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Blank and Steven G. Krantz " }{TEXT 295 3 " \n\n" }{TEXT 296 11 "Section 5.4" }{TEXT 297 1 "\n" } {TEXT 291 45 "The Fundamental Theorem \nof Calculus \nPart II" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "1. Differentiating Integrals" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Suppose \+ " }{TEXT 267 1 "f" }{TEXT -1 52 " is a continuous function defined \+ on an interval " }{TEXT 268 1 "I" }{TEXT -1 10 ". 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::;J:<::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::=ja^G>:;:::::::::N;?B:yyyyyy:>Z:>Z ::::::j;<:c:;::::::::::::vYxI:;Z::::::::::::::::::::yay=J:B::::::::::: ::::::::jysy:>:<::::::::1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "The second part of the Fundamental Theorem of Cal culus asserts that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 " \+ " }{XPPEDIT 329 0 "D(F)(x) = f(x);" "6#/--%\"DG6# %\"FG6#%\"xG-%\"fG6#F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 331 77 "Notice that you nee d not calculate the integral in order to differentiate it!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 45 "2. Examples of Functions Defined by Integ rals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "S everal important functions of mathematics, physics, and engineering ar e defined as area integrals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 332 18 "The Error Func tion" }}{PARA 0 "" 0 "" {TEXT -1 35 " \+ " }{XPPEDIT 18 0 "erf(x) = 2/sqrt(Pi)*Int(exp(-t^2),t = 0 .. x);" "6 #/-%$erfG6#%\"xG,$-%$IntG6$-%$expG6#,$*$)%\"tG\"\"#\"\"\"!\"\"/F2;\"\" !F'*&F3F4-%%sqrtG6#%#PiGF5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "is a fundamental concept of probability and stati stics." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(erf(x), x=-4..4,thi ckness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6&-% 'CURVESG6$7V7$$!\"%\"\"!$!23UFe%)*******!#<7$$!3ommmmFiDQF-$!2EIB1P*** ****F-7$$!35LLLo!)*Qn$F-$!2sqK-'z******F-7$$!3nmmmwxE.NF-$!2-y****FB7$$!3!******\\`oz$GF-$!3)QCDPt,%****F B7$$!3!omm;)3DoEF-$!3g\"fYm6!R)***FB7$$!3?+++:v2*\\#F-$!3WDG2R.\"f***F B7$$!3BLLL8>1DBF-$!3Ei0)3i9**)**FB7$$!3kmmmw))yr@F-$!3#>\\%eTBpy**FB7$ $!3;+++S(R#**>F-$!3-giC)GlI&**FB7$$!30++++@)f#=F-$!3i(H*))=R'=!**FB7$$ !3-+++gi,f;F-$!3zas'HSU.\")*FB7$$!3qmmm\"G&R2:F-$!3-(=p'*o\\(p'*FB7$$! 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They are defined as follows:" }}{PARA 0 "" 0 "" {TEXT -1 22 " " }{XPPEDIT 18 0 "FresnelS(x) = Int(sin(Pi*t ^2/2),t = 0 .. x);" "6#/-%)FresnelSG6#%\"xG-%$IntG6$-%$sinG6#*(%#PiG\" \"\"*$)%\"tG\"\"#F0F0F4!\"\"/F3;\"\"!F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " " }{XPPEDIT 18 0 "FresnelC(x) = Int(cos(Pi*t^2/2),t = 0 .. x);" "6#/-%)FresnelCG6# %\"xG-%$IntG6$-%$cosG6#*(%#PiG\"\"\"*$)%\"tG\"\"#F0F0F4!\"\"/F3;\"\"!F '" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The first of the two plots that follow shows th e graphs of the equations " }{XPPEDIT 18 0 "y = FresnelS(x);" "6#/% \"yG-%)FresnelSG6#%\"xG" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "y = F resnelC(x);" "6#/%\"yG-%)FresnelCG6#%\"xG" }{TEXT -1 97 ". \nThe secon d of the two plots shows the graph of the curve that is defined parame trically by " }{XPPEDIT 18 0 "x = FresnelS(u),y = FresnelC(u);" "6$ /%\"xG-%)FresnelSG6#%\"uG/%\"yG-%)FresnelCG6#F(" }{TEXT -1 50 ". Noti ce that the Maple syntax is rather similar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plot([FresnelS(x) , FresnelC(x)], x = -2*Pi..2*Pi, co lor = [NAVY,MAROON], thickness = 2);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6'-%'CURVESG6$7gz7$$!+&H&=$G'!\"*$!+^CvxgF*$!+-QZX\\F- 7$$!+yTF4gF*$!+:upxWF-7$$!+H?p\\fF*$!+AVH!p%F-7$$!+z)4,*eF*$!+=:la_F-7 $$!+0)=.'eF*$!+%fG&oaF-7$$!+Ix_IeF*$!+pttXbF-7$$!+bmt+eF*$!+q\"yVY&F-7 $$!+!eX4x&F*$!+8Bg[_F-7$$!++CWPdF*$!+%ybO#\\F-7$$!+@#RRq&F*$!+#=sOi%F- 7$$!+Iw=(o&F*$!+'o+u^%F-7$$!+TgVqcF*$!+gLz`WF-7$$!+^Wo`cF*$!+m6OQWF-7$ $!+iG$pj&F*$!+3jKsWF-7$$!+.l#*pbF*$!+'zWU)\\F-7$$!+X,#H]&F*$!+g)*)\\_& F-7$$!+9v0'[&F*$!+8WDubF-7$$!+%)[>paF*$!+a!>ad&F-7$$!+aAL_aF*$!+b\"G&G bF-7$$!+C'paV&F*$!+gZhPaF-7$$!+kVu,aF*$!+Z)[r:&F-7$$!+/\">!o`F*$!+r`XC [F-7$$!+WQHM`F*$!+/)>\\a%F-7$$!+%eo0I&F*$!+jc]0WF-7$$!+afq$G&F*$!+Ghs. 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The specification " }{XPPEDIT 18 0 "x = a .. b;" "6#/%\"xG;%\"aG%\"bG" }{TEXT -1 46 " of the range of the independent variable " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 113 ", that is, the range of the variable that is plotted along the horizontal axis, goes outside the expression list:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 354 6 " " }{TEXT 355 36 " plot( [ f(x) , g(x) ] , x = a .. b )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "In a parametric curve, the variable " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 53 " plotted along the horizo ntal axis and the variable " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 110 " plotted along the vertical axis are both given in terms of anoth er variable that is called a parameter. If " }{XPPEDIT 18 0 "u;" "6# %\"uG" }{TEXT -1 25 " is the parameter with " }{XPPEDIT 18 0 "x = f( u);" "6#/%\"xG-%\"fG6#%\"uG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = \+ g(u);" "6#/%\"yG-%\"gG6#%\"uG" }{TEXT -1 29 ", then the plotting synta x is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 356 7 " \+ " }{TEXT 357 45 "plot( [ f(x) , g(x) , u = alpha .. beta ] )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Notice that the specification " }{XPPEDIT 18 0 "u = alp ha .. beta;" "6#/%\"uG;%&alphaG%%betaG" }{TEXT -1 54 " of the range \+ of the parameter goes inside the list." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 334 17 "The Sine Integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " " }{XPPEDIT 18 0 "Si(x) = In t(sin(t)/t,t = 0 .. x);" "6#/-%#SiG6#%\"xG-%$IntG6$*&-%$sinG6#%\"tG\" \"\"F/!\"\"/F/;\"\"!F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(Si(x), x = -6*Pi .. 6*Pi, thic kness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6&-%' CURVESG6$7gr7$$!+)eb\\)=!\")$!+hR.=:!\"*7$$!+`Ay-=F*$!+GDYN:F-7$$!+tOG J8im:F-7$$!+i6OP8F*$!+o*=d^\"F-7$$!+#QstH\"F*$!+Ie`)\\\"F -7$$!+-OQd7F*$!+UM;#\\\"F-7$$!+nI_<7F*$!+v')H)\\\"F-7$$!+KDmx6F*$!+![Q n^\"F-7$$!+s5mO6F*$!+1>FY:F-7$$!+7'fc4\"F*$!+=&fAe\"F-7$$!+/baf5F*$!+u my9;F-7$$!+&RJM-\"F*$!+&4/Ok\"F-7$$!+HM5.5F*$!+G=ic;F-7$$!+DYvF)*F-$!+ 8o\\m;F-7$$!+i\\ZC'*F-$!+![#os;F-7$$!+*H&>@%*F-$!+=6wu;F-7$$!+_43<#*F- $!+HuWs;F-7$$!+0m'H,*F-$!+$)[il;F-7$$!+dA&)3))F-$!+kmPa;F-7$$!*\"zt/') F*$!+TK(*Q;F-7$$!+:RL6#)F-$!+%>&R*f\"F-7$$!*#*Hz\"yF*$!+r6B^:F-7$$!+59 oguF-$!+p#)y0:F-7$$!*!HV.rF*$!+)='yk9F-7$$!+#QJ5*oF-$!+n[&\\W\"F-7$$!+ l)H'ymF-$!+]L%*H9F-7$$!+Z$GiY'F-$!+d)f2U\"F-7$$!*$o#QD'F*$!+v.A=9F-7$$ !+$)[*Q2'F-$!+0Oq@9F-7$$!+NH'R*eF-$!+d*p0V\"F-7$$!+))4.9dF-$!+v!e[W\"F -7$$!*/*4MbF*$!+(yQWY\"F-7$$!+&y\"\\:^F-$!+'>a%G:F-7$$!*`%)op%F*$!+$>o ;h\"F-7$$!*#y!e&RF*$!+)QZkw\"F-7$$!+q\"QDv$F-$!+&HT3!=F-7$$!+?&o#\\NF- $!+wA!z#=F-7$$!+&pLwW$F-$!+50/Q=F-7$$!+q))*fM$F-$!+*H%eX=F-7$$!+XSOWKF -$!+*G$H]=F-7$$!*AHF9$F*$!+Yo$>&=F-7$$!+J)[f/$F-$!+S?X]=F-7$$!+U%o\"\\ HF-$!+\\3\"e%=F-7$$!+`!)Q_GF-$!+%\\Zy$=F-7$$!+lwgbFF-$!+)p4k#=F-7$$!+( )o/iDF-$!+*=wDz\"F-7$$!*6'[oBF*$!+r\\SVF-$!+)[H!*e\"F-7$ $!*mW1c\"F*$!+a&yUO\"F-7$$!+S$>(*=\"F-$!+3S/+6F-7$$!)-%z=)F*$!+A%Q!*)y !#57$$!+v[Z(='F`]l$!+!)HPdgF`]l7$$!+]&4q=%F`]l$!+ATWYTF`]l7$$!+DUa'=#F `]l$!+t[u!=#F`]l7$$!(*yg=F*$!+1Kvg=!#67$$\"+vB%=*=F`]l$\"+FZ3))=F`]l7$ $\"+]OwpRF`]l$\"+\">s^$RF`]l7$$\"+D\\oZgF`]l$\"+-59EfF`]l7$$\")igD\")F *$\"+D-RLyF`]l7$$\"+IzKu6F-$\"+7X*z3\"F-7$$\"*C&4O:F*$\"+4wU[8F-7$$\"+ ]Q\"o#>F-$\"+)[u4d\"F-7$$\"*YKvJ#F*$\"+3d#ys\"F-7$$\"+QxN>DF-$\"+Kj2$y \"F-7$$\"+:I=@FF-$\"+YqZ@=F-7$$\"+ac4AGF-$\"+\"yZY$=F-7$$\"+$H3I#HF-$ \"+lD*R%=F-7$$\"+J4#R-$F-$\"+*Gz'\\=F-7$$\"*dL[7$F*$\"+#>#*=&=F-7$$\"+ +pbBKF-$\"+am)3&=F-7$$\"+I-GALF-$\"+![Xp%=F-7$$\"+gN+@MF-$\"+%)yFS=F-7 $$\"+!*os>NF-$\"+5@5J=F-7$$\"+]N<u;#*F -$\"+R)RCn\"F-7$$\"*hLkS*F*$\"+CRuu;F-7$$\"+Pgr/'*F-$\"+A-2t;F-7$$\"+l %)*H!)*F-$\"+xqXn;F-7$$\"+*3G,+\"F*$\"+ZyFe;F-7$$\"+Kj&*>5F*$\"+F_-Y;F -7$$\"+'zF(e5F*$\"+2l\\:;F-7$$\"+f#*\\(4\"F*$\"+iEe!e\"F-7$$\"+)Q*3Q6F *$\"+_[5X:F-7$$\"+<&z'y6F*$\"+m&Gh^\"F-7$$\"+OEx<7F*$\"+s2A)\\\"F-7$$ \"+ad'oD\"F*$\"+M9;#\\\"F-7$$\"+9T%oH\"F*$\"+naP)\\\"F-7$$\"+uC#oL\"F* $\"+'oGa^\"F-7$$\"+7r6;9F*$\"+Hnin:F-7$$\"+!))[DX\"F*$\"+/nN#f\"F-7$$ \"+\\1)*)[\"F*$\"+r158;F-7$$\"+nbtI:F*$\"+Cs$)G;F-7$$\"+&[!\\s:F*$\"+< d&Rj\"F-7$$\"+#eO)4;F*$\"+$Q_#H;F-7$$\"+yE=Z;F*$\"+'[Doh\"F-7$$\"+N,#o s\"F*$\"+#p$pu:F-7$$\"+UL/.=F*$\"+/mNN:F-7$$\"+)eb\\)=F*$\"+hR.=:F--%' COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FdbmFcbm-%+AXESLABELSG6$Q\"x6\"Q!Fibm-% *THICKNESSG6#\"\"#-%%VIEWG6$;$!+#fb\\)=F*$\"+#fb\\)=F*%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 352 51 "The Elliptic Function of the First Kind (Ellip ticF)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " The function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " " }{XPPEDIT 353 0 "EllipticF(k,x) = In t(1/sqrt(1-t^2)/sqrt(1-k^2*t^2),t = 0 .. x);" "6#/-%*EllipticFG6$%\"kG %\"xG-%$IntG6$*(\"\"\"F--%%sqrtG6#,&F-F-*$)%\"tG\"\"#F-!\"\"F6-F/6#,&F -F-*&)F'F5F-F3F-F6F6/F4;\"\"!F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "is used to calculate the arc length of an ellip se." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "3. An Example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Suppose that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 20 " " }{XPPEDIT 18 0 "F(x) = Int(s qrt(t)*(1-t)^(3/2),t = 0 .. x);" "6#/-%\"FG6#%\"xG-%$IntG6$*&-%%sqrtG6 #%\"tG\"\"\"),&F0F0F/!\"\"*&\"\"$F0\"\"#F3F0/F/;\"\"!F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "It is known that " }{XPPEDIT 18 0 "F(1/2) = Pi/32+1/24;" "6#/-%\"FG6#*&\"\"\"F(\"\"#!\"\" ,&*&%#PiGF(\"#KF*F(*&F(F(\"#CF*F(" }{TEXT -1 66 " . Calculate the tang ent line to the graph of F at the point (" }{XPPEDIT 18 0 "1/2,Pi/ 32+1/24;" "6$*&\"\"\"F$\"\"#!\"\",&*&%#PiGF$\"#KF&F$*&F$F$\"#CF&F$" } {TEXT -1 3 " )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "F := x -> Int(sqrt(t)*(1-t)^(3/2),t=0..x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrow GF(-%$IntG6$*&-%%sqrtG6#%\"tG\"\"\"),&F4F4F3!\"\"#\"\"$\"\"#F4/F3;\"\" !9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(F)(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG#\"\"\"\"\"#),&F&F&F$!\"\"#\"\" $F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "m := D(F)(1/2);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "tangentLineEquation := y = m*(x-1/2)+Pi/3 2+1/24;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4tangentLineEquationG/%\" yG,(*&\"\"%!\"\"%\"xG\"\"\"F,#F,\"#7F**&\"#KF*%#PiGF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot([F(x),rhs(tangentLineEquation) ], x = 0 .. 1, color = [NAVY,PLUM],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 444 279 279 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"\"!F)F(7$$\"+; arz@!#6$\"+)zxM5#!#77$$\"+XTFwSF-$\"+O`z'G&F07$$\"+\"z_\"4iF-$\"+D%[Zu *F07$$\"+S&phN)F-$\"+)pn5\\\"F-7$$\"+*=)H\\5!#5$\"+\\!fg0#F-7$$\"+[!3u C\"FC$\"+Ip%[h#F-7$$\"+J$RDX\"FC$\"+87#4A$F-7$$\"+)R'ok;FC$\"+E5CqQF-7 $$\"+1J:w=FC$\"+m:IMXF-7$$\"+3En$4#FC$\"+6e\\H_F-7$$\"+/RE&G#FC$\"+2u> [eF-7$$\"+D.&4]#FC$\"+m42[lF-7$$\"+vB_CK&)F-7$$\"+347TLFC$\"+:ImX#*F-7$$\" +LY.KNFC$\"+juqS)*F-7$$\"+\"o7Tv$FC$\"+#H**>0\"FC7$$\"+$Q*o]RFC$\"+UU! 36\"FC7$$\"+\"=lj;%FC$\"+9Hst6FC7$$\"+V&RB\"FC7$$\"+Xh-'e %FC$\"+iAw!H\"FC7$$\"+R\"3Gy%FC$\"+NZ&HM\"FC7$$\"+.T1&*\\FC$\"+)yzrR\" FC7$$\"+(RQb@&FC$\"+(R@6X\"FC7$$\"+=>Y2aFC$\"+eK-'\\\"FC7$$\"+yXu9cFC$ \"+d/IU:FC7$$\"+\\y))GeFC$\"+5ai(e\"FC7$$\"+i_QQgFC$\"+Z!p%H;FC7$$\"+! y%3TiFC$\"+u9cn;FC7$$\"+O![hY'FC$\"+6812aDx\"FC7$$\"+(pe*zqFC$\"+j!f'*z\"FC7$$\"+C\\'QH(FC $\"+]irE=FC7$$\"+8S8&\\(FC$\"+u`t\\=FC7$$\"+0#=bq(FC$\"+[vHr=FC7$$\"+2 s?6zFC$\"+t\"[**)=FC7$$\"+IXaE\")FC$\"+G\\&p!>FC7$$\"+l*RRL)FC$\"+mi'4 #>FC7$$\"+`<.Y&)FC$\"+gi(H$>FC7$$\"+8tOc()FC$\"+cBnU>FC7$$\"+\\Qk\\*)F C$\"+C7u\\>FC7$$\"+p0;r\"*FC$\"+7C#e&>FC7$$\"+lxGp$*FC$\"+S/ff>FC7$$\" +!oK0e*FC$\"+8d2i>FC7$$\"+<5s#y*FC$\"+>#>K'>FC7$$\"\"\"F)$\"+3a\\j>FC- %'COLOURG6&%$RGBG$\")!\\DP\"!\")F[[l$\")viobF][l-F$6$7S7$F($\"3iqZ84P9 %[\"!#>7$$\"3emmm;arz@Ff[l$\"3EP9IjD2H?Ff[l7$$\"3[LL$e9ui2%Ff[l$\"3H.J fXA@.DFf[l7$$\"3nmmm\"z_\"4iFf[l$\"3HP902>VOIFf[l7$$\"3[mmmT&phN)Ff[l$ \"3CP9b%4'=tNFf[l7$$\"3CLLe*=)H\\5!#=$\"3O.J4$=*Q2TFf[l7$$\"3gmm\"z/3u C\"F^]l$\"36Pk#*GQm-YFf[l7$$\"3%)***\\7LRDX\"F^]l$\"3Bq(fs.#\\:^Ff[l7$ $\"3]mm\"zR'ok;F^]l$\"3)oVERqfek&Ff[l7$$\"3w***\\i5`h(=F^]l$\"3-q(fZZE X<'Ff[l7$$\"3WLLL3En$4#F^]l$\"3F/\"o*H_K=nFf[l7$$\"3qmm;/RE&G#F^]l$\"3 OP9bpMI(>(Ff[l7$$\"3\")*****\\K]4]#F^]l$\"39qZj@&>lt(Ff[l7$$\"3$****** \\PAvr#F^]l$\"3YqZjY'\\zF)Ff[l7$$\"3)******\\nHi#HF^]l$\"3aqZj'*yr*z)F f[l7$$\"3jmm\"z*ev:JF^]l$\"3?Pk#RXLNF*Ff[l7$$\"3?LLL347TLF^]l$\"3h.\"o *zf%p$)*Ff[l7$$\"3,LLLLY.KNF^]l$\"3J5oCHIUJ5F^]l7$$\"3w***\\7o7Tv$F^]l $\"3+xfATD%p3\"F^]l7$$\"3'GLLLQ*o]RF^]l$\"3G5ou;n3O6F^]l7$$\"3A++D\"=l j;%F^]l$\"35xfAmc+!>\"F^]l7$$\"31++vV&RY2aF^]l$\"3zw4g])z-]\"F^]l7$$\"39mm;z Xu9cF^]l$\"3uV^q:05_:F^]l7$$\"3l******\\y))GeF^]l$\"36xMTLjj0;F^]l7$$ \"3'*)***\\i_QQgF^]l$\"3%pZQlo5!e;F^]l7$$\"3@***\\7y%3TiF^]l$\"3+xfAmb o3F^]l7$$\"3%)*****\\#\\'QH(F^]l$\"3;xM;-1)=(> F^]l7$$\"3GKLe9S8&\\(F^]l$\"3**4$fX(y>A?F^]l7$$\"3R***\\i?=bq(F^]l$\"3 /xfZARzu?F^]l7$$\"3\"HLL$3s?6zF^]l$\"3V5o*H<;i7#F^]l7$$\"3a***\\7`Wl7) F^]l$\"35xfs.00!=#F^]l7$$\"3#pmmm'*RRL)F^]l$\"3mV,ei$**=B#F^]l7$$\"3Qm m;a<.Y&)F^]l$\"3_V^X4B#\\G#F^]l7$$\"3=LLe9tOc()F^]l$\"3\\5$f&*>1vL#F^] l7$$\"3u******\\Qk\\*)F^]l$\"39xMTL`#eQ#F^]l7$$\"3CLL$3dg6<*F^]l$\"3C5 =i8X?TCF^]l7$$\"3ImmmmxGp$*F^]l$\"3]V,e7jt!\\#F^]l7$$\"3A++D\"oK0e*F^] l$\"3ExfATvaVDF^]l7$$\"3A++v=5s#y*F^]l$\"3Cx4gDY4%f#F^]l7$Fcz$\"3?xM\" 4P9%[EF^]l-Fhz6&Fjz$\")1Zw\"*F][l$\")PJ%y'F][lFhjl-%+AXESLABELSG6$Q\"x 6\"Q!F`[m-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "4. Exercise 61, Page 384" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The funct ion C(x) defined by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 " " }{XPPEDIT 18 0 "C(x) = Int (t^4/sqrt(1+t^2),t = 0 .. x);" "6#/-%\"CG6#%\"xG-%$IntG6$*&%\"tG\"\"%- %%sqrtG6#,&\"\"\"F2*$)F,\"\"#F2F2!\"\"/F,;\"\"!F'" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "for " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{TEXT -1 138 " arises in the computation of pressure within a white dwarf. Show that C is an increasing func tion with a graph that is concave up." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 335 9 "Solut ion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "C := x -> Int( t^4/sqrt(1+t^2) , t = 0 .. x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CGf*6#%\"xG6\"6$%)operatorG%&arrow GF(-%$IntG6$*&%\"tG\"\"%-%%sqrtG6#,&\"\"\"F6*$)F0\"\"#F6F6!\"\"/F0;\" \"!9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(C)(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"%,&\"\"\"F'*$)F$\"\"#F'F'#! \"\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Since the derivative of C is posit ive everywhere, it is an increasing function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify(diff(D(C) (x), x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"xG\"\"$,&\"\"%\"\"\" *&F%F()F$\"\"#F(F(F(,&F(F(*$F*F(F(#!\"$F+" }}}{PARA 0 "" 0 "" {TEXT 305 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Because " }{XPPEDIT 18 0 "diff(C(x),`$`(x,2));" "6#-%%diffG6$-% \"CG6#%\"xG-%\"$G6$F)\"\"#" }{TEXT -1 49 " is positive, the graph \+ of C is concave up." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "plot( C(x), x = 0 .. 10);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)F(7$$\"+; arz@!#5$\"+M$o#y'*!#97$$\"+XTFwSF-$\"+\"f>*G@!#77$$\"+\"z_\"4iF-$\"+vF 6P;!#67$$\"+S&phN)F-$\"+'\\^Co'F<7$$\"+*=)H\\5!\"*$\"+d&Hs\">F-7$$\"+[ !3uC\"FE$\"+HVt,UF-7$$\"+J$RDX\"FE$\"+SZ/'G)F-7$$\"+)R'ok;FE$\"+$=\"R3 :FE7$$\"+1J:w=FE$\"+(HDV`#FE7$$\"+3En$4#FE$\"+L%4'eSFE7$$\"+/RE&G#FE$ \"+2qh\"*eFE7$$\"+D.&4]#FE$\"+126D')FE7$$\"+vB_Y2aFE$\"+H-Br?Fhr7$$\"+y Xu9cFE$\"+=yt7CFhr7$$\"+\\y))GeFE$\"+D\"Q!3GFhr7$$\"+i_QQgFE$\"+OJvRKF hr7$$\"+!y%3TiFE$\"+r&=Hq$Fhr7$$\"+O![hY'FE$\"+8[LtUFhr7$$\"+#Qx$omFE$ \"+hftR[Fhr7$$\"+u.I%)oFE$\"+vkd/bFhr7$$\"+(pe*zqFE$\"+$GhR;'Fhr7$$\"+ C\\'QH(FE$\"+x\\h]pFhr7$$\"+8S8&\\(FE$\"+1\")>dxFhr7$$\"+0#=bq(FE$\"+K u8t')Fhr7$$\"+2s?6zFE$\"+\"\\=Yk*Fhr7$$\"+IXaE\")FE$\"+)RjY2\"!\"'7$$ \"+l*RRL)FE$\"+<(e%*=\"F^x7$$\"+`<.Y&)FE$\"+OX5;8F^x7$$\"+8tOc()FE$\"+ WFT^9F^x7$$\"+\\Qk\\*)FE$\"+b#)o%e\"F^x7$$\"+p0;r\"*FE$\"+htZ[F^x7$$\"+!oK0e*FE$\"+.*QT3#F^x7$$\"+<5s#y*FE$\"+\\ JnmAF^x7$$\"#5F)$\"+0i!fZ#F^x-%'COLOURG6&%$RGBG$Fiz!\"\"F(F(-%+AXESLAB ELSG6$Q\"x6\"Q!Ff[l-%%VIEWG6$;F(Fhz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 53 "5. An Example \+ (Local Extrema of an Integral Function)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "What are the local extrema of" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ " }{XPPEDIT 18 0 "F(x) = Int((t^2-t-6)/(1+t^4),t = 0 . . x);" "6#/-%\"FG6#%\"xG-%$IntG6$*&,(*$)%\"tG\"\"#\"\"\"F1F/!\"\"\"\"' F2F1,&F1F1*$)F/\"\"%F1F1F2/F/;\"\"!F'" }{TEXT -1 4 " ?" }}{PARA 0 " " 0 "" {TEXT -1 1 "\n" }{TEXT 336 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "F := x -> In t((t^2-t-6)/(1+t^4),t = 0 .. x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$IntG6$*&,(*$)%\"tG\"\"#\"\" \"F5F3!\"\"\"\"'F6F5,&F5F5*$)F3\"\"%F5F5F6/F3;\"\"!9$F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor( D(F)(x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"xG\"\"\"\"\"#F&F&,&F%F&\"\"$!\"\"F&,& F&F&*$)F%\"\"%F&F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "cri ticalPoints := [solve( D(F)(x) = 0 , x)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/criticalPointsG7$\"\"$!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Notice that " }{XPPEDIT 18 0 "0 < D(F)(x);" "6#2\"\"!--%\"DG6#%\"FG6#%\"xG" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "x < -2;" "6#2%\"xG,$\"\"#!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "D(F)(x) < 0;" "6#2--%\"DG6#%\"FG6#%\"xG\"\"!" } {TEXT -1 7 " for " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 166 " be tween -2 and 3. Therefore, F is increasing on the interval to the \+ left of -2 and decreasing on an interval to the right of -2. It fo llows that F has a " }{TEXT 337 24 "local maximum at x = -2" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 151 "Similarly, we deduce th at F is decreasing on an interval to the left of 3 and increasing o n the interval to the right of 3. It follows that F has a " }{TEXT 338 22 "local minimum at x = 3" }{TEXT -1 20 ". Here is the plot:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(F(x), x = -10 .. 10);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6%-%'CURVESG6$7^ o7$$!#5\"\"!$\"+GM?r[!\"*7$$!+#f)[F-7$$!+#4m(G$)F-$\"+VB,#*[F-7$$!+@OS,zF -$\"+ZHr)*[F-7$$!+/R=0vF-$\"+ZPe0\\F-7$$!+P8#\\4(F-$\"+*zsM\"\\F-7$$!+ /siqmF-$\"+S&*fA\\F-7$$!+(y$pZiF-$\"+[9&G$\\F-7$$!+$yaE\"eF-$\"+1#G[% \\F-7$$!+\">s%HaF-$\"+*H3o&\\F-7$$!+]$*4)*\\F-$\"+iZ@s\\F-7$$!+]_&\\c% F-$\"+7n8!*\\F-7$$!+]1aZTF-$\"+19>5]F-7$$!+/#)[oPF-$\"+1(f6.&F-7$$!+$= exJ$F-$\"+mj%)f]F-7$$!+L2$f$HF-$\"+'Rtq3&F-7$$!+PYx\"\\#F-$\"+&ze%>^F- 7$$!+L7i)4#F-$\"+L5\")R^F-7$$!+Ma%H)=F-$\"+m*)yQ^F-7$$!+P'psm\"F-$\"+_ Lv;^F-7$$!+cCek:F-$\"+M.K$4&F-7$$!+u_*=Y\"F-$\"+2u)p0&F-7$$!+$43#f8F-$ \"+l?)G+&F-7$$!+74_c7F-$\"+5#\\X#\\F-7$$!+5VBU5F-$\"+Z*p]k%F-7$$!*3x%z #)F-$\"+wCLLTF-7$$!+Srl6jF)$\"+`e<3MF-7$$!*?PQM%F-$\"+!3SzY#F-7$$!+ytb #G$F)$\"+!f*e**=F-7$$!+bvF@AF)$\"+%f=QI\"F-7$$!+Lx**f6F)$\"+bhD()oF)7$ $!(\"zr)*F-$\"+3*p\"=f!#67$$\"+nNl.5F)$!+lOzogF)7$$\"+X]-1@F)$!+k&p@G \"F-7$$\"+AlR3KF)$!++8Jh>F-7$$\")!o2J%!\")$!+7I[MEF-7$$\"++K+IiF)$!+l= >VPF-7$$\")%Q#\\\")F`w$!+j1DzYF-7$$\"++l?A5F-$!+[#>eT&F-7$$\"*;*[H7F`w $!+)Qm_)eF-7$$\"+ICjV9F-$!+ZACshF-7$$\"*qvxl\"F`w$!+5:%oL'F-7$$\"+:JFn =F-$!+gYcHkF-7$$\"*`qn2#F`w$!+4=E$['F-7$$\"+X+ZzAF-$!+Y#3K^'F-7$$\"*cp @[#F`w$!+/^+IlF-7$$\"*3'HKHF`w$!+YoNUlF-7$$\"*xanL$F`w$!+\"Qw(RlF-7$$ \"*v+'oPF`w$!+DXlJlF-7$$\"*S<*fTF`w$!+GkiAlF-7$$\"*&)Hxe%F`w$!+'RXC^'F -7$$\"*.o-*\\F`w$!+At<.lF-7$$\"*TO5T&F`w$!+6&yS\\'F-7$$\"*U9C#eF`w$!+M %[e['F-7$$\"*1*3`iF`w$!+S(RzZ'F-7$$\"*$*zym'F`w$!+40(4Z'F-7$$\"*^j?4(F `w$!+&*zWkkF-7$$\"*jMF^(F`w$!+7p_ekF-7$$\"*q(G**yF`w$!+1'>NX'F-7$$\"*9 @BM)F`w$!+l!R#[kF-7$$\"*`v&Q()F`w$!+(y))QW'F-7$$\"*Ol5;*F`w$!+&y(fRkF- 7$$\"*/Uac*F`w$!+-HzNkF-7$$\"#5F*$!+*3+?V'F--%'COLOURG6&%$RGBG$F\\`l! \"\"$F*F*Fe`l-%+AXESLABELSG6$Q\"x6\"Q!Fj`l-%%VIEWG6$;F(F[`l%(DEFAULTG " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "6. A Variant (Variable " } {TEXT 342 5 "Lower" }{TEXT -1 25 " Endpoint of Integration)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "If " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " \+ " }{TEXT 339 4 " " }{XPPEDIT 340 0 "G(x) = Int(f(t),t = x .. \+ b);" "6#/-%\"GG6#%\"xG-%$IntG6$-%\"fG6#%\"tG/F.;F'%\"bG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ " }{XPPEDIT 341 0 "D(G)(x) = -f(x);" "6#/--%\"DG6#%\"GG6#%\"xG,$-% \"fG6#F*!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 60 "7. The Most General Form of the Fundamental Theorem, Part II" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "u;" "6#% \"uG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 21 " are functions and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 18 " " }{XPPEDIT 344 0 "F(x) = Int(f( t),t = v(x) .. u(x));" "6#/-%\"FG6#%\"xG-%$IntG6$-%\"fG6#%\"tG/F.;-%\" vG6#F'-%\"uG6#F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " " }{XPPEDIT 343 0 "D(F)(x) = f(u(x))*D(u) (x)-f(v(x))*D(v)(x);" "6#/--%\"DG6#%\"FG6#%\"xG,&*&-%\"fG6#-%\"uG6#F* \"\"\"--F&6#F16#F*F3F3*&-F.6#-%\"vG6#F*F3--F&6#F<6#F*F3!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "8. An \+ Example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Suppose that " }{XPPEDIT 18 0 "F(x) = Int(t/sqrt(1-t^2),t = cos (x) .. sin(x));" "6#/-%\"FG6#%\"xG-%$IntG6$*&%\"tG\"\"\"-%%sqrtG6#,&F- F-*$)F,\"\"#F-!\"\"F5/F,;-%$cosG6#F'-%$sinG6#F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Calculate " }{XPPEDIT 18 0 "D(F)(x);" "6#--% \"DG6#%\"FG6#%\"xG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 345 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := t -> t/sqrt(1-t^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%%s qrtG6#,&F.F.*$)F-\"\"#F.!\"\"F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "u := x -> sin(x); v := x -> cos(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG%$sinG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" vG%$cosG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "F := x -> Int(f (t), t = v(x) .. u(x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6# %\"xG6\"6$%)operatorG%&arrowGF(-%$IntG6$-%\"fG6#%\"tG/F2;-%\"vG6#9$-% \"uGF7F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "D(F)(x) = f (u(x))*D(u)(x) - f(v(x))*D(v)(x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /--%\"DG6#%\"FG6#%\"xG,&*(-%$sinGF)\"\"\",&F/F/*$)F-\"\"#F/!\"\"#F4F3- %$cosGF)F/F/*(F6F/,&F/F/*$)F6F3F/F4F5F-F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 57 "9. 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LKD:b=n;:oF?o>WZSrjb=:onFJAZ=a\\SZH::::::::::::[ZJZHJA:::::::::::::::: :::::Z:n;t:::::J:<:t::::::::::::L;crjZHJA::::VRcb=n;dc=]H:VRJAZ=e`b=:: ::::::::::[ZJZHJA:::::::::::::::::::::B:WZH:::::>Z:ZH:::::::::::Z=QmSF =n[SF=n;:::JG_\\N^>Z:ZH::::::::::::::::::VrS^\\T\\HjFLKAb=:A@b=B?m: W:::::::::::::B>>>Z:ZH::::::::::::::::::jFNMn;dCn;:Lk[u:WJDdCt::::::::::::::ZJ JBb=n;:::::::::::::::::::B:WZH:::::>Z:ZH::::::::::::::::::ZN^Z:ZH:::::::: ::::::::::Z=U@Axjb=Wb`HFt::HfT\\H:::::::::::::::B>>Z:ZH:::::::::::::::::::AX@LKGKAVRWZH:ZSVRn;:::::::::::: :::B>><<[HJA::::::::::::::::::t:W::::::::::::::::::< JAb=:::::;B:b=::::::::::::::::::::::::::::::::::::::::ZJJBn;t::::::::: :::::::::<<[HJA:::::::::::::::::>Z:ZH::::::::::::::::::::::::::::::::::::::::::JBB>t :W::::::::::::::::B:WZH:ZSn;t:::>Z:ZH::::::::::::::::::::::::::::::::: ::::::::::[ZJn;t::::::::::::::::B:WZH::m:W:::;B:b=:::::::::::::::::::: ::::::::::::::::::::::ZJJBn;t::::::::::::::::t:W:::::::::::::::Z:n;t::B?JA:: >Z:ZH:::::::::::::::::::::::::::::::::::::::::::JBB>WZH::::::::::::::: B:WZH::AH?cR@b=::;B:b=:::::::::::::::::::::::::::::::::::::::::::B>>W::>Z:ZH::::::::::::::::::::::::::::::::: :::::::::::[ZJb=n;::::::::::::::B:WZH::oF?mJG[H::;B:b=:::::::::::::::: ::::::::::::::::::::::::::::[ZJn;t:::::::::::::::Z:ZH::::::::::::::::::::::::::::::::::::::::::::JB:W::::: :::::::::Z:n;t::F]NNMNm=]H:J:<:t:::::::::::::::::::::::::::::::::::::: :::::::<[HJA::::::::::::::B:WZH::m:VRb=B:b=;B:b=;b=n;::::::::::::::::: ::::::::::::::::::::::::::JBt:W::::::::::::::Z:ZH:Z=aZ:B::WZH::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::B:WZH::::Z:>:;:t:W:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::Z:n;t::::::b=n;:: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::;:t:W:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :Z:n;t:::::>Z:B:n;t::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::Z:B:n;t::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::;b=n;::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::B:WZH:::::ZHJA:::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::Z:n;t:::::B:;J:t:W::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::;b=n ;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::B:WZH::::: >Z:n;t::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::B:WZ H:::::>Z:ZH::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::;:::::::::N[:NZ:vyyuy:>Z:>Z::::::j;:<:::::::::::::::::::vYxI: ;Z::::::::5:" }}{PARA 0 "" 0 "" {TEXT -1 65 "According to the Mean Val ue Theorem for Integrals, there is a " }{XPPEDIT 18 0 "c;" "6#%\"cG " }{TEXT -1 11 " between " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x+Delta*x;" "6#,&%\"xG\"\"\"*&%&DeltaGF%F$F% F%" }{TEXT -1 30 " such that the shaded area is " }{XPPEDIT 18 0 "f(c) *Delta*x;" "6#*(-%\"fG6#%\"cG\"\"\"%&DeltaGF(%\"xGF(" }{TEXT -1 45 ". \+ Then\n\n " }{TEXT 360 1 " " } {TEXT 362 1 " " }{XPPEDIT 361 0 "(F(x+Delta*x)-F(x))/(Delta*x) = f(c); " "6#/*&,&-%\"FG6#,&%\"xG\"\"\"*&%&DeltaGF+F*F+F+F+-F'6#F*!\"\"F+*&F-F +F*F+F0-%\"fG6#%\"cG" }{TEXT -1 8 " \n\nand," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " " } {XPPEDIT 347 0 "D(F)(x) = limit(f(c),Delta*x = 0);" "6#/--%\"DG6#%\"FG 6#%\"xG-%&limitG6$-%\"fG6#%\"cG/*&%&DeltaG\"\"\"F*F5\"\"!" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 5 "But " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 21 " is \+ continuous and " }{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 22 " is squ eezed between " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x+Delta*x;" "6#,&%\"xG\"\"\"*&%&DeltaGF%F$F%F%" }{TEXT -1 38 " . Therefore, \n\n " }{XPPEDIT 348 0 "limi t(f(c),Delta*x = 0) = f(x);" "6#/-%&limitG6$-%\"fG6#%\"cG/*&%&DeltaG\" \"\"%\"xGF.\"\"!-F(6#F/" }{TEXT 349 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " " }{TEXT 350 1 " " }{XPPEDIT 351 0 "D(F)(x) = f(x);" "6#/--%\"DG6#%\"FG6#%\"xG-%\"fG6 #F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information" }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 62 "Title: BlankKrantz-5_4R8.mws A Maple Release 8 works heet." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 81 "Author: Brian E. Blank \nCreated: 30 January 2000\nLast Revised: 5 September 2006" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 51 "This document may not be distributed by any medium," }} {PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and electronic tra nsfer, without" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permissio n of the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 49 "For more information, please contact the author:" }} {PARA 256 "" 0 "" {TEXT -1 4 " " }}{PARA 256 "" 0 "" {TEXT -1 32 " \+ Department of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " W ashington University in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ St. Louis, MO 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 " " 0 "" {TEXT -1 33 " Telephone: (314) 935-6763" }}{PARA 256 "" 0 "" {TEXT -1 44 " e-mail: brian@math.wustl.edu" }} {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 61 "Copyr ight: \251 2000-2006 Brian E. Blank, All Rights Reserved." }}}}} {MARK "9" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }