{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Heading 2" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 256 " " 0 "" {TEXT -1 49 "Lesson 19: Applications of Inverse Trig Functions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "1) A lighthouse is on a small island 3 km away from the nearest po int O " }}{PARA 0 "" 0 "" {TEXT -1 70 "on a straight shoreline and its light makes 4 revolutions per minute. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "a) How fast is the beam of light mov ing along the shoreline when it is 1 km from O? See diagram." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 73 "b) Plot the ra te of change of x (from the diagram) with respect to time. " }}{PARA 0 "" 0 "" {TEXT -1 11 "c) What is " }{XPPEDIT 18 0 "Limit(dx/dt,x = in finity)" "6#-%&LimitG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"xG%)infinityG" } {TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 11 "d) What is " }{XPPEDIT 18 0 "Limit(dx/dt,x = 0)" "6#-%&LimitG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"xG \"\"!" }{TEXT -1 2 "? " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "r estart; with(plottools): with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 " Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "PO := line([0,0],[0,3], linestyle=3, col or=brown):\nPX := line([0,3], [-1, 3], linestyle=1, thickness=2, color =blue):\nXP := line([-1, 3], [0,0], thickness=3, color=yellow):\nshore line := line([-3,3],[3,3], color=coral):\ntext:=textplot(\{[.1,0,`P`], [0,3.05,`O`],[-1.1,3.05,`x`],[-.4,.9,`alpha`],[.1,1.5,`3`],[-.5,3.05,` 1`]\},color=black,align=\{ABOVE, RIGHT\}):\na := arc([0,0], .8, Pi/2.. Pi/1.65, thickness=2):\ndisplay(PO, PX, XP, shoreline, text, a, axes=n one);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6.-%'CU RVESG6%7$7$$\"\"!F)F(7$F($\"\"$F)-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))e qk\"F3F4-%*LINESTYLEG6#F,-F$6&7$F*7$$!\"\"F)F+-F.6&F0F(F($\"*++++\"F3- %*THICKNESSG6#\"\"#-F76#\"\"\"-F$6%7$FF(Q\"P6\"%+ALIGNA BOVEG%+ALIGNRIGHTG-F.6&F0F)F)F)-Ffn6'7$F($\"$0$!\"#Q\"OF[oF\\oF]oF^o-F fn6'7$$!#6F>FcoQ\"xF[oF\\oF]oF^o-Ffn6'7$$!\"%F>$\"\"*F>Q&alphaF[oF\\oF ]oF^o-Ffn6'7$Fin$\"#:F>Q\"3F[oF\\oF]oF^o-Ffn6'7$$!\"&F>FcoQ\"1F[oF\\oF ]oF^o-F$6$7U7$$!+Yq#3k\"!#>$\"\")F>7$$!+1z9J`!#7$\"+mB#)**z!#57$$!+qf? m5F[p$\"+r%*G**zF`r7$$!+@(\\#*f\"F[p$\"+T8S)*zF`r7$$!+JDAK@F[p$\"+7!er *zF`r7$$!+t05lEF[p$\"+U&fb*zF`r7$$!+T.'y>$F[p$\"+,gg$*zF`r7$$!+&z%F[p$\"+)*eh&)zF` r7$$!+iDCF`F[p$\"+)4VA)zF`r7$$!+?e0feF[p$\"+=e^yzF`r7$$!+1)31R'F[p$\"+ CUVuzF`r7$$!+&*z(=#pF[p$\"+)\\)**pzF`r7$$!+t)RGX(F[p$\"+M)3_'zF`r7$$!+ -2Z$)zF[p$\"+\\a1gzF`r7$$!+!3ZP^)F[p$\"+o&oX&zF`r7$$!+)GXO/*F[p$\"+Q%= ([zF`r7$$!+o>9t&*F[p$\"+:`^UzF`r7$$!+V8A55F`r$\"+y%ff$zF`r7$$!+FO3j5F` r$\"+<70HzF`r7$$!+5()*e6\"F`r$\"+Q3z@zF`r7$$!+HUmo6F`r$\"+k'yT\"zF`r7$ $!+dyP@7F`r$\"+K]@1zF`r7$$!+Qs.u7F`r$\"+)H+z*yF`r7$$!+[+kE8F`r$\"+H[B* )yF`r7$$!+PR=z8F`r$\"+6!>-)yF`r7$$!+ylmJ9F`r$\"+YK&3(yF`r7$$!+[c3%[\"F `r$\"+Xz8hyF`r7$$!+0)Qk`\"F`r$\"+WN2^yF`r7$$!+QPs)e\"F`r$\"+)[g1%yF`r7 $$!+6\"Q4k\"F`r$\"+S#**)HyF`r7$$!+8'zIp\"F`r$\"+z-z=yF`r7$$!+Mf9XF`r$\"+&Q()3x(F`r7$$!+VJh_>F`r$\"+ev/exF`r7$$!+5*oU+#F`r$\"+5K '[u(F`r7$$!+pc$e0#F`r$\"+C\\LJxF`r7$$!+Q6J2@F`r$\"+)Hjur(F`r7$$!+>Ipe@ F`r$\"+a*[Kq(F`r7$$!+U!z*4AF`r$\"+=Dp)o(F`r7$$!+;p;hAF`r$\"+SYztwF`r7$ $!+uVD7BF`r$\"+\")fbewF`r7$$!+d\"RKO#F`r$\"+9s(Hk(F`r7$$!+%)*=TT#F`r$ \"+N!fqi(F`r7$$!+6;*[Y#F`r$\"+\\@!3h(F`r7$$!+oZb:DF`r$\"+!G2Uf(F`r7$$! +@i5mDF`r$\"+h^FxvF`r7$$!+5Pa;EF`r$\"+\\l+gvF`rFC-%*AXESSTYLEG6#%%NONE G" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" " Curve 9" "Curve 10" "Curve 11" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "In the above diagram, the length of segment OP is given to be 3km. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "alpha;" "6#%&alpha G" }{TEXT -1 19 " be the angle xPO. " }}{PARA 0 "" 0 "" {TEXT -1 18 "W e are given that " }{XPPEDIT 18 0 "d*alpha/dt" "6#*(%\"dG\"\"\"%&alpha GF%%#dtG!\"\"" }{TEXT -1 21 " = 4 rev/min = 4 ( 2 " }{XPPEDIT 18 0 "Pi ;" "6#%#PiG" }{TEXT -1 11 ") 60 = 480 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG " }{TEXT -1 15 " radians/hour. " }}{PARA 0 "" 0 "" {TEXT -1 21 "We are asked to find " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dtG!\"\"" } {TEXT -1 25 " when x = 1 km. As tan ( " }{XPPEDIT 18 0 "alpha;" "6#%&a lphaG" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "x/3" "6#*&%\"xG\"\"\"\"\"$! \"\"" }{TEXT -1 8 " we see " }}{PARA 0 "" 0 "" {TEXT -1 5 "that " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 " arctan(x/3)" "6#-%'arctanG6#*&%\"xG\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus, " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "d*alpha/dt" "6#*(%\"dG\"\"\"%&alphaGF%%#dtG!\"\"" } {TEXT -1 2 "= " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 7 " 480 = " } {XPPEDIT 18 0 "( 1 / 3*( 1 + (x/3)^2) )* (dx/dt) " "6#**\"\"\"F$\"\"$! \"\",&F$F$*$*&%\"xGF$F%F&\"\"#F$F$*&%#dxGF$%#dtGF&F$" }}{PARA 0 "" 0 " " {TEXT -1 14 "and therefore " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dtG!\"\"" }{TEXT -1 7 " = 480 \+ " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 10 " 3 ( 1 + (" }{XPPEDIT 18 0 "x/3)^2" "6#*$*&%\"xG\"\"\"\"\"$!\"\"\"\"#" }{TEXT -1 5 " ) . " } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dxt:= x -> 480 * Pi * 3 * ( 1 + (x/3)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dxtGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&%#PiG\"\" \",&F/F/*&#F/\"\"*F/)9$\"\"#F/F/F/\"%S9F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(dxt(1));" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+Y#[l-&!\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dtG!\"\"" }{TEXT -1 21 " when x = 1 is about " }{XPPEDIT 18 0 "5026.548246;" "6#$\"+Y#[l-& !\"'" }{TEXT -1 10 " km/hour. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 " b) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(dxt(x), x = 0.. 100);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)$\"3?-$p6U$*Q_%!#97$$\"3ymmm;ar z@!#<$\"3X.n#FS7$$\"3_+++D.&4]#FF$\"3]E())*o/A*=$FS7$$\"3 2+++vB_N[^&3tv$FS7$$\"3W+++v'Hi#HFF$\"3umJ*[x\"Q\\VFS7$$\"3 %pm;z*ev:JFF$\"3\"HAFe?z\\#\\FS7$$\"3)RLL$347TLFF$\"3oBV1R%>kl&FS7$$\" 3KLLLLY.KNFF$\"3mR9`4H*fJ'FS7$$\"3O++D\"o7Tv$FF$\"3A@#3)\\`LHrFS7$$\"3 kLLL$Q*o]RFF$\"3[[;?'f[1*yFS7$$\"3m++D\"=lj;%FF$\"3)o:62QA1x)FS7$$\"3A ++vV&RY2aFF$\"3y53p\"**=VZ\"Fir7$ $\"3%pmm\"zXu9cFF$\"3K\"o]q:h\"*e\"Fir7$$\"3F+++]y))GeFF$\"3NcB:'eSBr \"Fir7$$\"3I++]i_QQgFF$\"3b]HMp(3t$=Fir7$$\"3>++D\"y%3TiFF$\"3?te#**p@ C'>Fir7$$\"3m****\\P![hY'FF$\"3daOH@v<1@Fir7$$\"3MLLL$Qx$omFF$\"3JuIR; ?pRAFir7$$\"3%3++]P+V)oFF$\"3uv7EVcy'Q#Fir7$$\"3Amm\"zpe*zqFF$\"3:tAAw @7CDFir7$$\"3;,++D\\'QH(FF$\"3/oy&4'4nyEFir7$$\"3yKLe9S8&\\(FF$\"3+P(f _g*GGGFir7$$\"3!4+]i?=bq(FF$\"3!GK[&eu.*)HFir7$$\"34LLL3s?6zFF$\"39lfg r'*\\]JFir7$$\"3h++DJXaE\")FF$\"3h**[d[G4CLFir7$$\"3&zmmm'*RRL)FF$\"3k S1'yi!p&\\$Fir7$$\"3)ommTvJga)FF$\"3M?OE^ikvOFir7$$\"3'[L$e9tOc()FF$\" 3qdyS9zdeQFir7$$\"33,++]Qk\\*)FF$\"3.w0ihUfISFir7$$\"3uLL$3dg6<*FF$\"3 S_O:$)HOKUFir7$$\"3]nmmmxGp$*FF$\"3)*4Af\\l+A=YFir7$$\"3Y,+v=5s#y*FF$\"3!*3)4S_7]\"[Fir7$$\"$+\"F)$\"3,Q [;R@2J]Fir-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!6\" -%%VIEWG6$;F(Fhz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "c) and d) We see that " }{XPPEDIT 18 0 "Limit(dx/dt, x=infinity) = infinity " "6#/-%&LimitG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"xG%)infinityGF." }}{PARA 0 "" 0 "" {TEXT -1 9 "and that " }{XPPEDIT 18 0 "Limit(dx/dt,x = 0) = \+ 0;" "6#/-%&LimitG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"xG\"\"!F." }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "2) Where should the point P \+ be chosen on the line segment AB " }}{PARA 0 "" 0 "" {TEXT -1 36 "so a s to maximize the angle theta = " }{XPPEDIT 18 0 "theta;" "6#%&thetaG " }{TEXT -1 31 "? What is the maximal value of " }{XPPEDIT 18 0 "theta ;" "6#%&thetaG" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 71 "Assume that the segment AB has length 3 and the vertical line segments " }} {PARA 0 "" 0 "" {TEXT -1 35 "have lengths 5 and 2. See Diagram. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 543 "X := 1:\nground := line([0, 0],[3,0], color=brown, thickness=3):\nstake1 := line([0,0],[0,5], colo r=blue, thickness=2):\nstake2 := line([3,0],[3,2], color=blue, thickne ss=2):\nline1 := line([X,0],[0,5], color=red, linestyle=2, thickness=2 ):\nline2 := line([X,0],[3,2], color=red, linestyle=2, thickness=2):\n text := textplot(\{[-.1,-.1,`A`],[X,-.1,`P`],[3,-.1,`B`],[X+.4,1.3,`th eta`], [-.1,2.5,`5`],[3.1,1,`2`]\}):\na := arc([1,0], 1, arccot((3-X)/ 2)..Pi-arccot(X/5)):\ndisplay(ground, stake1, stake2, line1, line2, te xt,a,axes=none,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "60-%'CURVESG6%7$7$$\"\"!F)F(7$$\"\"$F)F(-%'CO LOURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"F3F4-%*THICKNESSG6#F,-F$6%7$F'7$F( $\"\"&F)-F.6&F0F(F($\"*++++\"F3-F76#\"\"#-F$6%7$F*7$F+$FEF)F?FC-F$6&7$ 7$$\"\"\"F)F(F<-F.6&F0FAF(F(FC-%*LINESTYLEGFD-F$6&7$FNFIFQFCFS-%%TEXTG 6$7$$!\"\"FgnFfnQ\"A6\"-FY6$7$F+FfnQ\"BFin-FY6$7$Ffn$\"#DFgnQ\"5Fin-FY 6$7$$\"#JFgnFOQ\"2Fin-FY6$7$FOFfnQ\"PFin-FY6$7$$\"#9Fgn$\"#8FgnQ&theta Fin-F$6#7U7$$\"+\"y1rq\"!\"*$\"+8y1rq!#57$$\"+#GsIp\"F\\q$\"+j4o3sF_q7 $$\"+?+xy;F\\q$\"+/\"4NM(F_q7$$\"+Ab?k;F\\q$\"+Z,]vuF_q7$$\"+9WQ\\;F\\ q$\"+)4.Yg(F_q7$$\"+ACJM;F\\q$\"+!3o2t(F_q7$$\"+q`**=;F\\q$\"+[j%R&yF_ q7$$\"+u\"RMg\"F\\q$\"+;.4uzF_q7$$\"+W)\\we\"F\\q$\"+nN:\"4)F_q7$$\"+# [L;d\"F\\q$\"+s340#)F_q7$$\"+uiRb:F\\q$\"+;#eeJ)F_q7$$\"+%\\W*Q:F\\q$ \"+/GTB%)F_q7$$\"+)\\%GA:F\\q$\"+zIrF&)F_q7$$\"+@FU0:F\\q$\"+a(=(G')F_ q7$$\"+zcO)[\"F\\q$\"+(z!RE()F_q7$$\"+i*>6Z\"F\\q$\"+x9p?))F_q7$$\"+JA p`9F\\q$\"+oVe6*)F_q7$$\"+?#*3O9F\\q$\"+\\V.***)F_q7$$\"+JxJ=9F\\q$\"+ Hw+$3*F_q7$$\"+HYQ+9F\\q$\"+s_oX8F\\q$\"+l`]$Q*F_q7$$\"+JbD\"F\\q$\"+zbTx'*F_q7$ $\"+Qd(GB\"F\\q$\"+\"*\\1D(*F_q7$$\"+=lr87F\\q$\"+5s&*o(*F_q7$$\"+JZZ% >\"F\\q$\"+$Gv!4)*F_q7$$\"+6y:v6F\\q$\"+1PSX)*F_q7$$\"+?Kxb6F\\q$\"+Y% Gz()*F_q7$$\"+Z%Gj8\"F\\q$\"+Spj1**F_q7$$\"+/5$o6\"F\\q$\"+$4=:$**F_q7 $$\"+E%)G(4\"F\\q$\"+%HiD&**F_q7$$\"+i#3x2\"F\\q$\"+79wp**F_q7$$\"+u!) 4e5F\\q$\"+.)3J)**F_q7$$\"+UaYQ5F\\q$\"+5$*f#***F_q7$$\"+[z\")=5F\\q$ \"+n#H#)***F_q7$$\"+\\=j\"***F_q$\"+*\\'******F_q7$$\"+Tu3&z*F_q$\"+A. !z***F_q7$$\"+4Ai)f*F_q$\"+Z:%>***F_q7$$\"+O?J-%*F_q$\"+vC7#)**F_q7$$ \"+%yKi?*F_q$\"++pWo**F_q7$$\"+'=g/,*F_q$\"+1,#4&**F_q7$$\"+**)p]\"))F _q$\"+l)[&H**F_q7$$\"+*QP,i)F_q$\"+J9M/**F_q7$$\"+')ztD%)F_q$\"+XvIv)* F_q7$$\"+&yY>B)F_q$\"+C%eC%)*F_q7$$\"+X'Q)Q!)F_q$\"+cn!e!)*F_q-%(SCALI NGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Let x denote t he length of the line segment AP. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Det ermine " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 37 " in terms \+ of x. Plot this function.; " }}{PARA 0 "" 0 "" {TEXT -1 41 "we are loo king for what value of x makes " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 10 " maximal. " }}{PARA 0 "" 0 "" {TEXT -1 17 " What happens to " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 30 " as x varies \+ between 0 and 3? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "We have: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Pi;" "6#%#Pi G" }{TEXT -1 3 " - " }{XPPEDIT 18 0 "arccot(x/5)" "6#-%'arccotG6#*&%\" xG\"\"\"\"\"&!\"\"" }{TEXT -1 3 " - " }{XPPEDIT 18 0 "arccot( (3-x)/2 \+ )" "6#-%'arccotG6#*&,&\"\"$\"\"\"%\"xG!\"\"F)\"\"#F+" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus, " }{XPPEDIT 18 0 "d*theta/dt" "6 #*(%\"dG\"\"\"%&thetaGF%%#dtG!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 " ( 1 / (5*( 1 + (x/5)^2)) ) -( 1 / (2*(1 + ( (3-x)/5 )^2 )) ) " "6#,&*& \"\"\"F%*&\"\"&F%,&F%F%*$*&%\"xGF%F'!\"\"\"\"#F%F%F,F%*&F%F%*&F-F%,&F% F%*$*&,&\"\"$F%F+F,F%F'F,F-F%F%F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 9 " Setting " }{XPPEDIT 18 0 "d*theta/dt" "6#*(%\"dG\"\"\"% &thetaGF%%#dtG!\"\"" }{TEXT -1 14 " = 0 we have: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "5* ( 1 + x^2/25 ) = 2 *( 1 + (9 -6*x + x^2)/4 )" "6#/*&\"\"&\"\"\",&F&F& *&%\"xG\"\"#\"#D!\"\"F&F&*&F*F&,&F&F&*&,(\"\"*F&*&\"\"'F&F)F&F,*$F)F*F &F&\"\"%F,F&F&" }{TEXT -1 5 " iff " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x^2 - 10 *x + 5 = 0" "6#/,(*$%\"xG\"\"#\"\"\"*&\"#5F(F& F(!\"\"\"\"&F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Hence we have two critical values; as x must be between 0 and 3 we " }}{PARA 0 "" 0 "" {TEXT -1 5 "have " } {XPPEDIT 18 0 "5 - 2 *sqrt(5)" "6#,&\"\"&\"\"\"*&\"\"#F%-%%sqrtG6#F$F% !\"\"" }{TEXT -1 24 " as our critical value. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf( 5 - 2* sqrt(5));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*WS'y_!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "th:= x -> Pi - arccot(x/5) - arccot((3-x)/2);" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#thGR6#%\"xG6\"6$%)o peratorG%&arrowGF(,(%#PiG\"\"\"-%'arccotG6#,$9$#F.\"\"&!\"\"-F06#,&#\" \"$\"\"#F.*&#F.F " 0 "" {MPLTEXT 1 0 6 "th(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arccotG 6#,$%\"xG#\"\"\"\"\"&!\"\"-F%6#,&#!\"$\"\"#F**&#F*F2F*F(F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(th(x), x = 0..3);" } {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6% -%'CURVESG6$7S7$$\"\"!F)$\"3l\"HtCBPz#)*!#=7$$\"3s******\\i9Rl!#>$\"3W Cf,AEdc)*F,7$$\"3/++vVA)GA\"F,$\"39ED9Y**))y)*F,7$$\"3+++]Peui=F,$\"3W %z)zL^&4!**F,7$$\"3A++]i3&o]#F,$\"3C(*46+fw>**F,7$$\"3%)***\\(oX*y9$F, $\"3*)R'pf?K\\$**F,7$$\"3z***\\P9CAu$F,$\"3yYGcjdnX**F,7$$\"3!)***\\P* zhdVF,$\"3'=eA:*QG`**F,7$$\"31++v$>fS*\\F,$\"39vwFtHAd**F,7$$\"3$)*** \\(=$f%GcF,$\"3o^]w,!)*p&**F,7$$\"3Q+++Dy,\"G'F,$\"3ml<`QCD_**F,7$$\"3 3++]7')*>)))* F,7$$\"3))***\\PpnsM*F,$\"3+%Rr,n^E')*F,7$$\"3,++]siL-5!#<$\"39ALHmbeE )*F,7$$\"3-+++!R5'f5Ffp$\"354sNzk.\"z*F,7$$\"3)***\\P/QBE6Ffp$\"3w)*z' ))\\@Pu*F,7$$\"3!******\\\"o?&=\"Ffp$\"3#QrK\\FAjp*F,7$$\"31+]Pa&4*\\7 Ffp$\"3U`aj*\\s\"Q'*F,7$$\"33+]7j=_68Ffp$\"33I5G%H`md*F,7$$\"33++vVy!e P\"Ffp$\"3?&fV7<'*e]*F,7$$\"34+](=WU[V\"Ffp$\"3'=&)\\\"Ffp$\"3uaXo!e)f^$*F,7$$\"3)***\\P>:mk:Ffp$\"3'f2r.3]wD*F,7$ $\"3'***\\iv&QAi\"Ffp$\"3eEA*)Hgcp\"*F,7$$\"31++vtLU%o\"Ffp$\"3QrI)G)= un!*F,7$$\"3!******\\Nm'[Ffp$\"3G[%3R/7^d)F,7$$\"3z*****\\@80+#Ffp$\"3Eocb7;PS%)F,7$$\"31++] 7,Hl?Ffp$\"3KwRzcc/*G)F,7$$\"3()**\\P4w)R7#Ffp$\"3cE)oMXwF,7$$\"3')****\\i@OtBFfp$\"3mwoeBz6ouF ,7$$\"3')**\\PfL'zV#Ffp$\"3#>s%4;JDwsF,7$$\"3>+++!*>=+DFfp$\"3%[.t\"fu p&3(F,7$$\"3-++DE&4Qc#Ffp$\"3#>e7D$oP&)oF,7$$\"3=+]P%>5pi#Ffp$\"3'R_fo w:/aF, -%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$ ;F(Fez%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(th(0 ));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Ns$z#)*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(th(.527864044));" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*PXw&**!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "From the plot of " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 15 " , we see that " }{XPPEDIT 18 0 "t heta;" "6#%&thetaG" }{TEXT -1 30 " has one maximum which occurs " }} {PARA 0 "" 0 "" {TEXT -1 3 "at " }{XPPEDIT 18 0 "x = 5 - 2 *sqrt(5)" " 6#/%\"xG,&\"\"&\"\"\"*&\"\"#F'-%%sqrtG6#F&F'!\"\"" }{TEXT -1 16 " whic h is about " }{XPPEDIT 18 0 ".527864044;" "6#$\"*WS'y_!\"*" }{TEXT -1 8 " units. " }}{PARA 0 "" 0 "" {TEXT -1 67 "You can also use the first derivative test to see that indeed this " }}{PARA 0 "" 0 "" {TEXT -1 43 "critical value leads to a maximal value of " }{XPPEDIT 18 0 "theta ;" "6#%&thetaG" }{TEXT -1 21 " . Make sure you can " }}{PARA 0 "" 0 " " {TEXT -1 18 "do this analysis. " }}{PARA 0 "" 0 "" {TEXT -1 18 " Fro m the plot of " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 43 " , \+ we see that as x increases from 0 to 3, " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 30 " increases fr om approximately " }{XPPEDIT 18 0 ".9827937235;" "6#$\"+Ns$z#)*!#5" } {TEXT -1 4 " to " }{XPPEDIT 18 0 ".995764537;" "6#$\"*PXw&**!\"*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "and then " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 27 " strictly decreases to 0 . " }}} }{MARK "14 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }