(*^ ::[ 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 = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, cellOutline, groupLikeTitle, center, M18, O486, R65535, e8, 24, "Calculus"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, bold, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Calculus"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, B65535, 12, "Calculus"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M36, N23, O486, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M36, N23, O486, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M18, O486, l19, o2, w378, h214, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = leftheader, inactive, M18, O486, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M18, O486, 12, "Times"; fontset = leftfooter, inactive, M18, O486, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, whiteBox, M18, O486, bold, R21845, G21845, B21845, 12, "Calculus"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, R21845, G21845, B21845, 12, "Calculus"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Mini-Project for Calculus and Mathematica Workshop ;[s] 3:0,0;30,1;41,0;51,-1; 2:2,45,32,Calculus,0,24,65535,0,0;1,45,32,Calculus,2,24,65535,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; noPageBreakBelow; preserveAspect; pictureLeft = 78; pictureTop = 2; pictureWidth = 319; pictureHeight = 264] %! %%Creator: Mathematica %%AspectRatio: .82055 MathPictureStart %% SurfaceGraphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0249355 0.99742 -0.0396341 0.99742 [ [ 0 0 0 0 ] [ 1 .82055 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 .82055 L 0 .82055 L closepath clip newpath p P p .727 .605 .721 r .0015 w .38753 .61321 .40752 .66524 .44401 .61996 .42389 .57791 Metetra .694 .617 .766 r .42389 .57791 .44401 .61996 .48009 .58581 .45991 .55748 Metetra .596 .647 .871 r .45991 .55748 .48009 .58581 .5161 .56667 .49608 .55355 Metetra .046 .518 .888 r .49608 .55355 .5161 .56667 .55258 .56399 .53297 .5652 Metetra .63 .081 0 r .53297 .5652 .55258 .56399 .59008 .57669 .57109 .58902 Metetra .699 .235 .145 r .57109 .58902 .59008 .57669 .62906 .6012 .61078 .61933 Metetra .685 .242 .192 r .61078 .61933 .62906 .6012 .66971 .63171 .65204 .64884 Metetra .624 .156 .12 r .65204 .64884 .66971 .63171 .71188 .66087 .69448 .66977 Metetra .384 0 0 r .69448 .66977 .71188 .66087 .75506 .68096 .7374 .6755 Metetra .552 .899 .977 r .7374 .6755 .75506 .68096 .7984 .68554 .77997 .66222 Metetra .79 .783 .846 r .77997 .66222 .7984 .68554 .84101 .67107 .82148 .63009 Metetra .791 .688 .75 r .82148 .63009 .84101 .67107 .8822 .63798 .86159 .58317 Metetra .778 .642 .711 r .86159 .58317 .8822 .63798 .92176 .59057 .90051 .52833 Metetra .763 .619 .7 r .90051 .52833 .92176 .59057 .96002 .5358 .93894 .47353 Metetra .721 .642 .77 r .36676 .57464 .38753 .61321 .42389 .57791 .40301 .54927 Metetra .642 .665 .857 r .40301 .54927 .42389 .57791 .45991 .55748 .43901 .54023 Metetra .296 .625 .961 r .43901 .54023 .45991 .55748 .49608 .55355 .47532 .54739 Metetra .488 0 0 r .47532 .54739 .49608 .55355 .53297 .5652 .51255 .56803 Metetra .685 .165 0 r .51255 .56803 .53297 .5652 .57109 .58902 .55114 .59711 Metetra .695 .202 .072 r .55114 .59711 .57109 .58902 .61078 .61933 .59133 .62768 Metetra .636 .12 0 r .59133 .62768 .61078 .61933 .65204 .64884 .63295 .65191 Metetra .382 0 0 r .63295 .65191 .65204 .64884 .69448 .66977 .67549 .66262 Metetra .458 .841 .985 r .67549 .66262 .69448 .66977 .7374 .6755 .71819 .65499 Metetra .769 .804 .88 r .71819 .65499 .7374 .6755 .77997 .66222 .76029 .62792 Metetra .8 .72 .776 r .76029 .62792 .77997 .66222 .82148 .63009 .80127 .58441 Metetra .798 .677 .73 r .80127 .58441 .82148 .63009 .86159 .58317 .84108 .5307 Metetra .788 .654 .714 r .84108 .5307 .86159 .58317 .90051 .52833 .88018 .47465 Metetra .773 .644 .719 r .88018 .47465 .90051 .52833 .93894 .47353 .91943 .42401 Metetra .721 .703 .832 r .34503 .54745 .36676 .57464 .40301 .54927 .38137 .52733 Metetra .567 .7 .931 r .38137 .52733 .40301 .54927 .43901 .54023 .41747 .52387 Metetra .092 .52 .906 r .41747 .52387 .43901 .54023 .47532 .54739 .45393 .53602 Metetra .452 0 0 r .45393 .53602 .47532 .54739 .51255 .56803 .49136 .56023 Metetra .617 .052 0 r .49136 .56023 .51255 .56803 .55114 .59711 .53024 .59067 Metetra .595 .023 0 r .53024 .59067 .55114 .59711 .59133 .62768 .57072 .61992 Metetra .368 0 0 r .57072 .61992 .59133 .62768 .63295 .65191 .61261 .64012 Metetra .235 .671 .957 r .61261 .64012 .63295 .65191 .67549 .66262 .65536 .64466 Metetra .67 .806 .948 r .65536 .64466 .67549 .66262 .71819 .65499 .69819 .62989 Metetra .791 .775 .838 r .69819 .62989 .71819 .65499 .76029 .62792 .74039 .59615 Metetra .824 .743 .774 r .74039 .59615 .76029 .62792 .80127 .58441 .78154 .54774 Metetra .831 .724 .746 r .78154 .54774 .80127 .58441 .84108 .5307 .82168 .49174 Metetra .826 .714 .741 r .82168 .49174 .84108 .5307 .88018 .47465 .86137 .4362 Metetra .807 .711 .759 r .86137 .4362 .88018 .47465 .91943 .42401 .90151 .38852 Metetra .746 .768 .87 r .32231 .52773 .34503 .54745 .38137 .52733 .35911 .50608 Metetra .579 .717 .937 r .35911 .50608 .38137 .52733 .41747 .52387 .39552 .50118 Metetra .264 .557 .931 r .39552 .50118 .41747 .52387 .45393 .53602 .43216 .51216 Metetra 0 .337 .813 r .43216 .51216 .45393 .53602 .49136 .56023 .46969 .5356 Metetra 0 .221 .727 r .46969 .5356 .49136 .56023 .53024 .59067 .50862 .5658 Metetra 0 .261 .759 r .50862 .5658 .53024 .59067 .57072 .61992 .54921 .59536 Metetra .078 .433 .873 r .54921 .59536 .57072 .61992 .61261 .64012 .59131 .61635 Metetra .407 .639 .95 r .59131 .61635 .61261 .64012 .65536 .64466 .63444 .62193 Metetra .661 .762 .925 r .63444 .62193 .65536 .64466 .69819 .62989 .67782 .60817 Metetra .797 .808 .861 r .67782 .60817 .69819 .62989 .74039 .59615 .72071 .57511 Metetra .861 .822 .808 r .72071 .57511 .74039 .59615 .78154 .54774 .76263 .52686 Metetra .888 .826 .78 r .76263 .52686 .78154 .54774 .82168 .49174 .80354 .4704 Metetra .891 .828 .779 r .80354 .4704 .82168 .49174 .86137 .4362 .84393 .41385 Metetra .87 .827 .803 r .84393 .41385 .86137 .4362 .90151 .38852 .88467 .36476 Metetra .812 .85 .878 r .29831 .51661 .32231 .52773 .35911 .50608 .33608 .48691 Metetra .676 .77 .922 r .33608 .48691 .35911 .50608 .39552 .50118 .37319 .47359 Metetra .471 .62 .915 r .37319 .47359 .39552 .50118 .43216 .51216 .41021 .47701 Metetra .276 .457 .858 r .41021 .47701 .43216 .51216 .46969 .5356 .44781 .49494 Metetra .168 .356 .807 r .44781 .49494 .46969 .5356 .50862 .5658 .48663 .52269 Metetra .159 .339 .795 r .48663 .52269 .50862 .5658 .54921 .59536 .52708 .5535 Metetra .236 .399 .822 r .52708 .5535 .54921 .59536 .59131 .61635 .56923 .57943 Metetra .384 .522 .872 r .56923 .57943 .59131 .61635 .63444 .62193 .61273 .59284 Metetra .573 .679 .912 r .61273 .59284 .63444 .62193 .67782 .60817 .65691 .58813 Metetra .749 .821 .908 r .65691 .58813 .67782 .60817 .72071 .57511 .70095 .56336 Metetra .865 .909 .86 r .70095 .56336 .72071 .57511 .76263 .52686 .74418 .52082 Metetra .917 .946 .803 r .74418 .52082 .76263 .52686 .80354 .4704 .78633 .46639 Metetra .929 .959 .774 r .78633 .46639 .80354 .4704 .84393 .41385 .82765 .40797 Metetra .919 .961 .793 r .82765 .40797 .84393 .41385 .88467 .36476 .86884 .35363 Metetra .882 .948 .849 r .2724 .51893 .29831 .51661 .33608 .48691 .3117 .47743 Metetra .797 .879 .907 r .3117 .47743 .33608 .48691 .37319 .47359 .35005 .45033 Metetra .639 .731 .915 r .35005 .45033 .37319 .47359 .41021 .47701 .38788 .44005 Metetra .465 .554 .862 r .38788 .44005 .41021 .47701 .44781 .49494 .4258 .4464 Metetra .344 .424 .8 r .4258 .4464 .44781 .49494 .48663 .52269 .46453 .46661 Metetra .29 .363 .765 r .46453 .46661 .48663 .52269 .52708 .5535 .50467 .49549 Metetra .293 .361 .762 r .50467 .49549 .52708 .5535 .56923 .57943 .54657 .52587 Metetra .346 .415 .791 r .54657 .52587 .56923 .57943 .61273 .59284 .59018 .54967 Metetra .452 .53 .847 r .59018 .54967 .61273 .59284 .65691 .58813 .63504 .5594 Metetra .612 .708 .913 r .63504 .5594 .65691 .58813 .70095 .56336 .68037 .55006 Metetra .782 .902 .926 r .68037 .55006 .70095 .56336 .74418 .52082 .72533 .52055 Metetra .858 .994 .823 r .72533 .52055 .74418 .52082 .78633 .46639 .76927 .47403 Metetra .839 .981 .69 r .76927 .47403 .78633 .46639 .82765 .40797 .81206 .41705 Metetra .816 .963 .638 r .81206 .41705 .82765 .40797 .86884 .35363 .85409 .35778 Metetra .89 .99 .772 r .24417 .53477 .2724 .51893 .3117 .47743 .28526 .48293 Metetra .874 .976 .844 r .28526 .48293 .3117 .47743 .35005 .45033 .32534 .44099 Metetra .794 .89 .913 r .32534 .44099 .35005 .45033 .38788 .44005 .36453 .41371 Metetra .635 .711 .902 r .36453 .41371 .38788 .44005 .4258 .4464 .40328 .40342 Metetra .477 .532 .835 r .40328 .40342 .4258 .4464 .46453 .46661 .44226 .40987 Metetra .376 .415 .774 r .44226 .40987 .46453 .46661 .50467 .49549 .48218 .43019 Metetra .332 .363 .744 r .48218 .43019 .50467 .49549 .54657 .52587 .52365 .45908 Metetra .332 .363 .743 r .52365 .45908 .54657 .52587 .59018 .54967 .56697 .48925 Metetra .374 .413 .773 r .56697 .48925 .59018 .54967 .63504 .5594 .61206 .51252 Metetra .465 .526 .836 r .61206 .51252 .63504 .5594 .68037 .55006 .65836 .52136 Metetra .617 .72 .92 r .65836 .52136 .68037 .55006 .72533 .52055 .70501 .51079 Metetra .778 .944 .932 r .70501 .51079 .72533 .52055 .76927 .47403 .75112 .47982 Metetra .785 .989 .746 r .75112 .47982 .76927 .47403 .81206 .41705 .79606 .43181 Metetra .699 .905 .548 r .79606 .43181 .81206 .41705 .85409 .35778 .83972 .37347 Metetra .891 .989 .752 r .21419 .55394 .24417 .53477 .28526 .48293 .25678 .49796 Metetra .89 .99 .774 r .25678 .49796 .28526 .48293 .32534 .44099 .29868 .44538 Metetra .873 .975 .847 r .29868 .44538 .32534 .44099 .36453 .41371 .33958 .40297 Metetra .79 .885 .914 r .33958 .40297 .36453 .41371 .40328 .40342 .37965 .37548 Metetra .629 .705 .9 r .37965 .37548 .40328 .40342 .44226 .40987 .41938 .36517 Metetra .473 .527 .832 r .41938 .36517 .44226 .40987 .48218 .43019 .45947 .37169 Metetra .374 .413 .773 r .45947 .37169 .48218 .43019 .52365 .45908 .50066 .39207 Metetra .331 .362 .743 r .50066 .39207 .52365 .45908 .56697 .48925 .54354 .42091 Metetra .333 .364 .744 r .54354 .42091 .56697 .48925 .61206 .51252 .58839 .45082 Metetra .376 .416 .775 r .58839 .45082 .61206 .51252 .65836 .52136 .63504 .47351 Metetra .469 .53 .838 r .63504 .47351 .65836 .52136 .70501 .51079 .68285 .48139 Metetra .622 .728 .922 r .68285 .48139 .70501 .51079 .75112 .47982 .7309 .46952 Metetra .781 .949 .929 r .7309 .46952 .75112 .47982 .79606 .43181 .77822 .43704 Metetra .783 .987 .739 r .77822 .43704 .79606 .43181 .83972 .37347 .82419 .38748 Metetra .911 .959 .807 r .18369 .56292 .21419 .55394 .25678 .49796 .22719 .50891 Metetra .918 .966 .788 r .22719 .50891 .25678 .49796 .29868 .44538 .27055 .45203 Metetra .907 .964 .81 r .27055 .45203 .29868 .44538 .33958 .40297 .31311 .40013 Metetra .866 .937 .866 r .31311 .40013 .33958 .40297 .37965 .37548 .35466 .35968 Metetra .761 .847 .916 r .35466 .35968 .37965 .37548 .41938 .36517 .39548 .33496 Metetra .592 .682 .904 r .39548 .33496 .41938 .36517 .45947 .37169 .43614 .3277 Metetra .429 .514 .844 r .43614 .3277 .45947 .37169 .50066 .39207 .47741 .33699 Metetra .328 .404 .788 r .47741 .33699 .50066 .39207 .54354 .42091 .52001 .35935 Metetra .29 .359 .762 r .52001 .35935 .54354 .42091 .58839 .45082 .56452 .38884 Metetra .305 .372 .767 r .56452 .38884 .58839 .45082 .63504 .47351 .61109 .41772 Metetra .371 .441 .803 r .61109 .41772 .63504 .47351 .68285 .48139 .65941 .43761 Metetra .49 .572 .865 r .65941 .43761 .68285 .48139 .7309 .46952 .70873 .44122 Metetra .66 .763 .926 r .70873 .44122 .7309 .46952 .77822 .43704 .75799 .4243 Metetra .814 .943 .91 r .75799 .4243 .77822 .43704 .82419 .38748 .80624 .38693 Metetra .883 .869 .82 r .15353 .55577 .18369 .56292 .22719 .50891 .19761 .50511 Metetra .906 .883 .799 r .19761 .50511 .22719 .50891 .27055 .45203 .24203 .44768 Metetra .906 .888 .802 r .24203 .44768 .27055 .45203 .31311 .40013 .28587 .39164 Metetra .881 .884 .831 r .28587 .39164 .31311 .40013 .35466 .35968 .3287 .34429 Metetra .813 .854 .88 r .3287 .34429 .35466 .35968 .39548 .33496 .37057 .31105 Metetra .677 .769 .92 r .37057 .31105 .39548 .33496 .43614 .3277 .41195 .29488 Metetra .481 .621 .912 r .41195 .29488 .43614 .3277 .47741 .33699 .45354 .29615 Metetra .3 .467 .859 r .45354 .29615 .47741 .33699 .52001 .35935 .49617 .31256 Metetra .2 .373 .812 r .49617 .31256 .52001 .35935 .56452 .38884 .54051 .33927 Metetra .19 .356 .8 r .54051 .33927 .56452 .38884 .61109 .41772 .58698 .36926 Metetra .258 .409 .824 r .58698 .36926 .61109 .41772 .65941 .43761 .6355 .39418 Metetra .392 .524 .871 r .6355 .39418 .65941 .43761 .70873 .44122 .68549 .40589 Metetra .572 .678 .912 r .68549 .40589 .70873 .44122 .75799 .4243 .73598 .39842 Metetra .748 .825 .912 r .73598 .39842 .75799 .4243 .80624 .38693 .78587 .36966 Metetra .843 .783 .793 r .12396 .53371 .15353 .55577 .19761 .50511 .16876 .48312 Metetra .861 .782 .773 r .16876 .48312 .19761 .50511 .24203 .44768 .21404 .42514 Metetra .861 .784 .774 r .21404 .42514 .24203 .44768 .28587 .39164 .25878 .36801 Metetra .844 .786 .796 r .25878 .36801 .28587 .39164 .3287 .34429 .30243 .31918 Metetra .796 .782 .841 r .30243 .31918 .3287 .34429 .37057 .31105 .34499 .28424 Metetra .687 .755 .903 r .34499 .28424 .37057 .31105 .41195 .29488 .38686 .26634 Metetra .475 .661 .943 r .38686 .26634 .41195 .29488 .45354 .29615 .42878 .26604 Metetra .191 .493 .903 r .42878 .26604 .45354 .29615 .49617 .31256 .47159 .2812 Metetra 0 .348 .824 r .47159 .2812 .49617 .31256 .54051 .33927 .51604 .30715 Metetra 0 .311 .801 r .51604 .30715 .54051 .33927 .58698 .36926 .56262 .33695 Metetra .051 .392 .851 r .56262 .33695 .58698 .36926 .6355 .39418 .61137 .36225 Metetra .284 .551 .924 r .61137 .36225 .6355 .39418 .68549 .40589 .66179 .37478 Metetra .545 .706 .945 r .66179 .37478 .68549 .40589 .73598 .39842 .71294 .36827 Metetra .734 .795 .901 r .71294 .36827 .73598 .39842 .78587 .36966 .76371 .3403 Metetra .815 .716 .755 r .0953 .49703 .12396 .53371 .16876 .48312 .14118 .44237 Metetra .817 .703 .739 r .14118 .44237 .16876 .48312 .21404 .42514 .1872 .38316 Metetra .809 .698 .743 r .1872 .38316 .21404 .42514 .25878 .36801 .23237 .32757 Metetra .789 .701 .767 r .23237 .32757 .25878 .36801 .30243 .31918 .27623 .28254 Metetra .742 .708 .819 r .27623 .28254 .30243 .31918 .34499 .28424 .31886 .25285 Metetra .627 .707 .903 r .31886 .25285 .34499 .28424 .38686 .26634 .3608 .24077 Metetra .334 .625 .958 r .3608 .24077 .38686 .26634 .42878 .26604 .40282 .24591 Metetra 0 .379 .802 r .40282 .24591 .42878 .26604 .47159 .2812 .44583 .26524 Metetra .377 0 0 r .44583 .26524 .47159 .2812 .51604 .30715 .4906 .2932 Metetra .4 0 0 r .4906 .2932 .51604 .30715 .56262 .33695 .53759 .32222 Metetra 0 .34 .725 r .53759 .32222 .56262 .33695 .61137 .36225 .58677 .34372 Metetra .218 .64 .954 r .58677 .34372 .61137 .36225 .66179 .37478 .63756 .34984 Metetra .61 .795 .97 r .63756 .34984 .66179 .37478 .71294 .36827 .689 .33545 Metetra .774 .796 .871 r .689 .33545 .71294 .36827 .76371 .3403 .73998 .29971 Metetra .789 .659 .72 r .06834 .44323 .0953 .49703 .14118 .44237 .11524 .38403 Metetra .779 .645 .714 r .11524 .38403 .14118 .44237 .1872 .38316 .16152 .32611 Metetra .762 .641 .727 r .16152 .32611 .1872 .38316 .23237 .32757 .20638 .27691 Metetra .733 .648 .764 r .20638 .27691 .23237 .32757 .27623 .28254 .24966 .24193 Metetra .664 .666 .841 r .24966 .24193 .27623 .28254 .31886 .25285 .29169 .22422 Metetra .449 .662 .952 r .29169 .22422 .31886 .25285 .3608 .24077 .33319 .22418 Metetra 0 .415 .794 r .33319 .22418 .3608 .24077 .40282 .24591 .37511 .23954 Metetra .499 0 0 r .37511 .23954 .40282 .24591 .44583 .26524 .41837 .26547 Metetra .571 0 0 r .41837 .26547 .44583 .26524 .4906 .2932 .46369 .29489 Metetra .501 0 0 r .46369 .29489 .4906 .2932 .53759 .32222 .51137 .31934 Metetra .206 0 0 r .51137 .31934 .53759 .32222 .58677 .34372 .56117 .33049 Metetra .429 .815 .989 r .56117 .33049 .58677 .34372 .63756 .34984 .61232 .32216 Metetra .746 .827 .914 r .61232 .32216 .63756 .34984 .689 .33545 .66377 .29215 Metetra .807 .751 .801 r .66377 .29215 .689 .33545 .73998 .29971 .71455 .24289 Metetra .762 .618 .7 r .04324 .37461 .06834 .44323 .11524 .38403 .09036 .31597 Metetra .745 .61 .708 r .09036 .31597 .11524 .38403 .16152 .32611 .13591 .26561 Metetra .718 .614 .74 r .13591 .26561 .16152 .32611 .20638 .27691 .17954 .22922 Metetra .662 .633 .811 r .17954 .22922 .20638 .27691 .24966 .24193 .22146 .21 Metetra .478 .653 .936 r .22146 .21 .24966 .24193 .29169 .22422 .26237 .20855 Metetra 0 .414 .778 r .26237 .20855 .29169 .22422 .33319 .22418 .30323 .22278 Metetra .541 0 0 r .30323 .22278 .33319 .22418 .37511 .23954 .34508 .24803 Metetra .602 .056 0 r .34508 .24803 .37511 .23954 .41837 .26547 .38885 .27733 Metetra .553 .004 0 r .38885 .27733 .41837 .26547 .46369 .29489 .43506 .30225 Metetra .354 0 0 r .43506 .30225 .46369 .29489 .51137 .31934 .48373 .31436 Metetra .268 .766 .89 r .48373 .31436 .51137 .31934 .56117 .33049 .53429 .30725 Metetra .739 .861 .938 r .53429 .30725 .56117 .33049 .61232 .32216 .58576 .27837 Metetra .802 .749 .803 r .58576 .27837 .61232 .32216 .66377 .29215 .63711 .22986 Metetra .799 .68 .733 r .63711 .22986 .66377 .29215 .71455 .24289 .68763 .16795 Metetra P p P % End of Graphics MathPictureEnd :[font = section; inactive; preserveAspect] Name: Don Anglen :[font = section; inactive; dontNoPageBreakBelow; preserveAspect] Title: Area Under a Curve :[font = section; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; startGroup] Description: This lesson graphs trapezoids under a parabola to compare areas. It explains some Mathematica language,also. ;[s] 3:0,0;99,1;110,0;127,-1; 2:2,26,19,Calculus,1,12,21845,21845,21845;1,26,19,Calculus,3,12,21845,21845,21845; :[font = text; inactive; thinLines; preserveAspect; center] Area Under a Curve ;[s] 2:0,0;9,1;28,-1; 2:1,26,19,Calculus,0,12,0,0,0;1,38,27,Calculus,0,20,0,0,0; :[font = text; inactive; noPageBreak; preserveAspect] We will use the Package called (Needs Graphics Filled Plot). :[font = input; initialization; preserveAspect] *) Needs["Graphics`FilledPlot`"] (* :[font = text; inactive; preserveAspect] The equation of a parabola can be written in the form y = axÛ + bx + c. As a function it can be written f(x) = axÛ + bx + c. We will use the parabola y = -3xÛ + 7x most frequently unless we change the parameters "a", "b", and "c". This paragraph is in "text" style: no operations will perform. :[font = input; preserveAspect] This is an "input" cell: operations will occur. :[font = text; inactive; preserveAspect] The ] on the right side of the screen are called "cells". Each cell is used for different outputs or operations. :[font = input; preserveAspect] Now we will look at a program and then explain it. ;[s] 1:0,1;51,-1; 2:0,12,10,Courier,1,12,0,0,0;1,25,19,Calculus,0,12,0,0,0; :[font = input; preserveAspect] Clear[a,b,c,f,x,y] f[x_] = a x^2 + b x + c a = -3 b = 7 c = 0 Solve[f[x]==0,x] :[font = input; preserveAspect] {{x -> 0}, {x -> 7/3}} ;[o] 7 {{x -> 0}, {x -> -}} 3 :[font = text; inactive; preserveAspect] The "Clear" command cleared the variables. The second line is the command for a function. In Mathematica. in order to multipley b times x, a space is needed. The "^" is the command for a power of a number. The fifth line will solve for x in the function. Mathematica needs a "==" to solve the equation. The ",x" means with respect to x. The output is the ordered pair (0, 7/3). These are the two points where the parabola will cross the x-axis. ;[s] 5:0,1;105,2;117,1;291,2;302,1;506,-1; 3:0,26,19,Calculus,0,12,0,0,0;3,13,10,Courier,0,12,0,0,0;2,13,10,Courier,2,12,0,0,0; :[font = text; inactive; pageBreakBelow; preserveAspect] We will find the area between the parabola and the x-axis. :[font = text; inactive; preserveAspect] First let's look at the graph. :[font = input; preserveAspect] Plot[f[x],{x, -1, 3}] :[font = text; inactive; preserveAspect] The "Plot" command plots the curve of the function f(x) from x = -1 to x = 3. The commands in Mathematica start with a capital letter. ;[s] 3:0,0;105,1;116,0;153,-1; 2:2,13,10,Courier,0,12,0,0,0;1,13,10,Courier,2,12,0,0,0; :[font = input; preserveAspect] Plot[f[x],{x, -1, 3}, PlotRange->{-2,5}] :[font = text; inactive; pageBreakBelow; preserveAspect] The second command gives the range of y from -2 to 5. :[font = input; preserveAspect] Plot[f[x],{x, -1, 3}, PlotRange->{-2,5}, AspectRatio->Automatic] :[font = text; inactive; preserveAspect] To make the difference between the units of the x and y axis the same, Mathematica has a command for that. ;[s] 3:0,0;78,1;89,0;116,-1; 2:2,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,2,12,0,0,0; :[font = input; preserveAspect] Plot[f[x],{x, -1, 3}, PlotRange->{-2,5}, AspectRatio->Automatic, PlotStyle->{RGBColor[1,0,0]}] :[font = text; inactive; pageBreakBelow; preserveAspect] There are several additional commands you could use next if you wish. One would be color. :[font = input; preserveAspect] One way to find the area between the parabola and the positive x axis is to integrate the function between the two points on the x axis where the curve intersects it. These two points are the roots of the function. Solve for x. -3 xÛ + 7 x = 0. ;[s] 1:0,1;271,-1; 2:0,12,10,Courier,1,12,0,0,0;1,33,23,Calculus,64,12,0,0,0; :[font = input; preserveAspect] Solve[-3 x^2 + 7 x == 0,x] :[font = input; preserveAspect] x = 0, x = 7/3 :[font = text; inactive; noPageBreakBelow; preserveAspect] Connect some points on the curve. :[font = input; preserveAspect] Clear[f,x,jump] f[x_]= -3 x^2 + 7 x; n=4; functionplot=Plot[f[x],{x,0,7/3}, DisplayFunction->Identity]; jump := (7/3)/n; points = Table[{x,f[x]},{x,0,7/3,jump}]; pointplot = ListPlot[points, PlotStyle->{RGBColor[1,0,0]}, DisplayFunction->Identity]; brokenlineplot = Graphics[Line[points]]; Show[functionplot,pointplot,brokenlineplot, DisplayFunction->$DisplayFunction]; :[font = text; inactive; preserveAspect] Would you like to see a trapezoid under the parabola? Of course! :[font = input; preserveAspect] Clear[trap] trap[x_]:=Graphics[{GrayLevel[.75], Polygon[{{x,0},{x,f[x]}, {x+jump,f[x+jump]},{x+jump,0}}]}] Show[trap[2jump],functionplot,brokenlineplot, Ticks->{{2jump,3jump},Automatic}, Axes->True]; :[font = text; inactive; pageBreak; preserveAspect] You're just dying to see several trapezoids under the curve; right? :[font = input; preserveAspect] Clear[trap] trap[x_]:=Graphics[{GrayLevel[.75], Polygon[{{x,0},{x,f[x]}, {x+jump,f[x+jump]},{x+jump,0}}]}] Show[Table[trap[x],{x,0,7/3 - jump, jump}]]; :[font = text; inactive; pageBreak; preserveAspect] How about finding the area of the sum of these trapezoids? Sure! :[font = input; preserveAspect] Clear[n,area] jump := (7/3)/n; area[n_] = Sum[jump (f[x] + f[x + jump])/2, {x,0,7/3-jump,jump}] :[font = input; preserveAspect] area[4] :[font = text; inactive; preserveAspect] The command "N" will approximate this to a decimal. :[font = input; preserveAspect] N[area[4]] :[font = text; inactive; preserveAspect] 5 trapezoids? Substitute "5" in for "n". :[font = input; preserveAspect] N[area[5]] :[font = text; inactive; preserveAspect] The area is getting closer to the area of the parabola. :[font = text; inactive; pageBreak; preserveAspect] Let's find the area under the parabola. It should be a little more than the sum of the areas of the trapezoids. :[font = text; inactive; preserveAspect; plain; subscript] Integrate: ¦ Ö £ #¼ (-3xÛ + 7x) dx = ;[s] 21:0,8;3,9;14,0;19,1;20,0;37,1;40,0;125,2;126,0;169,2;178,0;225,3;226,4;227,5;230,3;231,6;232,7;241,6;242,0;244,2;246,0;272,-1; 10:7,34,23,Calculus,64,12,0,0,0;2,34,23,Calculus,32,12,0,0,0;3,34,23,Calculus,96,12,0,0,0;2,48,32,Calculus,64,18,0,0,0;1,48,32,Calculus,96,18,0,0,0;1,38,26,Calculus,64,14,0,0,0;2,48,32,Calculus,32,18,0,0,0;1,38,26,Calculus,32,14,0,0,0;1,13,10,Courier,1,12,0,0,0;1,26,19,Calculus,0,12,0,0,0; :[font = text; inactive; preserveAspect] ¦ Ö £ [ -xÜ + (7/2)xÛ ] = 0 ;[s] 11:0,0;35,3;40,0;46,4;66,0;81,1;87,0;157,2;158,0;201,2;210,0;249,-1; 5:6,34,23,Calculus,64,12,0,0,0;1,34,23,Calculus,32,12,0,0,0;2,34,23,Calculus,96,12,0,0,0;1,38,26,Calculus,64,14,0,0,0;1,48,32,Calculus,64,18,0,0,0; :[font = text; inactive; preserveAspect] [ -(7/3)Ü + (7/2)(7/3)Û ] - [ -(0)Ü + (7/2)(0)Û ] = [ -(343/27) + (343/18) ] = ( -686 + 1029 ) / (54) = 343/54 :[font = text; inactive; preserveAspect] Mathematica can integrate for you. ;[s] 2:0,1;14,0;38,-1; 2:1,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,2,12,0,0,0; :[font = input; preserveAspect] Integrate[-3 x^2 + 7 x, {x, 0, 7/3}] :[font = text; inactive; pageBreakBelow; preserveAspect; fontSize = 20] Voila! :[font = text; inactive; preserveAspect] Now let's compare areas. The areas of 100 trapezoids and the area under the curve. :[font = input; preserveAspect] N[area[100]] :[font = input; preserveAspect] NIntegrate[-3 x^2 + 7 x, {x, 0, 7/3}] :[font = text; inactive; preserveAspect] With 100 trapezoids, it's getting closer. :[font = input; preserveAspect] :[font = input; preserveAspect; endGroup; endGroup] Try different values for a, b, and c in the parabola axÛ + bx + c = 0. ;[s] 1:0,1;71,-1; 2:0,12,10,Courier,1,12,0,0,0;1,25,19,Calculus,0,12,0,0,0; ^*)