{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 "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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 272 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 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 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time s" 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 Plot " -1 13 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" -1 256 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 42 "The derivative of the Dirac delta function" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C ., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }} {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 42 "The derivative of the Dirac delta funct ion" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 38 "As a first approach to the derivative " }{XPPEDIT 18 0 "d elta*`'`(x)" "6#*&%&deltaG\"\"\"-%\"'G6#%\"xGF%" }{TEXT -1 29 " of the Dirac delta function " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG " }{TEXT -1 52 ", consider any representative sequence of functions " }{XPPEDIT 18 0 "g[n](x) = n*g(n*x);" "6#/-&%\"gG6#%\"nG6#%\"xG*&F(\"\" \"-F&6#*&F(F,F*F,F," }{TEXT -1 5 " for " }{XPPEDIT 18 0 "delta(x)" "6# -%&deltaG6#%\"xG" }{TEXT -1 22 ", based on a function " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 11 ", such that" }}{PARA 0 "" 0 "" {TEXT -1 4 "(i) " }{XPPEDIT 18 0 "g(x)>=0" "6#1\"\"!-%\"gG6#%\"xG" }{TEXT -1 9 " for all " }{TEXT 271 1 "x" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 5 "(ii) " }{XPPEDIT 18 0 "Int(g(x),x=-infinity..infini ty)= 1" "6#/-%$IntG6$-%\"gG6#%\"xG/F*;,$%)infinityG!\"\"F.\"\"\"" } {TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "(iii) " }{XPPEDIT 18 0 "g(-x)=g(x)" "6#/-%\"gG6#,$%\"xG!\"\"-F%6#F(" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 10 " is even, " } }{PARA 0 "" 0 "" {TEXT -1 5 "(iv) " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#% \"xG" }{TEXT -1 4 " is " }{TEXT 261 25 "differentiable everywhere" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Then the sequence of functions " }{XPPEDIT 18 0 "g[n]*`'` (x)" "6#*&&%\"gG6#%\"nG\"\"\"-%\"'G6#%\"xGF(" }{TEXT -1 12 " represent s " }{XPPEDIT 18 0 "delta*`'`(x)" "6#*&%&deltaG\"\"\"-%\"'G6#%\"xGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG " }{TEXT -1 64 " is a differentiable bounded function, such that the d erivative " }{XPPEDIT 18 0 "phi*`'`(x)" "6#*&%$phiG\"\"\"-%\"'G6#%\"xG F%" }{TEXT -1 17 " is also bounded." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 39 "The integration by parts formula, gives " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(g[n]*`'`(x)* phi(x),x = -infinity .. infinity) = Limit(``,R = infinity);" "6#/-%$In tG6$*(&%\"gG6#%\"nG\"\"\"-%\"'G6#%\"xGF,-%$phiG6#F0F,/F0;,$%)infinityG !\"\"F7-%&LimitG6$%!G/%\"RGF7" }{XPPEDIT 18 0 "phi(x)*g[n](x);" "6#*&- %$phiG6#%\"xG\"\"\"-&%\"gG6#%\"nG6#F'F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``],[-R, ``])" "6#-%*PIECEWISEG6$7$%\"RG%!G7$,$F'!\" \"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-Int(g[n](x)*phi*`'`(x),x = -inf inity .. infinity);" "6#,$-%$IntG6$*(-&%\"gG6#%\"nG6#%\"xG\"\"\"%$phiG F/-%\"'G6#F.F//F.;,$%)infinityG!\"\"F7F8" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0-Int(g[n](x)*phi*`'`( x),x = -infinity .. infinity);" "6#/%!G,&\"\"!\"\"\"-%$IntG6$*(-&%\"gG 6#%\"nG6#%\"xGF'%$phiGF'-%\"'G6#F2F'/F2;,$%)infinityG!\"\"F:F;" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 29 "since the condition (ii ) for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 14 " implies that " }{XPPEDIT 18 0 "g[n](x)->0" "6#f*6#-&%\"gG6#%\"nG6#%\"xG7\"6$% )operatorG%&arrowG6\"\"\"!F0F0F0" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "a bs(x)->infinity" "6#f*6#-%$absG6#%\"xG7\"6$%)operatorG%&arrowG6\"%)inf inityGF-F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(``,n=infinity)" "6#-%&LimitG6$%!G/%\"nG%)infinity G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(g[n]*`'`(x)*phi(x),x = -infinit y .. infinity);" "6#-%$IntG6$*(&%\"gG6#%\"nG\"\"\"-%\"'G6#%\"xGF+-%$ph iG6#F/F+/F/;,$%)infinityG!\"\"F6" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Li mit(``,n = infinity)" "6#-%&LimitG6$%!G/%\"nG%)infinityG" }{XPPEDIT 18 0 "``(-Int(g[n](x)*phi*`'`(x),x = -infinity .. infinity));" "6#-%!G 6#,$-%$IntG6$*(-&%\"gG6#%\"nG6#%\"xG\"\"\"%$phiGF2-%\"'G6#F1F2/F1;,$%) infinityG!\"\"F:F;" }{XPPEDIT 18 0 "`` = -phi*`'`(0);" "6#/%!G,$*&%$ph iG\"\"\"-%\"'G6#\"\"!F(!\"\"" }{TEXT -1 13 " ------- (i)," }}{PARA 0 " " 0 "" {TEXT -1 28 "by the sampling property of " }{XPPEDIT 18 0 "delt a(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Thus we may write" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(delta*`'`(x)*phi(x),x = -infi nity .. infinity) = -phi*`'`(0);" "6#/-%$IntG6$*(%&deltaG\"\"\"-%\"'G6 #%\"xGF)-%$phiG6#F-F)/F-;,$%)infinityG!\"\"F4,$*&F/F)-F+6#\"\"!F)F5" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT 265 16 " _______________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "The relation (i), now be comes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(delta*`'`(x)*phi(x),x = -infinity .. infinity) = -Int(delta(x)*phi*`'`(x),x = -infinity .. infinity);" "6#/-%$IntG6$ *(%&deltaG\"\"\"-%\"'G6#%\"xGF)-%$phiG6#F-F)/F-;,$%)infinityG!\"\"F4,$ -F%6$*(-F(6#F-F)F/F)-F+6#F-F)/F-;,$F4F5F4F5" }{XPPEDIT 18 0 "`` = -phi *`'`(0);" "6#/%!G,$*&%$phiG\"\"\"-%\"'G6#\"\"!F(!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The 1 st derivative of the Dirac delta function is denoted in Maple by " } {TEXT 272 10 "Dirac(1,x)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "With the alias \"" }{TEXT 272 11 "delta=Dirac" }{TEXT -1 11 "\", w e have " }{XPPEDIT 18 0 "delta*`'`(x) = delta(1,x);" "6#/*&%&deltaG\" \"\"-%\"'G6#%\"xGF&-F%6$F&F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "alias(delta=Dirac ): phi := 'phi':\nDiff(delta(x),x);\nvalue(%);\nInt(%*phi(x),x=-infini ty..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6 $-%&deltaG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&deltaG6$\"\" \"%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%&deltaG6$\"\" \"%\"xGF*-%$phiG6#F+F*/F+;,$%)infinityG!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$--%\"DG6#%$phiG6#\"\"!!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The derivative of the del ta function may be thought of as a distribution, or linear mapping " } {XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 47 " ' from functions to real numbers, defined by; " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "delta*`'`(phi)=-phi*`'`(0)" "6#/*&%&deltaG\"\"\"-%\"'G6 #%$phiGF&,$*&F*F&-F(6#\"\"!F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "1st description of " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 3 " '(" }{TEXT 262 1 "x" }{TEXT -1 28 ") as a sequence of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {XPPEDIT 18 0 "delta(x)" "6#-%&d eltaG6#%\"xG" }{TEXT -1 49 " can be represented by the sequence of fun ctions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h[n](x) = \+ n/(Pi*(1+n^2*x^2));" "6#/-&%\"hG6#%\"nG6#%\"xG*&F(\"\"\"*&%#PiGF,,&F,F ,*&F(\"\"#F*F1F,F,!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 42 "Differentiating with respect to x, we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h[n]*`'`(x) = -n^3*x/sqrt(2*Pi);" "6#/* &&%\"hG6#%\"nG\"\"\"-%\"'G6#%\"xGF),$*(F(\"\"$F-F)-%%sqrtG6#*&\"\"#F)% #PiGF)!\"\"F7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "h := (n,x) -> n/(Pi*(1+n^2*x ^2));\nhp := unapply(diff(h(n,x),x),n,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)*&9$\"\" \"*&%#PiGF/,&F/F/*&)F.\"\"#F/)9%F5F/F/F/!\"\"F)F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#hpGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF),$*&**) 9$\"\"$\"\"\"9%F3-%$expG6#,$*&)F1\"\"#F3)F4F;F3#!\"\"F;F3-%%sqrtG6#F;F 3F3*$-F@6#%#PiGF3F>F=F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "plot([hp(1,x),hp(2,x),hp(3, x),hp(4,x)],x=-4..4,\n color=[red,green,blue,magen ta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 442 257 257 {PLOTDATA 2 "6(-%'CURV ESG6$7go7$$!\"%\"\"!$\"1ZrMIjM6))!#=7$$!1nmmmFiDQ!#:$\"19dHk,Ui**F-7$$ !1LLLo!)*Qn$F1$\"13%)fyd!G6\"!#<7$$!1nmmwxE.NF1$\"1EC#4F97$$!1+++N&oz$GF1$\"1M-XNM%R?#F97$$!1nmm\")3DoEF1$\"17qS TtcwDF97$$!1+++:v2*\\#F1$\"1C)3X-b1.$F97$$!1LLL8>1DBF1$\"1)*e$)>N02OF9 7$$!1nmmw))yr@F1$\"1%**)pcspIUF97$$!1+++S(R#**>F1$\"19mwo%>s4&F97$$!1+ +++@)f#=F1$\"13Tmz)*3)='F97$$!1+++gi,f;F1$\"14yVL^9,vF97$$!1nmm\"G&R2: F1$\"1*f/hXXA'*)F97$$!1LLLtK5F8F1$\"1t2PMM736!#;7$$!1MLL$HsV<\"F1$\"1! Q**H`:3K\"Fjp7$$!1-++]&)4n**Fjp$\"1PJS-ey'f\"Fjp7$$!1PLLL\\[%R)Fjp$\"1 Fjp7$$!1)*****\\&y!pmFjp$\"1V# o&o'**R.#Fjp7$$!1****\\(=/PY'Fjp$\"12J,S.3Z?Fjp7$$!1*****\\#)H$eiFjp$ \"1&ohwZfr0#Fjp7$$!1)***\\ia&H0'Fjp$\"1/.P$onR1#Fjp7$$!1*******4\"eZeF jp$\"1D#[lEIs1#Fjp7$$!1++]Pn?UcFjp$\"1BOm*Gpm1#Fjp7$$!1*****\\PKoV&Fjp $\"1[ayNf+i?Fjp7$$!1****\\7!e9B&Fjp$\"1x;v.K'H0#Fjp7$$!1******\\O3E]Fj p$\"1\"4<&H*o#R?Fjp7$$!1mmmTs$*oTFjp$\"1.slX#pi#>Fjp7$$!1KLLL3z6LFjp$ \"1;*p@tt@r\"Fjp7$$!1LLLeGmCDFjp$\"1O!*y#=3/U\"Fjp7$$!1MLL$)[`P&\\2zF97$$\"1FLL$=2Vs\"Fjp$!1\\SI?>DN5Fjp7$$\"1hmmm7+# \\#Fjp$!1#)GPl:O19Fjp7$$\"1'*****\\`pfKFjp$!1\\/FHso&p\"Fjp7$$\"1imm\" *f#))3%Fjp$!1A8D:-t5>Fjp7$$\"1HLLLm&z\"\\Fjp$!1A8^L,5I?Fjp7$$\"1jm;/** 4K^Fjp$!1z\\d:a$p/#Fjp7$$\"1'****\\Fjp7$$\"1#******4#32$)Fjp$!1Qj[&[n9&=Fjp7$$\"1%***** \\#y'G**Fjp$!1O#*Rd+!Hg\"Fjp7$$\"1******H%=H<\"F1$!1rCG#)H(HK\"Fjp7$$ \"1mmm1>qM8F1$!1?BnK(=$)4\"Fjp7$$\"1++++.W2:F1$!1'G6Dop<'*)F97$$\"1LLL ep'Rm\"F1$!1CL6D5*zX(F97$$\"1+++S>4N=F1$!1URr^eMChF97$$\"1mmm6s5'*>F1$ !1G<^MA#[6&F97$$\"1+++lXTk@F1$!1Nvrd#**QE%F97$$\"1mmmmd'*GBF1$!1a(zH_, Ff$F97$$\"1+++DcB,DF1$!1Q^[moDCIF97$$\"1MLLt>:nEF1$!17;7g(H#zDF97$$\"1 LLL.a#o$GF1$!1&)R^lf@1AF97$$\"1nmm^Q40IF1$!1s%*Q\\;X,>F97$$\"1+++!3:(f JF1$!1I[\"*z\\Ln;F97$$\"1nmmc%GpL$F1$!1\\kG>LeU9F97$$\"1LLL8-V&\\$F1$! 10sF38kt7F97$$\"1+++XhUkOF1$!1H:CWpl?6F97$$\"1+++:o\"e\\&F-7$F5$\"1U8%G9$o(='F-7$F;$\"1ST hnyt5rF-7$F@$\"1S\"o\\UDMB)F-7$FE$\"1)y&z!4@ff*F-7$FJ$\"1R4$\\2LQ6\"F9 7$FO$\"1TSCOZ,58F97$FT$\"1t%\\!*pVQc\"F97$FY$\"1g\\t%))pa)=F97$Fhn$\"1 &*\\XyBPnWeeF97$Fap$\"1OOCL#3Ba(F97$Ffp$\"1jhb)eUV/\"Fjp7$F\\q $\"1T-%=oC%39Fjp7$Faq$\"1KTAcF*>0#Fjp7$Ffq$\"1&*GNP!*zJHFjp7$F[r$\"1#o [L-$H*e$Fjp7$F`r$\"1cCaKI$yR%Fjp7$Fds$\"19C77:77`Fjp7$Fht$\"1`b&p)y)HL 'Fjp7$F]u$\"1**H@^4W)Q(Fjp7$Fbu$\"1C#Rl\"4d[\")Fjp7$$!1L$ek$)*R8KFjp$ \"1%yf)p<['>)Fjp7$$!1LLeR)3]6$Fjp$\"159w;U>L#)Fjp7$$!1K$3F%yh;IFjp$\"1 6YR!)ovd#)Fjp7$$!1LL$e%oA=HFjp$\"1e')p89?p#)Fjp7$$!1L$e*[e$)>GFjp$\"1< anE#\\lE)Fjp7$$!1LL3_[W@FFjp$\"1o&3L@>)[#)Fjp7$$!1L$3_&Q0BEFjp$\"1l)oL \\O]@)Fjp7$Fgu$\"1pm9k@Ck\")Fjp7$$!1LL$3())4J@Fjp$\"1n54Qh&Hx(Fjp7$F\\ v$\"1H]<1S%\\/(Fjp7$Fav$\"15iMD?>_eFjp7$Ffv$\"1g)[oGZBD%Fjp7$$!1tm;/1b inF9$\"1$*3pzw]@LFjp7$F[w$\"1&4Vr=YHK#Fjp7$$!12+]78VL$3Fr)4=F9$!1!Q&H)H\"\\$>*F97$Few$!1(4 Lw=;&=?Fjp7$$\"1/L$e9d$>iF9$!1sUG7aqrIFjp7$Fjw$!1%\\JJBxn0%Fjp7$$\"1HL 3-V)G1\"Fjp$!1Za4#pg_&\\Fjp7$F_x$!1vb!Q[!4`dFjp7$$\"1GLe*)G$Q]\"Fjp$!1 7hboy%4W'Fjp7$Fdx$!1qk-o(>U,(Fjp7$$\"1%****\\Aa\"3@Fjp$!1?z,,V:SxFjp7$ Fix$!1yH&=eeM9)Fjp7$$\"1GL3FI'ze#Fjp$!1Vg0i:#*)>)Fjp7$$\"1&***\\(yCRo# Fjp$!1>n]VF)yB)Fjp7$$\"1im\"za'))zFFjp$!1%p&QoQAh#)Fjp7$$\"1HLL3$[e(GF jp$!1FwnHZ%)p#)Fjp7$$\"1im;H=xnIFjp$!1i&Q5CilC)Fjp7$F^y$!1r!fa6w_<)Fjp 7$Fcy$!1%)Gh\\F5yuFjp7$Fhy$!10XUu)H1Z'Fjp7$F\\[l$!1:.jBx`*R&Fjp7$F`\\l $!1_!e!z&pqV%Fjp7$Fe\\l$!1&Q(*pSmFk$Fjp7$Fj\\l$!1q_`8d2#*HFjp7$F_]l$!1 [Eip'[%p?Fjp7$Fd]l$!1zs$Q)Gf79Fjp7$Fi]l$!1_[[/6^H5Fjp7$F^^l$!1m!ob\"=s TvF97$Fc^l$!1#**p%\\417eF97$Fh^l$!18(phc.NY%F97$F]_l$!1%R#*o_rNa$F97$F b_l$!1a)[PFY#HGF97$Fg_l$!1uV\"[,>EI#F97$F\\`l$!1z\\WrL%3)=F97$Fa`l$!1K cKE**ol:F97$Ff`l$!1z'3WPN:J\"F97$F[al$!1I&\\&)\\)f56F97$F`al$!1\\?^XIQ .'*F-7$Feal$!1hY^0&GX>)F-7$Fjal$!1Pm'e#RRdrF-7$F_bl$!1'>\"[9ehMiF-7$Fd bl$!1mE_Q#yM\\&F-7$Fibl$!1_m-abt@[F--F^cl6&F`clF*FaclF*-F$6$7bo7$F($\" 1Xu41::qKF-7$F/$\"15Sm:b@LPF-7$F5$\"1nFDOKv4UF-7$F;$\"1^8AA#>u%[F-7$F@ $\"1y3AQb!ei&F-7$FE$\"1G4!=oGWd'F-7$FJ$\"1'fTomaLl(F-7$FO$\"1a:q(=oI.* F-7$FT$\"1a*))\\\"))*H3\"F97$FY$\"1t#3L!)4DJ\"F97$Fhn$\"1ob&3M-5i\"F97 $F]o$\"1l`z?CMx>F97$Fbo$\"1C>EW'**Q^#F97$Fgo$\"1ARz1)Fjp7$F]u$\"1l0`7R%)*3\"F17$Fbu$ \"1*e?bI^;W\"F17$Fgu$\"1**QlRTQ_(Q=O 8p\"F17$Ffv$\"1%fC\"RHLJ8F17$Fj[m$\"1i\\/0,Js5F17$F[w$\"1%pgP)eBvwFjp7 $Fb\\m$\"1*RoV?.#yUFjp7$F`w$\"1^1]K%Ray'F97$$\"1BK$3-)*\\2(F-$!1/:i5w+ :7Fjp7$Fj\\m$!1!H7W\"yn#4$Fjp7$$\"1:LeRFC7HF9$!1un#32C-$\\Fjp7$Few$!1Z n5nSp/nFjp7$Fb]m$!1&)*[>8qah)4kP)Fjp7$ F\\[l$!1DE6eQC+iFjp7$F`\\l$!1Fns()G6v\"Fjp7$Fd]l$!1@UufD)e7\"Fjp7$Fi]l$!1k` 6#4\\x!zF97$F^^l$!1u$Rs?T3j&F97$Fc^l$!1YtiJ0XdUF97$Fh^l$!15k3F97$Fg_l$!1JS+q@38;F97$F\\`l$!1#Q3 +J%>48F97$Fa`l$!1j)yx`5V3\"F97$Ff`l$!1Dd\\?TzV!*F-7$F[al$!17jC\"4v1j(F -7$F`al$!1l1%plN'zlF-7$Feal$!1?>%47(y)f&F-7$Fjal$!1@D>vjqz[F-7$F_bl$!1 Vl58O7UUF-7$Fdbl$!1-jst-hJPF-7$Fibl$!1Xu41::qKF--F^cl6&F`clF*F*Facl-F$ 6$7cp7$F($\"1A5kQ6[nCF-7$F/$\"1-d)o&GY=GF-7$F5$\"1eOk7F**zJF-7$F;$\"1? l%y$[FkOF-7$F@$\"1tN;xq=cUF-7$FE$\"1&y@G0H'y\\F-7$FJ$\"1$)))yl\\n,eF-7 $FO$\"1Vw5*Q%>coF-7$FT$\"1&fCo$e\"GB)F-7$FY$\"1g49=!Hh***F-7$Fhn$\"12# zP_iuB\"F97$F]o$\"1hD6Q*GL^\"F97$Fbo$\"136xk,&3$>F97$Fgo$\"1B?%***R$)= DF97$F\\p$\"1o'fpofCL$F97$Fap$\"1_M!f<*=,WF97$Ffp$\"1Yi5Jmi]jF97$F\\q$ \"1uo_#)3.$**)F97$Faq$\"1rr$R!frA9Fjp7$$!1qmmT9Fjp$\"1aJ;1nE2LF17$Fav$\"16PfDR_'G$F17 $$!1MLe*)pw+6Fjp$\"1rNA>()fYJF17$Ffv$\"1OLTz*zO&GF17$Fj[m$\"1;U?7GR#R# F17$F[w$\"1!y#f@3pmF17$Fb]m$!1KT%**fIsC#F17$$\"1+L e9'G'>zS/`#F17$Fjw$!1\\67WR,oFF17$Fj]m$!14`(3cvh5$F17$F_x$!1 *)HxazBvKF17$$\"1G$3Fu&f$R\"Fjp$!1Z_2EX&\\I$F17$Fb^m$!1\"HI\">@u.LF17$ $\"1F$ek.qSh\"Fjp$!1!G$\\Hx*fF$F17$Fdx$!1!3f6ZGgA$F17$Fj^m$!1M'>wE1P$H F17$Fix$!1lHg[kiaDF17$Fa`m$!1eQ_1Z\"3<#F17$F^y$!1b=(ou/<#=F17$Fcy$!1.* =G%4aL7F17$Fhy$!1xyW$))3$\\%)Fjp7$F\\[l$!1,R=')43ieFjp7$F`\\l$!1L\"Hfw JV=%Fjp7$Fe\\l$!1?M'exT()3$Fjp7$Fj\\l$!1tzq$efVL#Fjp7$$\"1$****\\')H;RwC'F97$F^^l$!1#p5y%*33S%F97$Fc^l$!1ZBIQ^n.LF97$Fh^l$!1)Q)oxAT# [#F97$F]_l$!12yitrwR>F97$Fb_l$!1BJ*R%odG:F97$Fg_l$!1D*o'eWMJ7F97$F\\`l $!1/m&\\j<1(**F-7$Fa`l$!1!*)GT$z(GC)F-7$Ff`l$!1?b-i3SkoF-7$F[al$!1H(Rn dfVy&F-7$F`al$!1r5M&o(f#)\\F-7$Feal$!1./JaxiNUF-7$Fjal$!1XN4#y5))o$F-7 $F_bl$!1TdpFGc/KF-7$Fdbl$!1xYq#fXs\"GF-7$Fibl$!1A5kQ6[nCF--F^cl6&F`clF aclF*Facl-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fibl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "diff(h(n,x),x$2):\nfactor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)%\"nG\"\"$\"\"\",&*&)F'\"\"#F))%\"xGF -F)F(!\"\"F)F)F)*&%#PiGF)),&F)F)F+F)F(F)F0F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The maximum and minimum p oints on the graph of " }{XPPEDIT 18 0 "h[n]*`'`(x);" "6#*&&%\"hG6#%\" nG\"\"\"-%\"'G6#%\"xGF(" }{TEXT -1 13 " occur where " }{XPPEDIT 18 0 " x = -1/(sqrt(3)*n);" "6#/%\"xG,$*&\"\"\"F'*&-%%sqrtG6#\"\"$F'%\"nGF'! \"\"F." }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 1/(sqrt(3)*n);" "6#/% \"xG*&\"\"\"F&*&-%%sqrtG6#\"\"$F&%\"nGF&!\"\"" }{TEXT -1 14 " respecti vely." }}{PARA 0 "" 0 "" {TEXT -1 49 "The corresponding maximum and mi nimum values are " }{XPPEDIT 18 0 "3*n^2*sqrt(3)/(8*Pi);" "6#**\"\"$\" \"\"*$%\"nG\"\"#F%-%%sqrtG6#F$F%*&\"\")F%%#PiGF%!\"\"" }{TEXT -1 6 " \+ and " }{XPPEDIT 18 0 "-3*n^2*sqrt(3)/(8*Pi);" "6#,$**\"\"$\"\"\"*$%\"n G\"\"#F&-%%sqrtG6#F%F&*&\"\")F&%#PiGF&!\"\"F0" }{TEXT -1 15 " respect ively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "hp(n,-1/(sqrt(3)*n));\nhp(n,1/(sqrt(3)*n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)%\"nG\"\"#\"\"\"-%%sqrtG6#\"\"$F)F)%# PiG!\"\"#F-\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)%\"nG\"\"# \"\"\"-%%sqrtG6#\"\"$F)F)%#PiG!\"\"#!\"$\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "These maximum and minimum values tend to " } {XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT -1 4 " as " } {XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)i nfinityGF*F*F*" }{TEXT -1 57 ", at the same time as there horizontal l ocation tends to " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The following numerical example illustrates that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(h[n]*`'`(x)*f(x),x = -infinity .. i nfinity);" "6#-%$IntG6$*(&%\"hG6#%\"nG\"\"\"-%\"'G6#%\"xGF+-%\"fG6#F/F +/F/;,$%)infinityG!\"\"F6" }{TEXT -1 1 " " }{TEXT 276 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "- `f '`(0)" "6#,$-%$f~'G6#\"\"!!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "when " }{TEXT 273 1 "n" }{TEXT -1 10 " is large." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "f := x -> x*exp(Pi-x^2):\n'f(x)'=f(x);\nhp : = (n,x) -> -2*n^3*x/(Pi*(1+n^2*x^2)^2):\n'D(f)'(0)=evalf(D(f)(0));\nIn t('hp(10^6,x)*f(x)',x=-infinity..infinity);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&F'\"\"\"-%$expG6#,&%#PiGF)*$)F'\" \"#F)!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--%\"DG6#%\"fG6#\"\"! $\"+k#pSJ#!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%#hpG6$ \"(+++\"%\"xG\"\"\"-%\"fG6#F+F,/F+;,$%)infinityG!\"\"F3" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!+TS19B!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "2nd description of " }{XPPEDIT 18 0 "delta;" "6#%&delt aG" }{TEXT -1 3 " '(" }{TEXT 263 1 "x" }{TEXT -1 28 ") as a sequence o f functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 49 " c an be represented by the sequence of functions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[n](x) = n/sqrt(2*Pi);" "6#/-&%\"bG6#% \"nG6#%\"xG*&F(\"\"\"-%%sqrtG6#*&\"\"#F,%#PiGF,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-n^2*x^2/2);" "6#-%$expG6#,$*(%\"nG\"\"#%\"xGF)F)! \"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "Differentiating with respect to " }{TEXT 274 1 "x" }{TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[n]" "6#&%\"bG6#%\"nG" } {TEXT -1 8 " '(x) = " }{XPPEDIT 18 0 "-n^3*x/(sqrt(2*Pi))" "6#,$*(%\"n G\"\"$%\"xG\"\"\"-%%sqrtG6#*&\"\"#F(%#PiGF(!\"\"F/" }{TEXT -1 1 " " } {XPPEDIT 18 0 "exp(-n^2*x^2/2)" "6#-%$expG6#,$*(%\"nG\"\"#%\"xGF)F)!\" \"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "b := (n,x) -> n*exp(-n^2*x^2/2)/sqrt(2*Pi );\nbp := unapply(diff(b(n,x),x),n,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)*&*&9$\"\"\"-%$exp G6#,$*&)F/\"\"#F0)9%F7F0#!\"\"F7F0F0-%%sqrtG6#,$%#PiGF7F;F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bpGf*6$%\"nG%\"xG6\"6$%)operatorG%& arrowGF),$*&**)9$\"\"$\"\"\"9%F3-%$expG6#,$*&)F1\"\"#F3)F4F;F3#!\"\"F; F3-%%sqrtG6#F;F3F3*$-F@6#%#PiGF3F>F=F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot([bp(1,x),bp(2 ,x),bp(3,x),bp(4,x)],x=-4..4,\n color=[red,green,blue,m agenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 442 257 257 {PLOTDATA 2 "6(-%' CURVESG6$7go7$$!\"%\"\"!$\"1;afI!4KN&!#>7$$!1nmmmFiDQ!#:$\"18%)zmg#H, \"!#=7$$!1LLLo!)*Qn$F1$\"1/Mxb-D=_E\"*yXAIF47$$ !1mmmOk]JLF1$\"1Qfhv&)[p^F47$$!1MLL[9cgJF1$\"1P6'yst>a)F47$$!1nmmhN2-I F1$\"1\\o$\\))>AK\"!#<7$$!1+++N&oz$GF1$\"1srd@VO=?FN7$$!1nmm\")3DoEF1$ \"1u*yQY7z-$FN7$$!1+++:v2*\\#F1$\"1YV9@-d!R%FN7$$!1LLL8>1DBF1$\"1h9&R& *[`@'FN7$$!1nmmw))yr@F1$\"1_Ey)FN7$$!1+++S(R#**>F1$\"1)=(Hd50\"3 \"!#;7$$!1++++@)f#=F1$\"1@j;#e#Gv8Fgo7$$!1+++gi,f;F1$\"1gVL)pq9n\"Fgo7 $$!1nmm\"G&R2:F1$\"10'el#=vI>Fgo7$$!1LLLtK5F8F1$\"1%f%)R5'p%>#Fgo7$$!1 MLL$yP2D\"F1$\"1t??vYN#G#Fgo7$$!1MLL$HsV<\"F1$\"1Hi4)Q;4N#Fgo7$$!1++v$ oc*H6F1$\"1)[h4+s2Q#Fgo7$$!1nm;u5a&3\"F1$\"1\\nCAt`-CFgo7$$!1MLeka7T5F 1$\"1#Q;'yDn:CFgo7$$!1-++]&)4n**Fgo$\"1&ztI-\"o>CFgo7$$!1OL$e9XRd*Fgo$ \"1k*4bea_T#Fgo7$$!1qmmTFgo7$$!1******\\O3E]Fgo$\"1DSZ6v>n gv*FN7$$!1MLL$)[`P/E)[LFN7$$\"1FLL$ =2Vs\"Fgo$!1(\\53]#[xnFN7$$\"1hmmm7+#\\#Fgo$!1sQ%yK)pP'*FN7$$\"1'***** \\`pfKFgo$!1G.=jT9L7Fgo7$$\"1imm\"*f#))3%Fgo$!19pO#o#R+:Fgo7$$\"1HLLLm &z\"\\Fgo$!1isf_2]QFgo7$$\"1(****** z-6j'Fgo$!1^]C1uIB@Fgo7$$\"1%*****\\C4puFgo$!1,^A\\CVaAFgo7$$\"1#***** *4#32$)Fgo$!1#Q?ao&)pM#Fgo7$$\"1#***\\P6[7()Fgo$!1:jmk90yBFgo7$$\"1$** **\\CFgo7$$\"1**\\i]TP:5F1$!1)o#zw#Q\">CFgo7$$ \"1***\\iZ!)y.\"F1$!1+FY&>!G;CFgo7$$\"1**\\(=!oQg5F1$!1sgM:!p5T#Fgo7$$ \"1****\\FJ*G3\"F1$!1j5$pYnNS#Fgo7$$\"1***\\(yd!z7\"F1$!1$>HKZf>Q#Fgo7 $$\"1******H%=H<\"F1$!1?kHUe,_BFgo7$$\"1LLLo,\"QD\"F1$!1syiR_qM8F1$!1J@D(p_]=#Fgo7$$\"1++++.W2:F1$!1ZxOl%y1$>Fgo7$$\"1LLLep 'Rm\"F1$!1+Z)zLDFm\"Fgo7$$\"1+++S>4N=F1$!16>j*\\'Gf8Fgo7$$\"1mmm6s5'*> F1$!1%\\\\uuKh3\"Fgo7$$\"1+++lXTk@F1$!11\"\\,D\"=)H)FN7$$\"1mmmmd'*GBF 1$!1D^O8Q[phFN7$$\"1+++DcB,DF1$!1vc\\H`rqVFN7$$\"1MLLt>:nEF1$!10y,)H^b .$FN7$$\"1LLL.a#o$GF1$!1'46[\"f5C?FN7$$\"1nmm^Q40IF1$!1+sL$e(f68FN7$$ \"1+++!3:(fJF1$!1Ck_Nubi&)F47$$\"1nmmc%GpL$F1$!1](=A%e8&3&F47$$\"1LLL8 -V&\\$F1$!1QV/[-b*4$F47$$\"1+++XhUkOF1$!1Y=4w&HWx\"F47$$\"1+++:op#>[\"*Fcel7$FU$\"1?G^3_5vb!#@7$FZ$\"12*y+,$)) **H!#?7$Fin$\"1LY>uT&f\\\"F-7$F^o$\"13;_+d&>G\"FN7$F^p$\"1C6&FN7$$!1++]x#\\sT\"F1 $\"1!4!egoLV\")FN7$Fhp$\"1)[;RMr1D\"Fgo7$F]q$\"1PnRsZSZ-$Qgcy'*Fgo7$$!1mm\"z/(z6[Fgo$\" 1c$\\ux[\\m*Fgo7$$!1LL$eW5vf%Fgo$\"1\"yR6+MXh*Fgo7$$!1***\\P%QA$Q%Fgo$ \"1*)3y\"=\"4E&*Fgo7$Fcu$\"1_Syt\\i)R*Fgo7$$!1****\\PSOSPFgo$\"1tf;\\f #R-*Fgo7$Fhu$\"1'RMB9Ey[)Fgo7$F]v$\"18%f.;oJ4(Fgo7$Fbv$\"13r;=D[?_Fgo7 $$!1nm;aH-88Fgo$\"1C3hBE`[SFgo7$Fgv$\"1*=urG&H\"z#Fgo7$$!1SLLe4**RYFN$ \"1jF:\"33XZ\"Fgo7$F\\w$\"1o*\\as3-E\"FN7$$\"16LL3Uh9SFN$!105H=g:x7Fgo 7$Faw$!1Br(H'Rp]EFgo7$$\"1GL$efeLG\"Fgo$!1F'z(fm;jRFgo7$Ffw$!1ok.oj[&= &Fgo7$F[x$!19`y7`QCqFgo7$F`x$!1.Ug&R21ISNj*Fgo7$$\"1IL$eW'QgbFgo$!1rYX+2:i&*Fgo7$F_y$!1h/aAY%)f%*Fgo7$$ \"1ILLei\"G?'Fgo$!1uF5=qkq\"*Fgo7$Fdy$!1zK)=&\\A$y)Fgo7$Fiy$!1jf&G*R(4 \"yFgo7$F^z$!1/fbX#G*omFgo7$Fhz$!1cK&\\qb!=bFgo7$Fb[l$!1rB%4)*G@T%Fgo7 $Ff\\l$!1pM_Kn_6LFgo7$F`]l$!1-KTRmb*Q#Fgo7$Fe]l$!1Z]%z=X\\s\"Fgo7$Fj]l $!1)*)Rerlz?\"Fgo7$$\"1LLL.62@9F1$!1qyf(>d+*zFN7$F_^l$!1bzAEko5^FN7$$ \"1mm;HOq&e\"F1$!1.B1#p&y7LFN7$Fd^l$!1tgSLeH!4#FN7$$\"1mm;\\%H&\\MBHARE*Fcel7$F\\al$!1]^aGV+u8Fcel7$Faal$!1Rn!)\\**)p9#F_el7$F fal$!1wj'pC1uE#F[el7$F[bl$!1sY`N6.BFFgdl7$F`bl$!1w35u')\\QDFcdl7$Febl$ !1fn[qYL\\BF_dl7$Fjbl$!11=tYns;;F[dl-F_cl6&FaclF*FbclF*-F$6$7io7$F($\" 11h^MW4=B!#X7$F/$\"1c/NiNfH5!#U7$F6$\"1M_$Q'fb9E/R!#Q 7$F@$\"1=U2m&e6J(!#O7$FE$\"1eO3ruFB5!#L7$FJ$\"1:D>N-ayy!#K7$FP$\"1$G;Z h\"Gfb!#I7$FU$\"1w8r7_'R]$F[dl7$FZ$\"1!=;ks-qn\"Fcdl7$Fin$\"1JrGEP>?oF gdl7$F^o$\"1a(H:6:kT\"F_el7$Fco$\"1]MLY9\"\\K$Fcel7$Fio$\"1-5W'e@H*fFj el7$F^p$\"1q,Lk&etY(F^fl7$Fcp$\"1bV0:7y&)eF-7$Fhp$\"1RkYz4sm^F47$F]q$ \"1#)f0mWG7Fgo7$F`s$\"17*zC^*)zA#F go7$Fjs$\"1C$*Hlu2%z$Fgo7$F_t$\"1U&>cTOsJ'Fgo7$Fdt$\"1a7-0(Gyq*Fgo7$Fi t$\"1A$o6L $3Fr)4=FN$!1H5*3w?m%>Fgo7$Ff^m$!1u@5#ooIH%Fgo7$$\"1/L$e9d$>iFN$!1y+-lm a$e'Fgo7$Faw$!1CrOjex)y)Fgo7$F^_m$!1&ecu\\7OG\"F17$Ffw$!1ok4!=KZi\"F17 $F[x$!1F0n.\"=)H?F17$F`x$!1SMWDgmw@F17$Fex$!1LCq?gdv?F17$Fjx$!1[\"R`YP Ry\"F17$F_y$!1'>86HHrQ\"F17$Fdy$!1Bd`jCXu)*Fgo7$Fiy$!1P(3Ea^a`'Fgo7$F^ z$!1P4x[/\\4SFgo7$Fhz$!1:**yh3PIBFgo7$Fb[l$!1*=*f\\p`m7Fgo7$F\\\\l$!1c v(p&plu()FN7$Ff\\l$!1YFg=g1efFN7$F[]l$!1F4iw(4c'RFN7$F`]l$!1T*=P`Mwe#F N7$Fe]l$!1f9XX&zM9\"FN7$Fj]l$!1ZZ(>RNUu%F47$F_^l$!1%\\%*eAjB)eF-7$Fd^l $!1#*ovM1@bpF^fl7$Fi^l$!1\\EOe(eM=&Fjel7$F^_l$!1V20&HK?^$Fcel7$Fc_l$!1 )R%H%oo+j\"F_el7$Fh_l$!18`9>2K&H'Fgdl7$F]`l$!173AH=*))f\"Fcdl7$Fb`l$!1 -a#plihf$F[dl7$Fg`l$!13Fb#f_;s&Fjjm7$F\\al$!1j86x%e\"osFfjm7$Faal$!1$f R;tFz/\"Fbjm7$Ffal$!1mqAe&[MA'F^jm7$F[bl$!1yqrF3;')\\Fjim7$F`bl$!11>7! [qrD#Ffim7$Febl$!1\"!# g7$F;$\"1?t]:[NZ?!#c7$F@$\"1!p0'o,*QL#!#_7$FE$\"14=^'=O&)e\"!#[7$FJ$\" 1t\\JMf3LPF^im7$FP$\"1Hc'H(=kSvFbim7$FU$\"1!pwG\"[Z^7Fjim7$FZ$\"1#R??W KpF\"!#N7$Fin$\"1++p_pE0)*Fbjm7$F^o$\"1KIu0xisAFjjm7$Fco$\"1o*ps#fhBmF [dl7$Fio$\"1j=SUOP97Fgdl7$F^p$\"18>x31rf6F_el7$Fcp$\"1ctd4q>1\\Fcel7$F hp$\"1`,#QX:gd#F^fl7$Fbq$\"15+/)HmV%[F-7$F\\r$\"1ZCx/pHJAF47$Ffr$\"1TC [K*=w**)F47$F[s$\"1tTj\"*4)zf\"FN7$F`s$\"1t[&e\\DSw#FN7$Fes$\"1-rzb**e bYFN7$Fjs$\"1_7hl?yMwFN7$F_t$\"15w7/_>c?Fgo7$Fdt$\"1qdNsma^[Fgo7$F\\jl $\"1\"QHf-)QipFgo7$Fit$\"1+Wi![COo*Fgo7$Fijl$\"17JV=f]/8F17$F^u$\"1@r& 3xm2q\"F17$F[\\m$\"1P@#e<()Q;#F17$Fcu$\"1)[gqb`,l#F17$Fh\\m$\"1$4#e%3* Q=JF17$Fhu$\"1&)z\"[)*3j^$F17$$!1LL$e%oA=HFgo$\"1op:rD\"*pPF17$F]v$\"1 #>&*)eg:rQF17$$!1LL$3())4J@Fgo$\"1n\\5e)zNy$F17$Fbv$\"1()pkj0V%[$F17$$ !1++v=*y__\"Fgo$\"1t>Mok-LKF17$Ff]m$\"1(otA$>a?HF17$$!1MLe*)pw+6Fgo$\" 1As16F'3b#F17$Fgv$\"1&z%\\N(H(H@F17$F]`n$\"1NuJ*f1Ym\"F17$F^^m$\"1@/^Z mYk6F17$Fe`n$\"1d51wu2&R'Fgo7$F\\w$\"1Qn`mE235Fgo7$F]an$!1#4N19D*3YFgo 7$Ff^m$!1mzhs?*=,\"F17$Fean$!1Q#o/*)f&R:F17$Faw$!1FDSY(e@.#F17$$\"1HL3 -V)G1\"Fgo$!132V%=z#zCF17$F^_m$!1;]H#z-A(GF17$$\"1GLe*)G$Q]\"Fgo$!15^+ Cx\\rQF17$$\"1HLL3$[e(GFgo$!1'\\KISm))y$F17$F`x$!1NUkDY4dNF17$F\\`m$! 10***yawd=$F17$Fex$!1m@0,@\\SFF17$Fi`m$!1[*\\onE*pAF17$Fjx$!18X?#Fgo7$F^z$!1_sUK))=\"\\)FN7$Fcz$!1RZ KR'>z7&FN7$Fhz$!1X\\XgI!*4IFN7$F][l$!1Om&f&3X<>5!\\Fcel7$Fd^l$!1!Q=llL(>5F_el7$Fi^l$!1Qr'Q-biM*Fcdl7$F^_l$! 1^Znhkm4tF[dl7$Fc_l$!17*e@Mb^#HFjjm7$Fh_l$!1D>Tj*\\I\\)Fbjm7$F]`l$!1\" *4#z(4Ls6F[[o7$Fb`l$!1F4%pXL5J\"Fjim7$Fg`l$!1lQVjj5RzFbim7$F\\al$!1kda *)e(>B$F^im7$Faal$!1-axpzad;F^jn7$Ffal$!1))yCjw^]t$47'4 oJFfin7$F`bl$!1*GH/EGn2#Fbin7$Febl$!1m(*)=3k$Q8F^in7$Fjbl$!1tsMmu(pi#F jhn-F_cl6&FaclFbclF*Fbcl-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fjbl%(D EFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "diff(b(n,x),x$2):\nfactor(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*,)%\"nG\"\"$\"\"\"-%$expG6# ,$*&)F'\"\"#F))%\"xGF0F)#!\"\"F0F)-%%sqrtG6#F0F),&*&F'F)F2F)F)F4F)F),& F9F)F)F)F)F)*$-F66#%#PiGF)F4#F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The maximum and minimum points on the graph of " } {XPPEDIT 18 0 "b[n]*`'`(x);" "6#*&&%\"bG6#%\"nG\"\"\"-%\"'G6#%\"xGF(" }{TEXT -1 13 " occur where " }{XPPEDIT 18 0 "x = -1/n" "6#/%\"xG,$*&\" \"\"F'%\"nG!\"\"F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 1/n;" "6#/ %\"xG*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 14 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 49 "The corresponding maximum and minimum values are " } {XPPEDIT 18 0 "n^2/sqrt(2*Pi)" "6#*&%\"nG\"\"#-%%sqrtG6#*&F%\"\"\"%#Pi GF*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-1/2)" "6#-%$expG6#,$*& \"\"\"F(\"\"#!\"\"F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-n^2/sqrt(2* Pi)" "6#,$*&%\"nG\"\"#-%%sqrtG6#*&F&\"\"\"%#PiGF+!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-1/2)" "6#-%$expG6#,$*&\"\"\"F(\"\"#!\"\"F*" } {TEXT -1 14 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "bp(n,-1/n);\nbp(n,1/n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*()%\"nG\"\"#\"\"\"-%$expG6##!\"\"F(F)-% %sqrtG6#F(F)F)*$-F06#%#PiGF)F.#F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&*()%\"nG\"\"#\"\"\"-%$expG6##!\"\"F(F)-%%sqrtG6#F(F)F)*$-F06#%#Pi GF)F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "These maximum and mi nimum values tend to " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" } {TEXT -1 4 " as " }{XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%)oper atorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 57 ", at the same time as there horizontal location tends to " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG \"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The following numerical example illustrates that" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(b[n]*`'`(x)*f(x), x = -infinity .. infinity);" "6#-%$IntG6$*(&%\"bG6#%\"nG\"\"\"-%\"'G6# %\"xGF+-%\"fG6#F/F+/F/;,$%)infinityG!\"\"F6" }{TEXT -1 1 " " }{TEXT 264 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "- `f '`(0)" "6#,$-%$f~'G6# \"\"!!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "when " } {TEXT 275 1 "n" }{TEXT -1 10 " is large." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "b := (n,x) -> n*exp( -n^2*x^2/2)/sqrt(2*Pi):\nbp := unapply(diff(b(n,x),x),n,x):\nf := x -> (x*arctan(x+2))/(x^4+5):\n'f(x)'=f(x);\n'D(f)(0)'=evalf(D(f)(0));\nIn t('bp(500,x)*f(x)',x=-infinity..infinity);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*(F'\"\"\"-%'arctanG6#,&F'F)\"\"#F)F ),&*$)F'\"\"%F)F)\"\"&F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--% \"DG6#%\"fG6#\"\"!$\"+OuH9A!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I ntG6$*&-%#bpG6$\"$+&%\"xG\"\"\"-%\"fG6#F+F,/F+;,$%)infinityG!\"\"F3" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+:bH9A!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "3rd description of " }{XPPEDIT 18 0 "de lta;" "6#%&deltaG" }{TEXT -1 3 " '(" }{TEXT 266 1 "x" }{TEXT -1 28 ") \+ as a sequence of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" } {TEXT -1 49 " can be represented by the sequence of functions " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[n](x)=PIECEWISE([n* cos(n*Pi*x/2)^2 , abs(x)<1/n],[0 , abs(x)>=1/n])" "6#/-&%\"cG6#%\"nG6# %\"xG-%*PIECEWISEG6$7$*&F(\"\"\"*$-%$cosG6#**F(F0%#PiGF0F*F0\"\"#!\"\" F7F02-%$absG6#F**&F0F0F(F87$\"\"!1*&F0F0F(F8-F;6#F*" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 42 "Differentiating with respect to x, we ha ve" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[n];" "6#&%\"c G6#%\"nG" }{TEXT -1 3 " '(" }{TEXT 267 1 "x" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "PIECEWISE ([-n^2/Pi*sin(n*Pi*x) , abs(x)<1/n],[0 , abs( x)>=1/n])" "6#-%*PIECEWISEG6$7$,$*(%\"nG\"\"#%#PiG!\"\"-%$sinG6#*(F)\" \"\"F+F1%\"xGF1F1F,2-%$absG6#F2*&F1F1F)F,7$\"\"!1*&F1F1F)F,-F56#F2" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "cp := (n,x) -> piecewise(abs(x)<1/n,-n^2/Pi*sin(n*Pi*x),n*0.01):\nplot([cp(1,x),c p(2,x),cp(3,x),cp(4,x)],x=-2..2,\n color=[red,gree n,blue,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 350 283 283 {PLOTDATA 2 "6(-%'CURVESG6$7gq7$$!\"#\"\"!$\"1+++++++5!#<7$$!1LLL$Q6G\">!#:F+7$$ !1nm;M!\\p$=F1F+7$$!1LLL))Qj^.\"F1F+7$$!1LL37&)=@5F1F+7$$!1+D c^jz:5F1F+7$$!1n;/\">//,\"F1F+7$$!1]7y5\"3x+\"F1F+7$$!1L3_I?,05F1F+7$$ !1'***!#;$\"1[ZY'4*H,Q!#>7$$!1,+++Y0j&*F[p$\"1 EZ#\\_WdN%F-7$$!1++++0\"*H\"*F[p$\"1GeLVi%Hf)F-7$$!1++++f\\7()F[p$\"19 ,-F[p7$$!1LLL3k(p`(F[p$\"1!fQsC$\\CAF[p7$$!1nmmmj^NmF[p$\"1[PCzt3sFF[ p7$$!1omm;*)o`iF[p$\"1#*>fEiQRHF[p7$$!1ommm9'=(eF[p$\"12W:$F[p7$$!1,](o MmmJ&F[p$\"1i1(RGgt;$F[p7$$!1om\"HKFc?&F[p$\"19;hO%fk<$F[p7$$!1N$e*)H) e%4&F[p$\"1aD@#e$p\"=$F[p7$$!1,++v#\\N)\\F[p$\"1*RW?NcI=$F[p7$$!1M$eR# 4E&)[F[p$\"1G-lB6.\"=$F[p7$$!1om\"Hdspy%F[p$\"1*Gk29tf<$F[p7$$!1,](=A% o)o%F[p$\"1RHgDs)y;$F[p7$$!1NL$3(eR!f%F[p$\"1QX<(3\"ycJF[p7$$!1-+vo\"> QR%F[p$\"1&zmOO`b7$F[p7$$!1pmmmCC(>%F[p$\"1X&4..4C3$F[p7$$!1ML$3(3*ew$ F[p$\"1!3\"))*QWo%HF[p7$$!1*****\\FRXL$F[p$\"1zG4]sCdFF[p7$$!1******\\ 0zBHF[p$\"1A^)=`_'HDF[p7$$!1+++D=/8DF[p$\"10SNNQ**fAF[p7$$!1LL$3ioW3#F [p$\"1G?ns$['Q>F[p7$$!1mmm;a*el\"F[p$\"14OUg@@#e\"F[p7$$!1nm;H9Li7F[p$ \"1&)zo$y-&H7F[p7$$!1pmm;Wn(o)F-$\"19^'>w:-e)F-7$$!1.++D^bUWF-$\"1UX7_ k9GWF-7$$!1qLLL$eV(>!#=$\"1<*pNnXV(>Fhy7$$\"1[mmT+07UF-$!1&eXZho(*>%F- 7$$\"1Mmm;f`@')F-$!1)Gk5\"p];&)F-7$$\"1JLLL1+Y7F[p$!1dw[/LU97F[p7$$\"1 )****\\nZ)H;F[p$!1!RY%*[d&f:F[p7$$\"1JL$e*HTW?F[p$!1dkm_hs1>F[p7$$\"1l mm;$y*eCF[p$!1#zPd(zf@AF[p7$$\"1KLLe[E()GF[p$!1>$\\M&fJ2DF[p7$$\"1**** ***R^bJ$F[p$!1YM$Ha5xu#F[p7$$\"1(****\\AYXt$F[p$!124)*\\8&[$HF[p7$$\"1 '*****\\5a`TF[p$!19I_3T@rIF[p7$$\"1'**\\(o0CcVF[p$!1P#44`A#=JF[p7$$\"1 '***\\(3S*eXF[p$!1x#)R;/f_JF[p7$$\"1'*\\(o%)*GgYF[p$!1Ue--'))\\;$F[p7$ $\"1(**\\igR;w%F[p$!1#)zqX%yT<$F[p7$$\"1(*\\il$*)H'[F[p$!1&[?I&F[p$!1Yi8l&*y oJF[p7$$\"1(***\\PcY9aF[p$!151GZN:cJF[p7$$\"1(**\\P*)G&RcF[p$!1g*pCbp! >JF[p7$$\"1'*****\\@fkeF[p$!1^/k(*))RmIF[p7$$\"1jmmT30piF[p$!1Dee?oXLH F[p7$$\"1JLLL&4Nn'F[p$!1o1gwm@`FF[p7$$\"1*******\\,s`(F[p$!1E'*oRALCAF [p7$$\"1KL$e9=&GzF[p$!1_n%=*>LG>F[p7$$\"1lmm\"zM)>$)F[p$!1!f5^;DKg\"F[ p7$$\"1KL$eCZwu)F[p$!109[EDH?7F[p7$$\"1*******pfa<*F[p$!1\"*)zMd+N:)F- 7$$\"1km;zy*zd*F[p$!1())[P'*pw?%F-7$$\"1HLLeg`!)**F[p$!12iCP?QY>Fhy7$$ \"1m\"H#3Mo+5F1F+7$$\"1+]i5KJ.5F1F+7$$\"1L3-8I%f+\"F1F+7$$\"1mmT:Gd35F 1F+7$$\"1L$3-UKQ,\"F1F+7$$\"1+++D?4>5F1F+7$$\"1LLeM7hH5F1F+7$$\"1nm;W/ 8S5F1F+7$$\"1LLLj)o61\"F1F+7$$\"1++]#G2A3\"F1F+7$$\"1mm\"H3XL7\"F1F+7$ $\"1LLL$)G[k6F1F+7$$\"1++]7yh]7F1F+7$$\"1nmm')fdL8F1F+7$$\"1nmm,FT=9F1 F+7$$\"1LL$e#pa-:F1F+7$$\"1+++Sv&)z:F1F+7$$\"1LLLGUYo;F1F+7$$\"1nmm1^r ZF1F+7$$\"\"#F*F+-%'COLOURG6&%$R GBG$\"*++++\"!\")F*F*-F$6$7\\q7$F($\"1+++++++?F-7$F/Fchl7$F3Fchl7$F6Fc hl7$F9Fchl7$F)zn7Fg(F[p7$Fiv$\"1 ??M_N#G\"*)F[p7$$!1mm\"H2:-b$F[p$\"1scFV6&f+\"F17$F^w$\"1.ro(4v@5\"F17 $$!1****\\7\\;HJF[p$\"1CAia3/v6F17$Fcw$\"1vws1wOG7F17$$!1++voL5@GF[p$ \"1i\"4!=OTZ7F17$$!1++](=;%=FF[p$\"1p!fuho7E\"F17$$!1***\\i+Hdh#F[p$\" 19sg[\\()p7F17$Fhw$\"1Ug*oz'>t7F17$$!1L$eR_)*eS#F[p$\"1x3qmY,r7F17$$!1 mm\"HAb()H#F[p$\"1gjtuW2j7F17$$!1+](=#>h\">#F[p$\"1[VN#>7%\\7F17$F]x$ \"1**4i8(*3I7F17$$!1++v=?=q=F[p$\"1y=Qe&R[<\"F17$Fbx$\"1(fF92C$)4\"F17 $Fgx$\"1u6=@$\\E2*F[p7$F\\y$\"18@f3W45mF[p7$$!1OL$3x9^c'F-$\"1C=]Y!4W5 &F[p7$Fay$\"1i\"p_)323NF[p7$$!1qm;za**>BF-$\"1UX!)>/V\\=F[p7$Ffy$\"1Iu lbhWz:F-7$$\"1cm;/rI2?F-$!1ByT7!y#F[p$!1ls/\"=fND\"F17$Fj[l$!1$G)QcDtN7F17$$\"1lm;H\"395$F[p$!1:CxK IT$=\"F17$F_\\l$!1?&HZf,(46F17$$\"1)***\\7)[]_$F[p$!1I^h$om\"=5F17$Fd \\l$!1EZxDr:!4*F[p7$$\"1(***\\PO/WRF[p$!1%>1%Q=PTyF[p7$Fi\\l$!1K=:Qr\" pX'F[p7$F^]l$!1hL3`ny5]F[p7$Fc]l$!14O4J&)[$[$F[p7$F]^l$!1b,m(fi(**=F[p 7$Fg^l$!1:&yQHJE&GF-7$$\"1Z7.K?Z#*\\F[p$!1$*31>>NAgFhy7$$\"1'\\i!R\\g? ]F[pFchl7$$\"1ZP4Yyt[]F[pFchl7$F\\_lFchl7$$\"1(\\(=nl8L^F[pFchl7$Fa_lF chl7$Ff_lFchl7$F[`lFchl7$F``lFchl7$Fe`lFchl7$Fj`lFchl7$F_alFchl7$FdalF chl7$F^blFchl7$FhblFchl7$FbclFchl7$FbelFchl7$FhelFchl7$F[flFchl7$F^flF chl7$FaflFchl7$FdflFchl7$FgflFchl7$FjflFchl7$F]glFchl7$F`glFchl7$FcglF chl7$FfglFchl-Figl6&F[hlF*F\\hlF*-F$6$7hp7$F($\"1+++++++IF-7$F/F^hm7$F 3F^hm7$F6F^hm7$F9F^hm7$F\\aB^y5F1 7$Fd\\m$\"1&G*erU))o:F17$Fhw$\"1(4SR,g1+#F17$Ff]m$\"1z.9&y)HrBF17$F]x$ \"1&)3Ha`aXEF17$$!1m;z>`Kx>F[p$\"1WO:&=rGu#F17$Fc^m$\"1\"GH[s^A\"GF17$ $!1m\"H#o.h;=F[p$\"1/D2u-BOGF17$$!1L$3xrQIw\"F[p$\"1B'>65!)H&GF17$$!1+ v=nqY49iF[p7$Ffy$\"18B[s)f/L&F-7$$\"1$fmTNc$\\!*Fhy$!1GHPP^OSCF[p7$Fa`m$ !1XVQ\"pduQ&F[p7$$\"1_m\"Hdy'4JF-$!1cKL3#[kF)F[p7$F\\z$!1&e'R$)zh26F17 $Fi`m$!17j[xx$)G;F17$Faz$!1o[)y+\")*z?F17$Ffz$!1\"Hc2yoDk#F17$F[[l$!17 xF4V1jGF17$$\"1lT5S$o;o\"F[p$!1OV\"*QE]kGF17$$\"1J$3_+*[LF[p$!1L)**[lB(pFF17$F`[l$!1LV32Z9&o#F17$$\"1)**\\i l&p^AF[p$!1qYN%))Q-W#F17$Fe[l$!1J'\\+a'\\-@F17$Fbbm$!1Mvbd)\\(p;F17$Fj [l$!1jj3u!>#p6F17$$\"1)**\\P\\OV*HF[p$!1*H\\rZ))z**)F[p7$F_cm$!1([;I=M A@'F[p7$$\"1KLek(z%3KF[p$!1d4pL@FjLF[p7$F_\\l$!1vgHds*3![F-7$$\"1*\\il dQGu7F17$F[^m$\"1+7?:F17$F]x$\"1WI9)4%=SDF17$Fhjm$\" 10N>evt4JF17$Fc^m$\"1&pf5#G+BOF17$Fe[n$\"1%emSY*oqSF17$Fbx$\"1b_iySpWW F17$$!1mm\"HU8\"f9F[p$\"18@p+`7=\\F17$Fgx$\"1)pm\\qYB4&F17$$!1nmTN%\\b 1\"F[p$\"1kIRx*en&\\F17$F\\y$\"1e(\\#H5h>XF17$$!1-+v$f%REwF-$\"1Kj:YaJ nTF17$Fa_m$\"1U4()p&45u$F17$$!1pm\"z%\\$Q]&F-$\"1I$>qYk#[KF17$Fay$\"1 \"4&HX6$yp#F17$$!1OL3-`F\"Q$F-$\"12T9Q^[*4#F17$Fi_m$\"1)ztT*G&QY\"F17$ $!1.+Dccre7F-$\"1z2gX0BA!)F[p7$Ffy$\"1=h4,(fME\"F[p7$Fd]n$!1C=kOO6zdF[ p7$Fa`m$!1MLh%G'4r7F17$F\\^n$!1)=R@\"y#*R>F17$F\\z$!1o`6h:frDF17$$\"1X mT5:U9`F-$!1zoKd_)R:$F17$Fi`m$!1<(=$41&fn$F17$$\"1Qm\"zWk\">vF-$!1!QPT 4([FTF17$Faz$!1Op'4cV**\\%F17$$\"1(***\\7r2a5F[p$!1V*fV&zPR\\F17$Ffz$! 1]6jfQ*G4&F17$$\"1km;aT#zV\"F[p$!1bS%z#\\g^\\F17$F[[l$!1%)\\MV(*oBXF17 $Fe_n$!1m\"H$F17$F`[l$ !1!fy(Rk1fFF17$$\"1l;/EV0[@F[p$!1dG^Aotz@F17$F\\an$!1lRM3g[j:F17$$\"1J $ek)pLbBF[p$!17NlQG_2#*F[p7$Fe[l$!1:Q-J^ACEF[p7$$\"1J3_Diu&[#F[p$!1GIk yy#>7*F-7$$\"1)*\\PMT^7DF[pFien7$$\"1l\"HK/#GRDF[pFien7$F]bmFien7$$\"1 l;zpde>EF[pFien7$FbbmFien7$FgbmFien7$Fj[lFien7$F_cmFien7$F_\\lFien7$Fd \\lFien7$Fi\\lFien7$Fg^lFien7$Fe`lFien7$F_alFien7$FdalFien7$F^blFien7$ FhblFien7$FbclFien7$FbelFien7$FhelFien7$F[flFien7$F^flFien7$FaflFien7$ FdflFien7$FgflFien7$FjflFien7$F]glFien7$F`glFien7$FcglFien7$FfglFien-F igl6&F[hlF\\hlF*F\\hl-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Ffgl%(DEFA ULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The maximum and minimum points on the graph of " }{XPPEDIT 18 0 "c [n]*`'`(x);" "6#*&&%\"cG6#%\"nG\"\"\"-%\"'G6#%\"xGF(" }{TEXT -1 13 " o ccur where " }{XPPEDIT 18 0 "x = -1/(2*n);" "6#/%\"xG,$*&\"\"\"F'*&\" \"#F'%\"nGF'!\"\"F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 1/(2*n); " "6#/%\"xG*&\"\"\"F&*&\"\"#F&%\"nGF&!\"\"" }{TEXT -1 14 " respectivel y." }}{PARA 0 "" 0 "" {TEXT -1 49 "The corresponding maximum and minim um values are " }{XPPEDIT 18 0 "n^2/Pi;" "6#*&%\"nG\"\"#%#PiG!\"\"" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "-n^2/Pi;" "6#,$*&%\"nG\"\"#%#PiG! \"\"F(" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "These maximum and minimum values tend to " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT -1 4 " as " } {XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)i nfinityGF*F*F*" }{TEXT -1 57 ", at the same time as there horizontal l ocation tends to " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The following numerical example illustrates that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(c[n]*`'`(x)*f(x),x = -infinity .. i nfinity);" "6#-%$IntG6$*(&%\"cG6#%\"nG\"\"\"-%\"'G6#%\"xGF+-%\"fG6#F/F +/F/;,$%)infinityG!\"\"F6" }{TEXT -1 1 " " }{TEXT 277 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "- `f '`(0)" "6#,$-%$f~'G6#\"\"!!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "when " }{TEXT 278 1 "n" }{TEXT -1 10 " is large." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "cp := (n,x) -> -n^2*Pi/2*sin(n*Pi*x):\nf := \+ x -> (sqrt(x^2+7)+x*arctan(x+2))/(x^2+5):\n'f(x)'=f(x);\n'D(f)(0)'=eva lf(D(f)(0));\nInt('cp(100,x)*f(x)',x=-1/100..1/100);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&*$,&*$)F'\"\"#\"\"\" F/\"\"(F/#F/F.F/*&F'F/-%'arctanG6#,&F'F/F.F/F/F/F/,&F,F/\"\"&F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--%\"DG6#%\"fG6#\"\"!$\"+OuH9A!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%#cpG6$\"$+\"%\"xG\"\"\" -%\"fG6#F+F,/F+;#!\"\"F*#F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+*z tU@#!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "4th descri ption of " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 3 " '(" } {TEXT 268 1 "x" }{TEXT -1 28 ") as a sequence of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 49 " can be represented by t he sequence of functions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "v[n](x) = PIECEWISE([[n/A]*exp(1/(n^2*x^2-1)), abs(x) < 1/n],[0, 1/n <= abs(x)]);" "6#/-&%\"vG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$ *&7#*&F(\"\"\"%\"AG!\"\"F2-%$expG6#*&F2F2,&*&F(\"\"#F*F;F2F2F4F4F22-%$ absG6#F**&F2F2F(F47$\"\"!1*&F2F2F(F4-F>6#F*" }{TEXT -1 3 " , " }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A=Int(exp(1/(n^2*x ^2-1)),x=-1..1)" "6#/%\"AG-%$IntG6$-%$expG6#*&\"\"\"F,,&*&%\"nG\"\"#% \"xGF0F,F,!\"\"F2/F1;,$F,F2F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "Differentiating with respect to x, we have" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v[n];" "6#&%\"vG6#%\"nG" }{TEXT -1 3 " '(" }{TEXT 269 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "PIECEWI SE([[2*n^3/(A*(n^2*x^2-1)^2)]*exp(1/(n^2*x^2-1)), abs(x) < 1/n],[0, 1/ n <= abs(x)]);" "6#-%*PIECEWISEG6$7$*&7#*(\"\"#\"\"\"*$%\"nG\"\"$F+*&% \"AGF+*$,&*&F-F*%\"xGF*F+F+!\"\"F*F+F5F+-%$expG6#*&F+F+,&*&F-F*F4F*F+F +F5F5F+2-%$absG6#F4*&F+F+F-F57$\"\"!1*&F+F+F-F5-F>6#F4" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "Int(exp(1/(x^2-1)),x=-1..1) ;\nA := evalf(%);\nvp := (n,x) -> piecewise(abs(x)<1/n,-2*n^3*x/(A*(n^ 2*x^2-1)^2)*\n exp(1/(n^2*x^2-1)),n*0.01):\nplot([vp(1,x),vp( 2,x),vp(3,x),vp(4,x)],x=-1.5..1.5,\n xtickmarks=3,col or=[red,green,blue,magenta]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$-%$expG6#*&\"\"\"F*,&*$)%\"xG\"\"#F*F*!\"\"F*F0/F.;F0F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$\"+i\"Q*RW!#5" }}{PARA 13 "" 1 "" {GLPLOT2D 308 300 300 {PLOTDATA 2 "6)-%'CURVESG6$7ep7$$!1+++++++:!#:$ \"1+++++++5!#<7$$!1++]P&3YV\"F*F+7$$!1+]ivrL%*Fao7$$!1+](opS3X*FL$\"1%p-A(*Hl?$F-7$$!1+ +D\"oS:P*FL$\"1!pBu\\VNr(F-7$$!1,voa5S3#*FL$\"1,T/$oE$)\\#FL7$$!1+]7G9 EX!*FL$\"1q#olsw%Q]FL7$$!1*\\i:!=7#)))FL$\"1S>Wuz_myFL7$$!1+++v@)*=()F L$\"1U!e+L/KXg7F*7$$!1++DJ_fJ%)FL$\"1 $*fv4aaH9F*7$$!1+]Pfc\"F*7$$!1++](G3U9)FL$\"15Bh:sGg ;F*7$$!1+](=ZVC)zFL$\"19TcE%)QN!\\z\"F*7$$!1+ ++]4HsrFL$\"1Rev;sXZmPPFL$\"1%*R&R*oO9rFL 7$$!1,++]=$z9$FL$\"13wgKsyddFL7$$!1***\\iX/4]#FL$\"16[CPMV6WFL7$$!1*** \\(o8y%)=FL$\"11q26\"RiB$FL7$$!1****\\i:#>C\"FL$\"1?(3U&4-!4#FL7$$!1!* **\\7ev:lF-$\"1Hx>_UM%3\"FL7$$!1G++](o2[\"Fao$\"1\\N.rI%QX#Fao7$$\"1)* **\\P>:mkF-$!1B.KV*>g2\"FL7$$\"1++DcdQA7FL$!1N7\\.W:c?FL7$$\"1,+]PPBW= FL$!1gw>V\"R=;$FL7$$\"1******\\Nm'[#FL$!1jh@5F@$Q%FL7$$\"1****\\(yb^6$ FL$!1Z2eX5H'o&FL7$$\"1++vVVDBPFL$!1pDB#*z_zqFL7$$\"1++]7TW)R%FL$!1AD?$ ym_\"))FL7$$\"1)*****\\@80]FL$!1y^+GU-d5F*7$$\"1,++D6!Hl&FL$!1TmB/)*) \\E\"F*7$$\"1***\\P4w)RiFL$!1c`_[aql9F*7$$\"1-++vZf\")oFL$!19,B%p3Pn\" F*7$$\"1+]P4%)\\$=(FL$!1dJVwFL$!1ICKTEd(z\"F *7$$\"1**\\PfU3AxFL$!1ydl0Rf#z\"F*7$$\"1***\\7Ly4!yFL$!1R1vbtO#y\"F*7$ $\"1+++vkwezFL$!1nHTj3eVk(FL7$$\"1**\\P4wic!*FL$!1&*4AXk6Y[FL7$$\"1*\\i:gI\"=#* FL$!1*R&\\')psnBFL7$$\"1***\\PfL'z$*FL$!19^cUNnOrF-7$$\"1[7.#)oSd%*FL$ !12`A\"QSA%HF-7$$\"1)\\7.=+5F*F+7$$\"1+]7ed*>.\"F*F+7$$\"1++DE&4Q1\"F*F+7$$\"1 +]P%>5p7\"F*F+7$$\"1+++bJ*[=\"F*F+7$$\"1++Dr\"[8D\"F*F+7$$\"1+++Ijy58F *F+7$$\"1+]P/)fTP\"F*F+7$$\"1+]i0j\"[V\"F*F+7$$\"1+++++++:F*F+-%'COLOU RG6&%$RGBG$\"*++++\"!\")\"\"!Fhel-F$6$7gp7$F($\"1+++++++?F-7$F/F]fl7$F 2F]fl7$F5F]fl7$F8F]fl7$F;F]fl7$F>F]fl7$FAF]fl7$FGF]fl7$F]pF]fl7$FaqF]f l7$FerF]fl7$FctF]fl7$F]uF]fl7$FbuF]fl7$FguF]fl7$$!1**\\i!z%o9`FLF]fl7$ F\\v$\"1\"o))[#R&QS&!#d7$$!1+]7GY/0\\FL$\"191)*Q`ZqN!#B7$$!1++D\")>XL[ FL$\"1OM_dQ=E&*!#>7$$!1+D\"ylbwz%FL$\"1djAu!)Rv\"*Fao7$$!1+]PM$f=w%FL$ \"1RD.<_gQUF-7$$!1+v$4,jgs%FL$\"1>VF=)H9E\"FL7$$!1++](om-p%FL$\"1Hvq9* *oHGFL7$$!1+]iSSn=YFL$\"1&z#3asmv%)FL7$$!1++v$R\"3ZXFL$\"1?GAS33*o\"F* 7$$!1+](ou)[vWFL$\"1w!GZ(*3zm#F*7$Fav$\"1')R&G^R0l$F*7$$!1]7.Koh?VFL$ \"1&\\?]97@o%F*7$$!1+D1kvLPUFL$\"1!3C['>/JbF*7$$!1]P4'HeS:%FL$\"1JjBI> \"=='F*7$$!1+]7G!z22%FL$\"1c\"*RhYAZmF*7$$!1]i:g(*\\()RFL$\"1-Z%G(Q9_p F*7$$!1+v=#\\?U!RFL$\"1tHBXGUCrF*7$$!1DJ?e3eiQFL$\"1%>S0N*=prF*7$$!1]( =UAT4#QFL$\"1Fk>LuX!>(F*7$$!1vVB!f,$zPFL$\"1()=W%)=+\">(F*7$Ffv$\"1Vwm M#eL<(F*7$$!1,]7.pzUMFL$\"1cxTHBU*p'F*7$F[w$\"1!flU-1'RfF*7$$!1+]7`\"= W#GFL$\"1*z'o]%4X0&F*7$F`w$\"1k>D^8/CUF*7$$!1****\\7H%G>#FL$\"1$yO*[b; 7NF*7$Few$\"1B$4XLJn(GF*7$Fjw$\"1-e0S?0^F-7$Fix$!1iPt%)3!or)FL7$F^y$!1$>c*fb??MFL$!1U?h#[gX k'F*7$Fbz$!1'=jouVM;(F*7$$\"1]P%[1`w!QFL$!1*Hl-rcF>(F*7$$\"1+v$fy^?*QF L$!16!))\\I3,9(F*7$$\"1\\7.20XwRFL$!1SsK=I.#)pF*7$$\"1+]7G#\\31%FL$!1# [h3\"pY\"p'F*7$$\"1](=#\\zCXTFL$!1V)yI;T%RiF*7$$\"1+DJqmkHUFL$!1ac1\"e %R*f&F*7$$\"1\\iS\"RXSJ%FL$!18V,T')>cZF*7$Fgz$!1[7;hinAPF*7$$\"1*\\(=< ,GuWFL$!1wOn#p&y%o#F*7$$\"1**\\(=7;,b%FL$!1*)*['QUc\\;F*7$$\"1*\\il7_f i%FL$!1lbmJHubxFL7$$\"1***\\78)y,ZFL$!1s-h99jNAFL7$$\"1[PfLhqRZFL$!18r g1oMj')F-7$$\"1)\\Pf8Cwx%FL$!1a'GLsO0J#F-7$$\"1[7GQ@a:[FL$!1AOTgxyLLFa o7$$\"1)*\\iS,Y`[FL$!1(*HG3`y&e\"F`hl7$$\"1)\\7`9'HH\\FL$!1EGCol28x!#F 7$F\\[lF]fl7$$\"1****\\Pm,H`FLF]fl7$Fa[lF]fl7$Ff[lF]fl7$F[\\lF]fl7$Fe \\lF]fl7$Fc^lF]fl7$Fg_lF]fl7$F[alF]fl7$FdclF]fl7$FjclF]fl7$F]dlF]fl7$F `dlF]fl7$FcdlF]fl7$FfdlF]fl7$FidlF]fl7$F\\elF]fl7$F_elF]fl-Fbel6&FdelF helFeelFhel-F$6$7]q7$F($\"1+++++++IF-7$F/Fghm7$F2Fghm7$F5Fghm7$F8Fghm7 $F;Fghm7$F>Fghm7$FAFghm7$FGFghm7$F]pFghm7$FaqFghm7$FerFghm7$FctFghm7$F ]uFghm7$FbuFghm7$FguFghm7$F\\vFghm7$FavFghm7$Fg[mFghm7$FfvFghm7$$!1+vo H%H-f$FLFghm7$Fh]mFghm7$$!1^P%)R13pLFLFghm7$$!1,DcwVO&H$FL$\"1(G!oGjXZ _!#I7$$!1w=#\\C1&eKFL$\"1p>SYjx8L!#@7$$!1^7G8\"[;A$FL$\"1(p:/i7FJ#Fao7 $$!1Q4YZ!>K?$FL$\"1EF9?pW49F-7$$!1E1k\")**y%=$FL$\"1x]x$y$=%H&F-7$$!18 .#e\"4OmJFL$\"1XiC!z9VW\"FL7$F[w$\"1cUsW'en:$FL7$$!1E1*yj#\\2JFL$\"1&o q-\"\\#=1\"F*7$$!1^7yDM0nIFL$\"1;b_y!3xL#F*7$$!1v=n8UhEIFL$\"1*Q)G&fTB ,%F*7$$!1+Dc,]<')HFL$\"1]?r.g:Fi^n7$F`w$\"1q8:w*Hdh\"Fi^n7$Fh ^m$\"1%G&eZs)=U\"Fi^n7$Few$\"1u8JWi#*Q6Fi^n7$$!1**\\il9Nj:FL$\"1w,N6SF r')F*7$Fjw$\"1'H5=cKqP'F*7$$!1\"**\\(=d[n%*F-$\"1LX1$ez4f%F*7$F_x$\"1] Y7k`\")GIF*7$$!1(**\\PM;>L$F-$\"1/+$>Hfd]\"F*7$Fdx$\"1J7R4D\\DmF-7$$\" 1(**\\7`P!fJF-$!1gpUj4>E9F*7$Fix$!1!e*Rf%=S+$F*7$$\"1(*****\\Z+X$*F-$! 1S=7z&oA_%F*7$F^y$!1\\bm39E]iF*7$$\"1+](ou4L`\"FL$!1wj?B_3R%)F*7$Fcy$! 1'ymyhqC5\"Fi^n7$Fb`m$!1WfY-2Y(R\"Fi^n7$Fhy$!1&R7Y\\qGh\"Fi^n7$$\"1\\P %[JWf_#FL$!1h)*>K/L=;Fi^n7$$\"1*\\(oz]AlDFL$!1$Ht'eJL:;Fi^n7$$\"1\\7`W e]/EFL$!1[>.'R&>-;Fi^n7$$\"1**\\P4myVEFL$!1]'\\(y;-x:Fi^n7$$\"1*\\i!R \"[Bs#FL$!1O=`\"4'4#[\"Fi^n7$Fj`m$!1s*R6hIJJ\"Fi^n7$$\"1\\PfL/>SGFL$!1 %Q%)ou3j>\"Fi^n7$$\"1*\\P%)>r%zGFL$!1n#)3.U2d5Fi^n7$$\"1\\7Gj>v=HFL$!1 >:7!p]&p*)F*7$$\"1**\\7GF.eHFL$!1]%>>tvT?(F*7$$\"1\\(oH\\8t*HFL$!1\\) \\s)HJg`F*7$$\"1*\\7yD%fOIFL$!10Q\\D\"*ysNF*7$$\"1\\ilA](e2$FL$!1NGkU \\#*=?F*7$F]z$!1&zx:ykGy)FL7$$\"1CcEZ>;`JFL$!1oz\"oV!fvDFL7$$\"1\\7.2 \"o6>$FL$!1q5JCl([\\$F-7$$\"1uozmUVLFLFghm7$FbamFghm7$$\"1+vo/(H7d$F LFghm7$FbzFghm7$FibmFghm7$FgzFghm7$F\\[lFghm7$Fa[lFghm7$Ff[lFghm7$F[\\ lFghm7$Fe\\lFghm7$Fc^lFghm7$Fg_lFghm7$F[alFghm7$FdclFghm7$FjclFghm7$F] dlFghm7$F`dlFghm7$FcdlFghm7$FfdlFghm7$FidlFghm7$F\\elFghm7$F_elFghm-Fb el6&FdelFhelFhelFeel-F$6$7gp7$F($\"1+++++++SF-7$F/Fhjn7$F2Fhjn7$F5Fhjn 7$F8Fhjn7$F;Fhjn7$F>Fhjn7$FAFhjn7$FGFhjn7$F]pFhjn7$FaqFhjn7$FerFhjn7$F ctFhjn7$F]uFhjn7$FbuFhjn7$FguFhjn7$F\\vFhjn7$FavFhjn7$FfvFhjn7$F[wFhjn 7$F`^mFhjn7$F`wFhjn7$$!1\\7GjnRiCFL$\"1>eCF-7$$!1\\PMx8Q&Q#FL$\"1 zz-c@P<7FL7$$!1u$f3`FhO#FL$\"1zd:V*pwF%FL7$$!1**\\P%otoM#FL$\"1&ef5f#R c5F*7$$!1\\iS\"*fO3BFL$\"1'=zw\"e6pMF*7$$!1*\\P%)He)pAFL$\"1sghHs*Q9(F *7$$!1\\(oag]8B#FL$\"1w()y\\MaQ6Fi^n7$Fh^m$\"1kHa&R@db\"Fi^n7$$!1\\7`> _La@FL$\"1#4\"49wOE>Fi^n7$$!1*\\il_Fe6#FL$\"1tE')HRoKAFi^n7$$!1\\PfL)> t2#FL$\"1j,vNWFL$\"1>D')3/!z$GFi^n7$Few$\"1ux84lOvGFi^n7$$!1*\\(=^\\d6Fi^n7$F_x$\"1E)emXl_R(F*7$F\\an$ \"143Rzk/(f$F*7$Fdx$\"18'o%=7^q:FL7$Fdan$!1ntD=`H/MF*7$Fix$!1Tp\")*zZ9 L(F*7$F\\bn$!1FCd*RI%Q6Fi^n7$F^y$!1'\\*Q.d\"p[>Fi^n7$Fdbn$!1:4DUXT'G#Fi^n7$$\"1+v=UFL$!1jbO'Q49(GFi^n7$$\"1]PMF$*ok>FL$!1(ftT_#*Q$GFi^n7$$\"1+]i!>T[+#FL $!1.*eh>7Sv#Fi^n7$$\"1]i!R0$*\\/#FL$!1Z\"z:2\\?i#Fi^n7$$\"1+v=<\\9&3#F L$!17@6H_1GCFi^n7$$\"1](o/y'HD@FL$!1-4FQnoj@Fi^n7$Fb`m$!1ag&ol_b#=Fi^n 7$$\"1]7.20g0AFL$!1](e`^J7U\"Fi^n7$$\"1+DJqBvXAFL$!1sU;li\"zx*F*7$$\"1 ]PfLU!fG#FL$!1T\")y#RK5]&F*7$$\"1**\\(o4cgK#FL$!1t/z!*yir@F*7$$\"1\\i: gz?mBFL$!1(>\"\\(3k)eUFL7$$\"1*\\PM#)fjS#FL$!1p-m*[Yme\"F-7$$\"1\\(=no 6lW#FL$!1BN " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "v := (n,x) -> n/A_*exp(1/(n^2*x^2-1));\nd iff(v(n,x),x$2):\nfactor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG f*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)*&*&9$\"\"\"-%$expG6#*&F0F0,&* &)F/\"\"#F0)9%F8F0F0!\"\"F0F;F0F0%#A_GF;F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*()%\"nG\"\"$\"\"\"-%$expG6#*&F)F)*&,&*&F'F)%\"xGF) F)!\"\"F)F),&F0F)F)F)F)F2F),&*&)F'\"\"%F))F1F7F)F(F2F)F)F)*(%#A_GF))F/ F7F))F3F7F)F2\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The maxim um and minimum points on the graph of " }{XPPEDIT 18 0 "v[n]*`'`(x);" "6#*&&%\"vG6#%\"nG\"\"\"-%\"'G6#%\"xGF(" }{TEXT -1 13 " occur where " }{XPPEDIT 18 0 "x = -1/(3^(1/4)*n);" "6#/%\"xG,$*&\"\"\"F'*&)\"\"$*&F' F'\"\"%!\"\"F'%\"nGF'F-F-" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "x = 1/ (3^(1/4)*n);" "6#/%\"xG*&\"\"\"F&*&)\"\"$*&F&F&\"\"%!\"\"F&%\"nGF&F," }{TEXT -1 14 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "subs(x=-1/(3^(1/4)*n),diff(v (n,x),x));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*()%\"nG\"\"#\"\" \")\"\"$#F+\"\"%F)-%$expG6#*&F)F),&*$-%%sqrtG6#F+F)#F)F+!\"\"F)F8F)F)* &%#A_GF))F2F(F)F8#F(F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalf(subs(\{n=1,A_=0.4439938162\}, %));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+`-H)z\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The corresponding maximum and minimum v alues are " }{XPPEDIT 18 0 "-B*n^2;" "6#,$*&%\"BG\"\"\"*$%\"nG\"\"#F&! \"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "B*n^2;" "6#*&%\"BG\"\"\"*$% \"nG\"\"#F%" }{TEXT -1 22 " respectively, where " }{XPPEDIT 18 0 "B = 2*exp(-(sqrt(3)+3)/2)*3^(3/4)/(3*A*(1/sqrt(3)-1)^2);" "6#/%\"BG**\"\" #\"\"\"-%$expG6#,$*&,&-%%sqrtG6#\"\"$F'F1F'F'F&!\"\"F2F')F1*&F1F'\"\"% F2F'*(F1F'%\"AGF',&*&F'F'-F/6#F1F2F'F'F2F&F2" }{TEXT -1 2 " " }{TEXT 270 1 "~" }{TEXT -1 13 " 1.798290253." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "These maximum and minimum values tend to " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT -1 4 " as " } {XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)i nfinityGF*F*F*" }{TEXT -1 57 ", at the same time as there horizontal l ocation tends to " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The following numerical example illustrates that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(v[n]*`'`(x)*f(x),x = -infinity .. i nfinity);" "6#-%$IntG6$*(&%\"vG6#%\"nG\"\"\"-%\"'G6#%\"xGF+-%\"fG6#F/F +/F/;,$%)infinityG!\"\"F6" }{TEXT -1 1 " " }{TEXT 279 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "- `f '`(0)" "6#,$-%$f~'G6#\"\"!!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "when " }{TEXT 280 1 "n" }{TEXT -1 10 " is large." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "A := 0.4439938162:\nvp := (n,x) ->-2*n^3*x/( A*(n^2*x^2-1)^2)*exp(1/(n^2*x^2-1));\nf := x -> (sqrt(x^2+7)+x*arctan( x+2))/(x^2+5);\n'D(f)(0)'=evalf(D(f)(0));\nInt('vp(100,x)*f(x)',x=-1/1 00..1/100);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vpGf*6$% \"nG%\"xG6\"6$%)operatorG%&arrowGF),$*&*()9$\"\"$\"\"\"9%F3-%$expG6#*& F3F3,&*&)F1\"\"#F3)F4FF3F3*&%\"AGF3)F9F!\"#F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arr owGF(*&,&-%%sqrtG6#,&*$)9$\"\"#\"\"\"F6\"\"(F6F6*&F4F6-%'arctanG6#,&F4 F6F5F6F6F6F6,&F2F6\"\"&F6!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/--%\"DG6#%\"fG6#\"\"!$\"+OuH9A!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$IntG6$*&-%#vpG6$\"$+\"%\"xG\"\"\"-%\"fG6#F+F,/F+;#!\"\"F*#F,F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+S)oU@#!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }