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}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 1 "1" }{TEXT 264 2 ". " }{TEXT 317 16 " A Riemann su m " }{XPPEDIT 365 0 "sum(f(xi[j])*Delta*x,j = 1 .. N);" "6#-%$sumG6$* (-%\"fG6#&%#xiG6#%\"jG\"\"\"%&DeltaGF.%\"xGF./F-;F.%\"NG" }{TEXT 364 19 " for a function " }{XPPEDIT 362 0 "f;" "6#%\"fG" }{TEXT 360 18 " on an interval " }{XPPEDIT 363 0 "[a, b];" "6#7$%\"aG%\"bG" } {TEXT 361 17 " is said to be a" }{TEXT 410 19 "n upper Riemann sum" } {TEXT 411 16 " if, for each " }{XPPEDIT 367 0 "j;" "6#%\"jG" }{TEXT 366 14 ", the point " }{XPPEDIT 369 0 "xi[j];" "6#&%#xiG6#%\"jG" } {TEXT 368 10 " in the " }{XPPEDIT 488 0 "j;" "6#%\"jG" }{XPPEDIT 18 0 "` `^th;" "6#)%\"~G%#thG" }{TEXT 489 34 " subinterval is chosen so that " }{XPPEDIT 256 0 "f(xi[j])" "6#-%\"fG6#&%#xiG6#%\"jG" }{TEXT 370 55 " is maximized. Calculate the upper Riemann sum for " } {XPPEDIT 373 0 "f(x) = x^3-3*x+2;" "6#/-%\"fG6#%\"xG,(*$F'\"\"$\"\"\"* &F*F+F'F+!\"\"\"\"#F+" }{TEXT 372 6 " , " }{XPPEDIT 376 0 "[a, b] = [-1, 3];" "6#/7$%\"aG%\"bG7$,$\"\"\"!\"\"\"\"$" }{TEXT 375 11 " , a nd " }{XPPEDIT 374 0 "N = 4;" "6#/%\"NG\"\"%" }{TEXT 371 61 ". (Use a partition of [a,b] into equal length subintervals.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 81 "a) 14 b) \+ 16 c) 18 d) 20 e) 22 \nf) 24 " } {TEXT 318 56 " g) 26 h) 28 i) 30 j) 32 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 319 8 "Solut ion" }{TEXT 320 8 ": ( i )\n" }}{PARA 0 "" 0 "" {TEXT -1 47 "The nodes (points of the uniform partition) are" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a := -1; b := 3; N := 4; Delta := (b-a)/N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"bG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&DeltaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for j from 0 to N do\nx[j] := a + j*Delta;\nod;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"xG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\" \"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%\"\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> x^3-3*x+2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6# %\"xG6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"$\"\"\"F1*&F0F1F/F1!\"\"\"\" #F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff('f(x)',x) =factor(D(f)(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"f G6#%\"xGF*,$*(\"\"$\"\"\",&F*F.F.!\"\"F.,&F*F.F.F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "This calculation tells us that " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 29 " decreases from -1 to 1 ( " }{XPPEDIT 18 0 "D(f)(x);" "6#--%\"DG6#%\"fG6#%\"xG" } {TEXT -1 20 " = ( - )( + ) so " }{XPPEDIT 18 0 "D(f)(x) < 0;" "6#2- -%\"DG6#%\"fG6#%\"xG\"\"!" }{TEXT -1 9 " ) and " }{XPPEDIT 18 0 "f(x );" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " increases for " }{XPPEDIT 18 0 " x;" "6#%\"xG" }{TEXT -1 17 " > 1 . Therefore:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "xi[1] := x[0]; xi[2] := x[1]; xi[3] := x[3]; xi[4] \+ := x[4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#xiG6#\"\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#xiG6#\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%#xiG6#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%#xiG6#\"\"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "fn Graph := plot(f(x),x=-1..3, color=PLUM, thickness=2): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "nodes := plot( [ seq([xi[j],f(xi[j] )], j = 1..N)], style = POINT, symbol = CIRCLE, color = NAVY):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plots[display](fnGraph, node s);" }}{PARA 13 "" 1 "" {GLPLOT2D 374 374 374 {PLOTDATA 2 "6&-%'CURVES G6%7U7$$!\"\"\"\"!$\"\"%F*7$$!3PLLLLQ6G\"*!#=$\"3Y>?eIs&y(R!#<7$$!3imm mT.\\p$)F0$\"3%oWE#ezdCRF37$$!3LLLL$))Qj^(F0$\"3UVd\\nMEIQF37$$!3ULLL$ =Kvl'F0$\"3pp7!=iz@q$F37$$!3hnmmTs!G!eF0$\"39Pj_GnWXNF37$$!3iLLL3yO5]F 0$\"3o/))QV6LxLF37$$!3i+++vE%)*=%F0$\"3))z\"H#H5S$=$F37$$!3)RLL$3WDTLF 0$\"3-V$)p@Y2lHF37$$!3'4++]d(Q&\\#F0$\"3S%R3gfxIt#F37$$!3:mmmm&4`i\"F0 $\"3#RNf:S*H$[#F37$$!3GKLLLQW*e)!#>$\"3ut%G$)f\\qD#F37$$\"3HI#*******H ,Q!#@$\"3_G\\05'f))*>F37$$\"3Q(*******\\*3q)F^o$\"3ad&[i&=jR$)F07$ $\"3+*****\\s]k,&F0$\"3-aHA2l-8iF07$$\"3WJLLLvv-eF0$\"3)*fR%fiJca%F07$ $\"3'3++]sgam'F0$\"3MiAecJ(\\'HF07$$\"3G+++v\"ep[(F0$\"3;6'*>Vc!ft\"F0 7$$\"3#QLLLe/TM)F0$\"3?!Grl3C>x(F^o7$$\"39LLLeDBJ\"*F0$\"3A?pz$y*p)>#F ^o7$$\"3Immm;kD!)**F0$\"3_!=%pDwlo6!#A7$$\"3Mmm;f`@'3\"F3$\"3gT>Q$36SH #F^o7$$\"3y****\\nZ)H;\"F3$\"3KBowGO;-%)F^o7$$\"3YmmmJy*eC\"F3$\"3!4u4 5EcE'>F07$$\"3')******R^bJ8F3$\"3[$=h&[*QBm$F07$$\"3f*****\\5a`T\"F3$ \"3-^Jxfe8#*eF07$$\"3o****\\7RV'\\\"F3$\"3#e%RNHN%oh)F07$$\"3k*****\\@ fke\"F3$\"3Oh(yrt1NB\"F37$$\"3/LLL`4Nn;F3$\"3sV,3+:GL;F37$$\"3#******* \\,s`pq8mC8#F37$$\"3[mm;zM)>$=F3$\"3Wh![\"*e&[_EF37$$\"3$*** ****pfa<>F3$\"3W_9)[:X\")H$F37$$\"3#HLLeg`!)*>F3$\"3#3pFe<0D)RF37$$\"3 w****\\#G2A3#F3$\"3EYt:E$p4y%F37$$\"3;LLL$)G[k@F3$\"3SAO=&oBrk&F37$$\" 3#)****\\7yh]AF3$\"3o([-l;d\"[mF37$$\"3xmmm')fdLBF3$\"3m]-m$\\Rpq(F37$ $\"3bmmm,FT=CF3$\"3)p#*33i!Q*)))F37$$\"3FLL$e#pa-DF3$\"3\"z/j\\F;l,\"! #;7$$\"3!*******Rv&)zDF3$\"3fDZ[s%4J9\"F^y7$$\"3ILLLGUYoEF3$\"3%yq\"e6 Wf*H\"F^y7$$\"3_mmm1^rZFF3$\"3=\"fl@U$>]9F^y7$$\"34++]sI@KGF3$\"3D'=w. Aw@i\"F^y7$$\"33+++S2lsGF3$\"3oNTq$H^(3 " 0 "" {MPLTEXT 1 0 295 "rect[1] := plottools[rectangle]([x[0],f( x[0])], [x[1],0], color = wheat):\nrect[2] := plottools[rectangle]([x[ 1],f(x[1])], [x[2],0], color = wheat):\nrect[3] := plottools[rectangle ]([x[2],f(x[3])], [x[3],0], color = wheat):\nrect[4] := plottools[rect angle]([x[3],f(x[4])], [x[4],0], color = wheat):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plots[display](fnGraph, nodes,seq(rect[j],j=1 ..N));" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6*-%'CUR VESG6%7U7$$!\"\"\"\"!$\"\"%F*7$$!3PLLLLQ6G\"*!#=$\"3Y>?eIs&y(R!#<7$$!3 immmT.\\p$)F0$\"3%oWE#ezdCRF37$$!3LLLL$))Qj^(F0$\"3UVd\\nMEIQF37$$!3UL LL$=Kvl'F0$\"3pp7!=iz@q$F37$$!3hnmmTs!G!eF0$\"39Pj_GnWXNF37$$!3iLLL3yO 5]F0$\"3o/))QV6LxLF37$$!3i+++vE%)*=%F0$\"3))z\"H#H5S$=$F37$$!3)RLL$3WD TLF0$\"3-V$)p@Y2lHF37$$!3'4++]d(Q&\\#F0$\"3S%R3gfxIt#F37$$!3:mmmm&4`i \"F0$\"3#RNf:S*H$[#F37$$!3GKLLLQW*e)!#>$\"3ut%G$)f\\qD#F37$$\"3HI#**** ***H,Q!#@$\"3_G\\05'f))*>F37$$\"3Q(*******\\*3q)F^o$\"3ad&[i&=jR$) F07$$\"3+*****\\s]k,&F0$\"3-aHA2l-8iF07$$\"3WJLLLvv-eF0$\"3)*fR%fiJca% F07$$\"3'3++]sgam'F0$\"3MiAecJ(\\'HF07$$\"3G+++v\"ep[(F0$\"3;6'*>Vc!ft \"F07$$\"3#QLLLe/TM)F0$\"3?!Grl3C>x(F^o7$$\"39LLLeDBJ\"*F0$\"3A?pz$y*p )>#F^o7$$\"3Immm;kD!)**F0$\"3_!=%pDwlo6!#A7$$\"3Mmm;f`@'3\"F3$\"3gT>Q$ 36SH#F^o7$$\"3y****\\nZ)H;\"F3$\"3KBowGO;-%)F^o7$$\"3YmmmJy*eC\"F3$\"3 !4u45EcE'>F07$$\"3')******R^bJ8F3$\"3[$=h&[*QBm$F07$$\"3f*****\\5a`T\" F3$\"3-^Jxfe8#*eF07$$\"3o****\\7RV'\\\"F3$\"3#e%RNHN%oh)F07$$\"3k***** \\@fke\"F3$\"3Oh(yrt1NB\"F37$$\"3/LLL`4Nn;F3$\"3sV,3+:GL;F37$$\"3#**** ***\\,s`pq8mC8#F37$$\"3[mm;zM)>$=F3$\"3Wh![\"*e&[_EF37$$\"3$ *******pfa<>F3$\"3W_9)[:X\")H$F37$$\"3#HLLeg`!)*>F3$\"3#3pFe<0D)RF37$$ \"3w****\\#G2A3#F3$\"3EYt:E$p4y%F37$$\"3;LLL$)G[k@F3$\"3SAO=&oBrk&F37$ $\"3#)****\\7yh]AF3$\"3o([-l;d\"[mF37$$\"3xmmm')fdLBF3$\"3m]-m$\\Rpq(F 37$$\"3bmmm,FT=CF3$\"3)p#*33i!Q*)))F37$$\"3FLL$e#pa-DF3$\"3\"z/j\\F;l, \"!#;7$$\"3!*******Rv&)zDF3$\"3fDZ[s%4J9\"F^y7$$\"3ILLLGUYoEF3$\"3%yq \"e6Wf*H\"F^y7$$\"3_mmm1^rZFF3$\"3=\"fl@U$>]9F^y7$$\"34++]sI@KGF3$\"3D '=w.Aw@i\"F^y7$$\"33+++S2lsGF3$\"3oNTq$H^(3!\\(F]\\ l-Fj]l6$7&Fg\\l7$$\"\"\"F*Fi\\l7$Fj^lFh\\lF^^lF`^l-Fj]l6$7&7$Fj^lF+Fj \\l7$Fi\\lFh\\lF\\_lF`^l-Fj]l6$7&7$Fi\\lFe[lFb[l7$Fc[lFh\\lFa_lF`^l-%+ AXESLABELSG6%Q\"x6\"Q!F[`l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fc[lF``l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 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 54 "ANSWER := (f(xi[1])+f(xi[2])+f(xi[3])+f(xi[4]))*De lta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ANSWERG\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 409 15 "2. Calcula te " }{XPPEDIT 413 0 "int(sec(theta)*tan(theta),theta = 0 .. Pi/3); " "6#-%$intG6$*&-%$secG6#%&thetaG\"\"\"-%$tanG6#F*F+/F*;\"\"!*&%#PiGF+ \"\"$!\"\"" }{TEXT 412 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 424 28 "a) 1 \+ b) " }{XPPEDIT 261 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT 425 17 " c) " }{XPPEDIT 262 0 "sqrt(3);" "6#-%%sqrtG6#\" \"$" }{TEXT 426 37 " d) 2 e) " }{XPPEDIT 263 0 "sqrt(3)-1;" "6#,&-%%sqrtG6#\"\"$\"\"\"F(!\"\"" }{TEXT 427 9 " \+ \nf) " }{XPPEDIT 264 0 "2/sqrt(3)-1;" "6#,&*&\"\"#\"\"\"-%%sqrtG6# \"\"$!\"\"F&F&F+" }{TEXT 428 11 " g) " }{TEXT 435 1 " " } {XPPEDIT 265 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 429 12 " h) " }{XPPEDIT 266 0 "2*sqrt(3);" "6#*&\"\"#\"\"\"-%% sqrtG6#\"\"$F%" }{TEXT 430 13 " i) " }{TEXT 433 1 " " } {XPPEDIT 267 0 "3*sqrt(2);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 431 2 " " }{TEXT 434 12 " j) " }{XPPEDIT 268 0 "2/sqrt(3);" " 6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$!\"\"" }{TEXT 423 1 "\n" }{TEXT 432 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT 436 8 "Solution" }{TEXT 437 8 ": ( a )\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " F := unapply(int(sec(theta) *tan(theta), theta), theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG %$secG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "F(Pi/3) - F(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 262 3 "3. " }{TEXT 322 1 " " }{TEXT 323 21 " An antiderivative o f" }{TEXT 377 2 " " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 378 3 " " }{TEXT 380 17 "is the function " }{XPPEDIT 18 0 "proc (x) options ope rator, arrow; (x+exp(x))/(1+exp(x)) end proc;" "6#f*6#%\"xG7\"6$%)oper atorG%&arrowG6\"*&,&F%\"\"\"-%$expG6#F%F-F-,&F-F--F/6#F%F-!\"\"F*F*F* " }{TEXT 379 1 " " }{TEXT 490 1 "." }{TEXT 491 8 " If " }{XPPEDIT 18 0 "int(f(x)+c,x = 0 .. 1) = 5/2;" "6#/-%$intG6$,&-%\"fG6#%\"xG\"\" \"%\"cGF,/F+;\"\"!F,*&\"\"&F,\"\"#!\"\"" }{TEXT 381 14 " , what is \+ " }{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT 569 2 " ?" }}{PARA 3 "" 0 "" {TEXT 321 3 "a) " }{XPPEDIT 567 0 "1/exp(1);" "6#*&\"\"\"F$-%$expG6#F$ !\"\"" }{TEXT 565 14 " b) " }{XPPEDIT 568 0 "1/exp(2);" "6#* &\"\"\"F$-%$expG6#\"\"#!\"\"" }{TEXT 566 16 " c) " } {XPPEDIT 257 0 "1/(1+exp(1));" "6#*&\"\"\"F$,&F$F$-%$expG6#F$F$!\"\"" }{TEXT 570 15 " d) " }{XPPEDIT 257 0 "2/(1+exp(1));" "6#*& \"\"#\"\"\",&F%F%-%$expG6#F%F%!\"\"" }{TEXT 571 14 " e) " } {XPPEDIT 256 0 "exp(1)/(1+exp(1));" "6#*&-%$expG6#\"\"\"F',&F'F'-F%6#F 'F'!\"\"" }{TEXT 572 14 " \nf) " }{XPPEDIT 256 0 "1/2;" "6#*& \"\"\"F$\"\"#!\"\"" }{TEXT 573 34 " g) 1 h) \+ " }{XPPEDIT 575 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 574 48 " \+ i) 2 j) 2 " }{TEXT 324 7 " \n " }}{PARA 3 "" 0 "" {TEXT 325 8 "Solution" }{TEXT 326 8 ": ( i )\n" } {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F := x -> (x+exp(x))/(1+exp( x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operator G%&arrowGF(*&,&9$\"\"\"-%$expG6#F.F/F/,&F/F/F0F/!\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn1 := Int(f(x)+c,x = 0 .. \+ 1) = F(1) - F(0) + c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%$ IntG6$,&-%\"fG6#%\"xG\"\"\"%\"cGF./F-;\"\"!F.,&#F.\"\"#F.F/F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn2 := 5/2 = rhs(eqn1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/#\"\"&\"\"#,&#\"\"\"F(F+%\"cG F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(eqn2,c);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 17 "4. Calculate " }{XPPEDIT 383 0 "int(Diff(x*ln(x),x),x = 1 .. exp( 1));" "6#-%$intG6$-%%DiffG6$*&%\"xG\"\"\"-%#lnG6#F*F+F*/F*;F+-%$expG6# F+" }{TEXT 382 2 " ." }{TEXT 330 2 "\n\n" }{TEXT 331 48 "a) 1 \+ b) 2 c) " }{XPPEDIT 583 0 "exp(1);" "6#-%$e xpG6#\"\"\"" }{TEXT 576 18 " d) " }{XPPEDIT 584 0 "1+exp (1);" "6#,&\"\"\"F$-%$expG6#F$F$" }{TEXT 577 15 " e) " } {TEXT 384 2 " " }{XPPEDIT 585 0 "2+exp(1);" "6#,&\"\"#\"\"\"-%$expG6# F%F%" }{TEXT 332 1 " " }{TEXT 586 18 " " }}{PARA 0 " " 0 "" {TEXT 328 5 "f) " }{XPPEDIT 18 0 "exp(1)/2;" "6#*&-%$expG6#\" \"\"F'\"\"#!\"\"" }{TEXT 578 17 " g) " }{XPPEDIT 18 0 "1+ exp(1)/2;" "6#,&\"\"\"F$*&-%$expG6#F$F$\"\"#!\"\"F$" }{TEXT 579 10 " \+ " }{TEXT -1 1 " " }{TEXT 329 4 "h) " }{XPPEDIT 18 0 "exp(1)-1 ;" "6#,&-%$expG6#\"\"\"F'F'!\"\"" }{TEXT 580 16 " i) " } {XPPEDIT 18 0 "exp(exp(1));" "6#-%$expG6#-F$6#\"\"\"" }{TEXT 581 15 " \+ j) " }{XPPEDIT 18 0 "exp(exp(1))-1;" "6#,&-%$expG6#-F%6#\" \"\"F)F)!\"\"" }{TEXT 582 1 " " }{TEXT -1 4 " " }}{PARA 3 "" 0 "" {TEXT 327 1 " " }}{PARA 3 "" 0 "" {TEXT 333 8 "Solution" }{TEXT 334 8 ": ( c )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "F := x -> x*ln (x); #This is an antiderivative of the integrand " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-% #lnG6#F-F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "F(exp(1 )) - F(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 457 16 "5. Suppose that " }{XPPEDIT 458 0 "int(f(x),x = 0 .. 2) = 4;" "6#/-%$ intG6$-%\"fG6#%\"xG/F*;\"\"!\"\"#\"\"%" }{TEXT 459 7 " and " } {XPPEDIT 460 0 "int(`(`*x^2+f(x)*`)`,x = 0 .. 3) = 17;" "6#/-%$intG6$, &*&%\"(G\"\"\"*$%\"xG\"\"#F*F**&-%\"fG6#F,F*%\")GF*F*/F,;\"\"!\"\"$\"# <" }{TEXT 461 14 ". What is " }{XPPEDIT 462 0 "int(f(x),x = 2 .. 3 );" "6#-%$intG6$-%\"fG6#%\"xG/F);\"\"#\"\"$" }{TEXT 463 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 467 66 "a) 1 \+ b) 2 c) 3 d) 4 e)" }{TEXT 470 2 " \+ " }{TEXT 464 1 "5" }{TEXT 469 2 " " }{TEXT 468 19 " \+ " }}{PARA 0 "" 0 "" {TEXT 465 33 "f) 6 g) 7 " }{TEXT -1 1 " " }{TEXT 466 38 "h) 8 i) 9 j) 10 " }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 471 8 "Solution" }{TEXT 472 8 ": ( d )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqn1 := int(x^2+f(x),x = 0.. 3) = 17; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%$intG6$,&*$) %\"xG\"\"#\"\"\"F.-%\"fG6#F,F./F,;\"\"!\"\"$\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn2 := int(x^2,x = 0..3) + int(f(x),x = 0.. 3)= 17;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&\"\"*\"\"\"-%$in tG6$-%\"fG6#%\"xG/F/;\"\"!\"\"$F(\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "eqn3 := 9 + int(f(x),x = 0 .. 2) + int(f(x),x = 2 .. \+ 3) = 17;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,(\"\"*\"\"\"-%$i ntG6$-%\"fG6#%\"xG/F/;\"\"!\"\"#F(-F*6$F,/F/;F3\"\"$F(\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eqn4 := 9 + 4 + int(f(x),x = 2 .. 3 ) = 17;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/,&\"#8\"\"\"-%$int G6$-%\"fG6#%\"xG/F/;\"\"#\"\"$F(\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve( eqn4, int(f(x),x = 2 .. 3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 631 17 "6. Suppose that " }{XPPEDIT 257 0 "f(x) = x^2 +6*x;" "6#/-%\"fG6#%\"xG,&*$F'\"\"#\"\"\"*&\"\"'F+F'F+F+" }{TEXT 632 72 " . The Mean Value Theorem for Integrals asserts that there is a p oint " }{XPPEDIT 259 0 "c;" "6#%\"cG" }{TEXT 633 39 " in the interva l [-1,2] such that " }{XPPEDIT 261 0 "f(c) = f[ave];" "6#/-%\"fG6# %\"cG&F%6#%$aveG" }{TEXT 634 10 " where " }{XPPEDIT 263 0 "f[ave]; " "6#&%\"fG6#%$aveG" }{TEXT 635 27 " is the average value of " } {XPPEDIT 265 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 636 6 " for " } {XPPEDIT 267 0 "x;" "6#%\"xG" }{TEXT 637 34 " in the interval [1,7]. \+ What is " }{XPPEDIT 269 0 "c;" "6#%\"cG" }{TEXT 638 1 "?" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 641 1 "\n" }{TEXT 642 4 " a) " }{XPPEDIT 283 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT 644 18 " \+ b) " }{XPPEDIT 284 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" } {TEXT 645 20 " c) " }{XPPEDIT 18 0 "sqrt(5)-1;" "6#,&- %%sqrtG6#\"\"&\"\"\"F(!\"\"" }{TEXT 652 12 " d) " }{XPPEDIT 285 0 "sqrt(6)-1;" "6#,&-%%sqrtG6#\"\"'\"\"\"F(!\"\"" }{TEXT 649 14 " \+ e) " }{XPPEDIT 256 0 "sqrt(10)-2;" "6#,&-%%sqrtG6#\"#5\"\"\" \"\"#!\"\"" }{TEXT 653 3 " " }{TEXT 643 20 " \n" } {TEXT 639 3 "f) " }{XPPEDIT 18 0 "sqrt(13)-3;" "6#,&-%%sqrtG6#\"#8\"\" \"\"\"$!\"\"" }{TEXT 648 12 " g) " }{XPPEDIT 288 0 "sqrt(14)-2 ;" "6#,&-%%sqrtG6#\"#9\"\"\"\"\"#!\"\"" }{TEXT 647 6 " " }{TEXT 650 1 " " }{TEXT 640 5 "h) " }{XPPEDIT 18 0 "sqrt(17)-3;" "6#,&-%%sq rtG6#\"#<\"\"\"\"\"$!\"\"" }{TEXT 656 10 " i) " }{XPPEDIT 290 0 "sqrt(19)-3;" "6#,&-%%sqrtG6#\"#>\"\"\"\"\"$!\"\"" }{TEXT 646 14 " \+ j) " }{XPPEDIT 291 0 "sqrt(21)-4;" "6#,&-%%sqrtG6#\"#@\"\"\"\" \"%!\"\"" }{TEXT 651 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 654 8 "Solution" }{TEXT 655 8 ": ( f )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x \+ -> x^2+6*x; interval := -1 .. 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"F1*&\"\"'F1F/ F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)intervalG;!\"\"\"\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Ave := int(f(x), x=inte rval)/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AveG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(f(x) = Ave, x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"\"$!\"\"*$\"#8#\"\"\"\"\"#F),&F$F%F&F%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 338 19 "7. Calculate " }{XPPEDIT 387 0 "`d `/(d*x);" "6#*&%#d~G\"\"\"*&%\"dGF%%\"xGF%!\"\"" }{TEXT 386 1 " " }{XPPEDIT 257 0 "int((9*t+tan(Pi*t/4))/(t^2+4),t = -1 .. x);" "6 #-%$intG6$*&,&*&\"\"*\"\"\"%\"tGF*F*-%$tanG6#*(%#PiGF*F+F*\"\"%!\"\"F* F*,&*$F+\"\"#F*F1F*F2/F+;,$F*F2%\"xG" }{TEXT 385 11 " at " } {XPPEDIT 389 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 388 1 "." }}{PARA 3 " " 0 "" {TEXT 337 1 "\n" }}{PARA 3 "" 0 "" {TEXT 336 132 "a) 0 \+ b) 1 c) 2 d) 3 e) 4 \nf) 5 \+ g) 6 h) 7 i) 8 j) 9" }{TEXT 335 1 "\n" }}{PARA 3 "" 0 "" {TEXT 339 8 "Solution" }{TEXT 340 8 ": ( c )\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "simplify(subs(t=1,(9*t+tan(P i*t/4))/(t^2+4))); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 392 17 "8. Suppose that " }{XPPEDIT 257 0 "F(x) = int(sqrt(7/2+t^2),t = 0 .. ` `*sin(x));" "6#/-%\"FG6#%\"xG-%$intG6$-%%sqrtG6#,&*&\"\"(\"\"\"\"\" #!\"\"F1*$%\"tGF2F1/F5;\"\"!*&%\"~GF1-%$sinG6#F'F1" }{TEXT 393 15 ". \+ What is " }{XPPEDIT 396 0 "D(F)(Pi/4);" "6#--%\"DG6#%\"FG6#*&%#PiG \"\"\"\"\"%!\"\"" }{TEXT 394 35 "? (The derivative of F(x) at " }{XPPEDIT 397 0 "x = Pi/4;" "6#/%\"xG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 395 3 " )." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 399 26 "a) 1 b) " }{XPPEDIT 414 0 "sqr t(2);" "6#-%%sqrtG6#\"\"#" }{TEXT 400 16 " c) " }{XPPEDIT 415 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT 401 37 " d) 2 e) " }{XPPEDIT 416 0 "3*sqrt(2)/2;" "6#*(\"\"$\"\"\"-% %sqrtG6#\"\"#F%F)!\"\"" }{TEXT 402 9 " \nf) " }{XPPEDIT 417 0 "3*s qrt(3)/2;" "6#*(\"\"$\"\"\"-%%sqrtG6#F$F%\"\"#!\"\"" }{TEXT 403 14 " \+ g) " }{XPPEDIT 418 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#F $F%" }{TEXT 404 11 " h) " }{XPPEDIT 419 0 "2*sqrt(3);" "6#*&\" \"#\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 405 14 " i) " }{XPPEDIT 420 0 "3*sqrt(2);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 406 14 " \+ j) " }{XPPEDIT 421 0 "3*sqrt(3);" "6#*&\"\"$\"\"\"-%%sqrtG6# F$F%" }{TEXT 398 1 "\n" }{TEXT 422 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT 407 8 "Solution" }{TEXT 408 7 ": ( b )" } {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart: \+ with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "F := (x) \+ -> Int(sqrt(7/2+t^2),t = 0..sin(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$IntG6$-%%sqrtG6#,&#\"\"( \"\"#\"\"\"*$)%\"tGF5F6F6/F9;\"\"!-%$sinG6#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "D(F)(x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&-%$cosG6#%\"xGF&,&\"#9F&*&\"\"%F&)-%$sinGF+F 'F&F&F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(D(F )(Pi/4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"##\"\"\"F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Where ans wer comes from:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "derivative : = subs(t=sin(x), integrand(F(x)))*D(sin)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+derivativeG,$*&#\"\"\"\"\"#F(*&-%$cosG6#%\"xGF(,&\"# 9F(*&\"\"%F()-%$sinGF-F)F(F(F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify(subs(x=Pi/4, derivative));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"##\"\"\"F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 260 1 " " }{TEXT 630 1 "9" }{TEXT 587 16 ". Suppose that " }{XPPEDIT 390 0 "F(x) = int(sqrt(144+t^2),t = ` \+ `*x .. ` `*(3*x+1));" "6#/-%\"FG6#%\"xG-%$intG6$-%%sqrtG6#,&\"$W\"\"\" \"*$%\"tG\"\"#F0/F2;*&%\"~GF0F'F0*&F7F0,&*&\"\"$F0F'F0F0F0F0F0" } {TEXT 391 60 ". What is D(F)(2)? (The derivative of F(x) at x = 5)." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 438 171 "a) 13 \+ b) 20 c) 27 d) 33 e) 40 \nf) \+ 47 g) 53 h) 60 i) 67 \+ j) 73 " }{TEXT 439 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 351 8 "Solution " }{TEXT 352 8 ": ( f )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "F := (x) -> Int(sqrt(144+t^2),t = x .. (3*x+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$% )operatorG%&arrowGF(-%$IntG6$-%%sqrtG6#,&\"$W\"\"\"\"*$)%\"tG\"\"#F4F4 /F7;9$,&*&\"\"$F4F;F4F4F4F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(D(F)(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"#Z" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "Where answer comes from:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "h := x -> 3*x+1; g := x -> x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\" \"$\"\"\"9$F/F/F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "derivative := subs(t=h(x), integrand(F(x)))*D(h)(x) - subs(t=g(x), integrand(F(x)))*D(g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+derivativeG,&*&\"\"$\"\"\",&\"$W\"F(*$),&*&F'F(%\"xGF(F(F(F( \"\"#F(F(#F(F0F(*$,&F*F(*$)F/F0F(F(F1!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "simplify(subs(x=5, derivative));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#Z" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 354 17 "10. Calculate " }{XPPEDIT 441 0 "int(4*cos(x)*(sin(x)+1)^3, x = 0 .. Pi/2);" "6#-%$intG6$*(\"\"%\"\"\"-%$cosG6#%\"xGF(,&-%$sinG6#F ,F(F(F(\"\"$/F,;\"\"!*&%#PiGF(\"\"#!\"\"" }{TEXT 440 2 " ." }}{PARA 0 "" 0 "" {TEXT 353 17 " " }}{PARA 0 "" 0 "" {TEXT 442 158 "a) 3 b) 4 c) 6 d) 8 \+ e) 9 \nf) 8 g) 12 h) 15 \+ i) 16 j) " }{TEXT 443 3 "20\n" }{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 355 8 "Solution" }{TEXT 356 8 ": ( h )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "J1 := Int(4*cos(x)*(sin(x)+1)^3,x = 0 .. Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G-%$IntG6$,$*(\" \"%\"\"\"-%$cosG6#%\"xGF+),&-%$sinGF.F+F+F+\"\"$F+F+/F/;\"\"!,$*&\"\"# !\"\"%#PiGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "J2 := cha ngevar(u = sin(x) + 1, J1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J 2G-%$IntG6$,$*&\"\"%\"\"\")%\"uG\"\"$F+F+/F-;F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value(J2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 1 " " }{TEXT 259 19 "11. Calculate " }{XPPEDIT 258 0 " int(x*sqrt(x-3),x = 3 .. 4);" "6#-%$intG6$*&%\"xG\"\"\"-%%sqrtG6#,&F'F (\"\"$!\"\"F(/F';F-\"\"%" }{TEXT 444 2 " ." }}{PARA 0 "" 0 "" {TEXT 269 4 "a) " }{XPPEDIT 446 1 "8/15;" "6#*&\"\")\"\"\"\"#:!\"\"" } {TEXT 341 15 " b) " }{XPPEDIT 447 1 "8/3;" "6#*&\"\")\"\"\" \"\"$!\"\"" }{TEXT 342 16 " c) " }{XPPEDIT 448 1 "4/5;" "6 #*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT 343 17 " d) " }{XPPEDIT 449 1 "1;" "6#\"\"\"" }{TEXT 344 14 " e) " }{XPPEDIT 450 1 " 7/3;" "6#*&\"\"(\"\"\"\"\"$!\"\"" }{TEXT 345 11 " \n f) " } {XPPEDIT 451 1 "16/15;" "6#*&\"#;\"\"\"\"#:!\"\"" }{TEXT 346 15 " \+ g) " }{XPPEDIT 452 1 "6/5;" "6#*&\"\"'\"\"\"\"\"&!\"\"" }{TEXT 347 16 " h) " }{XPPEDIT 453 1 "5/3;" "6#*&\"\"&\"\"\"\"\"$ !\"\"" }{TEXT 348 17 " i) " }{XPPEDIT 454 0 "12/5;" "6#*& \"#7\"\"\"\"\"&!\"\"" }{TEXT 445 15 " j) " }{XPPEDIT 455 0 "8/5;" "6#*&\"\")\"\"\"\"\"&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT 349 8 "Solution" }{TEXT 350 8 ": ( i )\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "J1 := Int(x*sqrt(x-3),x = 3..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#J1G-%$IntG6$*&%\"xG\"\"\",&F)F*\"\"$!\"\"#F*\"\"#/F);F,\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "J2 := changevar(u = x-3, J1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J2G-%$IntG6$*&,&\"\"$\"\"\" %\"uGF+F+F,#F+\"\"#/F,;\"\"!F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "J3 := Int(expand(integrand(J2)), u = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J3G-%$IntG6$,&*&\"\"$\"\"\"%\"uG#F+\"\"#F+*$)F, #F*F.F+F+/F,;\"\"!F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "val ue(J3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#7\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 357 17 "12. Calculate " }{XPPEDIT 256 0 "int(1/(x+x*ln(x)),x = 1 .. exp(1));" "6#-%$intG6$*&\"\"\"F',&%\"xGF'*&F)F'-%#lnG6#F)F'F'!\"\"/F) ;F'-%$expG6#F'" }{TEXT 456 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "ln(exp(1)/2);" "6# -%#lnG6#*&-%$expG6#\"\"\"F*\"\"#!\"\"" }{TEXT -1 18 " b) \+ " }{XPPEDIT 18 0 "ln((1+exp(1))/2);" "6#-%#lnG6#*&,&\"\"\"F(-%$expG6# F(F(F(\"\"#!\"\"" }{TEXT -1 15 " c) " }{XPPEDIT 18 0 "ln(1+ exp(1)/2);" "6#-%#lnG6#,&\"\"\"F'*&-%$expG6#F'F'\"\"#!\"\"F'" }{TEXT -1 12 " d) " }{XPPEDIT 18 0 "ln(1/2+exp(1));" "6#-%#lnG6#,&*& \"\"\"F(\"\"#!\"\"F(-%$expG6#F(F(" }{TEXT -1 12 " e) " } {XPPEDIT 18 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT -1 20 " \+ \nf) " }{XPPEDIT 18 0 "exp(1)/((1+exp(1))^2);" "6#*&-%$expG6#\"\"\"F' *$,&F'F'-F%6#F'F'\"\"#!\"\"" }{TEXT -1 14 " g) " }{XPPEDIT 18 0 "exp(1)/(1+exp(1));" "6#*&-%$expG6#\"\"\"F',&F'F'-F%6#F'F'!\"\"" }{TEXT -1 22 " h) " }{XPPEDIT 18 0 "1/(1+exp(1));" " 6#*&\"\"\"F$,&F$F$-%$expG6#F$F$!\"\"" }{TEXT -1 22 " \+ i) " }{XPPEDIT 18 0 "1-ln(2);" "6#,&\"\"\"F$-%#lnG6#\"\"#!\"\"" } {TEXT -1 15 " j) " }{XPPEDIT 18 0 "exp(1)-1;" "6#,&-%$expG6 #\"\"\"F'F'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT 358 8 "Solution" }{TEXT 359 8 ": ( e )\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "J1 := Int(1/(x+x*ln(x)),x = \+ 1 .. exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G-%$IntG6$*&\"\" \"F),&%\"xGF)*&F+F)-%#lnG6#F+F)F)!\"\"/F+;F)-%$expG6#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "J2 := Int(factor(integrand(J1)), x \+ = 1 .. exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J2G-%$IntG6$*&\" \"\"F)*&%\"xGF),&F)F)-%#lnG6#F+F)F)!\"\"/F+;F)-%$expG6#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "J3 := changevar(u=1+ln(x), J2, u); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J3G-%$IntG6$*&\"\"\"F)%\"uG!\" \"/F*;F)\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value(J3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 300 161 "13. Find the solutions x = a and x = b of th e equation sin(x) = cos(x) in the first and third quadrants respec tively. Calculate the area between " }{XPPEDIT 475 0 "y = sin(x); " "6#/%\"yG-%$sinG6#%\"xG" }{TEXT 473 12 " and " }{XPPEDIT 476 0 "y = cos(x);" "6#/%\"yG-%$cosG6#%\"xG" }{TEXT 474 7 " for " } {XPPEDIT 589 0 "x;" "6#%\"xG" }{TEXT 588 11 " in [a,b]." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 26 "a) 1 \+ b) " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT 590 44 " c) 2 d) " }{XPPEDIT 18 0 "1+sq rt(2);" "6#,&\"\"\"F$-%%sqrtG6#\"\"#F$" }{TEXT 591 11 " e) " } {XPPEDIT 18 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 477 18 " " }}{PARA 0 "" 0 "" {TEXT 302 4 "f) " } {XPPEDIT 18 0 "2+sqrt(2);" "6#,&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 592 12 " g) " }{XPPEDIT 18 0 "1+2*sqrt(2);" "6#,&\"\"\"F$*&\"\"#F$ -%%sqrtG6#F&F$F$" }{TEXT 593 6 " " }{TEXT -1 1 " " }{TEXT 305 3 " h) " }{XPPEDIT 18 0 "2*(1+sqrt(2));" "6#*&\"\"#\"\"\",&F%F%-%%sqrtG6#F $F%F%" }{TEXT 595 7 " i) " }{XPPEDIT 18 0 "3*sqrt(2);" "6#*&\"\"$\" \"\"-%%sqrtG6#\"\"#F%" }{TEXT 594 17 " j) " }{TEXT -1 3 " 4 " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 303 8 "So lution" }{TEXT 304 8 ": ( e )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve( sin(x) = cos(x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&\"\"%!\"\"%#PiG\"\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot([sin(x) , cos(x)], x = Pi/4 .. 5*Pi/4, color = [NAVY,MAROON]) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6&-%'CURVESG6 $7S7$$\"3o***z'\\;)R&y!#=$\"3E_4(H#y1rqF*7$$\"3gI)=;Wf(Q&)F*$\"3(fjAM8 G$QvF*7$$\"3B\\y*o$4eM\"*F*$\"3YEai^S@;zF*7$$\"3Q.SOBXk/)*F*$\"3j)p,L \"*fvI)F*7$$\"31TNis\\\"z/\"!#<$\"3uSK)f\"*QQm)F*7$$\"3f*zhB\"\\/:6F?$ \"3g;a%Hc0$z*)F*7$$\"3G&fH(fHGx6F?$\"3-#[Et7wcB*F*7$$\"3S^0\"G&osT7F?$ \"3uN2uVYVj%*F*7$$\"3==GA>[P38F?$\"3a5IJHikd'*F*7$$\"3kW:wW!4[P\"F?$\" 3-p\"eB*)e&3)*F*7$$\"3cSC`6[9V9F?$\"3!R+5#Ggj=**F*7$$\"3iy,5#*\\L.:F?$ \"3'>'3DqMDx**F*7$$\"3?dRTz[4r:F?$\"25nJVb*******F?7$$\"3B#GN`*H8R;F?$ \"3=5K!Refm(**F*7$$\"3*>y\"=F.q/]*= F?$\"3SJ&\\km%)*y%*F*7$$\"3?#y[MW(yk>F?$\"3:jVO6K%QB*F*7$$\"3ETodNQaE? F?$\"35N))4`FKz*)F*7$$\"33MHg&Q+V4#F?$\"3_X#[HeF2m)F*7$$\"3M&3cUk?)e@F ?$\"3_[aH%\\#Q?$)F*7$$\"3cY5al29EAF?$\"3$)3+lhzG#F? $\"3%R#zgs%=n`(F*7$$\"3#)f(\\'HQkaBF?$\"3sR8'>8C?3(F*7$$\"3Phrfry!RU#F ?$\"3'\\.p87Bkd'F*7$$\"3M?lsBC?%[#F?$\"3>>6U#oI06'F*7$$\"3'3Y\\=>A$\\D F?$\"35pH$H1iCe&F*7$$\"3!RN%H%G(f;EF?$\"3I5!*3w**37]F*7$$\"3ygEIPGT#o# F?$\"3t$F*7$$\"3G#Hs&)pI.)GF?$\"3mN&fa())*He#F*7$$\"3%R00o!\\; [HF?$\"3S!o;vSQA#>F*7$$\"3'ol4;yK'4IF?$\"3BJ,znAx:8F*7$$\"3Y:Wc/M$o2$F ?$\"3y8ouQ%*Rrk!#>7$$\"3`Myp`R1SJF?$\"3t&p\\lH*pG:!#?7$$\"3PUof'4eh?$F ?$!37%y)4S$f?X'Fgv7$$\"3')\\+>-sxqKF?$!3UIKu!Hb#)G\"F*7$$\"31>\"frTF%Q LF?$!3oJxK$)>mb>F*7$$\"32/Jf9Ee.MF?$!3W2>\")*>J+f#F*7$$\"3!H.VX@8-Z$F? $!3uBEdFrPFKF*7$$\"3!HJ%[z?HONF?$!37'>IcU2`%QF*7$$\"3o<*zOp6qf$F?$!3Cs V$f#oQ)R%F*7$$\"3%z-`PA.mm$F?$!3$p66E^@A,&F*7$$\"36qq&yrY)GPF?$!3GOe_& =s2a&F*7$$\"3qWW$[E6_z$F?$!3iu7y\\:j!3'F*7$$\"3;\"=,6'F*7$F8$\"3gnlc-wOmbF*7$F=$\"3gV(*)>m&y$*\\F*7$FC$\"3P h>yf:P,WF*7$FH$\"3l'*f%*pgNMQF*7$FM$\"3VIQ=7diJKF*7$FR$\"3S&>P`;*>%f#F *7$FW$\"3avY36,NZ>F*7$Ffn$\"3%o0fH>^IF\"F*7$F[o$\"3!>Bnel<5u'Fgv7$F`o$ !3e2SDYh_&)H!#@7$Feo$!3YyCpQ$\\$GoFgv7$Fjo$!3Kk=8H?/N8F*7$F_p$!3#)[rKN =TA>F*7$Fdp$!3zt3)oT9=h#F*7$Fip$!3O@kwGfs&=$F*7$F^q$!3FC-pl))QzF*7$Faz$!3*R_,\"*Q)*o`(F*7$Ffz$!3'H&QNny1rqF*-F[[ l6&F][lFa[lF^[l$\")%yg>%F`[l-%+AXESLABELSG6$Q\"x6\"Q!Fbel-%%VIEWG6$;$ \"+N;)R&y!#5$\"+=3*p#R!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(sin(x) - cos(x), x = Pi/4 .. 5*Pi/4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"F%#F&F%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 1 " " }{TEXT 270 110 "14. At irregular intervals during the first 10 se conds of a race, a radar gun recorded the following speeds " } {XPPEDIT 608 0 "v(t);" "6#-%\"vG6#%\"tG" }{TEXT 607 25 " (in m/s) of a runner: " }}{PARA 3 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 478 0 "v(0) \+ = 0,` `*v(1) = 6.8,` `*v(3) = 10.2,` `*v(5) = 11.0,` `*v(6) = 11.6 ,` `*v(10) = 11.8;" "6(/-%\"vG6#\"\"!F'/*&%#~~G\"\"\"-F%6#F+F+-%&Floa tG6$\"#o!\"\"/*&F*F+-F%6#\"\"$F+-F/6$\"$-\"F2/*&F*F+-F%6#\"\"&F+-F/6$ \"$5\"F2/*&F*F+-F%6#\"\"'F+-F/6$\"$;\"F2/*&F*F+-F%6#\"#5F+-F/6$\"$=\"F 2" }{TEXT 479 1 "." }}{PARA 3 "" 0 "" {TEXT 480 21 "\nGiven that dista nce " }{XPPEDIT 612 0 "s(t);" "6#-%\"sG6#%\"tG" }{TEXT 611 10 " at tim e " }{XPPEDIT 610 0 "t;" "6#%\"tG" }{TEXT 609 31 " is expressed by t he formula " }{XPPEDIT 614 0 "s(t) = int(v(tau),tau = 0 .. t);" "6#/- %\"sG6#%\"tG-%$intG6$-%\"vG6#%$tauG/F.;\"\"!F'" }{TEXT 613 100 ", est imate the distance (in m) the runner has covered during those 10 secon ds. Use trapezoids and " }{TEXT 481 3 "all" }{TEXT 482 17 " the give n data." }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 483 82 "a) 99.4 b) 99.5 c) 99.6 d) 99.7 \+ e) 99.8 " }{TEXT 486 1 " " }{TEXT 487 18 " " }}{PARA 0 "" 0 "" {TEXT 484 37 "f) 99.9 g) 100.0 \+ " }{TEXT -1 1 " " }{TEXT 485 43 "h) 100.1 i) 100.2 j) 1 00.3 " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 306 8 "Solution" }{TEXT 307 8 ": ( d )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 83 "(0+6.8)/2*1 + (6.8+10.2)/2*2 + (10.2+11.0)/2*2 + (11.0+11.6)/2*1 + (11.6+11.8)/2*4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++q**!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 359 "15. A swimmin g pool has the shape of a rectanle with width 12 feet and length 20 fe et. Measured at 5 foot intervals along its length, starting at the sha llow end and ending at the deepest end, the depths in feet are 1, 3, 7 , 9, 10. By applying Simpson's Rule with four subintervals, what appr oximation to the volume of the pool (in cubic feet) is obtained? " }} {PARA 0 "" 0 "" {TEXT 309 82 "a) 1410 b) 1420 \+ c) 1430 d) 1440 e) " }{TEXT 311 1 "1" }{TEXT 492 6 "450 " }{TEXT 313 18 " " }}{PARA 0 "" 0 "" {TEXT 310 40 "f) 1460 g) 1470 " }{TEXT -1 1 " \+ " }{TEXT 312 46 "h) 1480 i) 1490 j) 1500" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 314 8 "Solution " }{TEXT 315 8 ": ( f )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "12*5/3*(1+4*3+2*7+4*9+1*10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"%g9" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 11 "16. If " }{XPPEDIT 500 0 "y(x);" "6#-%\"yG6#%\"xG" }{TEXT 499 58 " is the unique solution of the initial value problem " } {XPPEDIT 495 0 "diff(y(x),x) = 2*x/(y(x)+x^2*y(x));" "6#/-%%diffG6$-% \"yG6#%\"xGF**(\"\"#\"\"\"F*F-,&-F(6#F*F-*&F*F,-F(6#F*F-F-!\"\"" } {TEXT 493 6 " , " }{XPPEDIT 496 0 "y(0) = 2;" "6#/-%\"yG6#\"\"!\"\" #" }{TEXT 494 39 " \n then what for what positive value " }{XPPEDIT 616 0 "x;" "6#%\"xG" }{TEXT 615 1 " " }{TEXT 617 7 " is " } {XPPEDIT 498 0 "y(x) = 4;" "6#/-%\"yG6#%\"xG\"\"%" }{TEXT 497 3 " ? " }{TEXT 273 2 " \n" }{TEXT 271 4 "\na) " }{XPPEDIT 257 0 "exp(1)-1;" "6 #,&-%$expG6#\"\"\"F'F'!\"\"" }{TEXT 501 16 " b) " } {XPPEDIT 503 0 "exp(2)-1;" "6#,&-%$expG6#\"\"#\"\"\"F(!\"\"" }{TEXT 502 15 " c) " }{XPPEDIT 511 0 "exp(4)-1;" "6#,&-%$expG6#\" \"%\"\"\"F(!\"\"" }{TEXT 504 18 " d) " }{XPPEDIT 512 0 " exp(6)-1;" "6#,&-%$expG6#\"\"'\"\"\"F(!\"\"" }{TEXT 505 15 " \+ e) " }{XPPEDIT 513 0 "exp(8)-1;" "6#,&-%$expG6#\"\")\"\"\"F(!\"\"" } {TEXT 506 9 " \nf) " }{XPPEDIT 514 0 "sqrt(exp(1)-1);" "6#-%%sqrtG 6#,&-%$expG6#\"\"\"F*F*!\"\"" }{TEXT 507 14 " g) " } {XPPEDIT 515 0 "sqrt(exp(2)-1);" "6#-%%sqrtG6#,&-%$expG6#\"\"#\"\"\"F+ !\"\"" }{TEXT 508 10 " h) " }{XPPEDIT 516 0 "sqrt(exp(4)-1);" "6 #-%%sqrtG6#,&-%$expG6#\"\"%\"\"\"F+!\"\"" }{TEXT 509 14 " i) " }{XPPEDIT 517 0 "sqrt(exp(6)-1);" "6#-%%sqrtG6#,&-%$expG6#\"\"'\"\" \"F+!\"\"" }{TEXT 510 12 " j) " }{XPPEDIT 518 0 "sqrt(exp(8)-1 );" "6#-%%sqrtG6#,&-%$expG6#\"\")\"\"\"F+!\"\"" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 308 0 "" }{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 298 8 "Solution" }{TEXT 299 8 ": ( i )\n" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqn1 \+ := int(y,y) = int(2*x/(1+x^2) , x) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,$*&\"\"#!\"\"%\"yGF(\"\"\",&-%#lnG6#,&F+F+*$)%\"xGF(F+ F+F+%\"CGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "eqn2 := C = \+ simplify(solve(subs(\{y=2,x=0\}, eqn1),C));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"CG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn3 := subs(eqn2, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,$*&\"\"#!\"\"%\"yGF(\"\"\",&-%#lnG6#,&F+F+*$) %\"xGF(F+F+F+F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eqn4 : = x = solve(subs(y=4,eqn3), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% eqn4G/%\"xG6$,$*$,&-%$expG6#\"\"'\"\"\"F/!\"\"#F/\"\"#F0F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 274 4 "17. " }{TEXT 295 44 " The differential equation for the current " } {XPPEDIT 623 0 "y;" "6#%\"yG" }{TEXT 622 28 " in a certain circuit i s " }{XPPEDIT 620 0 "dy/dt = 60-r*y;" "6#/*&%#dyG\"\"\"%#dtG!\"\",&\" #gF&*&%\"rGF&%\"yGF&F(" }{TEXT 618 10 " where " }{XPPEDIT 621 0 "r; " "6#%\"rG" }{TEXT 619 47 " is the resistance (measured in ohms). If " }{XPPEDIT 627 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT 624 11 " and " }{XPPEDIT 628 0 "y(1) = 5*(1-1/exp(12));" "6#/-%\"yG6#\" \"\"*&\"\"&F',&F'F'*&F'F'-%$expG6#\"#7!\"\"F0F'" }{TEXT 626 20 ", th en what is " }{XPPEDIT 629 0 "r;" "6#%\"rG" }{TEXT 625 2 " ?" }} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 294 138 "a) 1 \+ b) 2 c) 3 d) 4 e) 5 \nf) 6 \+ g) 10 h) 12 i) 15 j) 20 " }} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 296 8 "Solution " }{TEXT 297 8 ": ( h )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eqn := dsolve( \{diff(y(t),t) = 60- r*y(t), y(0)=0\} , y(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/ -%\"yG6#%\"tG,&*&\"#g\"\"\"%\"rG!\"\"F-*(F,F--%$expG6#,$*&F.F-F)F-F/F- F.F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "eqn1 := 5*(1-1/ex p(12)) = simplify(rhs(subs(t=1, eqn)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&\"\"&\"\"\"*&F'F(-%$expG6#\"#7!\"\"F.,$*(\"#gF(,&F(F. -F+6#,$%\"rGF.F(F(F6F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "testeq( subs(r=12,eqn1) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 276 82 "18. Find the solutions a and b, with a < b < 0 , of the equation " }{TEXT 597 1 " " }{XPPEDIT 598 0 "1/x = -(4*x+5) ;" "6#/*&\"\"\"F%%\"xG!\"\",$,&*&\"\"%F%F&F%F%\"\"&F%F'" }{TEXT 596 31 " . Calculate the area under " }{XPPEDIT 600 0 "y = 1/x;" "6#/% \"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT 599 13 " and over " }{XPPEDIT 602 0 "y = -(4*x+5);" "6#/%\"yG,$,&*&\"\"%\"\"\"%\"xGF)F)\"\"&F)!\"\"" } {TEXT 601 9 " for " }{XPPEDIT 603 0 "a <= x;" "6#1%\"aG%\"xG" } {XPPEDIT 604 0 "` ` <= b;" "6#1%\"~G%\"bG" }{TEXT 519 1 "." }}{PARA 3 "" 0 "" {TEXT 277 1 "a" }{TEXT 275 3 ") " }{XPPEDIT 524 0 "ln(2);" "6 #-%#lnG6#\"\"#" }{TEXT 520 19 " b) " }{XPPEDIT 525 0 "l n(1+sqrt(2));" "6#-%#lnG6#,&\"\"\"F'-%%sqrtG6#\"\"#F'" }{TEXT 521 12 " c) " }{XPPEDIT 606 0 "7/4-ln(2);" "6#,&*&\"\"(\"\"\"\"\"%!\" \"F&-%#lnG6#\"\"#F(" }{TEXT 605 12 " d) " }{XPPEDIT 256 0 "12/ 5-ln(2);" "6#,&*&\"#7\"\"\"\"\"&!\"\"F&-%#lnG6#\"\"#F(" }{TEXT 530 11 " e) " }{XPPEDIT 526 0 "15/8-2*ln(2);" "6#,&*&\"#:\"\"\"\"\")! \"\"F&*&\"\"#F&-%#lnG6#F*F&F(" }{TEXT 522 12 " \nf) " } {XPPEDIT 256 0 "16/5-ln(2);" "6#,&*&\"#;\"\"\"\"\"&!\"\"F&-%#lnG6#\"\" #F(" }{TEXT 523 10 " g) " }{XPPEDIT 532 0 "21/8-ln(2);" "6#,&*& \"#@\"\"\"\"\")!\"\"F&-%#lnG6#\"\"#F(" }{TEXT 527 11 " h) " } {XPPEDIT 533 0 "24/5-2*ln(2);" "6#,&*&\"#C\"\"\"\"\"&!\"\"F&*&\"\"#F&- %#lnG6#F*F&F(" }{TEXT 531 10 " i) " }{XPPEDIT 534 0 "5/3-ln(2); " "6#,&*&\"\"&\"\"\"\"\"$!\"\"F&-%#lnG6#\"\"#F(" }{TEXT 528 12 " \+ j) " }{XPPEDIT 535 0 "9/4-2*ln(2);" "6#,&*&\"\"*\"\"\"\"\"%!\"\"F&* &\"\"#F&-%#lnG6#F*F&F(" }{TEXT 529 8 " " }{TEXT 291 3 "\n " }} {PARA 3 "" 0 "" {TEXT 292 8 "Solution" }{TEXT 293 8 ": ( e )\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(1/x = -(4*x+5), x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#!\" \"\"\"%F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int( 1/x+(4*x+ 5), x=-1..-1/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"-%# lnG6#F%F&!\"\"#\"#:\"\")F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 278 16 "19. Calculate " }{XPPEDIT 18 0 "int(cot(x),x = Pi/6 .. Pi/2);" "6#-%$in tG6$-%$cotG6#%\"xG/F);*&%#PiG\"\"\"\"\"'!\"\"*&F-F.\"\"#F0" }{TEXT 536 9 ".\n\n\na) " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT 546 1 " " }{XPPEDIT 18 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 545 14 " b) " }{XPPEDIT 265 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 537 16 " c) " }{XPPEDIT 266 0 "2*ln(2);" "6#*&\"\"#\"\"\"- %#lnG6#F$F%" }{TEXT 538 12 " d) " }{XPPEDIT 18 0 "1/2;" "6#*& \"\"\"F$\"\"#!\"\"" }{TEXT 547 1 " " }{XPPEDIT 256 0 "ln(3);" "6#-%#ln G6#\"\"$" }{TEXT 543 14 " e) " }{XPPEDIT 267 0 "ln(3);" "6#- %#lnG6#\"\"$" }{TEXT 539 12 " \nf) " }{XPPEDIT 268 0 "2*ln(3); " "6#*&\"\"#\"\"\"-%#lnG6#\"\"$F%" }{TEXT 540 17 " g) " } {XPPEDIT 273 0 "3*ln(3);" "6#*&\"\"$\"\"\"-%#lnG6#F$F%" }{TEXT 541 12 " h) " }{XPPEDIT 274 0 "1;" "6#\"\"\"" }{TEXT 544 23 " \+ i) " }{XPPEDIT 275 0 "2;" "6#\"\"#" }{TEXT 542 25 " \+ j) " }{XPPEDIT 276 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 289 8 "Solution " }{TEXT 290 8 ": ( b )\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Integrand := int(cot( x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*IntegrandG-%#lnG6#-%$sinG 6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "answer := subs(x =Pi/2, Integrand) - subs(x=Pi/6, Integrand);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG,&-%#lnG6#-%$sinG6#,$*&\"\"#!\"\"%#PiG\"\"\"F 1F1-F'6#-F*6#,$*&\"\"'F/F0F1F1F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(answer);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%#lnG6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 281 16 "20. Calculate " }{XPPEDIT 549 0 "int((sin(x)+1)/c os(x),x = 0 .. Pi/3);" "6#-%$intG6$*&,&-%$sinG6#%\"xG\"\"\"F,F,F,-%$co sG6#F+!\"\"/%\"xG;\"\"!*&%#PiGF,\"\"$F0" }{TEXT 548 2 ".\n" }}{PARA 260 "" 0 "" {TEXT 280 5 "a) " }{XPPEDIT 555 1 "ln(sqrt(2))+Pi/6;" "6 #,&-%#lnG6#-%%sqrtG6#\"\"#\"\"\"*&%#PiGF+\"\"'!\"\"F+" }{TEXT 550 10 " b) " }{XPPEDIT 556 1 "ln(sqrt(3))+Pi/6;" "6#,&-%#lnG6#-%%sqrtG6 #\"\"$\"\"\"*&%#PiGF+\"\"'!\"\"F+" }{TEXT 282 12 " c) " } {XPPEDIT 557 1 "ln(1+sqrt(3));" "6#-%#lnG6#,&\"\"\"F'-%%sqrtG6#\"\"$F' " }{TEXT 283 12 " d) " }{XPPEDIT 558 1 "ln(2+sqrt(3));" "6#-%# lnG6#,&\"\"#\"\"\"-%%sqrtG6#\"\"$F(" }{TEXT 284 17 " e) \+ " }{XPPEDIT 559 0 "ln(1+sqrt(2));" "6#-%#lnG6#,&\"\"\"F'-%%sqrtG6#\"\" #F'" }{TEXT 551 10 " \nf) " }{XPPEDIT 560 1 "ln(2+sqrt(3));" "6#- %#lnG6#,&\"\"#\"\"\"-%%sqrtG6#\"\"$F(" }{TEXT 552 12 " g) " } {XPPEDIT 561 1 "ln(2)+Pi/3;" "6#,&-%#lnG6#\"\"#\"\"\"*&%#PiGF(\"\"$!\" \"F(" }{TEXT 285 15 " h) " }{XPPEDIT 562 0 "ln(2*sqrt(2)); " "6#-%#lnG6#*&\"\"#\"\"\"-%%sqrtG6#F'F(" }{TEXT 286 16 " \+ i) " }{XPPEDIT 563 0 "ln(2*sqrt(3))+Pi/3;" "6#,&-%#lnG6#*&\"\"#\"\"\"- %%sqrtG6#\"\"$F)F)*&%#PiGF)F-!\"\"F)" }{TEXT 553 14 " j) " } {XPPEDIT 564 1 "ln(4+2*sqrt(3));" "6#-%#lnG6#,&\"\"%\"\"\"*&\"\"#F(-%% sqrtG6#\"\"$F(F(" }{TEXT 554 2 " " }{TEXT 279 1 "\n" }}{PARA 3 "" 0 " " {TEXT 287 8 "Solution" }{TEXT 288 8 ": ( j )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "J := Int((si n(x)+1)/cos(x),x = 0 .. Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" JG-%$IntG6$*&,&-%$sinG6#%\"xG\"\"\"F.F.F.-%$cosGF,!\"\"/F-;\"\"!,$*&\" \"$F1%#PiGF.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "eqn := J \+ = Int(expand((sin(x)+1)/cos(x)),x = 0 .. Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$IntG6$*&,&-%$sinG6#%\"xG\"\"\"F/F/F/-%$cosGF -!\"\"/F.;\"\"!,$*&\"\"$F2%#PiGF/F/-F'6$,&*&F0F2F+F/F/*&F/F/F0F2F/F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "K := rhs(eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$IntG6$,&*&-%$cosG6#%\"xG!\"\"-%$sin GF,\"\"\"F1*&F1F1F*F.F1/F-;\"\"!,$*&\"\"$F.%#PiGF1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "L := Int(op(1, student[integrand](K)), x = 0 .. Pi/3) + Int(op(2, student[integrand](K)), x = 0 .. Pi/3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,&-%$IntG6$*&-%$cosG6#%\"xG!\"\" -%$sinGF,\"\"\"/F-;\"\"!,$*&\"\"$F.%#PiGF1F1F1-F'6$*&F1F1F*F.F2F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "antiderivative := int( op(1, student[integrand](L)) , x)+int( op(2,student[integrand](L)) , x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/antiderivativeG,&-%#lnG6#,&-%$secG6 #%\"xG\"\"\"-%$tanGF,F.F.-F'6#-%$cosGF,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "answer := subs(x=Pi/3, antiderivative) - subs(x= 0, antiderivative);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG,*-%# lnG6#,&-%$secG6#,$*&\"\"$!\"\"%#PiG\"\"\"F2F2-%$tanGF,F2F2-F'6#-%$cosG F,F0-F'6#,&-F+6#\"\"!F2-F4F=F2F0-F'6#-F8F=F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(answer);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#,&\"\"#\"\"\"*$\"\"$#F)F(F)F)-F%6#F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "testeq(simplify(answer) = ln(4 + 2* sqrt(3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}}{MARK "19 3 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }