(*^ ::[ 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, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Times"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = leftheader, L0, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, L0, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; 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; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] MAXAREA :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] Determination of maximal inscribed rectangle in a region which does not permit functional representation of an area function as an explicit function of one variable. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsubsection; inactive; preserveAspect; endGroup] FileName: MAXAREA Full title: Maximum Area Rectangle in Function Boundary Last Revision Date: 2 June 1996. Developer: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Contact: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Aaron D. Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@Rose-Hulman.Edu. FAX: 812-877-3198. Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; preserveAspect] 1. Determine the rectangle of largest area which can be inscribed in the region bounded above by the graph of y = cos(x) and below by the x-axis which lies between x = 0 and x = Pi/2 if one side of the rectangle must lie on the x-axis. Be sure you make clear your strategy for solving this problem, even if you do not complete the analysis. :[font = subsection; inactive; preserveAspect; endGroup] 2. Determine the rectangle of largest area which can be inscribed in the region bounded above by the graph of y = x^2*cos(x) and below by the x-axis which lies between x = 0 and x = Pi/2 if one side of the rectangle must lie on the x-axis and another side must lie on the y-axis. Be sure you make clear your strategy for solving this problem, even if you do not complete the analysis. As a prelude to the above problem answer the following question: Pick the line x = 0.3 as the left hand edge of the rectangle. Compute the area of the resulting rectangle. Show your method. :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Optimization, area, algorithm, data fitting. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect] ISSUES RELATED TO THE PROBLEM :[font = subsection; inactive; preserveAspect] This (2) is an extension of a traditional problem (1) in which students are asked to find the area of a rectangle inscribed in a geometric boundary. :[font = subsection; inactive; preserveAspect] The first problem permits ready formulation of an objective function (area) to be maximized. :[font = subsection; inactive; preserveAspect] The second problem does not permit formulation of a closed form objective function and the student may resort to trial and error or to the construction of a function of the area given, say the left hand end point as an independent variable. :[font = subsection; inactive; preserveAspect] In both problems it is presumed that the objective function, however found, will yield up an optimal solution either through setting its derivative (formal or numerical) equal to 0 OR examining the plot of the function for optimal values. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] Area, optimization of objective function using derivatives or plots. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] This problem can be assigned as a homework problem. We have used it as an examination question during first semester calculus equivalent course. The problem is a good one for groups of students for the difficulty in the second problem is usually overcome by groups more easily than individuals, for the discussion which ensues in groups does not get stymied as one individual might. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect] We would expect students to understand and formulate problem 1 quite routinely. As to problem 2 we expect students to appear to be frustrated at first, especially when these problems are juxtaposed and they thus see that the simple strategy which worked in problem 1 will not work in problem 2 as they cannot get their hands on the objective function so readily. :[font = subsubsection; inactive; preserveAspect; endGroup] But we expect the students to examine the possible range of values of the independent variable, to test out some values per the suggestion in the statement of the problem, and then to either make list, trial and error, or fit function to several data points and optimize either by inspection of table or plot of data, or fit a function to the data and proceed to optimize based on that fitted function. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] Students will see that even though they cannot formulate a function in closed form they may be able to optimize it. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] Use different shapes in different regions. :[font = subsection; inactive; preserveAspect; endGroup] References and Sources :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Solution to Problem 1. :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We seek to determine the rectangle of largest area which can be inscribed in the region bounded above by the graph of y = cos(x) and below by the x-axis which lies between x = 0 and x = Pi/2 if one side of the rectangle must lie on the x-axis and another side must lie on the y-axis. Be sure you make clear your strategy for solving this problem, even if you do not complete the analysis. :[font = input; preserveAspect] g[x_] = Cos[x]; :[font = input; preserveAspect; startGroup] Plot[Cos[x],{x,0, Pi/2}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.606305 0.0147151 0.588604 [ [(0.25)] .17539 .01472 0 2 Msboxa [(0.5)] .32696 .01472 0 2 Msboxa [(0.75)] .47854 .01472 0 2 Msboxa [(1)] .63011 .01472 0 2 Msboxa [(1.25)] .78169 .01472 0 2 Msboxa [(1.5)] .93327 .01472 0 2 Msboxa [(0.2)] .01131 .13244 1 0 Msboxa [(0.4)] .01131 .25016 1 0 Msboxa [(0.6)] .01131 .36788 1 0 Msboxa [(0.8)] .01131 .4856 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 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.02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .02505 .60332 L .02629 .60331 L .02753 .60331 L .02877 .6033 L .03001 .60329 L .03125 .60327 L .03373 .60324 L .03621 .6032 L .03869 .60314 L .04365 .603 L .04861 .60283 L .05357 .60261 L .06349 .60206 L .07341 .60135 L .08333 .60048 L .10317 .59828 L .12302 .59546 L .14286 .59201 L .18254 .58326 L .22222 .57208 L .2619 .55851 L .30159 .54262 L .34127 .52446 L .38095 .50412 L .42063 .48169 L .46032 .45725 L .5 .43092 L .53968 .40281 L .57937 .37303 L .61905 .34173 L .65873 .30902 L .69841 .27505 L .7381 .23996 L .77778 .20392 L .81746 .16706 L .85714 .12955 L .89683 .09154 L .93651 .05321 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Thus if we let our rectangle have sides of length x and g(x) we have a function of one variable x we need to optimize. :[font = input; preserveAspect] area[x_] = x g[x]; :[font = input; preserveAspect; startGroup] Plot[area[x],{x,0, Pi/2}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.606305 0.0147151 1.04903 [ [(0.25)] .17539 .01472 0 2 Msboxa [(0.5)] .32696 .01472 0 2 Msboxa [(0.75)] .47854 .01472 0 2 Msboxa [(1)] .63011 .01472 0 2 Msboxa [(1.25)] .78169 .01472 0 2 Msboxa [(1.5)] .93327 .01472 0 2 Msboxa [(0.1)] .01131 .11962 1 0 Msboxa [(0.2)] .01131 .22452 1 0 Msboxa [(0.3)] .01131 .32942 1 0 Msboxa [(0.4)] .01131 .43433 1 0 Msboxa [(0.5)] .01131 .53923 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.25)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(0.5)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(0.75)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(1)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(1.25)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(1.5)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .11962 m .03006 .11962 L s P [(0.1)] .01131 .11962 1 0 Mshowa p .002 w .02381 .22452 m .03006 .22452 L s P [(0.2)] .01131 .22452 1 0 Mshowa p .002 w .02381 .32942 m .03006 .32942 L s P [(0.3)] .01131 .32942 1 0 Mshowa p .002 w .02381 .43433 m .03006 .43433 L s P [(0.4)] .01131 .43433 1 0 Mshowa p .002 w 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L s P p .001 w .02381 .58119 m .02756 .58119 L s P p .001 w .02381 .60217 m .02756 .60217 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .06349 .08323 L .10317 .15086 L .14286 .21673 L .18254 .27999 L .22222 .33979 L .2619 .39531 L .30159 .44576 L .34127 .4904 L .38095 .5285 L .42063 .55942 L .44048 .57199 L .46032 .58254 L .48016 .591 L .49008 .59442 L .5 .5973 L .50992 .59963 L .51488 .60058 L .51984 .6014 L .5248 .60207 L .52976 .6026 L .53472 .60298 L .5372 .60312 L .53844 .60317 L .53968 .60322 L .54092 .60326 L .54216 .60329 L .5434 .60331 L .54464 .60332 L .54588 .60332 L .54712 .60331 L .54836 .60329 L .5496 .60327 L .55084 .60323 L .55208 .60319 L .55456 .60307 L .55704 .60292 L .55952 .60273 L .56448 .60224 L .56944 .6016 L .57937 .59987 L .58929 .59754 L .59921 .5946 L .61905 .58688 L .63889 .57669 L .65873 .56398 L .69841 .53095 L .7381 .48766 L .77778 .43404 L .81746 .37012 L Mistroke .85714 .296 L .89683 .21187 L .93651 .118 L .97619 .01472 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] And in order to determine where the maximum area is (near x = .9 it would appear) we determine what the derivative of the area function is and set it equal to 0. :[font = input; preserveAspect; startGroup] sol = FindRoot[area'[x]==0,{x,.9}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {x -> 0.860333589019381} ;[o] {x -> 0.860334} :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Thus we see the maximum area is 0.561096 square units. :[font = input; preserveAspect; startGroup] maxArea = area[x]/.sol[[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] 0.561096338191045 ;[o] 0.561096 :[font = subsection; inactive; preserveAspect] Solutions to Problem 2. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Brian J. Winkel's Grunt solution :[font = subsubsection; inactive; preserveAspect] Fitting Polynomials to solution data, point by point :[font = subsubsection; inactive; preserveAspect; startGroup] We enter the function: :[font = input; preserveAspect] f[x_] = x^2 Cos[x]; :[font = input; preserveAspect; startGroup] Plot[f[x],{x,0, Pi/2}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.606305 0.0147151 1.07063 [ [(0.25)] .17539 .01472 0 2 Msboxa [(0.5)] .32696 .01472 0 2 Msboxa [(0.75)] .47854 .01472 0 2 Msboxa [(1)] .63011 .01472 0 2 Msboxa [(1.25)] .78169 .01472 0 2 Msboxa [(1.5)] .93327 .01472 0 2 Msboxa [(0.1)] .01131 .12178 1 0 Msboxa [(0.2)] .01131 .22884 1 0 Msboxa [(0.3)] .01131 .3359 1 0 Msboxa [(0.4)] .01131 .44297 1 0 Msboxa [(0.5)] .01131 .55003 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.25)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(0.5)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(0.75)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(1)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(1.25)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(1.5)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .12178 m .03006 .12178 L s P [(0.1)] .01131 .12178 1 0 Mshowa p .002 w .02381 .22884 m .03006 .22884 L s P [(0.2)] .01131 .22884 1 0 Mshowa p .002 w .02381 .3359 m .03006 .3359 L s P [(0.3)] .01131 .3359 1 0 Mshowa p .002 w .02381 .44297 m .03006 .44297 L s P [(0.4)] .01131 .44297 1 0 Mshowa p .002 w .02381 .55003 m .03006 .55003 L s P [(0.5)] .01131 .55003 1 0 Mshowa p .001 w .02381 .03613 m .02756 .03613 L s P p .001 w .02381 .05754 m .02756 .05754 L s P p .001 w .02381 .07895 m .02756 .07895 L s P p .001 w .02381 .10037 m .02756 .10037 L s P p .001 w .02381 .14319 m .02756 .14319 L s P p .001 w .02381 .1646 m .02756 .1646 L s P p .001 w .02381 .18602 m .02756 .18602 L s P p .001 w .02381 .20743 m .02756 .20743 L s P p .001 w .02381 .25025 m .02756 .25025 L s P p .001 w .02381 .27167 m .02756 .27167 L s P p .001 w .02381 .29308 m .02756 .29308 L s P p .001 w .02381 .31449 m .02756 .31449 L s P p .001 w .02381 .35732 m .02756 .35732 L s P p .001 w .02381 .37873 m .02756 .37873 L s P p .001 w .02381 .40014 m .02756 .40014 L s P p .001 w .02381 .42156 m .02756 .42156 L s P p .001 w .02381 .46438 m .02756 .46438 L s P p .001 w .02381 .48579 m .02756 .48579 L s P p .001 w .02381 .50721 m .02756 .50721 L s P p .001 w .02381 .52862 m .02756 .52862 L s P p .001 w .02381 .57144 m .02756 .57144 L s P p .001 w .02381 .59286 m .02756 .59286 L s P p .001 w .02381 .61427 m .02756 .61427 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .02505 .01472 L .02629 .01473 L .02753 .01476 L .02877 .01479 L .03001 .01483 L .03125 .01488 L .03373 .015 L .03621 .01516 L .03869 .01536 L .04365 .01586 L .04861 .01651 L .05357 .01729 L .06349 .01929 L .07341 .02186 L .08333 .02498 L .10317 .0329 L .14286 .0552 L .18254 .08559 L .22222 .12329 L .2619 .16725 L .30159 .21627 L .34127 .26891 L .38095 .32359 L .42063 .37857 L .46032 .43194 L .5 .4817 L .53968 .52576 L .57937 .56193 L .59921 .57637 L .61905 .58801 L .62897 .59269 L .63889 .59656 L .64881 .5996 L .65377 .60079 L .65873 .60175 L .66121 .60215 L .66369 .60249 L .66617 .60278 L .66865 .603 L .66989 .60309 L .67113 .60317 L .67237 .60323 L .67361 .60327 L .67485 .6033 L .67609 .60332 L .67733 .60332 L .67857 .6033 L .67981 .60327 L .68105 .60323 L Mistroke .68229 .60317 L .68353 .60309 L .68849 .60263 L .69097 .6023 L .69345 .60191 L .69841 .60094 L .70833 .5982 L .71825 .59438 L .72817 .58944 L .7381 .58336 L .75794 .56762 L .77778 .5469 L .81746 .48952 L .85714 .40929 L .89683 .30445 L .93651 .17339 L .97619 .01472 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We find out the maximum value of f so that we do not consider any starting values (for our left hand side of the rectangle) values to the right of that x value. :[font = input; preserveAspect; startGroup] FindRoot[f'[x]==0,{x,1}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {x -> 1.076873986311803} ;[o] {x -> 1.07687} :[font = subsubsection; inactive; preserveAspect] Here is our basic algorithm to pick a left hand end point xl = a for our rectangle, compute the height, f[a], determine the corresponding right hand end point for which f[xr] = f[a], and then the area Area = (xr - xl) f(xr). :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Start of algorithm for determining area for a given starting left hand end point a. :[font = input; preserveAspect] Remove[a,xl,lr,Area]; :[font = input; preserveAspect] a = .3; :[font = input; preserveAspect; startGroup] sol = FindRoot[f[x] == f[a],{x,1.4}] :[font = output; output; inactive; preserveAspect; endGroup] {x -> 1.534262413045758} ;[o] {x -> 1.53426} :[font = input; preserveAspect] xl = a; :[font = input; preserveAspect] xr = x/.sol[[1]]; :[font = input; preserveAspect; startGroup] Area = (xr - xl) f[xr] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0.1061222328304952 ;[o] 0.106122 :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We perform this algorithm a number of times and collect the data pairs (left hand point, Area) in a data set for different values of a: :[font = input; preserveAspect; endGroup] data = {{.3,.106122},{.4,.162954},{.5,.212582}, {.6,.244731}, {.7,.250974},{.8,.22585}, {.9,.167938}}; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We plot the data to get an idea where the maximum area might occur. :[font = input; preserveAspect; startGroup] lp = ListPlot[data,PlotStyle->{PointSize[.02]}, PlotRange->{{0,1.1},{0,.3}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -2.3744e-017 0.909091 -2.18738e-017 2.06011 [ [(0.2)] .18182 0 0 2 Msboxa [(0.4)] .36364 0 0 2 Msboxa [(0.6)] .54545 0 0 2 Msboxa [(0.8)] .72727 0 0 2 Msboxa [(1)] .90909 0 0 2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.05)] -0.0125 .10301 1 0 Msboxa [(0.1)] -0.0125 .20601 1 0 Msboxa [(0.15)] -0.0125 .30902 1 0 Msboxa [(0.2)] -0.0125 .41202 1 0 Msboxa [(0.25)] -0.0125 .51503 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .18182 0 m .18182 .00625 L s P [(0.2)] .18182 0 0 2 Mshowa p .002 w .36364 0 m .36364 .00625 L s P [(0.4)] .36364 0 0 2 Mshowa p .002 w .54545 0 m .54545 .00625 L s P [(0.6)] .54545 0 0 2 Mshowa p .002 w .72727 0 m .72727 .00625 L s P [(0.8)] .72727 0 0 2 Mshowa p .002 w .90909 0 m .90909 .00625 L s P [(1)] .90909 0 0 2 Mshowa p .001 w .03636 0 m .03636 .00375 L s P p .001 w .07273 0 m .07273 .00375 L s P p .001 w .10909 0 m .10909 .00375 L s P p .001 w .14545 0 m .14545 .00375 L s P p .001 w .21818 0 m .21818 .00375 L s P p .001 w .25455 0 m .25455 .00375 L s P p .001 w .29091 0 m .29091 .00375 L s P p .001 w .32727 0 m .32727 .00375 L s P p .001 w .4 0 m .4 .00375 L s P p .001 w .43636 0 m .43636 .00375 L s P p .001 w .47273 0 m .47273 .00375 L s P p .001 w .50909 0 m .50909 .00375 L s P p .001 w .58182 0 m .58182 .00375 L s P p .001 w .61818 0 m .61818 .00375 L s P p .001 w .65455 0 m .65455 .00375 L s P p .001 w .69091 0 m .69091 .00375 L s P p .001 w .76364 0 m .76364 .00375 L s P p .001 w .8 0 m .8 .00375 L s P p .001 w .83636 0 m .83636 .00375 L s P p .001 w .87273 0 m .87273 .00375 L s P p .001 w .94545 0 m .94545 .00375 L s P p .001 w .98182 0 m .98182 .00375 L s P p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .10301 m .00625 .10301 L s P [(0.05)] -0.0125 .10301 1 0 Mshowa p .002 w 0 .20601 m .00625 .20601 L s P [(0.1)] -0.0125 .20601 1 0 Mshowa p .002 w 0 .30902 m .00625 .30902 L s P [(0.15)] -0.0125 .30902 1 0 Mshowa p .002 w 0 .41202 m .00625 .41202 L s P [(0.2)] -0.0125 .41202 1 0 Mshowa p .002 w 0 .51503 m .00625 .51503 L s P [(0.25)] -0.0125 .51503 1 0 Mshowa p .001 w 0 .0206 m .00375 .0206 L s P p .001 w 0 .0412 m .00375 .0412 L s P p .001 w 0 .0618 m .00375 .0618 L s P p .001 w 0 .0824 m .00375 .0824 L s P p .001 w 0 .12361 m .00375 .12361 L s P p .001 w 0 .14421 m .00375 .14421 L s P p .001 w 0 .16481 m .00375 .16481 L s P p .001 w 0 .18541 m .00375 .18541 L s P p .001 w 0 .22661 m .00375 .22661 L s P p .001 w 0 .24721 m .00375 .24721 L s P p .001 w 0 .26781 m .00375 .26781 L s P p .001 w 0 .28842 m .00375 .28842 L s P p .001 w 0 .32962 m .00375 .32962 L s P p .001 w 0 .35022 m .00375 .35022 L s P p .001 w 0 .37082 m .00375 .37082 L s P p .001 w 0 .39142 m .00375 .39142 L s P p .001 w 0 .43262 m .00375 .43262 L s P p .001 w 0 .45322 m .00375 .45322 L s P p .001 w 0 .47383 m .00375 .47383 L s P p .001 w 0 .49443 m .00375 .49443 L s P p .001 w 0 .53563 m .00375 .53563 L s P p .001 w 0 .55623 m .00375 .55623 L s P p .001 w 0 .57683 m .00375 .57683 L s P p .001 w 0 .59743 m .00375 .59743 L s P p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .02 w .27273 .21862 Mdot .36364 .3357 Mdot .45455 .43794 Mdot .54545 .50417 Mdot .63636 .51703 Mdot .72727 .46528 Mdot .81818 .34597 Mdot P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We put a polynomial through the data points and plot it to estimate the value of x which would give maximum area: :[font = input; preserveAspect; startGroup] g[x_] = InterpolatingPolynomial[data,x] :[font = output; output; inactive; preserveAspect; endGroup] 0.106122 + (0.5683199999999998 + (-0.3601999999999977 + (-1.712500000000008 + (0.7700000000000009 + (0.931666666666817 - 0.0611111111119685*(-0.8 + x))* (-0.7 + x))*(-0.6 + x))*(-0.5 + x))* (-0.4 + x))*(-0.3 + x) ;[o] 0.106122 + (0.56832 + (-0.3602 + (-1.7125 + (0.77 + (0.931667 - 0.0611111 (-0.8 + x)) (-0.7 + x)) (-0.6 + x)) (-0.5 + x)) (-0.4 + x)) (-0.3 + x) :[font = input; preserveAspect; startGroup] gp = Plot[g[x],{x,0, 1.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.865801 0.0689489 2.11868 [ [(0.2)] .19697 .06895 0 2 Msboxa [(0.4)] .37013 .06895 0 2 Msboxa [(0.6)] .54329 .06895 0 2 Msboxa [(0.8)] .71645 .06895 0 2 Msboxa [(1)] .88961 .06895 0 2 Msboxa [(0.05)] .01131 .17488 1 0 Msboxa [(0.1)] .01131 .28082 1 0 Msboxa [(0.15)] .01131 .38675 1 0 Msboxa [(0.2)] .01131 .49268 1 0 Msboxa [(0.25)] .01131 .59862 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .19697 .06895 m .19697 .0752 L s P [(0.2)] .19697 .06895 0 2 Mshowa p .002 w .37013 .06895 m .37013 .0752 L s P [(0.4)] .37013 .06895 0 2 Mshowa p .002 w .54329 .06895 m .54329 .0752 L s P [(0.6)] .54329 .06895 0 2 Mshowa p .002 w .71645 .06895 m .71645 .0752 L s P [(0.8)] .71645 .06895 0 2 Mshowa p .002 w .88961 .06895 m .88961 .0752 L s P [(1)] .88961 .06895 0 2 Mshowa p .001 w .05844 .06895 m .05844 .0727 L s P p .001 w .09307 .06895 m .09307 .0727 L s P p .001 w .12771 .06895 m .12771 .0727 L s P p .001 w .16234 .06895 m .16234 .0727 L s P p .001 w .2316 .06895 m .2316 .0727 L s P p .001 w .26623 .06895 m .26623 .0727 L s P p .001 w .30087 .06895 m .30087 .0727 L s P p .001 w .3355 .06895 m .3355 .0727 L s P p .001 w .40476 .06895 m .40476 .0727 L s P p .001 w .43939 .06895 m .43939 .0727 L s P p .001 w .47403 .06895 m .47403 .0727 L s P p .001 w .50866 .06895 m .50866 .0727 L s P p .001 w .57792 .06895 m .57792 .0727 L s P p .001 w .61255 .06895 m .61255 .0727 L s P p .001 w .64719 .06895 m .64719 .0727 L s P p .001 w .68182 .06895 m .68182 .0727 L s P p .001 w .75108 .06895 m .75108 .0727 L s P p .001 w .78571 .06895 m .78571 .0727 L s P p .001 w .82035 .06895 m .82035 .0727 L s P p .001 w .85498 .06895 m .85498 .0727 L s P p .001 w .92424 .06895 m .92424 .0727 L s P p .001 w .95887 .06895 m .95887 .0727 L s P p .001 w .99351 .06895 m .99351 .0727 L s P p .002 w 0 .06895 m 1 .06895 L s P p .002 w .02381 .17488 m .03006 .17488 L s P [(0.05)] .01131 .17488 1 0 Mshowa p .002 w .02381 .28082 m .03006 .28082 L s P [(0.1)] .01131 .28082 1 0 Mshowa p .002 w .02381 .38675 m .03006 .38675 L s P [(0.15)] .01131 .38675 1 0 Mshowa p .002 w .02381 .49268 m .03006 .49268 L s P [(0.2)] .01131 .49268 1 0 Mshowa p .002 w .02381 .59862 m .03006 .59862 L s P [(0.25)] .01131 .59862 1 0 Mshowa p .001 w .02381 .09014 m .02756 .09014 L s P p .001 w .02381 .11132 m .02756 .11132 L s P p .001 w .02381 .13251 m .02756 .13251 L s P p .001 w .02381 .1537 m .02756 .1537 L s P p .001 w .02381 .19607 m .02756 .19607 L s P p .001 w .02381 .21726 m .02756 .21726 L s P p .001 w .02381 .23844 m .02756 .23844 L s P p .001 w .02381 .25963 m .02756 .25963 L s P p .001 w .02381 .302 m .02756 .302 L s P p .001 w .02381 .32319 m .02756 .32319 L s P p .001 w .02381 .34438 m .02756 .34438 L s P p .001 w .02381 .36556 m .02756 .36556 L s P p .001 w .02381 .40794 m .02756 .40794 L s P p .001 w .02381 .42912 m .02756 .42912 L s P p .001 w .02381 .45031 m .02756 .45031 L s P p .001 w .02381 .4715 m .02756 .4715 L s P p .001 w .02381 .51387 m .02756 .51387 L s P p .001 w .02381 .53506 m .02756 .53506 L s P p .001 w .02381 .55624 m .02756 .55624 L s P p .001 w .02381 .57743 m .02756 .57743 L s P p .001 w .02381 .04776 m .02756 .04776 L s P p .001 w .02381 .02658 m .02756 .02658 L s P p .001 w .02381 .00539 m .02756 .00539 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .06505 m .02629 .06522 L .02877 .06544 L .03373 .06603 L .03869 .06682 L .04365 .0678 L .05357 .07034 L .06349 .07364 L .08333 .08245 L .10317 .09408 L .14286 .12505 L .18254 .16496 L .22222 .21196 L .2619 .26403 L .30159 .31899 L .34127 .37461 L .38095 .42864 L .42063 .47884 L .46032 .52306 L .5 .55926 L .51984 .57377 L .53968 .58561 L .5496 .59046 L .55952 .59456 L .56944 .59791 L .5744 .59929 L .57937 .60047 L .58433 .60145 L .58929 .60222 L .59177 .60254 L .59425 .6028 L .59673 .603 L .59797 .60309 L .59921 .60316 L .60045 .60322 L .60169 .60326 L .60293 .60329 L .60417 .60331 L .60541 .60332 L .60665 .60331 L .60789 .60329 L .60913 .60326 L .61037 .60321 L .61161 .60315 L .61409 .60298 L .61657 .60277 L .61905 .6025 L .62401 .60179 L .62897 .60087 L .63889 .59837 L Mistroke .64881 .59497 L .65873 .59067 L .67857 .57933 L .69841 .56432 L .7381 .52319 L .77778 .46749 L .81746 .39792 L .85714 .31572 L .89683 .22271 L .93651 .12134 L .97619 .01472 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] From the combination of the data and the interpolating polynomial graphs we see the polynomial is a good estimate of the actual data in the maximum region which interests us especially. :[font = input; preserveAspect; startGroup] Show[lp,gp] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -2.3744e-017 0.909091 -2.18738e-017 2.06011 [ [(0.2)] .18182 0 0 2 Msboxa [(0.4)] .36364 0 0 2 Msboxa [(0.6)] .54545 0 0 2 Msboxa [(0.8)] .72727 0 0 2 Msboxa [(1)] .90909 0 0 2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.05)] -0.0125 .10301 1 0 Msboxa [(0.1)] -0.0125 .20601 1 0 Msboxa [(0.15)] -0.0125 .30902 1 0 Msboxa [(0.2)] -0.0125 .41202 1 0 Msboxa [(0.25)] -0.0125 .51503 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .18182 0 m .18182 .00625 L s P [(0.2)] .18182 0 0 2 Mshowa p .002 w .36364 0 m .36364 .00625 L s P [(0.4)] .36364 0 0 2 Mshowa p .002 w .54545 0 m .54545 .00625 L s P [(0.6)] .54545 0 0 2 Mshowa p .002 w .72727 0 m .72727 .00625 L s P [(0.8)] .72727 0 0 2 Mshowa p .002 w .90909 0 m .90909 .00625 L s P [(1)] .90909 0 0 2 Mshowa p .001 w .03636 0 m .03636 .00375 L s P p .001 w .07273 0 m .07273 .00375 L s P p .001 w .10909 0 m .10909 .00375 L s P p .001 w .14545 0 m .14545 .00375 L s P p .001 w .21818 0 m .21818 .00375 L s P p .001 w .25455 0 m .25455 .00375 L s P p .001 w .29091 0 m .29091 .00375 L s P p .001 w .32727 0 m .32727 .00375 L s P p .001 w .4 0 m .4 .00375 L s P p .001 w .43636 0 m .43636 .00375 L s P p .001 w .47273 0 m .47273 .00375 L s P p .001 w .50909 0 m .50909 .00375 L s P p .001 w .58182 0 m .58182 .00375 L s P p .001 w .61818 0 m .61818 .00375 L s P p .001 w .65455 0 m .65455 .00375 L s P p .001 w .69091 0 m .69091 .00375 L s P p .001 w .76364 0 m .76364 .00375 L s P p .001 w .8 0 m .8 .00375 L s P p .001 w .83636 0 m .83636 .00375 L s P p .001 w .87273 0 m .87273 .00375 L s P p .001 w .94545 0 m .94545 .00375 L s P p .001 w .98182 0 m .98182 .00375 L s P p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .10301 m .00625 .10301 L s P [(0.05)] -0.0125 .10301 1 0 Mshowa p .002 w 0 .20601 m .00625 .20601 L s P [(0.1)] -0.0125 .20601 1 0 Mshowa p .002 w 0 .30902 m .00625 .30902 L s P [(0.15)] -0.0125 .30902 1 0 Mshowa p .002 w 0 .41202 m .00625 .41202 L s P [(0.2)] -0.0125 .41202 1 0 Mshowa p .002 w 0 .51503 m .00625 .51503 L s P [(0.25)] -0.0125 .51503 1 0 Mshowa p .001 w 0 .0206 m .00375 .0206 L s P p .001 w 0 .0412 m .00375 .0412 L s P p .001 w 0 .0618 m .00375 .0618 L s P p .001 w 0 .0824 m .00375 .0824 L s P p .001 w 0 .12361 m .00375 .12361 L s P p .001 w 0 .14421 m .00375 .14421 L s P p .001 w 0 .16481 m .00375 .16481 L s P p .001 w 0 .18541 m .00375 .18541 L s P p .001 w 0 .22661 m .00375 .22661 L s P p .001 w 0 .24721 m .00375 .24721 L s P p .001 w 0 .26781 m .00375 .26781 L s P p .001 w 0 .28842 m .00375 .28842 L s P p .001 w 0 .32962 m .00375 .32962 L s P p .001 w 0 .35022 m .00375 .35022 L s P p .001 w 0 .37082 m .00375 .37082 L s P p .001 w 0 .39142 m .00375 .39142 L s P p .001 w 0 .43262 m .00375 .43262 L s P p .001 w 0 .45322 m .00375 .45322 L s P p .001 w 0 .47383 m .00375 .47383 L s P p .001 w 0 .49443 m .00375 .49443 L s P p .001 w 0 .53563 m .00375 .53563 L s P p .001 w 0 .55623 m .00375 .55623 L s P p .001 w 0 .57683 m .00375 .57683 L s P p .001 w 0 .59743 m .00375 .59743 L s P p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .02 w .27273 .21862 Mdot .36364 .3357 Mdot .45455 .43794 Mdot .54545 .50417 Mdot .63636 .51703 Mdot .72727 .46528 Mdot .81818 .34597 Mdot P P p p .004 w s s s s s s .02554 0 m .03125 .00135 L s .03125 .00135 m .04167 .00456 L .0625 .01313 L .08333 .02444 L .125 .05455 L .16667 .09336 L .20833 .13906 L .25 .18968 L .29167 .24313 L .33333 .29722 L .375 .34975 L .41667 .39857 L .45833 .44156 L .5 .47676 L .52083 .49087 L .54167 .50238 L .55208 .50709 L .5625 .51109 L .57292 .51434 L .57813 .51568 L .58333 .51683 L .58854 .51778 L .59375 .51854 L .59635 .51884 L .59896 .51909 L .60156 .51929 L .60286 .51938 L .60417 .51944 L .60547 .5195 L .60677 .51954 L .60807 .51958 L .60938 .51959 L .61068 .5196 L .61198 .51959 L .61328 .51957 L .61458 .51954 L .61589 .51949 L .61719 .51943 L .61979 .51927 L .6224 .51906 L .625 .5188 L .63021 .51812 L .63542 .51722 L .64583 .51478 L .65625 .51148 L .66667 .5073 L .6875 .49628 L .70833 .48167 L .75 .44168 L .79167 .38752 L Mistroke .83333 .31988 L .875 .23995 L .91667 .14951 L .95833 .05094 L Mfstroke .97881 0 m .95833 .05094 L s s P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We could have gone after a cubic polynomial just as easily. :[font = input; preserveAspect] h[x_] = Fit[data,{1,x,x^2,x^3},x]; :[font = input; preserveAspect; startGroup] hp = Plot[h[x],{x,0, 1.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.865801 0.170108 1.72296 [ [(0.2)] .19697 .17011 0 2 Msboxa [(0.4)] .37013 .17011 0 2 Msboxa [(0.6)] .54329 .17011 0 2 Msboxa [(0.8)] .71645 .17011 0 2 Msboxa [(1)] .88961 .17011 0 2 Msboxa [(-0.05)] .01131 .08396 1 0 Msboxa [(0.05)] .01131 .25626 1 0 Msboxa [(0.1)] .01131 .3424 1 0 Msboxa [(0.15)] .01131 .42855 1 0 Msboxa [(0.2)] .01131 .5147 1 0 Msboxa [(0.25)] .01131 .60085 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .19697 .17011 m .19697 .17636 L s P [(0.2)] .19697 .17011 0 2 Mshowa p .002 w .37013 .17011 m .37013 .17636 L s P [(0.4)] .37013 .17011 0 2 Mshowa p .002 w .54329 .17011 m .54329 .17636 L s P [(0.6)] .54329 .17011 0 2 Mshowa p .002 w .71645 .17011 m .71645 .17636 L s P [(0.8)] .71645 .17011 0 2 Mshowa p .002 w .88961 .17011 m .88961 .17636 L s P [(1)] .88961 .17011 0 2 Mshowa p .001 w .05844 .17011 m .05844 .17386 L s P p .001 w .09307 .17011 m .09307 .17386 L s P p .001 w .12771 .17011 m .12771 .17386 L s P p .001 w .16234 .17011 m .16234 .17386 L s P p .001 w .2316 .17011 m .2316 .17386 L s P p .001 w .26623 .17011 m .26623 .17386 L s P p .001 w .30087 .17011 m .30087 .17386 L s P p .001 w .3355 .17011 m .3355 .17386 L s P p .001 w .40476 .17011 m .40476 .17386 L s P p .001 w .43939 .17011 m .43939 .17386 L s P p .001 w .47403 .17011 m .47403 .17386 L s P p .001 w .50866 .17011 m .50866 .17386 L s P p .001 w .57792 .17011 m .57792 .17386 L s P p .001 w .61255 .17011 m .61255 .17386 L s P p .001 w .64719 .17011 m .64719 .17386 L s P p .001 w .68182 .17011 m .68182 .17386 L s P p .001 w .75108 .17011 m .75108 .17386 L s P p .001 w .78571 .17011 m .78571 .17386 L s P p .001 w .82035 .17011 m .82035 .17386 L s P p .001 w .85498 .17011 m .85498 .17386 L s P p .001 w .92424 .17011 m .92424 .17386 L s P p .001 w .95887 .17011 m .95887 .17386 L s P p .001 w .99351 .17011 m .99351 .17386 L s P p .002 w 0 .17011 m 1 .17011 L s P p .002 w .02381 .08396 m .03006 .08396 L s P [(-0.05)] .01131 .08396 1 0 Mshowa p .002 w .02381 .25626 m .03006 .25626 L s P [(0.05)] .01131 .25626 1 0 Mshowa p .002 w .02381 .3424 m .03006 .3424 L s P [(0.1)] .01131 .3424 1 0 Mshowa p .002 w .02381 .42855 m .03006 .42855 L s P [(0.15)] .01131 .42855 1 0 Mshowa p .002 w .02381 .5147 m .03006 .5147 L s P [(0.2)] .01131 .5147 1 0 Mshowa p .002 w .02381 .60085 m .03006 .60085 L s P [(0.25)] .01131 .60085 1 0 Mshowa p .001 w .02381 .01504 m .02756 .01504 L s P p .001 w .02381 .03227 m .02756 .03227 L s P p .001 w .02381 .0495 m .02756 .0495 L s P p .001 w .02381 .06673 m .02756 .06673 L s P p .001 w .02381 .10119 m .02756 .10119 L s P p .001 w .02381 .11842 m .02756 .11842 L s P p .001 w .02381 .13565 m .02756 .13565 L s P p .001 w .02381 .15288 m .02756 .15288 L s P p .001 w .02381 .18734 m .02756 .18734 L s P p .001 w .02381 .20457 m .02756 .20457 L s P p .001 w .02381 .2218 m .02756 .2218 L s P p .001 w .02381 .23903 m .02756 .23903 L s P p .001 w .02381 .27349 m .02756 .27349 L s P p .001 w .02381 .29072 m .02756 .29072 L s P p .001 w .02381 .30794 m .02756 .30794 L s P p .001 w .02381 .32517 m .02756 .32517 L s P p .001 w .02381 .35963 m .02756 .35963 L s P p .001 w .02381 .37686 m .02756 .37686 L s P p .001 w .02381 .39409 m .02756 .39409 L s P p .001 w .02381 .41132 m .02756 .41132 L s P p .001 w .02381 .44578 m .02756 .44578 L s P p .001 w .02381 .46301 m .02756 .46301 L s P p .001 w .02381 .48024 m .02756 .48024 L s P p .001 w .02381 .49747 m .02756 .49747 L s P p .001 w .02381 .53193 m .02756 .53193 L s P p .001 w .02381 .54916 m .02756 .54916 L s P p .001 w .02381 .56639 m .02756 .56639 L s P p .001 w .02381 .58362 m .02756 .58362 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .05132 m .06349 .08912 L .10317 .1315 L .14286 .17738 L .18254 .22565 L .22222 .27524 L .2619 .32504 L .30159 .37397 L .34127 .42094 L .38095 .46485 L .42063 .50461 L .46032 .53913 L .5 .56733 L .51984 .57871 L .53968 .5881 L .5496 .59201 L .55952 .59537 L .56944 .59816 L .57937 .60037 L .58433 .60124 L .58929 .60197 L .59425 .60255 L .59673 .60277 L .59921 .60296 L .60169 .60311 L .60293 .60317 L .60417 .60322 L .60541 .60326 L .60665 .60329 L .60789 .60331 L .60913 .60332 L .61037 .60332 L .61161 .60331 L .61285 .60329 L .61409 .60325 L .61657 .60316 L .61781 .6031 L .61905 .60302 L .62153 .60285 L .62401 .60263 L .62897 .60206 L .63393 .60132 L .63889 .60041 L .64881 .59806 L .65873 .59499 L .67857 .58662 L .69841 .57516 L .71825 .56049 L .7381 .54246 L .77778 .49579 L Mistroke .81746 .43406 L .85714 .35618 L .89683 .26106 L .93651 .1476 L .97619 .01472 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Show[lp,hp] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -2.3744e-017 0.909091 -2.18738e-017 2.06011 [ [(0.2)] .18182 0 0 2 Msboxa [(0.4)] .36364 0 0 2 Msboxa [(0.6)] .54545 0 0 2 Msboxa [(0.8)] .72727 0 0 2 Msboxa [(1)] .90909 0 0 2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.05)] -0.0125 .10301 1 0 Msboxa [(0.1)] -0.0125 .20601 1 0 Msboxa [(0.15)] -0.0125 .30902 1 0 Msboxa [(0.2)] -0.0125 .41202 1 0 Msboxa [(0.25)] -0.0125 .51503 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .18182 0 m .18182 .00625 L s P [(0.2)] .18182 0 0 2 Mshowa p .002 w .36364 0 m .36364 .00625 L s P [(0.4)] .36364 0 0 2 Mshowa p .002 w .54545 0 m .54545 .00625 L s P [(0.6)] .54545 0 0 2 Mshowa p .002 w .72727 0 m .72727 .00625 L s P [(0.8)] .72727 0 0 2 Mshowa p .002 w .90909 0 m .90909 .00625 L s P [(1)] .90909 0 0 2 Mshowa p .001 w .03636 0 m .03636 .00375 L s P p .001 w .07273 0 m .07273 .00375 L s P p .001 w .10909 0 m .10909 .00375 L s P p .001 w .14545 0 m .14545 .00375 L s P p .001 w .21818 0 m .21818 .00375 L s P p .001 w .25455 0 m .25455 .00375 L s P p .001 w .29091 0 m .29091 .00375 L s P p .001 w .32727 0 m .32727 .00375 L s P p .001 w .4 0 m .4 .00375 L s P p .001 w .43636 0 m .43636 .00375 L s P p .001 w .47273 0 m .47273 .00375 L s P p .001 w .50909 0 m .50909 .00375 L s P p .001 w .58182 0 m .58182 .00375 L s P p .001 w .61818 0 m .61818 .00375 L s P p .001 w .65455 0 m .65455 .00375 L s P p .001 w .69091 0 m .69091 .00375 L s P p .001 w .76364 0 m .76364 .00375 L s P p .001 w .8 0 m .8 .00375 L s P p .001 w .83636 0 m .83636 .00375 L s P p .001 w .87273 0 m .87273 .00375 L s P p .001 w .94545 0 m .94545 .00375 L s P p .001 w .98182 0 m .98182 .00375 L s P p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .10301 m .00625 .10301 L s P [(0.05)] -0.0125 .10301 1 0 Mshowa p .002 w 0 .20601 m .00625 .20601 L s P [(0.1)] -0.0125 .20601 1 0 Mshowa p .002 w 0 .30902 m .00625 .30902 L s P [(0.15)] -0.0125 .30902 1 0 Mshowa p .002 w 0 .41202 m .00625 .41202 L s P [(0.2)] -0.0125 .41202 1 0 Mshowa p .002 w 0 .51503 m .00625 .51503 L s P [(0.25)] -0.0125 .51503 1 0 Mshowa p .001 w 0 .0206 m .00375 .0206 L s P p .001 w 0 .0412 m .00375 .0412 L s P p .001 w 0 .0618 m .00375 .0618 L s P p .001 w 0 .0824 m .00375 .0824 L s P p .001 w 0 .12361 m .00375 .12361 L s P p .001 w 0 .14421 m .00375 .14421 L s P p .001 w 0 .16481 m .00375 .16481 L s P p .001 w 0 .18541 m .00375 .18541 L s P p .001 w 0 .22661 m .00375 .22661 L s P p .001 w 0 .24721 m .00375 .24721 L s P p .001 w 0 .26781 m .00375 .26781 L s P p .001 w 0 .28842 m .00375 .28842 L s P p .001 w 0 .32962 m .00375 .32962 L s P p .001 w 0 .35022 m .00375 .35022 L s P p .001 w 0 .37082 m .00375 .37082 L s P p .001 w 0 .39142 m .00375 .39142 L s P p .001 w 0 .43262 m .00375 .43262 L s P p .001 w 0 .45322 m .00375 .45322 L s P p .001 w 0 .47383 m .00375 .47383 L s P p .001 w 0 .49443 m .00375 .49443 L s P p .001 w 0 .53563 m .00375 .53563 L s P p .001 w 0 .55623 m .00375 .55623 L s P p .001 w 0 .57683 m .00375 .57683 L s P p .001 w 0 .59743 m .00375 .59743 L s P p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .02 w .27273 .21862 Mdot .36364 .3357 Mdot .45455 .43794 Mdot .54545 .50417 Mdot .63636 .51703 Mdot .72727 .46528 Mdot .81818 .34597 Mdot P P p p .004 w s s s .1184 0 m .125 .00869 L s .125 .00869 m .16667 .06641 L .20833 .1257 L .25 .18525 L .29167 .24375 L .33333 .29991 L .375 .35241 L .41667 .39996 L .45833 .44124 L .5 .47495 L .52083 .48856 L .54167 .49979 L .55208 .50446 L .5625 .50848 L .57292 .51181 L .58333 .51445 L .58854 .5155 L .59375 .51637 L .59896 .51706 L .60156 .51733 L .60417 .51756 L .60677 .51774 L .60807 .51781 L .60938 .51787 L .61068 .51791 L .61198 .51795 L .61328 .51797 L .61458 .51798 L .61589 .51798 L .61719 .51797 L .61849 .51794 L .61979 .51791 L .6224 .51779 L .6237 .51772 L .625 .51763 L .6276 .51742 L .63021 .51716 L .63542 .51648 L .64063 .5156 L .64583 .51451 L .65625 .51169 L .66667 .50802 L .6875 .49801 L .70833 .48432 L .72917 .46678 L .75 .44522 L .79167 .38942 L .83333 .31561 L .875 .22249 L .91667 .10875 L Mistroke Mfstroke .95007 0 m .91667 .10875 L s s s P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We find the x value for our Interpolating Polynomial where the Area is greatest and it appears that for x = 0.671625 we get a maximum area of 0.255219 square units. :[font = input; preserveAspect; startGroup] sol = FindRoot[g'[x]==0,{x,.7}] :[font = output; output; inactive; preserveAspect; endGroup] {x -> 0.6716246156030936} ;[o] {x -> 0.671625} :[font = input; preserveAspect; startGroup] xsolve = x/.sol[[1]] :[font = output; output; inactive; preserveAspect; endGroup] 0.6716246156030936 ;[o] 0.671625 :[font = input; preserveAspect; startGroup] f[xsolve] :[font = output; output; inactive; preserveAspect; endGroup] 0.3531104383754368 ;[o] 0.35311 :[font = input; preserveAspect; startGroup] maxArea = g[xsolve] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0.2522189264633235 ;[o] 0.252219 :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We find the x value for our Fitted Cubic Polynomial where the Area is greatest and it appears that for x = 0.676623 we get a maximum area of 0.252181 square units - a little less than with the interpolating polynomial. :[font = input; preserveAspect; startGroup] sol = FindRoot[h'[x]==0,{x,.7}] :[font = output; output; inactive; preserveAspect; endGroup] {x -> 0.6766234092508125} ;[o] {x -> 0.676623} :[font = input; preserveAspect; startGroup] xsolve = x/.sol[[1]] :[font = output; output; inactive; preserveAspect; endGroup] 0.6766234092508125 ;[o] 0.676623 :[font = input; preserveAspect; startGroup] f[xsolve] :[font = output; output; inactive; preserveAspect; endGroup] 0.3569577494705046 ;[o] 0.356958 :[font = input; preserveAspect; startGroup] maxArea = g[xsolve] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] 0.2521807279340903 ;[o] 0.252181 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Ryan Easterhaus' Elegant Solution (done as a final exam problem!) :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We set up a function f(x) which finds the area obtained as a function of the variable y by first determining the "other" point on the curve corresponding to selected first independent point z which makes a rectangle. The area of this rectangle is then f[z] (x-z). :[font = input; preserveAspect; endGroup] f[x_] = x^2 Cos[x]; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We define a function which does our work for us: :[font = input; preserveAspect; endGroup] A[z_] := f[z]((x/.FindRoot[f[x]== f[z],{x,1.3}][[1]]) - z) :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We plot this function and see an obvious maximum on which we now concentrate: :[font = input; preserveAspect; startGroup] Plot[A[x],{x,0,1.07}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.890076 0.0147151 2.3337 [ [(0.2)] .20182 .01472 0 2 Msboxa [(0.4)] .37984 .01472 0 2 Msboxa [(0.6)] .55785 .01472 0 2 Msboxa [(0.8)] .73587 .01472 0 2 Msboxa [(1)] .91389 .01472 0 2 Msboxa [(0.05)] .01131 .1314 1 0 Msboxa [(0.1)] .01131 .24809 1 0 Msboxa [(0.15)] .01131 .36477 1 0 Msboxa [(0.2)] .01131 .48146 1 0 Msboxa [(0.25)] .01131 .59814 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .20182 .01472 m .20182 .02097 L s P [(0.2)] .20182 .01472 0 2 Mshowa p .002 w .37984 .01472 m .37984 .02097 L s P [(0.4)] .37984 .01472 0 2 Mshowa p .002 w .55785 .01472 m .55785 .02097 L s P [(0.6)] .55785 .01472 0 2 Mshowa p .002 w .73587 .01472 m .73587 .02097 L s P [(0.8)] .73587 .01472 0 2 Mshowa p .002 w .91389 .01472 m .91389 .02097 L s P [(1)] .91389 .01472 0 2 Mshowa p .001 w .05941 .01472 m .05941 .01847 L s P p .001 w .09502 .01472 m .09502 .01847 L s P p .001 w .13062 .01472 m .13062 .01847 L s P p .001 w .16622 .01472 m .16622 .01847 L s P p .001 w .23743 .01472 m .23743 .01847 L s P p .001 w .27303 .01472 m .27303 .01847 L s P p .001 w .30863 .01472 m .30863 .01847 L s P p .001 w .34424 .01472 m .34424 .01847 L s P p .001 w .41544 .01472 m .41544 .01847 L s P p .001 w .45105 .01472 m .45105 .01847 L s P p .001 w .48665 .01472 m .48665 .01847 L s P p .001 w .52225 .01472 m .52225 .01847 L s P p .001 w .59346 .01472 m .59346 .01847 L s P p .001 w .62906 .01472 m .62906 .01847 L s P p .001 w .66466 .01472 m .66466 .01847 L s P p .001 w .70027 .01472 m .70027 .01847 L s P p .001 w .77147 .01472 m .77147 .01847 L s P p .001 w .80708 .01472 m .80708 .01847 L s P p .001 w .84268 .01472 m .84268 .01847 L s P p .001 w .87828 .01472 m .87828 .01847 L s P p .001 w .94949 .01472 m .94949 .01847 L s P p .001 w .98509 .01472 m .98509 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .1314 m .03006 .1314 L s P [(0.05)] .01131 .1314 1 0 Mshowa p .002 w .02381 .24809 m .03006 .24809 L s P [(0.1)] .01131 .24809 1 0 Mshowa p .002 w .02381 .36477 m .03006 .36477 L s P [(0.15)] .01131 .36477 1 0 Mshowa p .002 w .02381 .48146 m .03006 .48146 L s P [(0.2)] .01131 .48146 1 0 Mshowa p .002 w .02381 .59814 m .03006 .59814 L s P [(0.25)] .01131 .59814 1 0 Mshowa p .001 w .02381 .03805 m .02756 .03805 L s P p .001 w .02381 .06139 m .02756 .06139 L s P p .001 w .02381 .08473 m .02756 .08473 L s P p .001 w .02381 .10806 m .02756 .10806 L s P p .001 w .02381 .15474 m .02756 .15474 L s P p .001 w .02381 .17807 m .02756 .17807 L s P p .001 w .02381 .20141 m .02756 .20141 L s P p .001 w .02381 .22475 m .02756 .22475 L s P p .001 w .02381 .27142 m .02756 .27142 L s P p .001 w .02381 .29476 m .02756 .29476 L s P p .001 w .02381 .3181 m .02756 .3181 L s P p .001 w .02381 .34143 m .02756 .34143 L s P p .001 w .02381 .38811 m .02756 .38811 L s P p .001 w .02381 .41144 m .02756 .41144 L s P p .001 w .02381 .43478 m .02756 .43478 L s P p .001 w .02381 .45812 m .02756 .45812 L s P p .001 w .02381 .50479 m .02756 .50479 L s P p .001 w .02381 .52813 m .02756 .52813 L s P p .001 w .02381 .55147 m .02756 .55147 L s P p .001 w .02381 .5748 m .02756 .5748 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .02505 .01472 L .02629 .01474 L .02753 .01478 L .02877 .01483 L .03125 .01497 L .03373 .01517 L .03621 .01542 L .03869 .01573 L .04365 .01651 L .05357 .01872 L .06349 .02178 L .08333 .03036 L .10317 .04204 L .14286 .07387 L .18254 .11548 L .22222 .16488 L .2619 .21994 L .30159 .27844 L .34127 .3381 L .38095 .39661 L .42063 .45169 L .46032 .50113 L .5 .54284 L .51984 .56018 L .53968 .57488 L .55952 .58672 L .56944 .59152 L .57937 .59552 L .58929 .59872 L .59921 .6011 L .60417 .60197 L .60665 .60232 L .60913 .60262 L .61161 .60287 L .61409 .60307 L .61657 .60321 L .61781 .60325 L .61905 .60329 L .62029 .60331 L .62153 .60332 L .62277 .60331 L .62401 .60329 L .62525 .60326 L .62649 .60321 L .62897 .60307 L .63145 .60288 L .63393 .60263 L .63889 .60196 L .64385 .60107 L Mistroke .64881 .59995 L .65873 .59701 L .66865 .59313 L .67857 .58832 L .69841 .57583 L .71825 .55951 L .7381 .53933 L .77778 .48746 L .81746 .4207 L .85714 .33999 L .89683 .24685 L .93651 .14338 L .97619 .03231 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We know we wish to examine when the derivative is zero, thus we take a numerical estimate of the derivative and plot it in the region [.6, .8] to ascertain just when we might expect A'(x) = 0. :[font = input; preserveAspect; startGroup] Plot[(A[x+.0001]-A[x])/.0001,{x,.6,.8}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -2.83333 4.7619 0.410068 0.948377 [ [(0.65)] .2619 .41007 0 2 Msboxa [(0.7)] .5 .41007 0 2 Msboxa [(0.75)] .7381 .41007 0 2 Msboxa [(0.8)] .97619 .41007 0 2 Msboxa [(-0.4)] .01131 .03072 1 0 Msboxa [(-0.3)] .01131 .12555 1 0 Msboxa [(-0.2)] .01131 .22039 1 0 Msboxa [(-0.1)] .01131 .31523 1 0 Msboxa [(0.1)] .01131 .50491 1 0 Msboxa [(0.2)] .01131 .59974 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .41007 m .2619 .41632 L s P [(0.65)] .2619 .41007 0 2 Mshowa p .002 w .5 .41007 m .5 .41632 L s P [(0.7)] .5 .41007 0 2 Mshowa p .002 w .7381 .41007 m .7381 .41632 L s P [(0.75)] .7381 .41007 0 2 Mshowa p .002 w .97619 .41007 m .97619 .41632 L s P [(0.8)] .97619 .41007 0 2 Mshowa p .001 w .07143 .41007 m .07143 .41382 L s P p .001 w .11905 .41007 m .11905 .41382 L s P p .001 w .16667 .41007 m .16667 .41382 L s P p .001 w .21429 .41007 m .21429 .41382 L s P p .001 w .30952 .41007 m .30952 .41382 L s P p .001 w .35714 .41007 m .35714 .41382 L s P p .001 w .40476 .41007 m .40476 .41382 L s P p .001 w .45238 .41007 m .45238 .41382 L s P p .001 w .54762 .41007 m .54762 .41382 L s P p .001 w .59524 .41007 m .59524 .41382 L s P p .001 w .64286 .41007 m .64286 .41382 L s P p .001 w .69048 .41007 m .69048 .41382 L s P p .001 w .78571 .41007 m .78571 .41382 L s P p .001 w .83333 .41007 m .83333 .41382 L s P p .001 w .88095 .41007 m .88095 .41382 L s P p .001 w .92857 .41007 m .92857 .41382 L s P p .002 w 0 .41007 m 1 .41007 L s P p .002 w .02381 .03072 m .03006 .03072 L s P [(-0.4)] .01131 .03072 1 0 Mshowa p .002 w .02381 .12555 m .03006 .12555 L s P [(-0.3)] .01131 .12555 1 0 Mshowa p .002 w .02381 .22039 m .03006 .22039 L s P [(-0.2)] .01131 .22039 1 0 Mshowa p .002 w .02381 .31523 m .03006 .31523 L s P [(-0.1)] .01131 .31523 1 0 Mshowa p .002 w .02381 .50491 m .03006 .50491 L s P [(0.1)] .01131 .50491 1 0 Mshowa p .002 w .02381 .59974 m .03006 .59974 L s P [(0.2)] .01131 .59974 1 0 Mshowa p .001 w .02381 .04968 m .02756 .04968 L s P p .001 w .02381 .06865 m .02756 .06865 L s P p .001 w .02381 .08762 m .02756 .08762 L s P p .001 w .02381 .10659 m .02756 .10659 L s P p .001 w .02381 .14452 m .02756 .14452 L s P p .001 w .02381 .16349 m .02756 .16349 L s P p .001 w .02381 .18246 m .02756 .18246 L s P p .001 w .02381 .20142 m .02756 .20142 L s P p .001 w .02381 .23936 m .02756 .23936 L s P p .001 w .02381 .25833 m .02756 .25833 L s P p .001 w .02381 .27729 m .02756 .27729 L s P p .001 w .02381 .29626 m .02756 .29626 L s P p .001 w .02381 .3342 m .02756 .3342 L s P p .001 w .02381 .35316 m .02756 .35316 L s P p .001 w .02381 .37213 m .02756 .37213 L s P p .001 w .02381 .3911 m .02756 .3911 L s P p .001 w .02381 .42904 m .02756 .42904 L s P p .001 w .02381 .448 m .02756 .448 L s P p .001 w .02381 .46697 m .02756 .46697 L s P p .001 w .02381 .48594 m .02756 .48594 L s P p .001 w .02381 .52387 m .02756 .52387 L s P p .001 w .02381 .54284 m .02756 .54284 L s P p .001 w .02381 .56181 m .02756 .56181 L s P p .001 w .02381 .58078 m .02756 .58078 L s P p .001 w .02381 .01175 m .02756 .01175 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .06349 .58241 L .10317 .56105 L .14286 .53924 L .18254 .51701 L .22222 .49437 L .2619 .47135 L .30159 .44796 L .34127 .42421 L .38095 .40014 L .42063 .37576 L .46032 .3511 L .5 .32616 L .53968 .30099 L .57937 .27559 L .61905 .24999 L .65873 .22422 L .69841 .1983 L .7381 .17225 L .77778 .1461 L .81746 .11987 L .85714 .09359 L .89683 .06729 L .93651 .04099 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] From this it appears that just above 0.67 there is a value of x which give maximum area of 0.252215. And so we estimate this area value. :[font = input; preserveAspect; startGroup] A[.67] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0.2522149298325205 ;[o] 0.252215 :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] And from the following diagram we see that perhaps x = 0.6715 would do better. :[font = input; preserveAspect; startGroup] Plot[(A[x+.0001]-A[x])/.0001,{x,.67,.675}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -127.595 190.476 0.418126 38.5482 [ [(0.671)] .21429 .41813 0 2 Msboxa [(0.672)] .40476 .41813 0 2 Msboxa [(0.673)] .59524 .41813 0 2 Msboxa [(0.674)] .78571 .41813 0 2 Msboxa [(0.675)] .97619 .41813 0 2 Msboxa [(-0.01)] .01131 .03264 1 0 Msboxa [(-0.0075)] .01131 .12901 1 0 Msboxa [(-0.005)] .01131 .22539 1 0 Msboxa [(-0.0025)] .01131 .32176 1 0 Msboxa [(0.0025)] .01131 .5145 1 0 Msboxa [(0.005)] .01131 .61087 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .41813 m .21429 .42438 L s P [(0.671)] .21429 .41813 0 2 Mshowa p .002 w .40476 .41813 m .40476 .42438 L s P [(0.672)] .40476 .41813 0 2 Mshowa p .002 w .59524 .41813 m .59524 .42438 L s P [(0.673)] .59524 .41813 0 2 Mshowa p .002 w .78571 .41813 m .78571 .42438 L s P [(0.674)] .78571 .41813 0 2 Mshowa p .002 w .97619 .41813 m .97619 .42438 L s P [(0.675)] .97619 .41813 0 2 Mshowa p .001 w .0619 .41813 m .0619 .42188 L s P p .001 w .1 .41813 m .1 .42188 L s P p .001 w .1381 .41813 m .1381 .42188 L s P p .001 w .17619 .41813 m .17619 .42188 L s P p .001 w .25238 .41813 m .25238 .42188 L s P p .001 w .29048 .41813 m .29048 .42188 L s P p .001 w .32857 .41813 m .32857 .42188 L s P p .001 w .36667 .41813 m .36667 .42188 L s P p .001 w .44286 .41813 m .44286 .42188 L s P p .001 w .48095 .41813 m .48095 .42188 L s P p .001 w .51905 .41813 m .51905 .42188 L s P p .001 w .55714 .41813 m .55714 .42188 L s P p .001 w .63333 .41813 m .63333 .42188 L s P p .001 w .67143 .41813 m .67143 .42188 L s P p .001 w .70952 .41813 m .70952 .42188 L s P p .001 w .74762 .41813 m .74762 .42188 L s P p .001 w .82381 .41813 m .82381 .42188 L s P p .001 w .8619 .41813 m .8619 .42188 L s P p .001 w .9 .41813 m .9 .42188 L s P p .001 w .9381 .41813 m .9381 .42188 L s P p .002 w 0 .41813 m 1 .41813 L s P p .002 w .02381 .03264 m .03006 .03264 L s P [(-0.01)] .01131 .03264 1 0 Mshowa p .002 w .02381 .12901 m .03006 .12901 L s P [(-0.0075)] .01131 .12901 1 0 Mshowa p .002 w .02381 .22539 m .03006 .22539 L s P [(-0.005)] .01131 .22539 1 0 Mshowa p .002 w .02381 .32176 m .03006 .32176 L s P [(-0.0025)] .01131 .32176 1 0 Mshowa p .002 w .02381 .5145 m .03006 .5145 L s P [(0.0025)] .01131 .5145 1 0 Mshowa p .002 w .02381 .61087 m .03006 .61087 L s P [(0.005)] .01131 .61087 1 0 Mshowa p .001 w .02381 .05192 m .02756 .05192 L s P p .001 w .02381 .07119 m .02756 .07119 L s P p .001 w .02381 .09047 m .02756 .09047 L s P p .001 w .02381 .10974 m .02756 .10974 L s P p .001 w .02381 .14829 m .02756 .14829 L s P p .001 w .02381 .16756 m .02756 .16756 L s P p .001 w .02381 .18684 m .02756 .18684 L s P p .001 w .02381 .20611 m .02756 .20611 L s P p .001 w .02381 .24466 m .02756 .24466 L s P p .001 w .02381 .26393 m .02756 .26393 L s P p .001 w .02381 .28321 m .02756 .28321 L s P p .001 w .02381 .30248 m .02756 .30248 L s P p .001 w .02381 .34103 m .02756 .34103 L s P p .001 w .02381 .3603 m .02756 .3603 L s P p .001 w .02381 .37958 m .02756 .37958 L s P p .001 w .02381 .39885 m .02756 .39885 L s P p .001 w .02381 .4374 m .02756 .4374 L s P p .001 w .02381 .45667 m .02756 .45667 L s P p .001 w .02381 .47595 m .02756 .47595 L s P p .001 w .02381 .49522 m .02756 .49522 L s P p .001 w .02381 .53377 m .02756 .53377 L s P p .001 w .02381 .55304 m .02756 .55304 L s P p .001 w .02381 .57232 m .02756 .57232 L s P p .001 w .02381 .59159 m .02756 .59159 L s P p .001 w .02381 .01337 m .02756 .01337 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .06349 .57889 L .10317 .55445 L .14286 .53 L .18254 .50554 L .22222 .48107 L .2619 .4566 L .30159 .43212 L .34127 .40763 L .38095 .38313 L .42063 .35863 L .46032 .33411 L .5 .30959 L .53968 .28506 L .57937 .26053 L .61905 .23598 L .65873 .21143 L .69841 .18687 L .7381 .1623 L .77778 .13772 L .81746 .11313 L .85714 .08854 L .89683 .06394 L .93651 .03933 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Indeed it is slightly better (0.252219 vs. 0.252215), but not enough to worry considering the numerical derivative we are estimating the area with in the first place. :[font = input; preserveAspect; startGroup] A[.6715] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0.2522189397200214 ;[o] 0.252219 :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Could we do better, knowing that we have an interpolating function and an approximate derivative? We could examine the graph of A(x) itself. But we get about the same results. :[font = input; preserveAspect; startGroup] Plot[A[x],{x,.65,.68}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -20.6111 31.746 -363.141 1442.18 [ [(0.655)] .18254 .28753 0 2 Msboxa [(0.66)] .34127 .28753 0 2 Msboxa [(0.665)] .5 .28753 0 2 Msboxa [(0.67)] .65873 .28753 0 2 Msboxa [(0.675)] .81746 .28753 0 2 Msboxa [(0.68)] .97619 .28753 0 2 Msboxa [(0.2519)] .01131 .14332 1 0 Msboxa [(0.2521)] .01131 .43175 1 0 Msboxa [(0.2522)] .01131 .57597 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .18254 .28753 m .18254 .29378 L s P [(0.655)] .18254 .28753 0 2 Mshowa p .002 w .34127 .28753 m .34127 .29378 L s P [(0.66)] .34127 .28753 0 2 Mshowa p .002 w .5 .28753 m .5 .29378 L s P [(0.665)] .5 .28753 0 2 Mshowa p .002 w .65873 .28753 m .65873 .29378 L s P [(0.67)] .65873 .28753 0 2 Mshowa p .002 w .81746 .28753 m .81746 .29378 L s P [(0.675)] .81746 .28753 0 2 Mshowa p .002 w .97619 .28753 m .97619 .29378 L s P [(0.68)] .97619 .28753 0 2 Mshowa p .001 w .05556 .28753 m .05556 .29128 L s P p .001 w .0873 .28753 m .0873 .29128 L s P p .001 w .11905 .28753 m .11905 .29128 L s P p .001 w .15079 .28753 m .15079 .29128 L s P p .001 w .21429 .28753 m .21429 .29128 L s P p .001 w .24603 .28753 m .24603 .29128 L s P p .001 w .27778 .28753 m .27778 .29128 L s P p .001 w .30952 .28753 m .30952 .29128 L s P p .001 w .37302 .28753 m .37302 .29128 L s P p .001 w .40476 .28753 m .40476 .29128 L s P p .001 w .43651 .28753 m .43651 .29128 L s P p .001 w .46825 .28753 m .46825 .29128 L s P p .001 w .53175 .28753 m .53175 .29128 L s P p .001 w .56349 .28753 m .56349 .29128 L s P p .001 w .59524 .28753 m .59524 .29128 L s P p .001 w .62698 .28753 m .62698 .29128 L s P p .001 w .69048 .28753 m .69048 .29128 L s P p .001 w .72222 .28753 m .72222 .29128 L s P p .001 w .75397 .28753 m .75397 .29128 L s P p .001 w .78571 .28753 m .78571 .29128 L s P p .001 w .84921 .28753 m .84921 .29128 L s P p .001 w .88095 .28753 m .88095 .29128 L s P p .001 w .9127 .28753 m .9127 .29128 L s P p .001 w .94444 .28753 m .94444 .29128 L s P p .002 w 0 .28753 m 1 .28753 L s P p .002 w .02381 .14332 m .03006 .14332 L s P [(0.2519)] .01131 .14332 1 0 Mshowa p .002 w .02381 .43175 m .03006 .43175 L s P [(0.2521)] .01131 .43175 1 0 Mshowa p .002 w .02381 .57597 m .03006 .57597 L s P [(0.2522)] .01131 .57597 1 0 Mshowa p .001 w .02381 .02794 m .02756 .02794 L s P p .001 w .02381 .05679 m .02756 .05679 L s P p .001 w .02381 .08563 m .02756 .08563 L s P p .001 w .02381 .11447 m .02756 .11447 L s P p .001 w .02381 .17216 m .02756 .17216 L s P p .001 w .02381 .201 m .02756 .201 L s P p .001 w .02381 .22985 m .02756 .22985 L s P p .001 w .02381 .25869 m .02756 .25869 L s P p .001 w .02381 .31638 m .02756 .31638 L s P p .001 w .02381 .34522 m .02756 .34522 L s P p .001 w .02381 .37406 m .02756 .37406 L s P p .001 w .02381 .40291 m .02756 .40291 L s P p .001 w .02381 .4606 m .02756 .4606 L s P p .001 w .02381 .48944 m .02756 .48944 L s P p .001 w .02381 .51828 m .02756 .51828 L s P p .001 w .02381 .54713 m .02756 .54713 L s P p .001 w .02381 .60481 m .02756 .60481 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w s s s s .18217 0 m .18254 .00087 L s .18254 .00087 m .22222 .08768 L .2619 .1678 L .30159 .24121 L .34127 .30789 L .38095 .36783 L .42063 .42102 L .46032 .46744 L .5 .50708 L .53968 .53992 L .57937 .56594 L .59921 .5764 L .61905 .58514 L .63889 .59218 L .64881 .59505 L .65873 .5975 L .66865 .59952 L .67857 .60111 L .68849 .60227 L .69345 .60269 L .69593 .60286 L .69841 .60301 L .70089 .60312 L .70337 .60321 L .70461 .60325 L .70585 .60327 L .70709 .6033 L .70833 .60331 L .70957 .60332 L .71081 .60332 L .71205 .60331 L .71329 .6033 L .71453 .60328 L .71577 .60326 L .71825 .60318 L .72073 .60309 L .72321 .60296 L .72817 .60263 L .73313 .60219 L .7381 .60164 L .74802 .60023 L .75794 .59838 L .77778 .5934 L .79762 .58669 L .81746 .57826 L .85714 .55622 L .89683 .52725 L .93651 .49135 L .97619 .4485 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Plot[A[x],{x,.67,.675}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -127.595 190.476 -14721.2 58369 [ [(0.671)] .21429 .08007 0 2 Msboxa [(0.672)] .40476 .08007 0 2 Msboxa [(0.673)] .59524 .08007 0 2 Msboxa [(0.674)] .78571 .08007 0 2 Msboxa [(0.675)] .97619 .08007 0 2 Msboxa [(0.252212)] .01131 .19681 1 0 Msboxa [(0.252214)] .01131 .31355 1 0 Msboxa [(0.252216)] .01131 .43029 1 0 Msboxa [(0.252218)] .01131 .54702 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .08007 m .21429 .08632 L s P [(0.671)] .21429 .08007 0 2 Mshowa p .002 w .40476 .08007 m .40476 .08632 L s P [(0.672)] .40476 .08007 0 2 Mshowa p .002 w .59524 .08007 m .59524 .08632 L s P [(0.673)] .59524 .08007 0 2 Mshowa p .002 w .78571 .08007 m .78571 .08632 L s P [(0.674)] .78571 .08007 0 2 Mshowa p .002 w .97619 .08007 m .97619 .08632 L s P [(0.675)] .97619 .08007 0 2 Mshowa p .001 w .0619 .08007 m .0619 .08382 L s P p .001 w .1 .08007 m .1 .08382 L s P p .001 w .1381 .08007 m .1381 .08382 L s P p .001 w .17619 .08007 m .17619 .08382 L s P p .001 w .25238 .08007 m .25238 .08382 L s P p .001 w .29048 .08007 m .29048 .08382 L s P p .001 w .32857 .08007 m .32857 .08382 L s P p .001 w .36667 .08007 m .36667 .08382 L s P p .001 w .44286 .08007 m .44286 .08382 L s P p .001 w .48095 .08007 m .48095 .08382 L s P p .001 w .51905 .08007 m .51905 .08382 L s P p .001 w .55714 .08007 m .55714 .08382 L s P p .001 w .63333 .08007 m .63333 .08382 L s P p .001 w .67143 .08007 m .67143 .08382 L s P p .001 w .70952 .08007 m .70952 .08382 L s P p .001 w .74762 .08007 m .74762 .08382 L s P p .001 w .82381 .08007 m .82381 .08382 L s P p .001 w .8619 .08007 m .8619 .08382 L s P p .001 w .9 .08007 m .9 .08382 L s P p .001 w .9381 .08007 m .9381 .08382 L s P p .002 w 0 .08007 m 1 .08007 L s P p .002 w .02381 .19681 m .03006 .19681 L s P [(0.252212)] .01131 .19681 1 0 Mshowa p .002 w .02381 .31355 m .03006 .31355 L s P [(0.252214)] .01131 .31355 1 0 Mshowa p .002 w .02381 .43029 m .03006 .43029 L s P [(0.252216)] .01131 .43029 1 0 Mshowa p .002 w .02381 .54702 m .03006 .54702 L s P [(0.252218)] .01131 .54702 1 0 Mshowa p .001 w .02381 .10342 m .02756 .10342 L s P p .001 w .02381 .12677 m .02756 .12677 L s P p .001 w .02381 .15011 m .02756 .15011 L s P p .001 w .02381 .17346 m .02756 .17346 L s P p .001 w .02381 .22016 m .02756 .22016 L s P p .001 w .02381 .24351 m .02756 .24351 L s P p .001 w .02381 .26685 m .02756 .26685 L s P p .001 w .02381 .2902 m .02756 .2902 L s P p .001 w .02381 .3369 m .02756 .3369 L s P p .001 w .02381 .36024 m .02756 .36024 L s P p .001 w .02381 .38359 m .02756 .38359 L s P p .001 w .02381 .40694 m .02756 .40694 L s P p .001 w .02381 .45363 m .02756 .45363 L s P p .001 w .02381 .47698 m .02756 .47698 L s P p .001 w .02381 .50033 m .02756 .50033 L s P p .001 w .02381 .52368 m .02756 .52368 L s P p .001 w .02381 .05672 m .02756 .05672 L s P p .001 w .02381 .03338 m .02756 .03338 L s P p .001 w .02381 .01003 m .02756 .01003 L s P p .001 w .02381 .57037 m .02756 .57037 L s P p .001 w .02381 .59372 m .02756 .59372 L s P p .001 w .02381 .61707 m .02756 .61707 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .36782 m .06349 .42424 L .10317 .47295 L .14286 .51394 L .18254 .54723 L .20238 .56098 L .22222 .5728 L .24206 .58269 L .2619 .59065 L .27183 .5939 L .28175 .59668 L .29167 .59897 L .30159 .60078 L .30655 .6015 L .31151 .6021 L .31647 .60258 L .31895 .60278 L .32143 .60294 L .32391 .60308 L .32639 .60318 L .32763 .60323 L .32887 .60326 L .33011 .60329 L .33135 .6033 L .33259 .60332 L .33383 .60332 L .33507 .60331 L .33631 .6033 L .33755 .60328 L .33879 .60326 L .34003 .60322 L .34127 .60318 L .34375 .60308 L .34623 .60294 L .35119 .60258 L .35615 .60209 L .36111 .60149 L .37103 .59992 L .38095 .59786 L .40079 .5923 L .42063 .58481 L .46032 .56403 L .5 .53552 L .53968 .49927 L .57937 .45528 L .61905 .40355 L .65873 .34408 L .69841 .27686 L .7381 .2019 L .77778 .11918 L Mistroke .81746 .02871 L Mfstroke .82906 0 m .81746 .02871 L s s s s s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Plot[A[x],{x,.67,.672}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -319.024 476.19 -36795.1 145888 [ [(0.6705)] .2619 .02495 0 2 Msboxa [(0.671)] .5 .02495 0 2 Msboxa [(0.6715)] .7381 .02495 0 2 Msboxa [(0.672)] .97619 .02495 0 2 Msboxa [(0.252216)] .01131 .17084 1 0 Msboxa [(0.252217)] .01131 .31673 1 0 Msboxa [(0.252218)] .01131 .46262 1 0 Msboxa [(0.252219)] .01131 .6085 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .02495 m .2619 .0312 L s P [(0.6705)] .2619 .02495 0 2 Mshowa p .002 w .5 .02495 m .5 .0312 L s P [(0.671)] .5 .02495 0 2 Mshowa p .002 w .7381 .02495 m .7381 .0312 L s P [(0.6715)] .7381 .02495 0 2 Mshowa p .002 w .97619 .02495 m .97619 .0312 L s P [(0.672)] .97619 .02495 0 2 Mshowa p .001 w .07143 .02495 m .07143 .0287 L s P p .001 w .11905 .02495 m .11905 .0287 L s P p .001 w .16667 .02495 m .16667 .0287 L s P p .001 w .21429 .02495 m .21429 .0287 L s P p .001 w .30952 .02495 m .30952 .0287 L s P p .001 w .35714 .02495 m .35714 .0287 L s P p .001 w .40476 .02495 m .40476 .0287 L s P p .001 w .45238 .02495 m .45238 .0287 L s P p .001 w .54762 .02495 m .54762 .0287 L s P p .001 w .59524 .02495 m .59524 .0287 L s P p .001 w .64286 .02495 m .64286 .0287 L s P p .001 w .69048 .02495 m .69048 .0287 L s P p .001 w .78571 .02495 m .78571 .0287 L s P p .001 w .83333 .02495 m .83333 .0287 L s P p .001 w .88095 .02495 m .88095 .0287 L s P p .001 w .92857 .02495 m .92857 .0287 L s P p .002 w 0 .02495 m 1 .02495 L s P p .002 w .02381 .17084 m .03006 .17084 L s P [(0.252216)] .01131 .17084 1 0 Mshowa p .002 w .02381 .31673 m .03006 .31673 L s P [(0.252217)] .01131 .31673 1 0 Mshowa p .002 w .02381 .46262 m .03006 .46262 L s P [(0.252218)] .01131 .46262 1 0 Mshowa p .002 w .02381 .6085 m .03006 .6085 L s P [(0.252219)] .01131 .6085 1 0 Mshowa p .001 w .02381 .05413 m .02756 .05413 L s P p .001 w .02381 .08331 m .02756 .08331 L s P p .001 w .02381 .11248 m .02756 .11248 L s P p .001 w .02381 .14166 m .02756 .14166 L s P p .001 w .02381 .20002 m .02756 .20002 L s P p .001 w .02381 .22919 m .02756 .22919 L s P p .001 w .02381 .25837 m .02756 .25837 L s P p .001 w .02381 .28755 m .02756 .28755 L s P p .001 w .02381 .34591 m .02756 .34591 L s P p .001 w .02381 .37508 m .02756 .37508 L s P p .001 w .02381 .40426 m .02756 .40426 L s P p .001 w .02381 .43344 m .02756 .43344 L s P p .001 w .02381 .49179 m .02756 .49179 L s P p .001 w .02381 .52097 m .02756 .52097 L s P p .001 w .02381 .55015 m .02756 .55015 L s P p .001 w .02381 .57933 m .02756 .57933 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .06349 .07343 L .10317 .12906 L .14286 .18161 L .18254 .23108 L .22222 .27746 L .2619 .32076 L .30159 .36098 L .34127 .39811 L .38095 .43216 L .42063 .46312 L .46032 .491 L .5 .51579 L .53968 .5375 L .57937 .55611 L .61905 .57164 L .65873 .58409 L .67857 .58915 L .69841 .59344 L .71825 .59696 L .7381 .59971 L .75794 .60168 L .76786 .60238 L .77282 .60266 L .77778 .60289 L .78274 .60307 L .78522 .60314 L .7877 .6032 L .79018 .60325 L .79142 .60327 L .79266 .60328 L .7939 .6033 L .79514 .60331 L .79638 .60331 L .79762 .60332 L .79886 .60332 L .8001 .60332 L .80134 .60331 L .80258 .6033 L .80382 .60329 L .80506 .60328 L .80754 .60324 L .81002 .60319 L .8125 .60313 L .81746 .60298 L .82242 .60277 L .82738 .60252 L .8373 .60186 L .84722 .60101 L .85714 .59997 L Mistroke .87698 .59732 L .89683 .59388 L .93651 .5847 L .97619 .57243 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] A[.6715] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0.2522189397200214 ;[o] 0.252219 :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Thus the maximum area appears to occur when we pick the left hand x value to be .6715 which yields an area of 0.252. This agrees with our previous solution using an interpolating function fit approach. :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; endGroup; endGroup] On a final exam (Fall 1992) in the first term of a three term course: Integrated First-Year Curriculum in Science, Engineering, and Mathematics I gave problem 2 and the students (even under exam conditions and given that this problem was significantly more complex than those they had been attempting thus far in the course) produced a number of very good solutions and attempts. Thus in addition to our own solution which was similar to what many of the students offered we give the solution of Ryan Easterhaus, a bright fellow in our course, who one-lined the function formulation and did a nice job of moving beyond this to a solution. ^*)