{VERSION 6 0 "IBM INTEL NT" "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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Tim es" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 271 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 290 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 295 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 298 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Headi ng 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 64 "A procedure which performs differ entiation from first principles" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Pet er Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Versio n: 14.7.2005" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "load calculus procedures " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 298 10 "cal culus.m" }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a Maple session by a command si milar to the one that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/proc drs/calculus.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 33 "Derivatives from first principles" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 15 "The derivati ve " }{XPPEDIT 18 0 "Diff([f(x)],x)=`` " "6#/-%%DiffG6$7#-%\"fG6#%\"xG F+%!G" }{TEXT -1 4 "f '(" }{TEXT 286 1 "x" }{TEXT -1 16 ") of a functi on " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 37 " is obtaine d from first principles as" }}{PARA 256 "" 0 "" {TEXT -1 5 " f '(" } {TEXT 284 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "`` = limit((f(x+h)-f(x ))/h,h = 0);" "6#/%!G-%&limitG6$*&,&-%\"fG6#,&%\"xG\"\"\"%\"hGF/F/-F+6 #F.!\"\"F/F0F3/F0\"\"!" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 438 324 324 {PLOTDATA 2 "67-%'CURVESG6$7S7$$\"3++++++ ++]!#=$\"3+++++++D@!#<7$$\"3>LLLe=HGeF*$\"3At&H*H\\%)p@F-7$$\"3mmmTvT) *[lF*$\"3m*G\\'ofW9AF-7$$\"3ALL$31y%ftF*$\"3cN%Rme43F#F-7$$\"3)HLLeUW` <)F*$\"3w?]S#G\"=MBF-7$$\"3AmmT?JL()*)F*$\"3#=p'3$ygQS#F-7$$\"3RLL3#e] ,u*F*$\"3O\"H3om_VZ#F-7$$\"3++](e%\\'>0\"F-$\"3m&*[nB^J`DF-7$$\"3?L$37 $3eK6F-$\"3Iz-hp'p8k#F-7$$\"3!***\\P!=QH@\"F-$\"3,3dq9&4ct#F-7$$\"3ymm ;\"f&f&H\"F-$\"3R__#z'RGRGF-7$$\"3KLLe$G+%o8F-$\"3yta0o'fi$HF-7$$\"3-+ +]B6O]9F-$\"3/\"4!G%pt<0$F-7$$\"3\"*****\\-&eE`\"F-$\"3-rFkU5_uJF-7$$ \"3$*****\\ws'>h\"F-$\"3W8MD]#>#*H$F-7$$\"3?L$37C()Ro\"F-$\"3QEiF9l!zT $F-7$$\"3smm;XfipF-$\"3y#zNv9Ae&QF-7$$\"3fmmml>E,?F-$\"3^0?hFZ_-SF-7 $$\"3o**\\()o(=K3#F-$\"3)[`'\\>-!*pTF-7$$\"33+]iE5Eh@F-$\"3^uSg7Y_NVF- 7$$\"3tmmTN**oUAF-$\"391'4K2H[^%F-7$$\"3am;/$4nuJ#F-$\"3K'\\lO'oK&o%F- 7$$\"3-LLefV7)R#F-$\"3a&[8?A+b([F-7$$\"33L$37f/>[#F-$\"3u,0$**>D*z]F-7 $$\"3++]7Hb$[b#F-$\"3y3(R/H#fj_F-7$$\"3=LL3SHgLEF-$\"3[)y2IAKzY&F-7$$ \"3\"******HQx\\r#F-$\"3%Rw+^4^bo&F-7$$\"3c***\\(*R'e%z#F-$\"3ZKQ$Gdc[ !fF-7$$\"3++](o@7;(GF-$\"3:wG0i$yI7'F-7$$\"3#****\\UDOr&HF-$\"3Sg)*48u KsjF-7$$\"38mmm0M)R.$F-$\"3$4NIHlFDg'F-7$$\"3x****\\UT.;JF-$\"3AaGh)QM [&oF-7$$\"39L$3_I%Q!>$F-$\"35H)fu+w#*3(F-7$$\"33++]r'o;F$F-$\"3a(GP![z !>N(F-7$$\"3Qm;aD4:[LF-$\"3)*eF55t00wF-7$$\"33+]P=p4GMF-$\"3m&3')3CCf( yF-7$$\"3Umm;R(ei]$F-$\"3q48*Hu#QF-$\"35`%H*=.dC$*F-7$$\"37+++jY'3!RF-$\"3'40@&fDP 3'*F-7$$\"3emm\"p,T])RF-$\"315TD`fFS**F-7$$\"3SLLL^$H.1%F-$\"3N[\\1APJ C5!#;7$$\"3\"***\\()=CgSTF-$\"33v'f&>%Hs0\"Fiy7$$\"3<+]7()RV " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "diffbylimit: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 262 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 23 " diffbylimit( fx, x ) " }{TEXT 264 1 "\n" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 " " {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 9 " fx - " }{TEXT -1 61 " an algebraic expression involving a single variable, say x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " \+ " }{TEXT 23 6 "x - " }{TEXT 265 46 "the variable to differentiate wi th respect to." }{TEXT -1 0 "" }{TEXT 266 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure \+ " }{TEXT 0 11 "diffbylimit" }{TEXT -1 89 " attempts to obtain the deri vative of fx with respect to x by using the limit definition " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff([phi(x)],x) = l imit((phi(x+h)-phi(x))/h,h = 0);" "6#/-%%DiffG6$7#-%$phiG6#%\"xGF+-%&l imitG6$*&,&-F)6#,&F+\"\"\"%\"hGF4F4-F)6#F+!\"\"F4F5F8/F5\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 263 8 "Options:" }{TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 85 "info=true/false\nWith the option \"info=t rue\" some of the steps involved will be shown." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To ma ke the procedure active, open the subsection, place the cursor anywher e after the prompt [ > and press [Enter].\nYou can then close up the \+ subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "diffbylimit: imple mentation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 6310 "diffbylimit := proc(ff::algebraic,x::name)\n lo cal phi,h,nq,fx,fxh,gx,gxh,rxh,qxh,nfx,dfx,nfxh,\n ex,exh,dfxh,enq,s nq,n,m,d,ans,ans2,prntflg,Options;\n\n # Get the option \"info\".\n \+ # Set the default values to start with.\n prntflg := false;\n if nargs>2 then\n Options:=[args[3..nargs]];\n if not type(Opt ions,list(equation)) then\n error \"each optional argument mus t be an equation\"\n end if;\n if hasoption(Options,info,'pr ntflg','Options') then\n if prntflg<>true then prntflg := fals e end if;\n end if;\n if nops(Options)>0 then\n erro r \"%1 is not a valid option for %2\",op(1,Options),procname;\n e nd if;\n end if;\n\n if type(ff,`^`) and type(op(2,ff),fraction) a nd numer(op(2,ff))>0 then\n d := denom(op(2,ff));\n n := num er(op(2,ff));\n m := d-1;\n gx := ff;\n gxh := subs(x=x +h,gx);\n fx := op(1,gx)^n;\n fxh := op(1,gxh)^n;\n if \+ prntflg then\n print(Diff([gx],x)=Limit((gxh-gx)/h,h=0));print (``);\n rxh := add(gxh^(m-i)*gx^i,i=0..m);\n print(``= Limit(((gxh-gx)*rxh)/(h*``(rxh)),h=0));print(``);\n print(``=L imit((``(gxh)^d-``(gx)^d)/(h*rxh),h=0));print(``);\n if not ty pe(fxh,Or(function,`^`)) or op(1,fxh)<>x+h then\n print(``= Limit((``(fxh)-``(fx))/(h*rxh),h=0));print(``);\n else\n \+ print(``=Limit((fxh-fx)/``(h*rxh),h=0));print(``);\n end if;\n if type(fx,polynom(anything,x)) then\n if de gree(fx,x)<>1 then \n print(``=Limit((``(expand(fxh))-`` (expand(fx)))/(h*rxh),h=0));\n else\n print(` `=Limit(``(fxh-fx)/(h*rxh),h=0));\n end if;\n pr int(``); \n elif type(fx,ratpoly(anything,x)) then\n \+ nfx := numer(fx);\n if nfx<>1 then \n \+ dfx := denom(fx);\n nfxh := numer(fxh);\n \+ dfxh := denom(fxh);\n print(``=Limit((``((nfxh*dfx-dfxh* nfx)/(dfxh*dfx)))/(h*rxh),h=0));\n print(``);\n \+ end if;\n print(``=Limit((``(normal(fxh-fx)))/(h*rxh),h= 0));print(``);\n end if;\n nq := (fxh-fx)/(h*rxh);\n \+ enq := simplify(fxh-fx)/(h*rxh);\n if enq<>nq then prin t(``=Limit(enq,h=0));print(``) end if;\n snq := simplify((fxh- fx)/h)/rxh;\n if snq<>enq and snq<>nq then print(``=Limit(snq, h=0));print(``) end if;\n end if;\n ans := limit((fxh-fx)/h, h=0)/(d*gx^m);\n ans2 := simplify(ans);\n if prntflg then\n \+ if ans2<>ans then print(``=ans);print(``) end if;\n end i f;\n ans := ans2;\n elif type(ff,`^`) and type(op(2,ff),fractio n) and numer(op(2,ff))<0 then\n ex := ff;\n exh := subs(x=x+ h,ex);\n d := denom(op(2,ff));\n n := -numer(op(2,ff));\n \+ m := d-1;\n gx := 1/ff;\n gxh := subs(x=x+h,gx);\n f x := op(1,gx)^n;\n fxh := op(1,gxh)^n;\n if prntflg then\n \+ print(Diff([(ex^(-d/n))^``(-n/d)],x)=Diff([ex],x));print(``);\n print(``=Limit(``(1/h)*(exh-ex),h=0));print(``);\n pr int(``=Limit(((gx-gxh))/(h*``(gxh*gx)),h=0));print(``); \n \+ rxh := add(gxh^(m-i)*gx^i,i=0..m);\n qxh := gxh*gx*rxh;\n print(``=Limit(-((gxh-gx)*rxh)/(h*``(qxh)),h=0));print(``);\n print(``=Limit(-(``(gxh)^d-``(gx)^d)/(h*qxh),h=0));print(``); \n if not type(fxh,Or(function,`^`)) or op(1,fxh)<>x+h then\n \+ print(``=Limit(-(``(fxh)-``(fx))/(h*qxh),h=0));print(``);\n else\n print(``=Limit(-(fxh-fx)/``(h*qxh),h=0));pr int(``);\n end if;\n if type(fx,polynom(anything,x)) t hen\n if degree(fx,x)<>1 then \n print(``=Lim it(-(``(expand(fxh))-``(expand(fx)))/(h*qxh),h=0));\n else \n print(``=Limit(-``(fxh-fx)/(h*qxh),h=0));\n \+ end if;\n print(``); \n elif type(fx,ratpoly(an ything,x)) then\n nfx := numer(fx);\n if nfx<>1 \+ then \n dfx := denom(fx);\n nfxh := num er(fxh);\n dfxh := denom(fxh);\n print(``= Limit(-(``((nfxh*dfx-dfxh*nfx)/(dfxh*dfx)))/(h*qxh),h=0));\n \+ print(``);\n end if;\n print(``=Limit(-(``( normal(fxh-fx)))/(h*qxh),h=0));print(``);\n end if;\n \+ nq := -(fxh-fx)/(h*qxh);\n enq := -simplify(fxh-fx)/(h*qxh);\n if enq<>nq then print(``=Limit(enq,h=0));print(``) end if;\n \+ snq := -simplify((fxh-fx)/h)/qxh;\n if snq<>enq and sn q<>nq then print(``=Limit(snq,h=0));print(``) end if;\n if (op (1,gx)=x and n<>1) or (type(op(1,gx),ratpoly(anything,x)) and \n \+ not type(op(1,gx),polynom(anything,x))) then\n print( ``=-``(limit((fxh-fx)/h,h=0))/``(d*gx^(m+2)))\n end if;\n \+ end if;\n ans := -limit((fxh-fx)/h,h=0)/(d*gx^(m+2));\n ans 2 := simplify(ans);\n if prntflg then\n if ans2<>ans then print(``=ans);print(``) end if;\n end if;\n ans := ans2;\n \+ else\n fx := ff;\n fxh := subs(x=x+h,fx);\n nq := (fx h-fx)/h;\n if prntflg then\n if not type(fxh,Or(function, `^`)) or op(1,fxh)<>x+h then\n print(Diff([fx],x)=Limit((`` (fxh)-``(fx))/h,h=0));\n else\n print(Diff([fx],x)= Limit(nq,h=0));\n end if;\n if type(fx,polynom(anythin g,x)) then\n if degree(fx,x)<>1 then \n print (``=Limit((``(expand(fxh))-``(expand(fx)))/h,h=0));\n else \n print(``=Limit(``(fxh-fx)/h,h=0));\n end i f;\n elif type(fx,ratpoly(anything,x)) then\n nfx : = numer(fx);\n if nfx<>1 then \n dfx := de nom(fx);\n nfxh := numer(fxh);\n dfxh := d enom(fxh);\n print(``=Limit((``((nfxh*dfx-dfxh*nfx)/(dfx h*dfx)))/h,h=0));\n end if;\n print(``=Limit((`` (normal(fxh-fx)))/h,h=0));\n else\n error \"unable \+ to obtain the derivative from the limit definition\"\n end if; \n enq := simplify(fxh-fx)/h;\n if enq<>nq then print( ``=Limit(enq,h=0)); end if;\n snq := simplify(enq);\n \+ if snq<>enq and snq<>nq then print(``=Limit(snq,h=0)) end if;\n e nd if;\n ans := limit((fxh-fx)/h,h=0); \n end if;\n if pr ntflg then print(``=ans); print(``) end if;\n ans;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 0 11 "diffbylimit" }{TEXT 270 25 " .. examples of the form " } {XPPEDIT 18 0 "Diff([x^r],x);" "6#-%%DiffG6$7#)%\"xG%\"rGF(" }{TEXT 271 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }{TEXT 272 4 " .. " }{XPPEDIT 18 0 "Diff([x^2],x)" "6#-%%DiffG6$7#*$%\"xG\"\"#F(" }{TEXT 273 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x^2],x) = 2*x; " "6#/-%%DiffG6$7#*$%\"xG\"\"#F)*&F*\"\"\"F)F," }{TEXT -1 2 ". " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "diffbylimit(x^2,x,info=true) :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*$)%\"xG\"\"#\"\"\"F *-%&LimitG6$*&,&*$),&F*F,%\"hGF,F+F,F,F(!\"\"F,F5F6/F5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&,&-F$6#,(*$)%\"xG\"\"#\"\" \"F1*(F0F1F/F1%\"hGF1F1*$)F3F0F1F1F1-F$6#F-!\"\"F1F3F8/F3\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&,&*(\"\"#\"\"\"%\"xGF ,%\"hGF,F,*$)F.F+F,F,F,F.!\"\"/F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&*&\"\"#\"\"\"%\"xGF+F+%\"hGF+/F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#\"\"\"%\"xGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }{TEXT 274 4 " .. " }{XPPEDIT 18 0 "Diff([x^3], x);" "6#-%%DiffG6$7#*$%\"xG\"\"$F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x^3],x) = 3*x^2;" "6#/-%%DiffG6 $7#*$%\"xG\"\"$F)*&F*\"\"\"*$F)\"\"#F," }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "diffbylimit(x^3,x,info=true):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*$)%\"xG\"\"$\"\"\"F*-%&L imitG6$*&,&*$),&F*F,%\"hGF,F+F,F,F(!\"\"F,F5F6/F5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&,&-F$6#,**$)%\"xG\"\"$\"\"\"F1*(F 0F1)F/\"\"#F1%\"hGF1F1*(F0F1F/F1)F5F4F1F1*$)F5F0F1F1F1-F$6#F-!\"\"F1F5 F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }{TEXT 275 4 " .. " }{XPPEDIT 18 0 "Diff([x^4], x);" "6#-%%DiffG6$7#*$%\"xG\"\"%F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x^4],x) = 4*x^3;" "6#/-%%DiffG6 $7#*$%\"xG\"\"%F)*&F*\"\"\"*$F)\"\"$F," }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "diffbylimit(x^4,x,info=true):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*$)%\"xG\"\"%\"\"\"F*-%&L imitG6$*&,&*$),&F*F,%\"hGF,F+F,F,F(!\"\"F,F5F6/F5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&,&-F$6#,,*$)%\"xG\"\"%\"\"\"F1*(F 0F1)F/\"\"$F1%\"hGF1F1*(\"\"'F1)F/\"\"#F1)F5F9F1F1*(F0F1F/F1)F5F4F1F1* $)F5F0F1F1F1-F$6#F-!\"\"F1F5FA/F5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&,**(\"\"%\"\"\")%\"xG\"\"$F,%\"hGF,F,*(\"\"'F,) F.\"\"#F,)F0F4F,F,*(F+F,F.F,)F0F/F,F,*$)F0F+F,F,F,F0!\"\"/F0\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,**&\"\"%\"\"\")%\"xG\" \"$F+F+*(\"\"'F+)F-\"\"#F+%\"hGF+F+*(F*F+F-F+)F3F2F+F+*$)F3F.F+F+/F3\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"%\"\"\")%\"xG\"\"$F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }{TEXT 276 4 " .. " }{XPPEDIT 18 0 "Diff([1/x],x);" "6#-%%DiffG6$7#*&\"\"\"F(%\"xG!\"\"F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([1/x] ,x) = -1/(x^2);" "6#/-%%DiffG6$7#*&\"\"\"F)%\"xG!\"\"F*,$*&F)F)*$F*\" \"#F+F+" }{TEXT -1 4 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "diffbylimit(1/x,x,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%%DiffG6$7#*&\"\"\"F)%\"xG!\"\"F*-%&LimitG6$*&,&*&F)F),&F*F)%\"hGF)F+ F)F(F+F)F3F+/F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6 $*&-F$6#,$*(%\"hG\"\"\",&%\"xGF.F-F.!\"\"F0F1F1F.F-F1/F-\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*&\"\"\"F**&,&%\"xGF* %\"hGF*F*F-F*!\"\"F//F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$ *&\"\"\"F'*$)%\"xG\"\"#F'!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }{TEXT 277 4 " .. " }{XPPEDIT 18 0 "Diff([1/(x^2)],x);" "6#-%%DiffG6$7 #*&\"\"\"F(*$%\"xG\"\"#!\"\"F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([1/(x^2)],x) = -2/(x^3);" "6#/-%%D iffG6$7#*&\"\"\"F)*$%\"xG\"\"#!\"\"F+,$*&F,F)*$F+\"\"$F-F-" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "diffbylimit(1/x^2,x ,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*&\"\"\"F )*$)%\"xG\"\"#F)!\"\"F,-%&LimitG6$*&,&*&F)F)*$),&F,F)%\"hGF)F-F)F.F)F( F.F)F8F./F8\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&- F$6#,$**%\"hG\"\"\",&*&\"\"#F.%\"xGF.F.F-F.F.,&F2F.F-F.!\"#F2F4!\"\"F. F-F5/F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*(,&* &\"\"#\"\"\"%\"xGF-F-%\"hGF-F-,&F.F-F/F-!\"#F.F1!\"\"/F/\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#\"\"\"%\"xG!\"$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }{TEXT 278 4 " .. " }{XPPEDIT 18 0 "Diff([ sqrt(x)],x);" "6#-%%DiffG6$7#-%%sqrtG6#%\"xGF*" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([sqrt(x)],x) = 1 /(2*sqrt(x));" "6#/-%%DiffG6$7#-%%sqrtG6#%\"xGF+*&\"\"\"F-*&\"\"#F--F) 6#F+F-!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diffbylimit(sqrt(x),x,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*$%\"xG#\"\"\"\"\"#F)-%&LimitG6$*&,&*$,&F)F+%\"hGF+F*F +F(!\"\"F+F4F5/F4\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$**,&*$,&%\"xG\"\"\"%\"h GF-#F-\"\"#F-*$F,F/!\"\"F-,&F*F-F1F-F-F.F2-F$6#F3F2/F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G- %&LimitG6$*(,&*$)-F$6#*$,&%\"xG\"\"\"%\"hGF1#F1\"\"#F4F1F1*$)-F$6#*$F0 F3F4F1!\"\"F1F2F:,&F.F1F9F1F:/F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(,&-F$6#,&% \"xG\"\"\"%\"hGF.F.-F$6#F-!\"\"F.F/F2,&*$F,#F.\"\"#F.*$F-F5F.F2/F/\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(-F$6#%\"hG\"\"\"F+!\"\",&*$,&%\"xGF,F+F,#F,\"\" #F,*$F1F2F,F-/F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"\"F'*&\"\"#F'%\"xG#F'F)!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 7" }{TEXT 279 4 " .. " }{XPPEDIT 18 0 "Diff ([1/sqrt(x)],x);" "6#-%%DiffG6$7#*&\"\"\"F(-%%sqrtG6#%\"xG!\"\"F," } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff ([1/sqrt(x)],x) = -1/(2*x^(3/2));" "6#/-%%DiffG6$7#*&\"\"\"F)-%%sqrtG6 #%\"xG!\"\"F-,$*&F)F)*&\"\"#F))F-*&\"\"$F)F2F.F)F.F." }{XPPEDIT 18 0 " ``=-1/(2*x*sqrt(x))" "6#/%!G,$*&\"\"\"F'*(\"\"#F'%\"xGF'-%%sqrtG6#F*F' !\"\"F." }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " diffbylimit(1/sqrt(x),x,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%%DiffG6$7#*&\"\"\"F)*$%\"xG#F)\"\"#!\"\"F+-%&LimitG6$,$**,&*$,&F+F )%\"hGF)#F)F-F)*$F+F8F.F),&F5F)F9F)F)F7F.-%!G6#*(F6F8F+F8F:F)F.F./F7\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*,,&*$)-F$6#*$,&%\"xG\"\"\"%\"hGF2#F2 \"\"#F5F2F2*$)-F$6#*$F1F4F5F2!\"\"F2F3F;F0#F;F5F1F<,&F/F2F:F2F;F;/F3\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*,,&-F$6#,&%\"xG\"\"\"%\"hGF/F/-F$6#F. !\"\"F/F0F3F-#F3\"\"#F.F4,&*$F-#F/F5F/*$F.F8F/F3F3/F0\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G- %&LimitG6$,$*,-F$6#%\"hG\"\"\"F,!\"\",&%\"xGF-F,F-#F.\"\"#F0F1,&*$F/#F -F2F-*$F0F5F-F.F./F,\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"\"F'*&\"\"#F')%\"xG#\"\"$F )F'!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }{TEXT 280 4 " .. " } {XPPEDIT 18 0 "Diff(x^(1/3),x);" "6#-%%DiffG6$)%\"xG*&\"\"\"F)\"\"$!\" \"F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x^(1/3)],x) = 1/(3*x^(2/3));" "6#/-%%DiffG6$7#)%\"xG*&\"\"\"F +\"\"$!\"\"F)*&F+F+*&F,F+)F)*&\"\"#F+F,F-F+F-" }{TEXT -1 2 ". " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diffbylimit(x^(1/3),x,info=t rue):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*$)%\"xG#\"\"\" \"\"$F,F*-%&LimitG6$*&,&*$),&F*F,%\"hGF,F+F,F,F(!\"\"F,F6F7/F6\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G-%&LimitG6$**,&*$),&%\"xG\"\"\"%\"hGF.#F.\"\"$F.F.*$)F-F0F.!\"\" F.,(*$)F,#\"\"#F1F.F.*&F+F.F3F.F.*$)F-F8F.F.F.F/F4-F$6#F5F4/F/\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G-%&LimitG6$*(,&*$)-F$6#*$),&%\"xG\"\"\"%\"hGF2#F2\"\"$F2F5F2F2*$ )-F$6#*$)F1F4F2F5F2!\"\"F2F3F<,(*$)F0#\"\"#F5F2F2*&F/F2F;F2F2*$)F1F@F2 F2F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "E xample 9" }{TEXT 281 4 " .. " }{XPPEDIT 18 0 "Diff(x^(2/3),x);" "6#-%% DiffG6$)%\"xG*&\"\"#\"\"\"\"\"$!\"\"F'" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x^(2/3)],x) = 2/(3*x^(1/3) );" "6#/-%%DiffG6$7#)%\"xG*&\"\"#\"\"\"\"\"$!\"\"F)*&F+F,*&F-F,)F)*&F, F,F-F.F,F." }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diffbylimit(x^(2/3),x,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*$)%\"xG#\"\"#\"\"$\"\"\"F*-%&LimitG6$*&,&*$),&F*F.%\" hGF.F+F.F.F(!\"\"F.F7F8/F7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$**,&*$),&%\"xG\"\" \"%\"hGF.#\"\"#\"\"$F.F.*$)F-F0F.!\"\"F.,(*$)F,#\"\"%F2F.F.*&F+F.F4F.F .*$)F-F9F.F.F.F/F5-F$6#F6F5/F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(,&*$)-F$6#*$) ,&%\"xG\"\"\"%\"hGF2#\"\"#\"\"$F2F6F2F2*$)-F$6#*$)F1F4F2F6F2!\"\"F2F3F =,(*$)F0#\"\"%F6F2F2*&F/F2F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 11" }{TEXT 282 4 " .. " } {XPPEDIT 18 0 "Diff(x^(-1/3),x);" "6#-%%DiffG6$)%\"xG,$*&\"\"\"F*\"\"$ !\"\"F,F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Diff([x^(-1/3)],x) = -1/(3*x^(4/3));" "6#/-%%DiffG6$7#) %\"xG,$*&\"\"\"F,\"\"$!\"\"F.F),$*&F,F,*&F-F,)F)*&\"\"%F,F-F.F,F.F." } {XPPEDIT 18 0 "``=-1/3" "6#/%!G,$*&\"\"\"F'\"\"$!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-4/3)" "6#)%\"xG,$*&\"\"%\"\"\"\"\"$!\"\"F*" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "diffbylimi t(x^(-1/3),x,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6 $7#)%\"xG-%!G6##!\"\"\"\"$F)-F%6$7#*&\"\"\"F4*$)F)#F4F/F4F.F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G-%&LimitG6$*&-F$6#*&\"\"\"F,%\"hG!\"\"F,,&*&F,F,*$),&%\"xGF,F-F,# F,\"\"$F,F.F,*&F,F,*$)F4#F,F6F,F.F.F,/F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$* (,&*$)%\"xG#\"\"\"\"\"$F.F.*$),&F,F.%\"hGF.F-F.!\"\"F.F3F4-F$6#*&F+F.F 1F.F4/F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$**,&*$),&%\"xG\"\"\"%\"hGF/#F/\"\"$ F/F/*$)F.F1F/!\"\"F/,(*$)F-#\"\"#F2F/F/*&F4F/F,F/F/*$)F.F9F/F/F/F0F5-F $6#*(F,F/F4F/F6F/F5F5/F0\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*,,&*$)-F$6#*$),&% \"xG\"\"\"%\"hGF3#F3\"\"$F3F6F3F3*$)-F$6#*$)F2F5F3F6F3!\"\"F3F4F=F1#F= F6F2F>,(*$)F1#\"\"#F6F3F3*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 " Example 12" }{TEXT 283 4 " .. " }{XPPEDIT 18 0 "Diff(x^(-3/4),x);" "6# -%%DiffG6$)%\"xG,$*&\"\"$\"\"\"\"\"%!\"\"F-F'" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x^(-3/4)],x) = \+ -3/(4*x^(7/4));" "6#/-%%DiffG6$7#)%\"xG,$*&\"\"$\"\"\"\"\"%!\"\"F/F),$ *&F,F-*&F.F-)F)*&\"\"(F-F.F/F-F/F/" }{XPPEDIT 18 0 "``=-3/4" "6#/%!G,$ *&\"\"$\"\"\"\"\"%!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-7/4)" " 6#)%\"xG,$*&\"\"(\"\"\"\"\"%!\"\"F*" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "diffbylimit(x^(-3/4),x,info=true):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#)%\"xG-%!G6##!\"$\"\"%F)- F%6$7#*&\"\"\"F4*$)F)#\"\"$F/F4!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&-F$6#*& \"\"\"F,%\"hG!\"\"F,,&*&F,F,*$),&%\"xGF,F-F,#\"\"$\"\"%F,F.F,*&F,F,*$) F4#F6F7F,F.F.F,/F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(,&*$)%\"xG#\"\"$\"\"% \"\"\"F0*$),&F,F0%\"hGF0F-F0!\"\"F0F4F5-F$6#*&F+F0F2F0F5/F4\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G-%&LimitG6$,$**,&*$),&%\"xG\"\"\"%\"hGF/#\"\"$\"\"%F/F/*$)F.F1F/! \"\"F/,**$)F-#\"\"*F3F/F/*&)F-#F2\"\"#F/F5F/F/*&F,F/)F.F>F/F/*$)F.F:F/ F/F/F0F6-F$6#*(F,F/F5F/F7F/F6F6/F0\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*,,&*$)- F$6#*$),&%\"xG\"\"\"%\"hGF3#\"\"$\"\"%F3F7F3F3*$)-F$6#*$)F2F5F3F7F3!\" \"F3F4F>F1#!\"$F7F2F?,**$)F1#\"\"*F7F3F3*&)F1#F6\"\"#F3F=F3F3*&F0F3)F2 FHF3F3*$)F2FDF3F3F>F>/F4\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*&,&*$),&%\"xG\"\" \"%\"hGF/\"\"$F/F/*$)F.F1F/!\"\"F/-F$6#**F0F/)F-#F1\"\"%F/)F.F9F/,**$) F-#\"\"*F:F/F/*&)F-#F1\"\"#F/F;F/F/*&F8F/)F.FCF/F/*$)F.F?F/F/F/F4F4/F0 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*,,&-F$6#,**$)%\"xG\"\"$\"\"\"F2*(F1F2 )F0\"\"#F2%\"hGF2F2*(F1F2F0F2)F6F5F2F2*$)F6F1F2F2F2-F$6#F.!\"\"F2F6F=, &F0F2F6F2#!\"$\"\"%F0F?,**$)F>#\"\"*FAF2F2*&)F>#F1F5F2)F0#F1FAF2F2*&)F >FKF2)F0FIF2F2*$)F0FEF2F2F=F=/F6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*,,(*(\"\"$ \"\"\")%\"xG\"\"#F-%\"hGF-F-*(F,F-F/F-)F1F0F-F-*$)F1F,F-F-F-F1!\"\",&F /F-F1F-#!\"$\"\"%F/F8,**$)F7#\"\"*F:F-F-*&)F7#F,F0F-)F/#F,F:F-F-*&)F7F DF-)F/FBF-F-*$)F/F>F-F-F6F6/F1\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$**,(*&\"\"$\" \"\")%\"xG\"\"#F-F-*(F,F-F/F-%\"hGF-F-*$)F2F0F-F-F-,&F/F-F2F-#!\"$\"\" %F/F6,**$)F5#\"\"*F8F-F-*&)F5#F,F0F-)F/#F,F8F-F-*&)F5FBF-)F/F@F-F-*$)F /F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " } {TEXT 0 11 "diffbylimit" }{TEXT 268 25 " .. examples of the form " } {XPPEDIT 18 0 "Diff([q(x)],x);" "6#-%%DiffG6$7#-%\"qG6#%\"xGF*" } {TEXT 269 8 ", where " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT 287 43 " is a polynomial or a rational function in " }{TEXT 288 1 "x" }{TEXT 289 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Example 13 .. " }{XPPEDIT 18 0 "D iff([3*x^2-2*x+4],x);" "6#-%%DiffG6$7#,(*&\"\"$\"\"\"*$%\"xG\"\"#F*F** &F-F*F,F*!\"\"\"\"%F*F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([3*x^2-2*x+4],x)=6*x-2" "6#/-%%DiffG6$7#,(* &\"\"$\"\"\"*$%\"xG\"\"#F+F+*&F.F+F-F+!\"\"\"\"%F+F-,&*&\"\"'F+F-F+F+F .F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "diffbylimit(3*x^2-2*x+4,x,info=true):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#,(*&\"\"$\"\"\")%\"xG\"\" #F+F+*&F.F+F-F+!\"\"\"\"%F+F--%&LimitG6$*&,&-%!G6#,**&F*F+),&F-F+%\"hG F+F.F+F+*&F.F+F-F+F0*&F.F+F>F+F0F1F+F+-F86#F(F0F+F>F0/F>\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&,&-F$6#,.*&\"\"$\"\" \")%\"xG\"\"#F/F/*(\"\"'F/F1F/%\"hGF/F/*&F.F/)F5F2F/F/*&F2F/F1F/!\"\"* &F2F/F5F/F9\"\"%F/F/-F$6#,(*&F.F/F0F/F/*&F2F/F1F/F9F;F/F9F/F5F9/F5\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&,(*(\"\"'\"\"\"% \"xGF,%\"hGF,F,*&\"\"$F,)F.\"\"#F,F,*&F2F,F.F,!\"\"F,F.F4/F.\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,(*&\"\"'\"\"\"%\"xGF+F +*&\"\"$F+%\"hGF+F+\"\"#!\"\"/F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"'\"\"\"%\"xGF(F(\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 " Example 14 .. " }{XPPEDIT 18 0 "Diff([x-2*x^2],x);" "6#-%%DiffG6$7#,&% \"xG\"\"\"*&\"\"#F)*$F(F+F)!\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x-2*x^2],x) = 6*x-2;" "6#/-%%Di ffG6$7#,&%\"xG\"\"\"*&\"\"#F**$F)F,F*!\"\"F),&*&\"\"'F*F)F*F*F,F." } {TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diffbyli mit(x-2*x^2,x,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG 6$7#,&%\"xG\"\"\"*&\"\"#F*)F)F,F*!\"\"F)-%&LimitG6$*&,&-%!G6#,(F)F*%\" hGF**&F,F*),&F)F*F8F*F,F*F.F*-F56#F(F.F*F8F./F8\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&,&-F$6#,,%\"xG\"\"\"%\"hGF.*&\"\"#F .)F-F1F.!\"\"*(\"\"%F.F-F.F/F.F3*&F1F.)F/F1F.F3F.-F$6#,&F-F.*&F1F.F2F. F3F3F.F/F3/F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$* &,(%\"hG\"\"\"*(\"\"%F+%\"xGF+F*F+!\"\"*&\"\"#F+)F*F1F+F/F+F*F//F*\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,(\"\"\"F)*&\"\"%F )%\"xGF)!\"\"*&\"\"#F)%\"hGF)F-/F0\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&\"\"\"F&*&\"\"%F&%\"xGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 " Example 15 .. " }{XPPEDIT 18 0 "Diff([x/(x+1)],x);" "6#-%%DiffG6$7#*&% \"xG\"\"\",&F(F)F)F)!\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x/(x+1)],x) = 1/((x+1)^2);" "6#/- %%DiffG6$7#*&%\"xG\"\"\",&F)F*F*F*!\"\"F)*&F*F**$,&F)F*F*F*\"\"#F," } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diffbylimi t(x/(x+1),x,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$ 7#*&%\"xG\"\"\",&F)F*F*F*!\"\"F)-%&LimitG6$*&,&-%!G6#*&,&F)F*%\"hGF*F* ,(F)F*F7F*F*F*F,F*-F36#F(F,F*F7F,/F7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&-F$6#*(,&*&,&%\"xG\"\"\"%\"hGF0F0,&F/F 0F0F0F0F0*&,(F/F0F1F0F0F0F0F/F0!\"\"F0F4F5F2F5F0F1F5/F1\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&-F$6#*(%\"hG\"\"\",(%\"xGF- F,F-F-F-!\"\",&F/F-F-F-F0F-F,F0/F,\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&\"\"\"F)*&,(%\"xGF)%\"hGF)F)F)F),&F,F)F)F)F)!\" \"/F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&*$),&%\"xG F&F&F&\"\"#F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Example 16 .. " }{XPPEDIT 18 0 "Diff([t^3+4/t],t);" "6#-%%DiffG6$7#,&*$%\"tG\"\"$\"\"\"*&\"\"%F+F)!\" \"F+F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([t^3+4/t],t) = 3*t^2-4/(t^2);" "6#/-%%DiffG6$7#,&*$%\"tG\" \"$\"\"\"*&\"\"%F,F*!\"\"F,F*,&*&F+F,*$F*\"\"#F,F,*&F.F,*$F*F3F/F/" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diffbylimit(t^3+4/t,t,info=true):" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#,&*$)%\"tG\"\"$\"\"\"F-*&\"\"%F-F+! \"\"F-F+-%&LimitG6$*&,&-%!G6#,&*$),&F+F-%\"hGF-F,F-F-*&F/F-F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 0 11 "diffbylimit" }{TEXT 267 25 " .. examples of the form " }{XPPEDIT 18 0 "Diff([q(x)^r],x);" "6#- %%DiffG6$7#)-%\"qG6#%\"xG%\"rGF+" }{TEXT 290 8 ", where " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT 291 43 " is a polynomial or a rat ional function in " }{TEXT 292 1 "x" }{TEXT 293 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 17" }{TEXT 297 4 " .. " }{XPPEDIT 18 0 "Diff([sqrt(x^2+1)],x) ;" "6#-%%DiffG6$7#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"F.F.F," }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([sqrt(x^2+1) ],x) = x/sqrt(x^2+1);" "6#/-%%DiffG6$7#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"F /F/F-*&F-F/-F)6#,&*$F-F.F/F/F/!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "diffbylimit(sqrt(x^2+1),x,info=true):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*$,&*$)%\"xG\"\"#\"\"\"F. F.F.#F.F-F,-%&LimitG6$*&,&*$,&*$),&F,F.%\"hGF.F-F.F.F.F.F/F.F(!\"\"F.F :F;/F:\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G-%&LimitG6$**,&*$,&*$),&%\"xG\"\"\"%\"hGF0\"\"#F0 F0F0F0#F0F2F0*$,&*$)F/F2F0F0F0F0F3!\"\"F0,&F*F0F4F0F0F1F8-F$6#F9F8/F1 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(,&*$)-F$6#*$,&*$),&%\"xG\"\"\"%\"hGF4 \"\"#F4F4F4F4#F4F6F6F4F4*$)-F$6#*$,&*$)F3F6F4F4F4F4F7F6F4!\"\"F4F5F@,& F.F4F " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 18" }{TEXT 294 4 " .. " }{XPPEDIT 18 0 "Diff((x^3-1)^(1/3),x);" "6#-%%DiffG6$),&*$%\"xG\"\"$\" \"\"F+!\"\"*&F+F+F*F,F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff((x^3-1)^(1/3),x) = x^2*(x^3-1)^(-2/3);" "6# /-%%DiffG6$),&*$%\"xG\"\"$\"\"\"F,!\"\"*&F,F,F+F-F**&F*\"\"#),&*$F*F+F ,F,F-,$*&F0F,F+F-F-F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "diffbylimit((x^3-1)^(1/3 ),x,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*$),&* $)%\"xG\"\"$\"\"\"F/F/!\"\"#F/F.F/F--%&LimitG6$*&,&*$),&*$),&F-F/%\"hG F/F.F/F/F/F0F1F/F/F(F0F/F=F0/F=\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$**,&*$),&*$),& %\"xG\"\"\"%\"hGF1\"\"$F1F1F1!\"\"#F1F3F1F1*$),&*$)F0F3F1F1F1F4F5F1F4F 1,(*$)F,#\"\"#F3F1F1*&F+F1F7F1F1*$)F8F>F1F1F1F2F4-F$6#F;F4/F2\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G-%&LimitG6$*(,&*$)-F$6#*$),&*$),&%\"xG\"\"\"%\"hGF5\"\"$F5F5F5!\" \"#F5F7F5F7F5F5*$)-F$6#*$),&*$)F4F7F5F5F5F8F9F5F7F5F8F5F6F8,(*$)F0#\" \"#F7F5F5*&F/F5F?F5F5*$)F@FFF5F5F8/F6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$* (,&-F$6#,&*$),&%\"xG\"\"\"%\"hGF1\"\"$F1F1F1!\"\"F1-F$6#,&*$)F0F3F1F1F 1F4F4F1F2F4,(*$)F,#\"\"#F3F1F1*&)F,#F1F3F1)F7FAF1F1*$)F7F=F1F1F4/F2\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(,&-F$6#,,*$)%\"xG\"\"$\"\"\"F1*(F0F1)F /\"\"#F1%\"hGF1F1*(F0F1F/F1)F5F4F1F1*$)F5F0F1F1F1!\"\"F1-F$6#,&F-F1F1F :F:F1F5F:,(*$),&*$),&F/F1F5F1F0F1F1F1F:#F4F0F1F1*&)FA#F1F0F1)F=FHF1F1* $)F=FEF1F1F:/F5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(,(*(\"\"$\"\"\")%\"xG\"\"#F ,%\"hGF,F,*(F+F,F.F,)F0F/F,F,*$)F0F+F,F,F,F0!\"\",(*$),&*$),&F.F,F0F,F +F,F,F,F5#F/F+F,F,*&)F9#F,F+F,),&*$)F.F+F,F,F,F5F@F,F,*$)FBF=F,F,F5/F0 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&,(*&\"\"$\"\"\")%\"xG\"\"#F,F,*(F+F,F. F,%\"hGF,F,*$)F1F/F,F,F,,(*$),&*$),&F.F,F1F,F+F,F,F,!\"\"#F/F+F,F,*&)F 7#F,F+F,),&*$)F.F+F,F,F,F;F?F,F,*$)FAF " 0 " " {MPLTEXT 1 0 44 "Diff((x^3-1)^(1/3),x)=diff((x^3-1)^(1/3),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*$),&*$)%\"xG\"\"$\"\"\"F.F .!\"\"#F.F-F.F,*&F,\"\"#F)#!\"#F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 19" }{TEXT 295 4 " .. " }{XPPEDIT 18 0 "Diff((1 +1/x)^(-3/4),x);" "6#-%%DiffG6$),&\"\"\"F(*&F(F(%\"xG!\"\"F(,$*&\"\"$F (\"\"%F+F+F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Diff((1+1/x)^(-3/4),x) = 3/(4*x^2*(1+1/x)^(7/4));" "6#/ -%%DiffG6$),&\"\"\"F)*&F)F)%\"xG!\"\"F),$*&\"\"$F)\"\"%F,F,F+*&F/F)*(F 0F)*$F+\"\"#F)),&F)F)*&F)F)F+F,F)*&\"\"(F)F0F,F)F," }{XPPEDIT 18 0 "`` =3/(4*x^2)" "6#/%!G*&\"\"$\"\"\"*&\"\"%F'*$%\"xG\"\"#F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1+1/x)^(-7/4)" "6#),&\"\"\"F%*&F%F%%\"xG!\" \"F%,$*&\"\"(F%\"\"%F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "diffbylimit((1+1/x)^( -3/4),x,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#), &\"\"\"F**&F*F*%\"xG!\"\"F*-%!G6##!\"$\"\"%F,-F%6$7#*&F*F**$)F)#\"\"$F 3F*F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&-F$6#*&\"\"\"F,%\"hG!\"\"F,,&*&F,F,*$) ,&F,F,*&F,F,,&%\"xGF,F-F,F.F,#\"\"$\"\"%F,F.F,*&F,F,*$),&F,F,*&F,F,F6F .F,#F8F9F,F.F.F,/F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*(,&*$),&\"\"\"F-*&F-F- %\"xG!\"\"F-#\"\"$\"\"%F-F-*$),&F-F-*&F-F-,&F/F-%\"hGF-F0F-F1F-F0F-F9F 0-F$6#*&F+F-F5F-F0/F9\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$**,&*$),&\"\"\"F.*&F. F.,&%\"xGF.%\"hGF.!\"\"F.#\"\"$\"\"%F.F.*$),&F.F.*&F.F.F1F3F.F4F.F3F., **$)F-#\"\"*F6F.F.*&)F-#F5\"\"#F.F8F.F.*&F,F.)F9FBF.F.*$)F9F>F.F.F.F2F 3-F$6#*(F,F.F8F.F;F.F3F3/F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*,,&*$)-F$6#*$), &\"\"\"F2*&F2F2,&%\"xGF2%\"hGF2!\"\"F2#\"\"$\"\"%F2F:F2F2*$)-F$6#*$),& F2F2*&F2F2F5F7F2F8F2F:F2F7F2F6F7F1#!\"$F:FAFC,**$)F1#\"\"*F:F2F2*&)F1# F9\"\"#F2F@F2F2*&F0F2)FAFLF2F2*$)FAFHF2F2F7F7/F6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&Limi tG6$,$*,,&-F$6#*$),&\"\"\"F0*&F0F0,&%\"xGF0%\"hGF0!\"\"F0\"\"$F0F0-F$6 #*$),&F0F0*&F0F0F3F5F0F6F0F5F0F4F5F/#!\"$\"\"%F;F=,**$)F/#\"\"*F?F0F0* &)F/#F6\"\"#F0)F;#F6F?F0F0*&)F/FJF0)F;FGF0F0*$)F;FCF0F0F5F5/F4\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G-%&LimitG6$,$*,-F$6#*(,&*&),(%\"xG\"\"\"%\"hGF2F2F2\"\"$F2)F1F4F 2F2*&),&F1F2F3F2F4F2),&F1F2F2F2F4F2!\"\"F2F8!\"$F1FF/F/*(\"\"*F/F 8F/F.F/F/*(F5F/F8F/F2F/F/*(F5F/F6F/F2F/F/F/,&F6F/F.F/!\"$F6FE!\"\"F/F. FF,&F/F/*&F/F/FDFFF/#FEF;,&F/F/*&F/F/F6FFF/FI,**$)FG#FAF;F/F/*&)FG#F5F 3F/)FJ#F5F;F/F/*&)FGFTF/)FJFRF/F/*$)FJFOF/F/FFFF/F.\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%!G-%&Li mitG6$*.,4*$)%\"hG\"\"#\"\"\"F.*(\"\"$F.%\"xGF.F,F.F.*&F0F.)F1F-F.F.*& F0F.)F1\"\"%F.F.*(\"\"'F.)F1F0F.F,F.F.*&F8F.F9F.F.*(\"\"*F.F3F.F,F.F.* (F0F.F3F.F+F.F.*(F0F.F1F.F+F.F.F.,&F1F.F,F.!\"$F1F@,&F.F.*&F.F.F?!\"\" F.#F@F6,&F.F.*&F.F.F1FCF.FD,**$)FA#F " 0 "" {MPLTEXT 1 0 67 "Diff((1+1/x)^(-3/4),x)=diff((1+1/x)^(-3/4),x);\n``=simplify(rhs( %));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&\"\"\"F(*$),&F(F( *&F(F(%\"xG!\"\"F(#\"\"$\"\"%F(F.F-,$**F0F(F1F.F+#!\"(F1F-!\"#F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*,\"\"$\"\"\"\"\"%!\"\"*&,&%\"xG F(F(F(F(F-F*#!\"$F)F,F*F-F*F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 20" }{TEXT 296 4 " .. " }{XPPEDIT 18 0 "Diff((x /(x+1))^(3/4),x);" "6#-%%DiffG6$)*&%\"xG\"\"\",&F(F)F)F)!\"\"*&\"\"$F) \"\"%F+F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Diff((x/(x+1))^(3/4),x) = 3/(4*(x+1)^2);" "6#/-%%DiffG6 $)*&%\"xG\"\"\",&F)F*F*F*!\"\"*&\"\"$F*\"\"%F,F)*&F.F**&F/F**$,&F)F*F* F*\"\"#F*F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "(x/(x+1))^(-1/4)" "6#)*&% \"xG\"\"\",&F%F&F&F&!\"\",$*&F&F&\"\"%F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "(x/(x+1) )^(3/4)=(1+1/x)^(-3/4)" "6#/)*&%\"xG\"\"\",&F&F'F'F'!\"\"*&\"\"$F'\"\" %F)),&F'F'*&F'F'F&F)F',$*&F+F'F,F)F)" }{TEXT -1 60 ", so the derivativ e is the same as in the previous example. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "diffbylimit((x/(x+ 1))^(3/4),x,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$ 7#*$)*&%\"xG\"\"\",&F+F,F,F,!\"\"#\"\"$\"\"%F,F+-%&LimitG6$*&,&*$)*&,& F+F,%\"hGF,F,,(F+F,F;F,F,F,F.F/F,F,F(F.F,F;F./F;\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&Limi tG6$**,&*$)*&,&%\"xG\"\"\"%\"hGF/F/,(F.F/F0F/F/F/!\"\"#\"\"$\"\"%F/F/* $)*&F.F/,&F.F/F/F/F2F3F/F2F/,**$)F,#\"\"*F5F/F/*&)F,#F4\"\"#F/F7F/F/*& F+F/)F8FAF/F/*$)F8F=F/F/F/F0F2-F$6#F:F2/F0\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$* (,&*$)-F$6#*$)*&,&%\"xG\"\"\"%\"hGF3F3,(F2F3F4F3F3F3!\"\"#\"\"$\"\"%F3 F9F3F3*$)-F$6#*$)*&F2F3,&F2F3F3F3F6F7F3F9F3F6F3F4F6,**$)F0#\"\"*F9F3F3 *&)F0#F8\"\"#F3F?F3F3*&F/F3)F@FIF3F3*$)F@FEF3F3F6/F4\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&L imitG6$*(,&-F$6#*&,&%\"xG\"\"\"%\"hGF/\"\"$,(F.F/F0F/F/F/!\"$F/-F$6#*& F.F1,&F.F/F/F/F3!\"\"F/F0F8,**$)*&F-F/F2F8#\"\"*\"\"%F/F/*&)F<#F1\"\"# F/)*&F.F/F7F8#F1F?F/F/*&)F " 0 "" {MPLTEXT 1 0 69 "Diff((x/(x+1))^(3/4),x)=diff((x/(x+1))^(3/4),x);\n``=simplify(rhs( %));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*$)*&%\"xG\"\"\",&F *F+F+F+!\"\"#\"\"$\"\"%F+F*,$**F/F+F0F-F)#F-F0,&*&F+F+F,F-F+*&F*F+F,! \"#F-F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"$\"\"\"\"\"%! \"\",&%\"xGF(F(F(!\"#*&F,F(F+F*#F*F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pictures" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 871 "fn := x -> x^2/2+2: x := 'x':\na := 1.5: b := 3.5:\nfa := fn(a): \+ fb := fn(b): \np1 := plot(fn(x),x=0.5..4.3,y=-0.5..10,color=red):\np2 \+ := plot([[[a,0],[a,fa]],[[b,0],[b,fb]],\n [[b,fb],[b,fb]],[[a,fa] ,[b,fa]],\n [[b,fa],[b,fb]]],linestyle=[2$2,1$3],\n color=[COLOR (RGB,.1,.1,.1)$2,COLOR(RGB,0,.7,0),tan,COLOR(RGB,0,.7,0)]):\np3 := plo t([[1,fa-1.25],[4,fb+1.25]],color=COLOR(RGB,.4,0,.9)):\np4 := plot([[ 0,0],[4,0]],color=black):\np5 := plot([[[a,fa],[b,fb]]$3],style=point, symbol=[circle,diamond,cross],color=black):\nt1 := plots[textplot]([3. 6,9.7,`y = f(x)`],color=red):\nt2 := plots[textplot]([3.85,6,`f(x+h)-f (x)`],color=COLOR(RGB,0,.7,0)):\nt3 := plots[textplot]([2.5,2.8,`h`],c olor=brown):\nt4 := plots[textplot]([[1.5,-.25,`x`],[3.5,-.25,`x+h`], \n[1.25,3.6,`P(x,f(x))`],[3.05,8.4,`Q(x+h,f(x+h))`]],color=black):\npl ots[display]([p1,p2,p3,p4,p5,t1,t2,t3,t4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}} {MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }