{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "M aple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 36 "Area between x/sqrt(1+x^2) and x^4-x" }}{PARA 0 "" 0 " " {TEXT -1 231 "The first curve is negative for x<0 while the second i s positive, so the bounded region between the curves begins at x=0. It continues to the right until the two curves cross, somewhere between \+ 1 and 4/3, by the look of the graph. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([x/sqrt(1+x^2),x^4-x],x=0..4/3,color=[blue,red]) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 397 397 397 {PLOTDATA 2 "6&-%'CURVESG6 $7S7$$\"\"!F)F(7$$\"_q@AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAsG 1H!$,\"$\"_qY:))3\"pZ5nTSWa-%G^O@rRr/4pU!Qy^!yn[KEt`'>)3sy<=!z1gg]!HF- 7$$\"_qVWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW%>K]V&F-$\"_qq,85Y !yB\\1Ik*)))\\N#*GL'e?u&GG3(R\"=HG$RlL2Y@\"\\>fWwnU#3D-FaF-7$$\"_q())) )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) )))))))))))))))))Qq)y#)F-$\"_q6%z/e'30e'**>yKZu@WY\")Q`z'>OCrE_\\3g!RMGL\"Q1T/t5\"F=7$$\"_qxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx_U1*R\"F=$\"_q%oF6Mx()\\^%\\y6$=S_ \\S6f2\"3/XYDpwTL;utCxgSP7XX(=yPE`pbQ\"F=7$$\"_qbbbbbbbbbbbbbbbbbbbbbb bbbbbbbbbbbbbbbbbbbbbbbI2@j;F=$\"_q-6c*3S**3oKnv;wBkLs$oS2`0m8;oO&*RA[ )ehRr*)yz/ubgSG\"GnS;F=7$$\"_qLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLL3\">n$>F=$\"_q$z%yF#z*4]F=7$$\"_q)))))))))))))))))))))))))))))))))))))))))))))))))))))))))) )))))))))))))))))))))))))))))))Q'=e>AF=$\"_q,)yCc)et:'R'yUXtk_%\\y3-Kh j&)\\-'\\dL*3n0\"[&f!=_C[Zii&>=[o;#F=7$$\"_q************************** ***************************************************************\\ZP:]# F=$\"_qz3bXjgchfM&yv$=IPV[DkBky^m-/oq#[B,n(Qi'fHXKF^88Y+gnU#F=7$$\"_qx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxMc\"z#F=$\"_qJ!ez!H,1'e<( R8$)f]KK4p^O:mh\\qCOLw)o#F=7$$\"_q)))))))))))))))) )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) ))))Q&=q/$F=$\"_qDqHJArWq8#yPGxq\\5n%\\>&*fu=a5S(Gb9/+%yw^t4\"z?at(ppm ar9HF=7$$\"_qKLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL/gMLF=$\"_q )oO8'y(Q\\lSR(oX?'oz?R%3oS8)G?^Ynr<;N@?FO;PKf5%4JUn%fL;$F=7$$\"_qmmmmm mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmJOBOF=$\"_q$p(ft$GQ#3g\"=r8c; **>au8R<7y)*G$HUX+QocMY\"[l#o<(z+$y=6Ij1MF=7$$\"_qmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmmmmmmmm&R;!RF=$\"_q>C16()=fN8_d'[&Ga$Rn`#yVUX(=13P` w/#\\KW`(*)pLn47]DoF]yZj$F=7$$\"_q@AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAA(>TV:%F=$\"_q%R&Qr&y(Q5NM3;*eDiK+0Z_`Ffrqq#= 01pJXOQF=7$$\"_qxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxy#[X%F=$ \"_q,#z\\*\\^[rmJ#Gj_)*QDpO&y7GQ![T9+$4X)>Z\\0C&QEDy>4$Q-vY.$pSF=7$$\" _q5666666666666666666666666666666666666666666666&z$4ZF=$\"_qk3Nk*zjE\" yE\")=AVknmx%4!fqxA*pV,[cV*3@\\v_IH/XXP/cgUF=7$$\"_q************** ********************************************************************** *****\\d$[0]F=$\"_q[.j41!)fO!H.(=-')4Il!4uYl]*\\&RsC+4qGR\"e\"yd[[(pl \"H'Q!4RegZ%F=7$$\"_qVWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWeen _F=$\"_qYLRXv*fhgRHc3qgl5A:[;#*fLlzlYty7TH\\u\\E![yq,fxbB*)=`gYF=7$$\" _q)******************************************************************* *********************\\d`^b&F=$\"_q&e\"3U&fQMQ9%pJ#[+JtF\\Qh+I)*fNFKVL Zf.s*[k\\.js^]jBo'ygh&[F=7$$\"_qKLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLeg)*GeF=$\"_q_ZZ4]_**e8-W!o?(z4@2\"p,FEZ#oi-d')*QX[S`())**\\(H+ UYJ2!Rx!f.&F=7$$\"_qwxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxF:q9h F=$\"_q?h$)[#\\VZZ?zV(oOxLw:l&*QFZ2MAU/3zkvj3d\\*RpEIWHk2M;In@&F=7$$\" _qVWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW>v2xjF=$\"_qO!GQ$)yF+ce JAupaS[4<9@^S&pF=$\"_q94*4:ijfzXNiekv2HQ(Q3$*fAXI:#[3p?]hFzrV2;#* R!*yFk1&zF4dF=7$$\"_q)************************************************ ****************************************\\A\\*4sF=$\"_qCIcui&=_/PC2xF= py7%GJ`w4R'p/GJe#Hwi%37x:%4Z_(pn!zE5O[eF=7$$\"_q`bbbbbbbbbbbbbbbbbbbbb bbbbbbbbbbbbbbbbbbbbbbb0hK'[(F=$\"_q'4$3K-/'HqJc+Xm9&*H?.HjWq3y]:_=g7q n<$)fkYb^N#zaU.q?*H*fF=7$$\"_qkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm mmmmmm/&=x(F=$\"_q^RUaM1wA?%Rln\">`pBYQA.BR,\\q;B]0Kc())*HD\"fsUWh:u`i m*[OhF=7$$\"_q)******************************************************* **********************************\\.=^!)F=$\"_q\"fTh20;\\T4*RtxvXSH@. :=z2\\()4*e&**H],k'>ygVl<'4fN;\\)QG7F'F=7$$\"_q)********************** ******************************************************************\\PY 9K)F=$\"_qFdLE#eJ9gUFvbruv^`g%)RK!))\\'Qst[fPpe91y`V\\Qa0$3A#4+]kR'F=7 $$\"_qkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm;2`@')F=$\"_qv0@(zC *G#oR'4(R%o&*))>qc1G$>-0K\\Jj$RL:8!)3#*=F!*Qs\"QFb=kh(HlF=7$$\"_qUWWWW WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW)p6*))F=$\"_q\"elnFd%*\\4,'z[ \"4g/))H!HuN_E#ePmyHn^>h/xzGuL/x-abFsc!fWmF=7$$\"_qkmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmmmmmmmmmr1z\"*F=$\"_qu$HWJLHp;#Gj_s?rWt4'fw2^;Xhnwp We(fNaSu-c$>%)\\F]\"*Q\"*3Aw'F=7$$\"_q`bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb bbbbbbbbbbbbbI\\%*R%*F=$\"_qH]I?NhW$RLQt6bxA*4#*o1?(oYlp:i'3!QRFO8\"H@ xVh!o:q:tv0X'oF=7$$\"_qJLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK :D(*F=$\"_q*R)=,0LLc&*\\)*G)>\\%HMMxLPm'y?^$***F=$\"_q (y4#=V_ReV)Re(fQh%pQry:M!pK>Px8A*)fM%4+Jq;M(3o/epzZGxoqF=7$$\"_q++++++ ++++++++++++++++++++++++++++++++++++++]FCSF5!#**$\"_q#=r_j*oQg)f;(F=7$$\"_qxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxFw#[0\"F]w$\"_qGpkU[Jm5GfmY;W/&H+fv-W2@r!G3\") ej'\\Q4II$*)zXn=&\\^[b**zXJoP.)3$GXGe7=P) \\o:]]Q?)zpn[tF=7$$\"_qbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb&* >>66F]w$\"_q*[MF[p-xK0bwY#fi`P!4'y'z_'3N*RT\">$HS=:N3+iL%=E[X(*R#f$=Lu F=7$$\"_qbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb+4ZR6F]w$\"_qSKf L5='opG\\EZ3_!z93yv-&*eR(R(zhhXF7*4;0#))*pM\"R@x$>cFtg^(F=7$$\"_qWWWWW WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW%>k:v;\"F]w$\"_q(*Q=='*egtR'>f0 '>>0J*4=')po$*pA?vH$)GPq'pPxbj#o]ALw'RwX!\\f(F=7$$\"_qmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmmmmmmmmmmmYeG$>\"F]w$\"_qE0kV,m7Ii>&Q*HV:.+-zbqB4Q[ ,0hox:cQd/'ejm'HOE;3ic#4Xm(F=7$$\"_qWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW WWWWWWWWWWW49#GA\"F]w$\"_q)>)=s,l))*>zG.7FqFz&R)30`JmaW'p&Q/4Z:?/)*G: \\-ah#*[S2z$4TxF=7$$\"_q*))))))))))))))))))))))))))))))))))))))))))))) ))))))))))))))))))))))))))))))))))))))))))))o$Q#\\7F]w$\"_qEZ@#\\!Gn`h h%G-/UpL4>X)e%QD5jq?2`([qnkQ]jY*G)=3-%*yG.$o!yF=7$$\"_q+++++++++++++++ +++++++++++++++++++++++++++++]dVSx7F]w$\"_q!p$\\,&)fLDR0:#oa(G0Zyb%=dL 7wx`\\#=kz%3XEjVV#>H.h#)e\\5cVX/RZix8%\\KN*ofD&=QlW?F.tMl6C[h' *\\a/o0=iWt4OzF=7$$\"_qLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LL8F]w$\"_q+++++++++++++++++++++++++++++++++++++++++++++++++!)F=-%'COL OURG6&%$RGBGF(F($\"*++++\"!\")-F$6$7UF'7$F+$!_qAdX5ue')zg^u^0eJ[I+!=z0 ?W+6)R34k&fs<x.X%o(ye@1HF-7$F1$!_q8d\"Hn&)R&yFdAz#))G0t([+)oE4s qg(pV`a_!4(H75t^)o2,K%f&zgfTV&F-7$F6$!_q0\"f-II#\\qv`$RQ@x%H1n+'>u7C(4 UeZ)**4/rusiB\"3Y7!=wq4\"*psTF)F-7$F;$!_qEp'p*erB/-10)Q9'HPtE'3%z!Rw3D 9lm2xj1#eE>hMK%=0Ut7`+:E6\"F=7$FA$!_qB>eiac!*yV2#*>8%\\y])Q\"\\;QOL^7R `xL%*ow2rh/#f+g!o;bYK\"HB&R\"F=7$FF$!_qR;%))f6*zP?$==(3[XcOTEZf=!)*H`B 'p7Z.1:fYjrW4%4VA]17[ebl\"F=7$FK$!_qaw)4Kaw)4Kaw)4Kaw)4Kaw)4Kaw)4Kaw)f pj(\\!ez>0'[^[kHx++lA>F=7$FP$!_qt\"y62`Q_$3K&pe46_@3,*3c?\"pl!H8[AD?l. g@4o*o,ARHc[:.6`>#F=7$FU$!_q****************************************** *****************\\PfelsNY#H!oS&\\D8%zyBY#F=7$FZ$!_qe')zg^u^0eJ[I+!=z0 ?W+6)R34k&fs<<$zOEq*Q.;WCM8^ht0LN3t#F=7$Fin$!_qP/L7JyU8zG^NqWZn$\\,#)z $Ge;#R2:^/k=21TX1!pm))yzw'GV!*>3'HF=7$F^o$!_q!pC!e8pC!e8pC!e8pC!e8pC!e 8pC!e8pC!e8pC*\\FQ9@Uw'p!Gw\"e&4@$F=7$Fco$!_qY-e8pC!e8pC!e8pC!e8pC!e8p C!e8pC!e8pC!3^o9R#zYg3%)y5))*4X$F=7$Fho$!_q!e8pC!e8pC!e8pC!e8pC!e8pC!e 8pC!e8pC!e8))>iD1Q!))4Ia9>1*pOF=7$F]p$!_q]d2BER%Q)fMIptln,!\\&=.JC0'*= '*\\-w#*\\QNBR+*3SV:B/)\\4AO[cQF=7$Fbp$!_q2&)Q\"\\;QOL^7R`xLWJV1)4Zt$p.xC26^U*\\/v@%F=7$F\\q$!_q+++++++++++++++++++++++++++ +++v$4Y7eAK9kth65iPFPxP%F=7$Faq$!_qiCWxz19d\"Hn&)R&yFdAz#))G0t([+)oE4s qg(pVP1=#4y7^&4#4JFkw\\%F=7$Ffq$!_q*********************************** ************************\\P4'*\\F]Ht;sGbp\"oi0$Gg%F=7$F[r$!_q!e8pC!e8p C!e8pC!e8pC!e8pC!e8pC!e8>iR>Q\"\\?YyfU9)4'4(>WXn%F=7$F`r$!_qOSKMoc@&)p (4>FcGm\"pUfT6\"H!p6`:@#4&>?]+)HE-!R4XZ1;,E7s;ZF=7$Fer$!_q0!)oE4sqg(pV `a-`h$[2p)=U7&e,O<,V)zfwyR+m:\")e^)4mJJeIEBZF=7$Fjr$!_q2vvIi#R%Q)fMIpt ln,!\\&=.JC0'*='*\\-w#\\Z>cfo^\\qRvOyQ5A_c#p%F=7$F_s$!_qiW$*Hwa(o\\SjS %p/s$>*)o!o%*3L*)*pG(zHXBb!HOz8&*y?wvV9Ug)[Kv*fm@!\\Fv_M%F=7$F^t$!_qg8pC!e8pC!e8pC!e8pC!e8 pC!e8pC!e8pC!e84rc'=R#pZQ$R-v**[BTF=7$Fct$!_q-++++++++++++++++++++++++ +++++++v$*\\h35cuw^bc(e$GN\\QF=7$Fht$!_q.+++++++++++++++++++++++++++++ v$4'*f*eCYgwfa,a6*zsj_$F=7$F]u$!_q=pC!e8pC!e8pC!e8pC!e8pC!e8pC!e8pC!ej 1C3R8!*GY\"[G%>C4kk4$F=7$Fbu$!_q7J,u/\\i+>t=61fDh@iQ1@Q8-gU0/%4.x'ye2' GeZ[-T#)Qrel3=k#F=7$Fgu$!_qXG<1&RG<1&RG<1&RG<1&RG<1&RG<1&RG<1&RGnV%RmF cdqjgRhf6!3#F=7$F\\v$!_qRt\\NJj'o&HgablX2cngaM 1#H0UxGl*))\\\"F=7$Fav$!^qo]RG<1&RG<1&RG<1&RG<1&RG<1&RG<1&RG<1&H3]3$>m \"G5w*oJ1/!yF=7$Ffv$!]q\"Gk.,7#p>G]Td`&)3z^#o-%y'4l*)QoL!R*\\,^Ms6x2j* oACbSD'pQ%>F=7$F[w$\"]q+++++++++++++++++++++++++++++]iS\">s8My#eb)z(z_ q4bz')F]w7$Faw$\"^qv(f)>xNtn'z$po8#=0gg%)49vqywUa*e7M,#R[]a\\ul2,#*))3 n[f(=$=F]w7$Ffw$\"^q:FQ\\gr#Q\\gr#Q\\gr#Q\\gr#Q\\gr#Q\\gr#)=60lwn#**>) f10DNKORr[HF]w7$F[x$\"^qkr=_;c:RrcRkvwTgwMza0w5a]WIlY)Ge'*yK7d'=vt2 4pi]!pF]w7$Fjx$\"^qkXB,zcM7!zcM7!zcM7!zcM7!zcM7!zcM7!zc%*HQ:2e!z'*G!>k kRHM)F]w7$F_y$\"_q!)HjFnE]#y1bn)GY$[[z&=@5IDS>2['oO8V&faz(3**egVf$y\"H @NxI,\"F]w7$Fdy$\"_qL#)fvSmjV<'4HJFGX*=Pg*f1Z#RD%pX#3>omD;Jo!R4A.$\\D^ 6uIA'=\"F]w7$Fiy$\"_q+++++++++++++++++++++++++++++]i!Rvo[J>X/w)=`4^;DC &Q\"F]w7$$\"_q+++++++++++++++++++++++++++++++++++++++++++++!e$)3H\"F]w $\"_q+++++++++++++++++++++++++++++++++g*[jG')H.x,vRX6Sf[\"F]w7$F^z$\"_ q+++++++++++++++++++++++++++++]i!R]nOXHF$zAt#\\sK%4F!f\"F]w7$$\"_qmmmm mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm\"z1[)=8F]w$\"_q2!zcM7!zcM7!zc M7!zcM7!zcM7!zcMi_3(G-7AH,9rc(=Mt^tGf_11 this is always mo re than 3. The apparent lower curve has derivative [from the quotient \+ rule!] " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "(well, you figure it out!) but once you have that formula and you simplify it, clearly the blue curve has derivative less than 1 when x >0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 " Thus, the red curve is rising faster than the blue, and so once it goe s ahead, it stays ahead. That is, the one crossover we see in the pict ure really is the only one. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 37 "fsolve(x/sqrt(1+x^2)=x^4-x,x,1..4/3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+.dx!=\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "?fsolve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " Digits:=100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"$+\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "fsolve(x/sqrt(1+x^2)=x^4-x,x ,1..4/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q,#p/!=/eb!4\"[s/?a_# Rnr'eq'e+?F1%eMPBJEdq2aJ8$3ZE1Jqv2=\"!#**" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "int(x/sqrt(1+x^2)-x^4+x,x=0..%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"_q6BN6!eKm_#4(f$\\vxLP(ohW'>'>#*RH4c\\Vn&e0$y*3mH8 _3Pc&\\]&)Q&y!$+\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "The crosso ver occurs very nearly at 1.18077etc, and as a result, the enclosed ar ea is very nearly 0.785etc. We can, without using much electricity, ge t the answer to more digits than we're ever going to really need. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 338 "Getting \+ an \"exact\" answer depends on the meaning of exact. Even if the answe r were sqrt(2), that's only an exact answer because of a societal conv ention which says that numbers such as 1.41421356etc. will be thought \+ of as exactly given, just because we know how to compute their digital approximations to many digits without much strain. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "As seen here, in that s ense, this answer too is \"exact\". " }}}}{MARK "9 4 0" 57 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }