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"" -1 -1 "T imes" 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 "The mean and root mean square val ue of a function on an interval" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Pet er Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Versio n: 23.3.2007" }}{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 44 "The mean value of a func tion on an interval " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 28 "The mean (or average) value " }{XPPEDIT 18 0 "y[M]" "6#&%\"yG6#%\"MG" }{TEXT -1 27 " of finitely many numbers \+ " }{XPPEDIT 18 0 "y[1],y[2],` . . . `,y[n]" "6&&%\"yG6#\"\"\"&F$6#\" \"#%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 34 " is calculated from the formul a: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[M]=(y[1]+y[2 ]+` . . .`+y[n])/n" "6#/&%\"yG6#%\"MG*&,*&F%6#\"\"\"F,&F%6#\"\"#F,%'~. ~.~.GF,&F%6#%\"nGF,F,F3!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 80 "Such a mean value is a \"typical\" or \"representative value\" \+ for the data values " }{XPPEDIT 18 0 "y[1],y[2],` . . . `,y[n]" "6&&% \"yG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "How do we calcul ate the mean value of a quantity such as temperature which varies cont inuously over a period of time?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "For example the following graph shows how the temperature T varies with time " }{TEXT 263 1 "t" }{TEXT -1 71 " \+ (measured in hours) at a particular location during a 24 hour period. \+ " }}{PARA 0 "" 0 "" {TEXT -1 246 "An mean value for the temperature is also shown by the horizontal blue line. The mean value could be estim ated by taking temperature readings at regular time intervals, for exa mple every 20 minutes, and then calculating the mean of these values. \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 371 295 295 {PLOTDATA 2 "6&-%'CURVESG6$7fn7$$\"\"!F)$\"#5F)7$$\"3x********pJJ_!#=$\"3=a;<'[) *GL*!#<7$$\"3M+++]z0$y*F/$\"3>l+!))F27$$\"3++++qm>!\\\"F2$\"3aUVO 3S\")f#)F27$$\"3=+++!p![0?F2$\"3'Rw%)3R4Hx(F27$$\"3))*****\\l:$=DF2$\" 3EkvNvG!pM(F27$$\"3#)*****\\JzP*HF2$\"3l=8*HU^T+(F27$$\"3')*****\\R%4' [$F2$\"3'o/]cE\">-nF27$$\"31+++btC&*RF2$\"3#zF[3CflW'F27$$\"3')*****\\ XnF]%F2$\"3Cw&yfOy!\\iF27$$\"3J+++gU\"[-&F2$\"3MF**\\Z_m0hF27$$\"31+++ qLj%[&F2$\"38&RY\"=9^HgF27$$\"3V+++!y!G-gF2$\"3w^$*[+8++gF27$$\"3%)*** ****p`?_'F2$\"37B4U;]8ogF27$$\"3Q+++?7&H-(F2$\"3=s[i*H2;E'F27$$\"3#*** ***\\:9yZ(F2$\"3S\"fz\"pO)fa'F27$$\"36+++!=!p=!)F2$\"3(o522^x(=qF27$$ \"35+++?J)oZ)F2$\"3qM_`(\\PP`(F27$$\"3%)*****\\Vq)4!*F2$\"3cs<()3+$[E) F27$$\"3K******>Xl\"[*F2$\"3A2\"f\"\\&z/.*F27$$\"3_+++NkF****F2$\"3Gxl )3+`&)***F27$$\"31++]!\\<#\\5!#;$\"37+++\")\\V)4\"F_r7$$\"32+++vik+6F_ r$\"3:+++]DH,7F_r7$$\"33++]`R(y9\"F_r$\"3;+++2zu&H\"F_r7$$\"3)******\\ Q:))>\"F_r$\"3'*******p2j(R\"F_r7$$\"3)*****\\:#HF_r7$$\"3)*******Gb(=b\"F_r$\"3;Cus\"QBq4#F_r7$$\"3$)***** >d5/g\"F_r$\"3M#ewlV:c<#F_r7$$\"3/+++!4KAl\"F_r$\"3)zdNplFlC#F_r7$$\"3 !*****\\(3!>*p\"F_r$\"3?bVgZ%)=*H#F_r7$$\"39+++#eF0v\"F_r$\"3t%3V1*\\9 WBF_r7$$\"3%)****\\j@$))z\"F_r$\"3!GxW@n7WP#F_r7$$\"3/++]pVK\\=F_r$\"3 opN=^*zNR#F_r7$$\"3)****\\(\\q+u=F_r$\"3i^KM;4J)R#F_r7$$\"3!*******H(* o)*=F_r$\"39<8)zq&***R#F_r7$$\"3;+]Pp*4;\">F_r$\"3!o%\\8J/#*)R#F_r7$$ \"32++v3-`C>F_r$\"3aG^lO`D&R#F_r7$$\"3)***\\7[/XP>F_r$\"3WrAO`&3\"*Q#F _r7$$\"3!*****\\(oq.&>F_r$\"3A$\\(f^#*e!Q#F_r7$$\"3?++vR\"e_(>F_r$\"3? \"yFo_gyN#F_r7$$\"3:+++#fX,+#F_r$\"3qO([Da`tK#F_r7$$\"3-+++@w/^?F_r$\" 3v\"=6;wqIC#F_r7$$\"39++]b\"G:5#F_r$\"3'=G3Y'3pL@F_r7$$\"37+++CX\"z9#F _r$\"3F;gFrqF8?F_r7$$\"32+++P&y5?#F_r$\"3C,;ZA1[a=F_r7$$\"3/++]+Q&[A#F _r$\"3ik9Y&H#fB\\g\"F_r7$$\"31++]VyK*H#F_r$\"3S*Q^81z.^\"F_r7$$ \"3))*****RW!fBBF_r$\"3T&z+(*HAkT\"F_r7$$\"31++]WI&yM#F_r$\"3Z(e![^oA> 8F_r7$$\"3A++DAl#RP#F_r$\"3I)*R^1VK67F_r7$$\"#CF)$\"#6F)-%'COLOURG6&%$ RGBG$\"*++++\"!\")F(F(-F$6$7S7$F($\"3'yxxxxFlR\"F_r7$F-Fa^l7$F4Fa^l7$F 9Fa^l7$F>Fa^l7$FCFa^l7$FHFa^l7$FMFa^l7$FRFa^l7$FWFa^l7$FfnFa^l7$F[oFa^ l7$F`oFa^l7$FeoFa^l7$FjoFa^l7$F_pFa^l7$FdpFa^l7$FipFa^l7$F^qFa^l7$FcqF a^l7$FhqFa^l7$F]rFa^l7$FcrFa^l7$FhrFa^l7$F]sFa^l7$FbsFa^l7$FgsFa^l7$F \\tFa^l7$FatFa^l7$FftFa^l7$F[uFa^l7$F`uFa^l7$FeuFa^l7$FjuFa^l7$F_vFa^l 7$FdvFa^l7$FivFa^l7$F^wFa^l7$FhwFa^l7$F\\yFa^l7$FfyFa^l7$F[zFa^l7$F`zF a^l7$FezFa^l7$FjzFa^l7$Fd[lFa^l7$F^\\lFa^l7$Fh\\lFa^l7$Fb]lFa^l-Fg]l6& Fi]lF(F(Fj]l-%+AXESLABELSG6'%2time,~t,~in~hoursG%:temperature,~T,~in~d eg.~CG-%%FONTG6#%(DEFAULTG%+HORIZONTALG%)VERTICALG-%%VIEWG6$;F(Fb]l;F( $\"#DF)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 43.000000 0 0 "Curv e 1" "Curve 2" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "Suppose that we have a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 24 " defined on an interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\" aG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 90 "To arrive at \+ a reasonable way of obtaining an mean value for the function on the in terval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 38 " we con sider subdividing the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"b G" }{TEXT -1 6 " into " }{TEXT 266 1 "n" }{TEXT -1 32 " of subinterval s of equal width " }{TEXT 264 1 "h" }{TEXT -1 19 " by equally spaced \+ " }{TEXT 265 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "x[0]=a,x[1], x[2],` . . . `,x[n]=b" "6'/&%\"xG6#\"\"!%\"aG&F%6#\"\"\"&F%6#\"\"#%(~. ~.~.~G/&F%6#%\"nG%\"bG" }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "h=(b-a)/ n" "6#/%\"hG*&,&%\"bG\"\"\"%\"aG!\"\"F(%\"nGF*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "n=(b-a)/h" "6#/%\"nG*&,&%\"bG\"\"\"%\"aG!\"\"F(%\"hGF* " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Pick " }{TEXT 267 1 " x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "x[1]*`*`,x[2]*`*`,` . . . `, x[n]*`*`" "6&*&&%\"xG6#\"\"\"F'%\"*GF'*&&F%6#\"\"#F'F(F'%(~.~.~.~G*&&F %6#%\"nGF'F(F'" }{TEXT -1 26 " in successive intervals " }{XPPEDIT 18 0 "[x[0],x[1]],[x[1],x[2]],` . . . `,[x[n-1],x[n]]" "6&7$&%\"xG6#\" \"!&F%6#\"\"\"7$&F%6#F*&F%6#\"\"#%(~.~.~.~G7$&F%6#,&%\"nGF*F*!\"\"&F%6 #F6" }{TEXT -1 91 ", in a systematic way. For example we could pick t he mid-points of the intervals so that " }{XPPEDIT 18 0 "x[i]*`*`=(x[ i-1]+x[i])/2" "6#/*&&%\"xG6#%\"iG\"\"\"%\"*GF)*&,&&F&6#,&F(F)F)!\"\"F) &F&6#F(F)F)\"\"#F0" }{TEXT -1 11 " for each " }{TEXT 268 1 "i" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 13 "" 1 "" {GLPLOT2D 587 83 83 {PLOTDATA 2 "6>-%'CURVESG6$7$7$$\"\"!F)F(7$$\"\"'F )F(-%'COLOURG6&%$RGBGF)F)F)-F$6$7$7$F($!\"\"F)7$F($\"\"\"F)F--F$6$7$7$ F8F57$F8F8F--F$6$7$7$$\"\"#F)F57$FCF8F--F$6$7$7$$\"\"&F)F57$FJF8F--F$6 $7$7$F+F57$F+F8F--F$6$7$7$$\"3++++++++]!#=$!3++++++++]FX7$FVFV-F.6&F0F (F($\"*++++\"!\")-F$6$7$7$$\"3++++++++:!#infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF *F*F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "delta" "6#%&deltaG" } {XPPEDIT 18 0 "x->0" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F *" }{TEXT -1 11 ", that is, " }}{PARA 0 "" 0 "" {TEXT -1 4 "the " } {TEXT 259 4 "mean" }{TEXT -1 5 " (or " }{TEXT 259 7 "average" }{TEXT -1 2 ") " }{TEXT 259 5 "value" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[ a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 2 " f" }{XPPEDIT 18 0 "``[m] = 1/(b-a);" "6#/&%!G6#%\"mG*&\"\"\"F), &%\"bGF)%\"aG!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x=a..b) " "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 2 ". 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"Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" }}{TEXT -1 2 " " }} {PARA 257 "" 0 "" {TEXT -1 7 "If the " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 38 " is greater than or equal to zero for " } {XPPEDIT 18 0 "a<=x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b" "6#1%!G%\" bG" }{TEXT -1 18 ", the mean value f" }{XPPEDIT 18 0 "``[m];" "6#&%!G6 #%\"mG" }{TEXT -1 71 " is such that area of the rectangular region und er the horizontal line " }{XPPEDIT 18 0 "y=``" "6#/%\"yG%!G" }{TEXT -1 1 "f" }{XPPEDIT 18 0 "``[m];" "6#&%!G6#%\"mG" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 38 " is equal to the area under the \+ graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 6 " \+ from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 238 "The limiting process in arriving at this integral defi nition may be illustrated with reference to the temperature example by considering the temperature readings to be made more and more frequen tly, say every minute, or every second etc. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 31 "The mean value of the function " }{XPPEDIT 18 0 "f(x) = x^2+1; " "6#/-%\"fG6#%\"xG,&*$F'\"\"#\"\"\"F+F+" }{TEXT -1 19 " over the inte rval " }{XPPEDIT 18 0 "[-1,2]" "6#7$,$\"\"\"!\"\"\"\"#" }{TEXT -1 6 " \+ is: " }}{PARA 256 "" 0 "" {TEXT -1 1 "f" }{XPPEDIT 18 0 "``[m] = 1/(2 -(-1));" "6#/&%!G6#%\"mG*&\"\"\"F),&\"\"#F),$F)!\"\"F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(x^2+1),x = -1 .. 2);" "6#-%$IntG6$-%!G6#,&* $%\"xG\"\"#\"\"\"F-F-/F+;,$F-!\"\"F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/3" "6#/%!G*&\"\"\"F&\"\"$!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(x^3/3+x);" "6#-%!G6#,&*&%\"xG\" \"$F)!\"\"\"\"\"F(F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2,``] ,[-1,`` ])" "6#-%*PIECEWISEG6$7$\"\"#%!G7$,$\"\"\"!\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``(8/3+2-(-1/3-1)) = 2;" "6#/-%!G6#,(*&\"\")\"\"\"\" \"$!\"\"F*\"\"#F*,&*&F*F*F+F,F,F*F,F,F-" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "f := x -> x^2+1:\na := -1: b := 2:\nInt(f( x),x=a..b)/(b-a);\nfm := value(%);\np1 := plot(f(x),x=a..b,filled=true ,color=COLOR(RGB,.9,.9,.95)):\np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b, f(b)]]],color=black):\np3 := plot([f(x),fav],x=-1.5..2.5,color=[red,bl ue]):\nplots[display]([p1,p2,p3],tickmarks=[4,7],view=[-1.5..2.5,0..6] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&-%$IntG6$,&*$) %\"xG\"\"#F&F&F&F&/F.;!\"\"F/F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#fmG\"\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 418 308 308 {PLOTDATA 2 "6*-% )POLYGONSG6U7&7$$!\"\"\"\"!F*7$$!3[*****\\P&3Y$*!#=F*7$F,$\"3o))yO=J\\ t=!#<7$F($\"\"#F*7&F+7$$!3C++Dcx6x()F.F*7$F8$\"3Q!zq5'zPq7$$!3A****\\P\"\\J\\(F.F*7 $FF$\"3))p\"o*RGZh:F2FA7&FE7$$!3g***\\7V0@&oF.F*7$FM$\"3Yd'4%)[8&p9F2F H7&FL7$$!3w++DcexdiF.F*7$FT$\"3Ca1n'e(f\"R\"F2FO7&FS7$$!3j***\\i+#QUcF .F*7$Fen$\"3mPX/ZZO=8F2FV7&FZ7$$!3$****\\i!3%f+&F.F*7$F\\o$\"37*ycNV%f ]7F2Fgn7&F[o7$$!3;++D\"oS:P%F.F*7$Fco$\"3AP#y#zO5\">\"F2F^o7&Fbo7$$!3h *****\\<#)*=PF.F*7$Fjo$\"3Ax'zTG3$Q6F2Feo7&Fio7$$!3#*****\\(G3U9$F.F*7 $Fap$\"3!o$=bd/'))4\"F2F\\p7&F`p7$$!3Y*****\\-\\r\\#F.F*7$Fhp$\"3_%eID `dB1\"F2Fcp7&Fgp7$$!3?+++vGVZ=F.F*7$F_q$\"3w2jF#3IT.\"F2Fjp7&F^q7$$!3_ *****\\(4J@7F.F*7$Ffq$\"3Val(\\+;\\,\"F2Faq7&Feq7$$!3;,+]iIKFl!#>F*7$F ]r$\"3TCij%fgU+\"F2Fhq7&F\\r7$$\"3(R,++]siL#!#?F*7$Fer$\"3<%>p\"ea++5F 2F`r7&Fdr7$$\"3K,+++!R5'fF_rF*7$F]s$\"3;_ff)R`N+\"F2Fhr7&F\\s7$$\"3!)* **\\P/QBE\"F.F*7$Fds$\"3z&)pOt\\$f,\"F2F_s7&Fcs7$$\"39******\\\"o?&=F. 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\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/x,x = 1 .. 5);" "6#-%$In tG6$*&\"\"\"F'%\"xG!\"\"/F(;F'\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = 1/4;" "6#/%!G*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "l n*x;" "6#*&%#lnG\"\"\"%\"xGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWI SE([5, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$\"\"&%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/4;" "6#/%!G*&\"\"\"F&\"\"%!\"\"" } {XPPEDIT 18 0 "``(ln*5-ln*1) = 1/4;" "6#/-%!G6#,&*&%#lnG\"\"\"\"\"&F*F **&F)F*F*F*!\"\"*&F*F*\"\"%F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln*5;" "6#*&%#lnG\"\"\"\"\"&F%" }{TEXT -1 1 " " }{TEXT 272 1 "~" }{TEXT -1 15 " 0.4023594781. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 424 "f := x -> 1/x:\na := 1: b := 5:\nInt(f(x),x=a..b)/(b-a);\nvalue(%);\nfm := eval f(evalf[14](%));\np1 := plot(f(x),x=a..b,filled=true,color=COLOR(RGB,. 9,.9,.95)):\np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=blac k):\np3 := plot([f(x),fav],x=0..5.5,y=0..1,color=[red,blue]):\np4 := p lot([[b,f(b)],[b,fav]],color=black,linestyle=2):\nplots[display]([p1,p 2,p3,p4],tickmarks=[4,4],view=[0..5.5,0..1.3],\n labels=[`x`,` y`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F&-%$IntG6$*& F&F&%\"xG!\"\"/F,;F&\"\"&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&# \"\"\"\"\"%F&-%#lnG6#\"\"&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f mG$\"+\"y%fBS!#5" }}{PARA 13 "" 1 "" {GLPLOT2D 275 299 299 {PLOTDATA 2 "6+-%)POLYGONSG6W7&7$$\"\"\"\"\"!F*7$$\"3ALLL3VfV5!#*F2F/7&F57$$ \"3&****\\7z>^7\"F.F*7$F=$\"3Knp*3\\Uz)))F2F87&F<7$$\"3WLL$e'40j6F.F*7 $FD$\"3'R\\hu[v!)f)F2F?7&FC7$$\"3ommm6hO[7F.F*7$FK$\"3sk!H:dq/,)F2FF7& FJ7$$\"3xmmm\"yYUL\"F.F*7$FR$\"3Er))yq`'[\\(F2FM7&FQ7$$\"3CLL$eF>(>9F. 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Fi_n7$$\"33nmm\")pm$e%F.Fi_n7$$\"3dm;zkuJ+ZF.Fi_n7$$\"3OL3-B?+;[F.Fi_n 7$$\"3T++] " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 31 "The mean value of the function " }{XPPEDIT 18 0 "f(x) = s in*x;" "6#/-%\"fG6#%\"xG*&%$sinG\"\"\"F'F*" }{TEXT -1 19 " over the in terval " }{XPPEDIT 18 0 "[0, Pi];" "6#7$\"\"!%#PiG" }{TEXT -1 7 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 "f" }{XPPEDIT 18 0 "``[m] = 1/(Pi-0 );" "6#/&%!G6#%\"mG*&\"\"\"F),&%#PiGF)\"\"!!\"\"F-" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sin*x,x = 0 .. Pi);" "6#-%$IntG6$*&%$sinG\"\"\"%\"x GF(/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*& \"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-cos*x);" "6#-% !G6#,$*&%$cosG\"\"\"%\"xGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIEC EWISE([Pi, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$%#PiG%!G7$\"\"!F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G*&\"\"\"F&%#PiG!\"\" " }{XPPEDIT 18 0 "``(-cos*Pi-(-cos*0)) = 2/Pi;" "6#/-%!G6#,&*&%$cosG\" \"\"%#PiGF*!\"\",$*&F)F*\"\"!F*F,F,*&\"\"#F*F+F," }{TEXT -1 1 " " } {TEXT 273 1 "~" }{TEXT -1 15 " 0.6366197724. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 430 "f := x -> sin(x):\na := 0: b := Pi:\nInt(f(x),x=a ..b)/(b-a);\nvalue(%);\nfm := evalf(evalf[14](%));\np1 := plot(f(x),x= a..b,filled=true,color=COLOR(RGB,.9,.9,.95)):\np2 := plot([f(x),fav],x =a..b,y=0..1,color=[red,blue]):\np3 := plot([[b,f(b)],[b,fav]],color=b lack,linestyle=2):\nplots[display]([p1,p2,p3],labels=[`x`,`y`],ytickma rks=2,\n xtickmarks=[0=`0`,evalf(Pi/2)=`p/2`,evalf(Pi)=`p`],\n fo nt=[SYMBOL,10],labelfont=[HELVETICA,10]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$-%$sinG6#%\"xG/F*;\"\"!%#PiG*$F.!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"%#PiG!\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fmG$\"+Cx>mj!#5" }}{PARA 13 "" 1 "" {GLPLOT2D 425 198 198 {PLOTDATA 2 "6*-%)POLYGONSG6&7^q7$$\"\"!F)F)7$$ \"3%)eD2LzxZo!#>F)7$$\"3)\\$px*G*f!G\"!#=F)7$$\"3+5@exGm]>F1F)7$$\"3[9 9!=3o^i#F1F)7$$\"35!\\D0[nkH$F1F)7$$\"37\"=Za&z%)=RF1F)7$$\"3edXa()oGj XF1F)7$$\"3W%3**Hbm(H_F1F)7$$\"37PRr4)3T*eF1F)7$$\"3y\"[)yykYxlF1F)7$$ \"3[s'ocGo$zrF1F)7$$\"3UQ0;gr'p&yF1F)7$$\"3vFMt?$[t`)F1F)7$$\"3u\"p30k @I>*F1F)7$$\"3#*R7\\HeV)y*F1F)7$$\"3#G[))*)3W'\\5!#%3*3tc9T7FfnF)7$$\"3Ey0DBA!*38FfnF)7$ $\"3qjE.#[AMP\"FfnF)7$$\"3*R:_MgU2W\"FfnF)7$$\"3\"Qu#)**[jD]\"FfnF)7$$ \"3=hw\"ymX#p:FfnF)7$$\"3mwM!*4(4&Q;FfnF)7$$\"3CDL:iU!))p\"FfnF)7$$\"3 o(R1/.CRw\"FfnF)7$$\"32Id)H7*>J=FfnF)7$$\"3Gfb7wY,(*=FfnF)7$$\"3cM5'zg %pg>FfnF)7$$\"3**oJ8:.SJ?FfnF)7$$\"3)**p!zPD$\\4#FfnF)7$$\"3kF17$Fis$\" 3!*=aVZ4buDF17$Ffs$\"3O2NfUFBSKF17$Fcs$\"3S5BwseM3QF17$F`s$\"33\"RnBm$ e5WF17$F]s$\"3o139l%\\$)*\\F17$Fjr$\"3O-G%)H)F17$Feq$\"3?%>'G5ccd')F17$Fbq$ \"3?af@Y>%y&*)F17$F_q$\"3ADf%RFx%\\#*F17$F\\q$\"3Sm0JN*4EZ*F17$Fip$\"3 ?VTu'[jGm*F17$Ffp$\"3Vz\\$>Q(39)*F17$Fcp$\"3[oG=d;==**F17$F`p$\"39vB0Z K3x**F17$F]p$\"274'Gxz)*****Ffn7$Fjo$\"3!>/b+VIn(**F17$Fgo$\"3!f%R/z# \\b\"**F17$Fdo$\"3W%*yQT$\\e!)*F17$Fao$\"3'*))o?tV#*F17$Fhn$\"3oX3_YFIb*)F17$FZ$\"3Y4SV'zgCn)F 17$FW$\"3=)oD&\\m_)H)F17$FT$\"3C$=wEj'y^zF17$FQ$\"3Cf#G%e3SPvF17$FN$\" 3S'e5DeyJ2(F17$FK$\"3iG>vb;KylF17$FH$\"32tCIi;N8hF17$FE$\"3g4tHMSrebF1 7$FB$\"3OWs$HJ6Y*\\F17$F?$\"33g-l[Vb1WF17$F<$\"3b2`lK+J>QF17$F9$\"3'yF M)o\")3PKF17$F6$\"3/)Q8J`>^f#F17$F3$\"3/`P()fcJQ>F17$F/$\"3q>Km$*>5x7F 17$F+$\"3v9-&QTFC%oF-7$F(F(7\"-%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&%$RG BG$\"\"*!\"\"Fb^l$\"#&*!\"#-%'CURVESG6$7SFh]lFe]lFb]lF_]lF\\]lFi\\lFf \\lFc\\lF`\\lF]\\lFj[lFg[lFd[lFa[lF^[lF[[lFhzFezFbzF_zF\\zFiyFfyFcyF`y F]yFjxFgxFdxFaxF^xF[xFhwFewFbwF_wF\\wFivFfvFcvF`vF]vFjuFguFduFauF^uF[u Fgt-%'COLOURG6&Fa^l$\"*++++\"!\")F(F(-Fi^l6$7S7$F($\"3%******fj6`6$Ffn 7$F+Ff_l7$F/Ff_l7$F3Ff_l7$F6Ff_l7$F9Ff_l7$F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example \+ 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 278 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 47 "(i) Show that the mean value of the derivative " }{XPPEDIT 18 0 "`f ' `(x);" "6#-%$f~'G6#%\"xG" }{TEXT -1 15 " of a function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 18 " over an interval " } {XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "(f(b)-f(a))/(b-a);" "6#*&,&-%\"fG6#%\"bG\"\"\"-F&6#%\"aG!\"\"F), &F(F)F,F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 "(ii) Find the mean value of the derivative of the function " }{XPPEDIT 18 0 "f( x) = x*sqrt(x^2+3)+ln(exp(x)+sin*Pi*x)/(2+cos*Pi*x);" "6#/-%\"fG6#%\"x G,&*&F'\"\"\"-%%sqrtG6#,&*$F'\"\"#F*\"\"$F*F*F**&-%#lnG6#,&-%$expG6#F' F**(%$sinGF*%#PiGF*F'F*F*F*,&F0F**(%$cosGF*F " 0 "" {MPLTEXT 1 0 481 "g := x -> (x*sqrt(x^2+3)+ln(exp(x)+sin(Pi* x)))/(2+cos(Pi*x)):\nf := D(g);\na := 0: b := 1:\nInt(f(x),x=a..b)/(b- a);\n#value(%);\nfm := evalf(evalf[14](%));\np1 := plot(f(x),x=a..b,fi lled=true,color=COLOR(RGB,.9,.9,.95)):\np2 := plot([[[a,0],[a,f(a)]],[ [b,0],[b,f(b)]]],color=black):\np3 := plot([f(x),fav],x=0..1.5,y=0..1, color=[red,blue]):\np4 := plot([[b,f(b)],[b,fav]],color=black,linestyl e=2):\nplots[display]([p1,p2,p3,p4],tickmarks=[5,5],view=[0..1,0..5.3] ,\n labels=[`x`,`y`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&,(-%%sqrtG6#,&*$)9$\"\"#\"\" \"F7\"\"$F7F7*&F5F6F/!\"\"F7*&,&-%$expG6#F5F7*&-%$cosG6#*&%#PiGF7F5F7F 7FEF7F7F7,&F=F7-%$sinGFCF7F:F7F7,&F6F7FAF7F:F7**,&*&F5F7F/F7F7-%#lnG6# FFF7F7FI!\"#FGF7FEF7F7F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int G6$,&*&,(*$,&*$)%\"xG\"\"#\"\"\"F/\"\"$F/#F/F.F/*&F-F.F*#!\"\"F.F/*&,& -%$expG6#F-F/*&-%$cosG6#*&%#PiGF/F-F/F/F?F/F/F/,&F7F/-%$sinGF=F/F4F/F/ ,&F.F/F;F/F4F/**,&*&F-F/F*F1F/-%#lnG6#F@F/F/FC!\"#FAF/F?F/F//F-;\"\"!F /" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fmG$\"+++++I!\"*" }}{PARA 13 " " 1 "" {GLPLOT2D 366 260 260 {PLOTDATA 2 "6+-%)POLYGONSG6D7&7$$\"\"!F) F)7$$\"3emmm;arz@!#>F)7$F+$\"3CE,I/&*R_=!#<7$F($\"3ic&>P:\")y&>F17&F*7 $$\"3[LL$e9ui2%F-F)7$F7$\"3WkJJ]2*zx\"F1F.7&F67$$\"3nmmm\"z_\"4iF-F)7$ F>$\"3.t)*[g:\"4r\"F1F97&F=7$$\"3[mmmT&phN)F-F)7$FE$\"3Wvp,g#R*e;F1F@7 &FD7$$\"3CLLe*=)H\\5!#=F)7$FL$\"3_0HoeM&4i\"F1FG7&FK7$$\"3gmm\"z/3uC\" FNF)7$FT$\"3%3l?(ex!of\"F1FO7&FS7$$\"3%)***\\7LRDX\"FNF)7$Fen$\"3M*4Vl hK@e\"F1FV7&FZ7$$\"3]mm\"zR'ok;FNF)7$F\\o$\"3/zCUraFx:F1Fgn7&F[o7$$\"3 w***\\i5`h(=FNF)7$Fco$\"31Z]T3GJ#e\"F1F^o7&Fbo7$$\"3WLLL3En$4#FNF)7$Fj 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iKF17$$\"3d*****\\2Mg#fFN$\"3.H`K*\\uz_$F17$$\"3K+](=xZ&\\iFN$\"334T+x n7LQF17$$\"3Q+]i:$4wb'FN$\"3?%3)R^JEHTF17$$\"3Y++v=#R!zoFN$\"3]k;?6\")FN$\" 3/V@z=wsL#F17$$ \"3&)**\\(=$3X;5F1$\"3)=Aq34_(*)=F17$$\"3.++Dc]kK5F1$\"3ep(480MgV\"F17 $$\"3')\\(o/V>t/\"F1$\"3%eI/(oi]E5F17$$\"3%**\\(o/Q*>1\"F1$\"3qu;ay$4q C'FN7$$\"3+](=C,$*RX\"F17$$\"3#****\\73\"o'=\"F1$!3O,V $*)4Yv(=F17$$\"3$**\\(oz;)*=7F1$!3AQQ%\\(4Sw@F17$$\"38]PMPj`M7F1$!3g3K \"H())prAF17$$\"35+++&*44]7F1$!3b,$)eOfUQBF17$$\"3;+D1zy*fE\"F1$!3%))[ #*fwb(zBF17$$\"3-+]7jZ!>G\"F1$!3W*3?&o>n'R#F17$$\"3/]i:I*zwH\"F1$!3]y( 4zB=@R#F17$$\"34+v=(4bMJ\"F1$!3MpRtA[,pBF17$$\"33]PMP3&zK\"F1$!3*e=,Vx DOL#F17$$\"32++]xlWU8F1$!3->fu!=!f'G#F17$$\"3/+]i&3ucP\"F1$!3TH2]JG%Q9 #F17$$\"3++++lJR09F1$!3ab2_*3'H')>F17$$\"3.+v=-*zqV\"F1$!3M%)o,6(G))z \"F17$$\"3/+D\"G:3uY\"F1$!3H@[D,wi3;F17$$\"3++++++++:F1$!3%*[xB^_\"*)R \"F1-F[fl6&F^el$\"*++++\"!\")F(F(-Ffel6$7S7$F($\"3%******fj6`6$F17$Fhf lFe[n7$F]glFe[n7$FbglFe[n7$FgglFe[n7$F\\hlFe[n7$FahlFe[n7$FfhlFe[n7$F[ ilFe[n7$F`ilFe[n7$FeilFe[n7$FjilFe[n7$F_jlFe[n7$FdjlFe[n7$FijlFe[n7$F^ [mFe[n7$Fc[mFe[n7$Fh[mFe[n7$F]\\mFe[n7$Fb\\mFe[n7$Fg\\mFe[n7$F\\]mFe[n 7$Fa]mFe[n7$Ff]mFe[n7$F[^mFe[n7$Fe^mFe[n7$F__mFe[n7$Fc`mFe[n7$F]amFe[n 7$FbamFe[n7$F\\bmFe[n7$FfbmFe[n7$F`cmFe[n7$FjcmFe[n7$FddmFe[n7$F^emFe[ n7$FhemFe[n7$F]fmFe[n7$FbfmFe[n7$FgfmFe[n7$FagmFe[n7$F[hmFe[n7$FehmFe[ n7$F_imFe[n7$FdimFe[n7$FiimFe[n7$F^jmFe[n7$FcjmFe[n7$FhjmFe[n-F[fl6&F^ elF(F(F^[n-Ffel6%7$Fafl7$Fg^lFe[nFjel-%*LINESTYLEG6#\"\"#-%*AXESTICKSG 6$\"\"&Fd_n-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F( Fg^l;F($\"#`Fael" 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" "Curve 5" "Curve 6" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 274 8 "Q uestion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 11 "The number " }{TEXT 281 1 "N" }{TEXT -1 74 " of atoms remaining in a sample of a ra dioactive material after a time of " }{TEXT 280 1 "t" }{TEXT -1 31 " s ecs. has elapsed is given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "N=N[0]*exp(-lambda*t)" "6#/%\"NG*&&F$6#\"\"!\"\"\"-%$ex pG6#,$*&%'lambdaGF)%\"tGF)!\"\"F)" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "N[0]" "6#&%\"NG6#\"\"!" }{TEXT -1 48 " is the number of atoms present initially (when " }{XPPEDIT 18 0 " t=0" "6#/%\"tG\"\"!" }{TEXT -1 6 ") and " }{XPPEDIT 18 0 "lambda" "6#% 'lambdaG" }{TEXT -1 16 " is a constant. " }}{PARA 0 "" 0 "" {TEXT -1 67 "Find the mean number of atoms in the sample over the interval from " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "t=1/lambda" "6#/%\"tG*&\"\"\"F&%'lambdaG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 8 "Solutio n" }{TEXT -1 3 ": " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "N[av]=1/(1/lam bda-0)" "6#/&%\"NG6#%#avG*&\"\"\"F),&*&F)F)%'lambdaG!\"\"F)\"\"!F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(N[0]*exp(-lambda*t),t=0..1/lambda) " "6#-%$IntG6$*&&%\"NG6#\"\"!\"\"\"-%$expG6#,$*&%'lambdaGF+%\"tGF+!\" \"F+/F2;F**&F+F+F1F3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = N[0]*lambda;" "6#/%!G*&&%\"NG6#\"\"!\"\"\"%'la mbdaGF*" }{XPPEDIT 18 0 " ``(-exp(-lambda*t)/lambda)" "6#-%!G6#,$*&-%$ expG6#,$*&%'lambdaG\"\"\"%\"tGF.!\"\"F.F-F0F0" }{TEXT -1 2 " " } {XPPEDIT 18 0 "PIECEWISE([1/lambda,``],[0,``])" "6#-%*PIECEWISEG6$7$*& \"\"\"F(%'lambdaG!\"\"%!G7$\"\"!F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -N[0]*exp(-lambda*t);" "6#/%!G,$ *&&%\"NG6#\"\"!\"\"\"-%$expG6#,$*&%'lambdaGF+%\"tGF+!\"\"F+F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1/lambda, ``],[0, ``])" "6#-%*PIEC EWISEG6$7$*&\"\"\"F(%'lambdaG!\"\"%!G7$\"\"!F+" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -N[0]*(exp(-1)-e xp(0));" "6#/%!G,$*&&%\"NG6#\"\"!\"\"\",&-%$expG6#,$F+!\"\"F+-F.6#F*F1 F+F1" }{XPPEDIT 18 0 " ``=-N[0]*(exp(-1)-1)" "6#/%!G,$*&&%\"NG6#\"\"! \"\"\",&-%$expG6#,$F+!\"\"F+F+F1F+F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= N[0]*(1-exp(-1))" "6#/%!G*&&%\"NG6#\"\"!\"\"\",&F*F*-%$expG6#,$F*!\"\" F0F*" }{TEXT -1 1 " " }{TEXT 282 1 "~" }{TEXT -1 2 " 0" }{XPPEDIT 18 0 ".63212*N[0]" "6#*&-%&FloatG6$\"&7K'!\"&\"\"\"&%\"NG6#\"\"!F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "f := x -> N [0]*exp(-lambda*t):\na := 0: b := 1/lambda:\n1/(b-a)*Int(f(t),t=a..b); \nN[av] = value(%);\n``=evalf(evalf(rhs(%),13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$IntG6$*&&%\"NG6#\"\"!\"\"\"-%$expG6#,$*&%'lambdaGF ,%\"tGF,!\"\"F,/F3;F+*&F,F,F2F4F,F2F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"NG6#%#avG,$*&&F%6#\"\"!\"\"\",&-%$expG6#!\"\"F-F-F2F-F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&$\"+)e07K'!#5\"\"\"&%\"NG6#\" \"!F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "The root mean square value of a \+ function on an interval " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 31 "root mean square (r.m.s.) value" }{TEXT -1 15 " of a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " on an interval " }{XPPEDIT 18 0 "[a,b ]" "6#7$%\"aG%\"bG" }{TEXT -1 48 " is the square root of the mean of t he function " }{XPPEDIT 18 0 "f(x)^2" "6#*$-%\"fG6#%\"xG\"\"#" }{TEXT -1 27 " on the interval, that is, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " f" }{XPPEDIT 18 0 "``[rms] = sqrt(``(1/ (b-a))*Int(f(x)^2,x = a .. b));" "6#/&%!G6#%$rmsG-%%sqrtG6#*&-F%6#*&\" \"\"F/,&%\"bGF/%\"aG!\"\"F3F/-%$IntG6$*$-%\"fG6#%\"xG\"\"#/F;;F2F1F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 276 19 "____ _______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 " " {TEXT -1 122 "The root mean square value of a periodic function is t he root mean square value calculated over a period of the function. " }}{PARA 15 "" 0 "" {TEXT -1 158 "The root mean square value of an alte rnating current (AC) is the direct current (DC) which produces the sam e heating effect as the given alternating current. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "We can calculate the ro ot mean square value of a sine wave " }{XPPEDIT 18 0 "f(t) = A*sin*ome ga*t;" "6#/-%\"fG6#%\"tG**%\"AG\"\"\"%$sinGF*%&omegaGF*F'F*" }{TEXT -1 16 " with amplitude " }{TEXT 277 1 "A" }{TEXT -1 12 " and period " }{XPPEDIT 18 0 "2*Pi/omega" "6#*(\"\"#\"\"\"%#PiGF%%&omegaG!\"\"" } {TEXT -1 13 " as follows. " }}{PARA 0 "" 0 "" {TEXT -1 32 "First we ca lculate the mean of " }{XPPEDIT 18 0 "f(t)^2=A^2*sin^2*omega*t" "6#/* $-%\"fG6#%\"tG\"\"#**%\"AGF)%$sinGF)%&omegaG\"\"\"F(F." }{TEXT -1 25 " over the interval from " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "t=2*Pi/omega" "6#/%\"tG*(\"\"#\"\"\" %#PiGF'%&omegaG!\"\"" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "omega/(2*Pi)" "6#*&%&omegaG\"\"\"*&\"\"#F%%#PiGF%! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)^2,t = 0 .. 2*Pi/omega) \+ = A^2*omega/(2*Pi);" "6#/-%$IntG6$*$-%\"fG6#%\"tG\"\"#/F+;\"\"!*(F,\" \"\"%#PiGF1%&omegaG!\"\"*(%\"AGF,F3F1*&F,F1F2F1F4" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sin^2*omega*t,t = 0 .. 2*Pi/omega)" "6#-%$IntG6$*(% $sinG\"\"#%&omegaG\"\"\"%\"tGF*/F+;\"\"!*(F(F*%#PiGF*F)!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = A^2*o mega/(2*Pi);" "6#/%!G*(%\"AG\"\"#%&omegaG\"\"\"*&F'F)%#PiGF)!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(1/2-cos*2*omega*t/2),t = 0 .. 2* Pi/omega);" "6#-%$IntG6$-%!G6#,&*&\"\"\"F+\"\"#!\"\"F+*,%$cosGF+F,F+%& omegaGF+%\"tGF+F,F-F-/F1;\"\"!*(F,F+%#PiGF+F0F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 20 "(using the formula: " }{XPPEDIT 18 0 "sin ^2*theta = 1/2-cos*2*theta/2;" "6#/*&%$sinG\"\"#%&thetaG\"\"\",&*&F(F( F&!\"\"F(**%$cosGF(F&F(F'F(F&F+F+" }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = A^2*omega/(2*Pi);" "6#/%!G*(%\" AG\"\"#%&omegaG\"\"\"*&F'F)%#PiGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(t/2-sin*2*omega*t/(4*omega))" "6#-%!G6#,&*&%\"tG\"\"\"\"\"#!\"\" F)*,%$sinGF)F*F)%&omegaGF)F(F)*&\"\"%F)F.F)F+F+" }{TEXT -1 2 " " } {XPPEDIT 18 0 "PIECEWISE([2*Pi/omega, ``],[0, ``]);" "6#-%*PIECEWISEG6 $7$*(\"\"#\"\"\"%#PiGF)%&omegaG!\"\"%!G7$\"\"!F-" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = A^2*omega/(2*Pi )" "6#/%!G*(%\"AG\"\"#%&omegaG\"\"\"*&F'F)%#PiGF)!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Pi/omega = A^2/2;" "6#/*&%#PiG\"\"\"%&omegaG!\"\"*&% \"AG\"\"#F+F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "The roo t mean square value is therefore: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "sqrt(A^2/2)=A/sqrt(2)" "6#/-%%sqrtG6#*&%\"AG\"\"#F)! \"\"*&F(\"\"\"-F%6#F)F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 54 "that is, the root mean square value of a sine wave is " }{XPPEDIT 18 0 "1/sqrt(2)" "6#*&\"\"\"F$-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 22 " tim es the amplitude. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A^2*omega/(2 *Pi)*Int(sin(omega*t)^2,t = 0 .. 2*Pi/omega);\nvalue(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&,$-%$IntG6$*$)-%$sinG6#*&%&ome gaGF&%\"tGF&F'F&/F3;\"\"!,$*(F'F&%#PiGF&F2!\"\"F&*(%\"AGF'F2F&F9F:F&F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"%\"AGF%\"\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 44 "We calculate the root mean square value o f " }{XPPEDIT 18 0 "f(x)=2*x^2" "6#/-%\"fG6#%\"xG*&\"\"#\"\"\"*$F'F)F *" }{TEXT -1 22 " on the interval from " }{XPPEDIT 18 0 "x=1" "6#/%\"x G\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "The mean of " } {XPPEDIT 18 0 "f(x)^2" "6#*$-%\"fG6#%\"xG\"\"#" }{TEXT -1 21 " on the \+ interval is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(4 -1);" "6#*&\"\"\"F$,&\"\"%F$F$!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((2*x^2)^2,x = 1 .. 4) = 1/3;" "6#/-%$IntG6$*$*&\"\"#\"\"\"*$%\"xG F)F*F)/F,;F*\"\"%*&F*F*\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int (4*x^4,x=1..4)" "6#-%$IntG6$*&\"\"%\"\"\"*$%\"xGF'F(/F*;F(F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/3" "6 #/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(4*x^5/5) " "6#-%!G6#*(\"\"%\"\"\"*$%\"xG\"\"&F(F+!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([4, ``],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6 %7$\"\"%%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``(4^6/5-4/5)=4/15" "6#/-%!G6#,&*&\"\"% \"\"'\"\"&!\"\"\"\"\"*&F)F-F+F,F,*&F)F-\"#:F," }{XPPEDIT 18 0 "``(4^5- 1)=4092/15" "6#/-%!G6#,&*$\"\"%\"\"&\"\"\"F+!\"\"*&\"%#4%F+\"#:F," } {XPPEDIT 18 0 "``=1364/5" "6#/%!G*&\"%k8\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The root mean square value is: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1364/5);" "6#-% %sqrtG6#*&\"%k8\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{TEXT 284 1 "~" } {TEXT -1 14 " 16.51665826. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "f := x -> 2*x^2:\na := 1: b := 4:\nInt(f(x)^2,x=a..b)/(b-a);\nvalue(%);\nf[rms]=sqrt(%);\nevalf(e valf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&-%$I ntG6$,$*&\"\"%F&)%\"xGF-F&F&/F/;F&F-F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%k8\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"fG6#%$rmsG,$ *(\"\"#\"\"\"\"\"&!\"\"\"%0<#F+F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"fG6#%$rmsG$\"+Eem^;!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 65 "We calculate the root mean square v alue of the periodic function " }{XPPEDIT 18 0 "f(x) = 1+2*cos*x;" "6# /-%\"fG6#%\"xG,&\"\"\"F)*(\"\"#F)%$cosGF)F'F)F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 14 "The period of " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\" \"%#PiGF%" }{TEXT -1 65 " so we calculate the root mean square value o n the interval from " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2*Pi;" "6#/%\"xG*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "The mean of " } {XPPEDIT 18 0 "f(x)^2" "6#*$-%\"fG6#%\"xG\"\"#" }{TEXT -1 21 " on the \+ interval is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(2 *Pi);" "6#*&\"\"\"F$*&\"\"#F$%#PiGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((1+2*cos*x)^2,x = 0 .. 2*Pi) = 1/(2*Pi);" "6#/-%$IntG6$*$,& \"\"\"F)*(\"\"#F)%$cosGF)%\"xGF)F)F+/F-;\"\"!*&F+F)%#PiGF)*&F)F)*&F+F) F2F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(1+4*cos*x+4*cos^2*x) ,x = 0 .. 2*Pi);" "6#-%$IntG6$-%!G6#,(\"\"\"F**(\"\"%F*%$cosGF*%\"xGF* F**(F,F**$F-\"\"#F*F.F*F*/F.;\"\"!*&F1F*%#PiGF*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 20 "(using the formula " }{XPPEDIT 18 0 "cos ^2*theta=1/2+1/2" "6#/*&%$cosG\"\"#%&thetaG\"\"\",&*&F(F(F&!\"\"F(*&F( F(F&F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*theta " "6#*(%$cosG\" \"\"\"\"#F%%&thetaGF%" }{TEXT -1 2 ") " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(2*Pi)" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(1+4*cos*x+2+2*cos*2*x),x \+ = 0 .. 2*Pi);" "6#-%$IntG6$-%!G6#,*\"\"\"F**(\"\"%F*%$cosGF*%\"xGF*F* \"\"#F***F/F*F-F*F/F*F.F*F*/F.;\"\"!*&F/F*%#PiGF*" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(2*Pi)" "6#/%!G* &\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(`` (3+4*cos*x+2*cos*2*x),x = 0 .. 2*Pi);" "6#-%$IntG6$-%!G6#,(\"\"$\"\"\" *(\"\"%F+%$cosGF+%\"xGF+F+**\"\"#F+F.F+F1F+F/F+F+/F/;\"\"!*&F1F+%#PiGF +" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=1/(2*Pi)" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(3*x+4*sin*x+sin*2*x)" "6#-%!G6#,(*&\"\"$\"\"\"%\"xGF )F)*(\"\"%F)%$sinGF)F*F)F)*(F-F)\"\"#F)F*F)F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([2*Pi, ``],[0, ``])" "6#-%*PIECEWISEG6$7$*&\" \"#\"\"\"%#PiGF)%!G7$\"\"!F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(2*Pi)" "6#/%!G*&\"\"\"F&*&\"\"#F& %#PiGF&!\"\"" }{XPPEDIT 18 0 "``(6*Pi-0) = 3;" "6#/-%!G6#,&*&\"\"'\"\" \"%#PiGF*F*\"\"!!\"\"\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The root mean square value is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT -1 1 " " } {TEXT 283 1 "~" }{TEXT -1 14 " 1.732050808. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "f := x -> 1+2*co s(x):\na := 0: b := 2*Pi:\nInt(f(x)^2,x=a..b)/(b-a);\nvalue(%);\nfrms \+ :=sqrt(%);\nevalf(evalf(%,15));\nplot([f(x),frms],x=a..b,y=-1..3,color =[red,blue],\n ytickmarks=2,xtickmarks=[0=`0`,evalf(Pi/2)=`p/2`,eva lf(Pi)=`p`,\n evalf(3*Pi/2)=`3p/2`,evalf(2*Pi)=`2p`],\n font=[ SYMBOL,10],labelfont=[HELVETICA,10]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&,$-%$IntG6$*$),&F&F&*&F'F&-%$cosG6#%\"xGF&F&F'F&/ F3;\"\"!,$*&F'F&%#PiGF&F&*$F9!\"\"F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%frmsG*$\"\"$#\"\"\" \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+330K$\"3;b)[ei7`*H!#<7$$\"3i]cC&eb&p8!#=$\"3)R&=([ Zs7)HF27$$\"3q5)>63x`'>F6$\"3=&fco5(\\hHF27$$\"3YqR*pd)>hDF6$\"3]c4/?/ wMHF27$$\"3Zs[E^dK,RF6$\"3'*>TS'H<(\\GF27$$\"3ab^NehL]_F6$\"3-0-)[=91t #F27$$\"35*QhW&\\$Hf'F6$\"3)y-&H3/&3e#F27$$\"3zS11.fpPyF6$\"3W!R%3O[^; CF27$$\"3upr'fwtl7*F6$\"3x\"**QB:\"HBAF27$$\"3!QRa&4L&f/\"F2$\"3#ff;^J a@+#F27$$\"3YjXwg<#)y6F2$\"3#>(RdLz-kF2$\"39_h3:Ay`CF67$$\"3#=uye<)G*4#F2$!3'[LsM+ZIY)!# ?7$$\"3Ua[#o!GC>AF2$!3m/(zt()y*y?F67$$\"3-YwJf&y(eBF2$!3K*HwEiK'yTF67$ $\"3oNrpV8H#[#F2$!3%oaWjM3%3eF67$$\"3EzY)QW/yh#F2$!3(QIcQU8'=tF67$$\"3 5v)>8'\\%ou#F2$!3%fI#G,D(=Y)F67$$\"3w$GCS?&[\")GF2$!3_;cC()pCF$*F67$$ \"3*Q3$\\!41L%HF2$!3:]wJF463'*F67$$\"3Z%)='p(p70IF2$!3&H:A9-gS\")*F67$ $\"3g`,ta\"4=2$F2$!3Cvyg\"RA8&**F67$$\"3sA%)\\K8\\QJF2$!3#R(>:=Q!***** F67$$\"3#*3]^u`v2KF2$!3qWT154Cc**F67$$\"3c&fJlT>qF$F2$!3D'f*Gcg(o\")*F 67$$\"3e!=@(oRJPLF2$!3)pg9A&H:='*F67$$\"39l2\"4_3wR$F2$!3Mq,:r48[$*F67 $$\"3MOnGd![y_$F2$!3<^!pE[Cl_)F67$$\"3#*=4JU#)RiOF2$!3A[%4=Ms$[tF67$$ \"3%G,f%[$HSz$F2$!3CwLmSc8#*eF67$$\"3KoE+7#*Q@RF2$!3<$>9;*>8@UF67$$\"3 y(f0ii+G1%F2$!3EH%G!)*H<(4#F67$$\"3;fORr]')*=%F2$\"3oQqr&)3Ii=Fcq7$$\" 3R&p%*z[LbK%F2$\"3Is!\\f'oVaCF67$$\"3M'pExBp%[WF2$\"3MFa^a'oEy%F67$$\" 3oh/x$[qGe%F2$\"3i\"eofDmoT(F67$$\"3e;O;#eJ$4ZF2$\"3:(zV38_)Q**F67$$\" 3&)oO4o)>:%[F2$\"3)y:y-jWvD\"F27$$\"3Ou\"4%z!e2(\\F2$\"3q%3t'[#35^\"F2 7$$\"3*zX#[4&eg5&F2$\"3S$RGqlfrw\"F27$$\"3n*e![/*ojB&F2$\"3g,+\"f4g1+# F27$$\"3#ob8X5I'p`F2$\"3TgDzh+(=A#F27$$\"3PA\"GX$yy,bF2$\"3D()4zg+')>C F27$$\"39L1/jqABcF2$\"3mv91kc.!e#F27$$\"3mweKB,TidF2$\"3jw5.\"\\n[t#F2 7$$\"3'oIe;6(*o)eF2$\"3Kr`'4=+]%GF27$$\"3nKcu0ii>gF2$\"36Kb;nx$4$HF27$ $\"3G:=>Xb9$3'F2$\"3!Q)z:bt6gHF27$$\"3w)*zj%)[mYhF2$\"3O*HIO5\"R\")HF2 7$$\"3P***G'*3D\\@'F2$\"3%G$3j^BM&*HF27$$\"3)****>YH&=$G'F2F*-%'COLOUR G6&%$RGBG$\"*++++\"!\")F(F(-F$6$7S7$F($\"3?x)ov!30K " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 44 "Find the mean or mean value of the functi on " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 24 " on the giv en interval. " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "f( x)=4*x-x^2" "6#/-%\"fG6#%\"xG,&*&\"\"%\"\"\"F'F+F+*$F'\"\"#!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "[0, 3];" "6#7$\"\"!\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(b) " }{XPPEDIT 18 0 "f(x)=sqrt(x )" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[4 ,9]" "6#7$\"\"%\"\"*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(c ) " }{XPPEDIT 18 0 "f(x)=x^3" "6#/-%\"fG6#%\"xG*$F'\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[-1,2]" "6#7$,$\"\"\"!\"\"\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(d) " }{XPPEDIT 18 0 "f(x)=1/x^2" "6 #/-%\"fG6#%\"xG*&\"\"\"F)*$F'\"\"#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[1,5]" "6#7$\"\"\"\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(e) " }{XPPEDIT 18 0 "f(x)=1/(1+x^2)" "6#/-%\"fG6#%\"xG*& \"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[-2, 2]" "6#7$,$\"\"#!\"\"F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(f) " }{XPPEDIT 18 0 "f(x)=x/(x^2+1)" "6#/-%\"fG6#%\"xG*&F'\"\"\",&* $F'\"\"#F)F)F)!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0,4]" "6#7$\" \"!\"\"%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(g) " } {XPPEDIT 18 0 "f(x)=sin^2*x*cos*x" "6#/-%\"fG6#%\"xG**%$sinG\"\"#F'\" \"\"%$cosGF+F'F+" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0, Pi/2];" "6#7$ \"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 6 " (a) " }{XPPEDIT 18 0 "3;" "6#\"\"$" }{TEXT -1 8 " (b) \+ " }{XPPEDIT 18 0 "38/15" "6#*&\"#Q\"\"\"\"#:!\"\"" }{TEXT -1 1 " " } {TEXT 285 1 "~" }{TEXT -1 19 " 2.533333333 (c) " }{XPPEDIT 18 0 "5/4 " "6#*&\"\"&\"\"\"\"\"%!\"\"" }{TEXT -1 8 " (d) " }{XPPEDIT 18 0 "1 /5;" "6#*&\"\"\"F$\"\"&!\"\"" }{TEXT -1 8 " (e) " }{XPPEDIT 18 0 "1 /2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(2) " "6#-%'arctanG6#\"\"#" }{TEXT -1 1 " " }{TEXT 286 1 "~" }{TEXT -1 14 " 0.5535743589 " }}{PARA 0 "" 0 "" {TEXT -1 6 " (f) " }{XPPEDIT 18 0 "1/8" "6#*&\"\"\"F$\"\")!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(17) " "6#-%#lnG6#\"#<" }{TEXT -1 1 " " }{TEXT 287 1 "~" }{TEXT -1 21 " 0.3 541516680 (g) " }{XPPEDIT 18 0 "2/(3*Pi)" "6#*&\"\"#\"\"\"*&\"\"$F% %#PiGF%!\"\"" }{TEXT -1 1 " " }{TEXT 288 1 "~" }{TEXT -1 14 " 0.212206 5908 " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }} {PARA 0 "" 0 "" {TEXT -1 48 "Find the root mean square value of the fu nction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 24 " on the given interval. " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "f(x)=4*x-x^2" "6#/-%\"fG6#%\"xG,&*&\"\"%\"\"\"F'F+F+*$F'\"\"#!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0,3]" "6#7$\"\"!\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(b) " }{XPPEDIT 18 0 "f(x)=sqrt(x )" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[4 ,9]" "6#7$\"\"%\"\"*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(c ) " }{XPPEDIT 18 0 "f(x)=x^3" "6#/-%\"fG6#%\"xG*$F'\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[-1,2]" "6#7$,$\"\"\"!\"\"\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(d) " }{XPPEDIT 18 0 "f(x)=1/x^2" "6 #/-%\"fG6#%\"xG*&\"\"\"F)*$F'\"\"#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[1,5]" "6#7$\"\"\"\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(e) " }{XPPEDIT 18 0 "f(x)=(1+sin*x)" "6#/-%\"fG6#%\"xG,& \"\"\"F)*&%$sinGF)F'F)F)" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0, 2*Pi]; " "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(f) " }{XPPEDIT 18 0 "f(x)=sin*x+cos*x" "6#/-%\"fG6#%\"xG ,&*&%$sinG\"\"\"F'F+F+*&%$cosGF+F'F+F+" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "1/5" "6#*&\" \"\"F$\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(255)" "6#-%%sqr tG6#\"$b#" }{TEXT -1 1 " " }{TEXT 289 1 "~" }{TEXT -1 20 " 3.193743885 (b) " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "sqrt(26)" "6#-%%sqrtG6#\"#E" }{TEXT -1 1 " " }{TEXT 290 1 "~" }{TEXT -1 19 " 2.549509757 (c) " }{XPPEDIT 18 0 "1/7" "6#* &\"\"\"F$\"\"(!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(301)" "6#-%% sqrtG6#\"$,$" }{TEXT -1 1 " " }{TEXT 291 1 "~" }{TEXT -1 13 " 2.478478 796 " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "1/75" "6#*& \"\"\"F$\"#v!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(465)" "6#-%%sq rtG6#\"$l%" }{TEXT -1 1 " " }{TEXT 292 1 "~" }{TEXT -1 20 " 0.28751811 54 (e) " }{XPPEDIT 18 0 "sqrt(3/2+4/Pi)" "6#-%%sqrtG6#,&*&\"\"$\"\" \"\"\"#!\"\"F)*&\"\"%F)%#PiGF+F)" }{TEXT -1 1 " " }{TEXT 293 1 "~" } {TEXT -1 19 " 1.665304640 (f) 1 " }}}{PARA 0 "" 0 "" {TEXT -1 37 "____ _________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find the mean va lue of the derivative of the function " }{XPPEDIT 18 0 "f(x) = ln(x*(3 -x)+exp(x))+3*x+sin*Pi*x/sqrt(4+3*cos*Pi*x);" "6#/-%\"fG6#%\"xG,(-%#ln G6#,&*&F'\"\"\",&\"\"$F.F'!\"\"F.F.-%$expG6#F'F.F.*&F0F.F'F.F.**%$sinG F.%#PiGF.F'F.-%%sqrtG6#,&\"\"%F.**F0F.%$cosGF.F8F.F'F.F.F1F." }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[0, 3];" "6#7$\"\"!\"\"$" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(f(3)-f(0))/(3-0) = 4;" "6#/*&,&-%\"fG6#\"\"$\"\"\"-F'6#\"\"!!\" \"F*,&F)F*F-F.F.\"\"%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 490 "f := x -> ln(x*(3-x)+ex p(x))+3*x+sin(Pi*x)/sqrt(4+3*cos(Pi*x)):\ndf := D(f);\na := 0: b := 3: \nInt(df(x),x=a..b)/(b-a);\n#value(%);\ndfm := evalf(evalf[14](%));\np 1 := plot(df(x),x=a..b,filled=true,color=COLOR(RGB,.9,.9,.95)):\np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=black):\np3 := plot([d f(x),dfm],x=0..3.1,y=0..1,color=[red,blue]):\np4 := plot([[b,f(b)],[b, dfm]],color=black,linestyle=2):\nplots[display]([p1,p2,p3,p4],tickmark s=[5,5],view=[0..3.1,0..8.5],\n labels=[`x`,`y`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,** &,(\"\"$\"\"\"*&\"\"#F09$F0!\"\"-%$expG6#F3F0F0,&*&F3F0,&F/F0F3F4F0F0F 5F0F4F0F/F0*(-%$cosG6#*&%#PiGF0F3F0F0F@F0-%%sqrtG6#,&\"\"%F0*&F/F0FF2FA!\"$F@F0F0F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&-%$IntG6$,**&,(F'F&*&\"\"#F&%\"xGF&! \"\"-%$expG6#F0F&F&,&*&F0F&,&F'F&F0F1F&F&F2F&F1F&F'F&*(-%$cosG6#*&%#Pi GF&F0F&F&F=F&,&\"\"%F&*&F'F&F9F&F&#F1F/F&*&#F'F/F&*(-%$sinGF;F/F>#!\"$ F/F=F&F&F&/F0;\"\"!F'F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dfmG$ \"+++++S!\"*" }}{PARA 13 "" 1 "" {GLPLOT2D 515 340 340 {PLOTDATA 2 "6+ -%)POLYGONSG6B7&7$$\"\"!F)F)7$$\"3')*****\\7t&pK!#>F)7$F+$\"3A`%3*y$\\ dp(!#<7$F($\"3=DPs6/T(=)F17&F*7$$\"3s******\\i9RlF-F)7$F7$\"3?5W2&=!>/ tF1F.7&F67$$\"33++vVV)RQ*F-F)7$F>$\"3(\\sNs]p6-(F1F97&F=7$$\"3/++vVA)G A\"!#=F)7$FE$\"3c\\!\\afS*\\FGF)7$$ \"3$)***\\(=$f%GcFGF)7$$\"3Q+++Dy,\"G'FGF)7$$\"33++]79C\"F1F]y7&Fay7$$\"3)***\\P/QBE6F1F)7$Fiy$ \"3kXTj\"*3WT;F1Fdy7\\oFhy7$$\"3%**\\(o4.sb6F1F)7$$\"3!******\\\"o?&= \"F1F)7$$\"31+]Pa&4*\\7F1F)7$$\"33+]7j=_68F1F)7$$\"33++vVy!eP\"F1F)7$$ 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an7$FfbmFdan7$F`cmFdan7$FjcmFdan7$F_dmFdan7$FddmFdan7$FidmFdan7$F^emFd an7$FcemFdan7$FhemFdan7$F]fmFdan7$FbfmFdan7$FgfmFdan7$F\\gmFdan7$FagmF dan7$FfgmFdan7$F[hmFdan7$F`hmFdan7$FehmFdan7$FjhmFdan7$F_imFdan7$FdimF dan7$FiimFdan7$F^jmFdan7$FcjmFdan7$FhjmFdan7$F][nFdan7$Fg[nFdan7$Fa\\n Fdan7$F[]nFdan7$Fe]nFdan7$F]`nFdan7$Fg`nFdan-Fchl6&FfglF(F(F]an-F^hl6% 7$Fihl7$FiflFdanFbhl-%*LINESTYLEG6#\"\"#-%*AXESTICKSG6$\"\"&Fcen-%+AXE SLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#JFigl;F($\"#& )Figl" 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" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 " " }}} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q4 " }} {PARA 0 "" 0 "" {TEXT -1 20 "The vertical height " }{TEXT 295 1 "h" } {TEXT -1 53 " km of a missile varies with its horizontal distance " } {TEXT 296 1 "s" }{TEXT -1 42 " km. from the launch pad and is given by " }{XPPEDIT 18 0 "h = 5*s-s^2;" "6#/%\"hG,&*&\"\"&\"\"\"%\"sGF(F(*$F )\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "Determine the mean height of the missile from " }{XPPEDIT 18 0 "s = 0;" "6#/%\" sG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "s = 5;" "6#/%\"sG\"\"&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/5" "6#*&\"\"\"F$\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(5 *s-s^2,s = 0 .. 5) = 25/6;" "6#/-%$IntG6$,&*&\"\"&\"\"\"%\"sGF*F**$F+ \"\"#!\"\"/F+;\"\"!F)*&\"#DF*\"\"'F." }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "f := x -> \+ 5*x-x^2:\na := 0: b := 5:\nInt(f(x),x=a..b)/(b-a);\nvalue(%);\nfav := \+ evalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\" &F&-%$IntG6$,&*&F'F&%\"xGF&F&*$)F-\"\"#F&!\"\"/F-;\"\"!F'F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"#D\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$favG$\"+nmmmT!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }}}{PARA 0 "" 0 "" {TEXT -1 37 "__________________________________ ___" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q5 " }} {PARA 0 "" 0 "" {TEXT -1 13 "The velocity " }{TEXT 298 1 "v" }{TEXT -1 52 " of an oscillating piston at any time t is given by " } {XPPEDIT 18 0 "v=c*sin*omega*t" "6#/%\"vG**%\"cG\"\"\"%$sinGF'%&omegaG F'%\"tGF'" }{TEXT -1 8 ", where " }{TEXT 297 1 "c" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 16 " are constants. " }}{PARA 0 "" 0 "" {TEXT -1 36 "Determine the mean velocity between " } {XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t=Pi/omega" "6#/%\"tG*&%#PiG\"\"\"%&omegaG!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "A ns " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*c/Pi" "6#*(\" \"#\"\"\"%\"cGF%%#PiG!\"\"" }{TEXT -1 2 " " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 13 "A cur rent of " }{TEXT 300 1 "i" }{TEXT -1 58 " amperes flowing through an e lectric circuit is given by " }{XPPEDIT 18 0 "i=30*sin*100*Pi*t" "6#/ %\"iG*,\"#I\"\"\"%$sinGF'\"$+\"F'%#PiGF'%\"tGF'" }{TEXT -1 8 ", where \+ " }{TEXT 299 1 "t" }{TEXT -1 25 " is the time in seconds. " }}{PARA 0 "" 0 "" {TEXT -1 55 "Calculate its mean and r.m.s. values over the int erval " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT -1 6 " to " } {XPPEDIT 18 0 "t=10" "6#/%\"tG\"#5" }{TEXT -1 5 " ms. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 12 "mean value: " }{XPPEDIT 18 0 "100*Int(30* sin*100*Pi*t,t = 0 .. 1/100) = 60/Pi;" "6#/*&\"$+\"\"\"\"-%$IntG6$*,\" #IF&%$sinGF&F%F&%#PiGF&%\"tGF&/F.;\"\"!*&F&F&F%!\"\"F&*&\"#gF&F-F3" } {TEXT -1 1 " " }{TEXT 302 1 "~" }{TEXT -1 10 " 19.099 A " }}{PARA 0 " " 0 "" {TEXT -1 15 " r.m.s. value: " }{XPPEDIT 18 0 "100*Int(900*sin^2 *100*Pi*t,t = 0 .. 1/100) = 15*sqrt(2);" "6#/*&\"$+\"\"\"\"-%$IntG6$*, \"$+*F&*$%$sinG\"\"#F&F%F&%#PiGF&%\"tGF&/F0;\"\"!*&F&F&F%!\"\"F&*&\"#: F&-%%sqrtG6#F.F&" }{TEXT -1 1 " " }{TEXT 301 1 "~" }{TEXT -1 10 " 21.2 13 A " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 17 "Code for pictures" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 29 "the mean value of a function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "bezier([[0,10],[3 ,6],[6,6]],animate=false,info=true):\nbezier([[6,6],[8,6],[10,10]],ani mate=false,info=true):\nbezier([[15,20],[17,24],[19,24]],animate=false ,info=true):\nbezier([[19,24],[21,24],[24,11]],animate=false,info=true ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 295 "p1 := plot([6*t,4*t^2-8*t+10,t=0..1],color=red):\np2 := plot([4*t+6,4*t^2+6,t=0..1],color=red):\np3 := plot([[10,10],[15,2 0]],color=red):\np4 := plot([4*t+15,-4*t^2+8*t+20,t=0..1],color=red): \np5 := plot([t^2+4*t+19,-14*t^2+24,t=0..1],color=red):\nplots[display ]([p1,p2,p3,p4,p5],view=[0..24,0..26]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "eliminate(\{x=6*t,y=4 *t^2-8*t+10\},t);\nsolve(9*y-x^2+12*x-90,y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eliminate(\{ x=4*t+6,y=4*t^2+6\},t);\nsolve(4*y-x^2+12*x-60,y);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "eliminate( \{x=4*t+15,y=-4*t^2+8*t+20\},t);\nsolve(4*y+265-38*x+x^2,y);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "eliminate(\{x=t^2+4*t+19,y=-13*t^2+24\},t);\nsolve(-334*y+68449+16 9*x^2-7046*x+26*x*y+y^2,y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 335 "f := t -> piecewise(t<6,t^2 /9-4/3*t+10,t<10,t^2/4-3*t+15,t<15,2*t-10,\n t<19,-265/4+19/2*t -t^2/4,167-13*t+52*sqrt(t-15)):\nInt('f(t)',t=0..24)/24:\nfav := value (%);\n'f(t)'=f(t):\nplot([f(t),fav],t=0..24,T=0..25,color=[red,blue], \n labels=[`time, t, in hours`,`temperature, T, in deg. C`],\n lab eldirections=[HORIZONTAL,VERTICAL]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 778 "p1 := plot([[[0,0],[6,0 ]],[[0,-1],[0,1]],[[1,-1],[1,1]],\n [[2,-1],[2,1]],[[5,-1],[5,1]],[ [6,-1],[6,1]]],color=black):\np2 := plot([[[.5,-.5],[.5,.5]],[[1.5,-.5 ],[1.5,.5]],\n [[5.5,-.5],[5.5,.5]]],color=blue):\nt 1 := plots[textplot]([[0,-2,`x = a`],[1,-2,`x`],[2,-2,`x`],\n [ 5,-2,`x`],[6,-2,`x = b`]],color=black,font=[HELVETICA,10]):\nt2 := pl ots[textplot]([[-.09,-2.4,`0`],[1.1,-2.4,`1`],[2.1,-2.4,`2`],\n [5.1 3,-2.3,`n-1`],[5.93,-2.3,`n`]],color=black,font=[HELVETICA,8]):\nt3 := plots[textplot]([[.5,1.5,`x *`],[1.5,1.5,`x *`],[5.5,1.5,`x *`]],\n \+ color=blue,font=[HELVETICA,10]):\nt4 := plots[textplot]([[.53,1,`1`], [1.55,1,`2`],\n [5.55,1.1,`n`]],color=blue,font=[HELVETICA,8]): \nplots[display]([p1,p2,t1,t2,t3,t4],axes=none,view=[-.3..6,-3..2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 798 "f := x -> 2+sin(2*x)/5-cos(4*x+1)/4+arctan(x):\na := 0: b := \+ 5.5:\nInt('f(t)',t=a..b)/(b-a):\nfav := evalf(%):\n\np1 := plot([f(x), fav],x=-1..7,y=0..4,color=[red,blue]):\np2 := plot(f(x),x=a..b,filled= true,color=COLOR(RGB,.9,.9,.95)):\np3 := plot([[[a,0],[a,f(a)]],\n \+ [[b,0],[b,f(b)]],[[a,f(a)],[a,fav]]],color=black,linestyle=[1$2,2]) :\np4 := plottools[arrow]([-1.2,0],[7,0],0,.07,.017,arrow,color=black) :\np5 := plottools[arrow]([-1,-.2],[-1,4.7],0,.12,.017,arrow,color=bla ck):\nt1 := plots[textplot]([[7,-.15,`x`],[-1.2,4.6,`y`],\n [a,-. 2,`x = a`],[b,-.2,`x = b`],[-1.4,fav,`f`]],color=black):\nt2 := plots[ textplot]([-1.23,fav-.05,`m`],color=black,font=[HELVETICA,8]):\nt3 := \+ plots[textplot]([6,4,`y = f(x)`],color=red):\nplots[display]([p1,p2,p3 ,p4,p5,t1,t2,t3],axes=none,view=[-1.4..7,-.4..4.7]);" }}}{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 "" }}{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 }