058 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.53932 moveto 0.02756 0.53932 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.55805 moveto 0.02756 0.55805 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.59553 moveto 0.02756 0.59553 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.61426 moveto 0.02756 0.61426 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave 0.004 setlinewidth 0.02381 0.01472 moveto 0.06349 0.05001 lineto 0.10317 0.08423 lineto 0.14286 0.11738 lineto 0.18254 0.14948 lineto 0.22222 0.18054 lineto 0.2619 0.21058 lineto 0.30159 0.23963 lineto 0.34127 0.26769 lineto 0.38095 0.29479 lineto 0.42063 0.32095 lineto 0.46032 0.34619 lineto 0.5 0.37054 lineto 0.53968 0.39402 lineto 0.57937 0.41666 lineto 0.61905 0.43849 lineto 0.65873 0.45954 lineto 0.69841 0.47982 lineto 0.7381 0.49939 lineto 0.77778 0.51826 lineto 0.81746 0.53647 lineto 0.85714 0.55405 lineto 0.89683 0.57103 lineto 0.93651 0.58745 lineto 0.97619 0.60332 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] pHyperbola = Plot[NIntegrate[dsH, {u, 0, ArcCosh[1+d]}], {d, 0, 1}, PlotStyle -> GrayLevel[0.3] ] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.014715 0.184549 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa [(0.5)] 0.01131 0.10699 1 0 Msboxa [(1)] 0.01131 0.19926 1 0 Msboxa [(1.5)] 0.01131 0.29154 1 0 Msboxa [(2)] 0.01131 0.38381 1 0 Msboxa [(2.5)] 0.01131 0.47609 1 0 Msboxa [(3)] 0.01131 0.56836 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.01472 moveto 0.21429 0.02097 lineto stroke grestore [(0.2)] 0.21429 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.01472 moveto 0.40476 0.02097 lineto stroke grestore [(0.4)] 0.40476 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.01472 moveto 0.59524 0.02097 lineto stroke grestore [(0.6)] 0.59524 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.01472 moveto 0.78571 0.02097 lineto stroke grestore [(0.8)] 0.78571 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(1)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.01472 moveto 0.0619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.01472 moveto 0.1 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.01472 moveto 0.1381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.01472 moveto 0.17619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.01472 moveto 0.25238 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.01472 moveto 0.29048 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.01472 moveto 0.32857 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.01472 moveto 0.36667 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.01472 moveto 0.44286 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.01472 moveto 0.48095 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.01472 moveto 0.51905 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.01472 moveto 0.55714 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.01472 moveto 0.63333 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.01472 moveto 0.67143 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.01472 moveto 0.70952 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.01472 moveto 0.74762 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.01472 moveto 0.82381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.01472 moveto 0.8619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.01472 moveto 0.9 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.01472 moveto 0.9381 0.01847 lineto stroke grestore gsave 0.002 setlinewidth 0 0.01472 moveto 1 0.01472 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.10699 moveto 0.03006 0.10699 lineto stroke grestore [(0.5)] 0.01131 0.10699 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.19926 moveto 0.03006 0.19926 lineto stroke grestore [(1)] 0.01131 0.19926 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.29154 moveto 0.03006 0.29154 lineto stroke grestore [(1.5)] 0.01131 0.29154 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.38381 moveto 0.03006 0.38381 lineto stroke grestore [(2)] 0.01131 0.38381 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.47609 moveto 0.03006 0.47609 lineto stroke grestore [(2.5)] 0.01131 0.47609 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.56836 moveto 0.03006 0.56836 lineto stroke grestore [(3)] 0.01131 0.56836 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.03317 moveto 0.02756 0.03317 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.05162 moveto 0.02756 0.05162 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.07008 moveto 0.02756 0.07008 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.08853 moveto 0.02756 0.08853 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.12544 moveto 0.02756 0.12544 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.1439 moveto 0.02756 0.1439 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.16235 moveto 0.02756 0.16235 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.18081 moveto 0.02756 0.18081 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.21772 moveto 0.02756 0.21772 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.23617 moveto 0.02756 0.23617 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.25463 moveto 0.02756 0.25463 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.27308 moveto 0.02756 0.27308 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.30999 moveto 0.02756 0.30999 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.32845 moveto 0.02756 0.32845 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.3469 moveto 0.02756 0.3469 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.36536 moveto 0.02756 0.36536 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.40227 moveto 0.02756 0.40227 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.42072 moveto 0.02756 0.42072 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.43918 moveto 0.02756 0.43918 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.45763 moveto 0.02756 0.45763 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.49454 moveto 0.02756 0.49454 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.513 moveto 0.02756 0.513 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.53145 moveto 0.02756 0.53145 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.54991 moveto 0.02756 0.54991 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58682 moveto 0.02756 0.58682 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.60527 moveto 0.02756 0.60527 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave 0.3 setgray gsave 0.004 setlinewidth 0.02381 0.01472 moveto 0.06349 0.03264 lineto 0.10317 0.0511 lineto 0.14286 0.07012 lineto 0.18254 0.08969 lineto 0.22222 0.10982 lineto 0.2619 0.13052 lineto 0.30159 0.15179 lineto 0.34127 0.17362 lineto 0.38095 0.19604 lineto 0.42063 0.21903 lineto 0.46032 0.24261 lineto 0.5 0.26677 lineto 0.53968 0.29152 lineto 0.57937 0.31687 lineto 0.61905 0.3428 lineto 0.65873 0.36934 lineto 0.69841 0.39647 lineto 0.7381 0.4242 lineto 0.77778 0.45254 lineto 0.81746 0.48148 lineto 0.85714 0.51103 lineto 0.89683 0.54118 lineto 0.93651 0.57194 lineto 0.97619 0.60332 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] Show[pEllipse, pHyperbola] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.014715 0.184549 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa [(0.5)] 0.01131 0.10699 1 0 Msboxa [(1)] 0.01131 0.19926 1 0 Msboxa [(1.5)] 0.01131 0.29154 1 0 Msboxa [(2)] 0.01131 0.38381 1 0 Msboxa [(2.5)] 0.01131 0.47609 1 0 Msboxa [(3)] 0.01131 0.56836 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.01472 moveto 0.21429 0.02097 lineto stroke grestore [(0.2)] 0.21429 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.01472 moveto 0.40476 0.02097 lineto stroke grestore [(0.4)] 0.40476 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.01472 moveto 0.59524 0.02097 lineto stroke grestore [(0.6)] 0.59524 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.01472 moveto 0.78571 0.02097 lineto stroke grestore [(0.8)] 0.78571 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(1)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.01472 moveto 0.0619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.01472 moveto 0.1 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.01472 moveto 0.1381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.01472 moveto 0.17619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.01472 moveto 0.25238 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.01472 moveto 0.29048 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.01472 moveto 0.32857 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.01472 moveto 0.36667 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.01472 moveto 0.44286 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.01472 moveto 0.48095 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.01472 moveto 0.51905 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.01472 moveto 0.55714 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.01472 moveto 0.63333 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.01472 moveto 0.67143 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.01472 moveto 0.70952 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.01472 moveto 0.74762 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.01472 moveto 0.82381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.01472 moveto 0.8619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.01472 moveto 0.9 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.01472 moveto 0.9381 0.01847 lineto stroke grestore gsave 0.002 setlinewidth 0 0.01472 moveto 1 0.01472 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.10699 moveto 0.03006 0.10699 lineto stroke grestore [(0.5)] 0.01131 0.10699 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.19926 moveto 0.03006 0.19926 lineto stroke grestore [(1)] 0.01131 0.19926 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.29154 moveto 0.03006 0.29154 lineto stroke grestore [(1.5)] 0.01131 0.29154 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.38381 moveto 0.03006 0.38381 lineto stroke grestore [(2)] 0.01131 0.38381 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.47609 moveto 0.03006 0.47609 lineto stroke grestore [(2.5)] 0.01131 0.47609 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.56836 moveto 0.03006 0.56836 lineto stroke grestore [(3)] 0.01131 0.56836 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.03317 moveto 0.02756 0.03317 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.05162 moveto 0.02756 0.05162 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.07008 moveto 0.02756 0.07008 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.08853 moveto 0.02756 0.08853 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.12544 moveto 0.02756 0.12544 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.1439 moveto 0.02756 0.1439 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.16235 moveto 0.02756 0.16235 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.18081 moveto 0.02756 0.18081 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.21772 moveto 0.02756 0.21772 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.23617 moveto 0.02756 0.23617 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.25463 moveto 0.02756 0.25463 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.27308 moveto 0.02756 0.27308 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.30999 moveto 0.02756 0.30999 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.32845 moveto 0.02756 0.32845 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.3469 moveto 0.02756 0.3469 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.36536 moveto 0.02756 0.36536 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.40227 moveto 0.02756 0.40227 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.42072 moveto 0.02756 0.42072 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.43918 moveto 0.02756 0.43918 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.45763 moveto 0.02756 0.45763 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.49454 moveto 0.02756 0.49454 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.513 moveto 0.02756 0.513 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.53145 moveto 0.02756 0.53145 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.54991 moveto 0.02756 0.54991 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58682 moveto 0.02756 0.58682 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.60527 moveto 0.02756 0.60527 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave gsave gsave 0.004 setlinewidth 0.02381 0.01472 moveto 0.06349 0.0321 lineto 0.10317 0.04895 lineto 0.14286 0.06528 lineto 0.18254 0.08109 lineto 0.22222 0.09638 lineto 0.2619 0.11118 lineto 0.30159 0.12548 lineto 0.34127 0.1393 lineto 0.38095 0.15265 lineto 0.42063 0.16553 lineto 0.46032 0.17797 lineto 0.5 0.18996 lineto 0.53968 0.20152 lineto 0.57937 0.21268 lineto 0.61905 0.22343 lineto 0.65873 0.23379 lineto 0.69841 0.24378 lineto 0.7381 0.25342 lineto 0.77778 0.26271 lineto 0.81746 0.27168 lineto 0.85714 0.28034 lineto 0.89683 0.2887 lineto 0.93651 0.29679 lineto 0.97619 0.3046 lineto stroke grestore grestore grestore gsave gsave 0.3 setgray gsave 0.004 setlinewidth 0.02381 0.01472 moveto 0.06349 0.03264 lineto 0.10317 0.0511 lineto 0.14286 0.07012 lineto 0.18254 0.08969 lineto 0.22222 0.10982 lineto 0.2619 0.13052 lineto 0.30159 0.15179 lineto 0.34127 0.17362 lineto 0.38095 0.19604 lineto 0.42063 0.21903 lineto 0.46032 0.24261 lineto 0.5 0.26677 lineto 0.53968 0.29152 lineto 0.57937 0.31687 lineto 0.61905 0.3428 lineto 0.65873 0.36934 lineto 0.69841 0.39647 lineto 0.7381 0.4242 lineto 0.77778 0.45254 lineto 0.81746 0.48148 lineto 0.85714 0.51103 lineto 0.89683 0.54118 lineto 0.93651 0.57194 lineto 0.97619 0.60332 lineto stroke grestore grestore grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = text; inactive; preserveAspect; endGroup; endGroup; endGroup] Answer: For any value of d between 0 and 1, the arclength is greater on the hyperbola. ;[s] 5:0,0;12,1;15,2;25,3;26,4;91,-1; 5:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0; ^*)