{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "2D Comment" -1 18 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1" -1 219 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle2" -1 220 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 221 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle4" -1 222 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle6" -1 224 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle7" -1 225 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle8" -1 226 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 } {CSTYLE "" 220 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple P lot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "_pstyle1" -1 201 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 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle4" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle 5" -1 205 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 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle6" -1 206 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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle1 " -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 2 0 2 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 201 "" 0 "" {TEXT 219 68 "C. Design and Thrill of One Coaster Drop using a Polynomial Function" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 220 169 "In this Maple workshe et, we build one hill of a roller coaster using one peak and one valle y point. Our approach is to determine a cubic polynomial of the form \+ f (x) = " }{XPPEDIT 18 0 "a*x^3+b*x^2+c*x+d;" "6#,**&%\"aG\"\"\"*$%\"x G\"\"$F&F&*&%\"bGF&*$F(\"\"#F&F&*&%\"cGF&F(F&F&%\"dGF&" }{TEXT 220 120 " that connects these points and then to calculate the angle of s teepest descent/ascent and the resulting thrill factor." }}{PARA 201 " " 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 221 19 "I. Getting Sta rted" }}{PARA 201 "" 0 "" {TEXT 220 113 "We clear all variables. To a void multiple solutions, we set the maximum number of solutions variab le equal to 1." }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 8 "restar t:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 12 "_MaxSols:=1:" }}} {EXCHG {PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 221 16 "II. Data Points" }}{PARA 201 "" 0 "" {TEXT 220 313 "We enter x coordinates and y coordinates and slope conditions for the peak and valley points using lists. These lists c an be easily extended to cover the case where more than 2 peak and val ley points are used and the more complicated case where general points (not necessarily peak and valley points) are used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "In this example, we ha ve an initial peak at (0,75) followed by the first valley at (50,0)." }}{PARA 256 "" 0 "" {TEXT -1 89 "In your work, use the collected peak \+ and valley data points from the Colossus (Module A)." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 14 "xd ata:=[0,50]:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 14 "ydata:= [75,0]:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 14 "slopes:=[0,0 ]:" }}}{EXCHG {PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 221 33 "III. Connecting Cubic Polynomial" }}{PARA 201 "" 0 "" {TEXT 220 50 "We determine cubic poly nomials of the form f(x) = " }{XPPEDIT 18 0 "a*x^3+b*x^2+c*x+d;" "6#,* *&%\"aG\"\"\"*$%\"xG\"\"$F&F&*&%\"bGF&*$F(\"\"#F&F&*&%\"cGF&F(F&F&%\"d GF&" }{TEXT 220 214 " that connect the peak and valley points (with \+ zero slope conditions). Note that the Maple code is designed for easy extension using a \"do\" loop for the case where more than 2 peaks an d valleys have been marked." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }} {PARA 201 "" 0 "" {TEXT 220 94 "We define a function f(x) with unknown coefficients a1, b1, c1, d1 and then determine f ' (x)." }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 36 "f:=x->a[1]*x^3+b[1]*x^2+c[1]*x+ d[1]:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 9 "fp:=D(f):" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 220 53 "Now, we \"fit\" f(x) to the pea k and valley conditions." }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 126 "s[1]:=fsolve(\{f(xdata[1])=ydata[1],fp(xdata[1])=slopes[1],f( xdata[2])=ydata[2],fp(xdata[2])=slopes[2]\},\{a[1],b[1],c[1],d[1]\}): " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 13 "assign(s[1]):" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 36 "f:=x->a[1]*x^3+b[1]*x^2+ c[1]*x+d[1]:" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 220 61 "We plot our fu nction f(x) for the given interval and display." }}}{EXCHG {PARA 201 " > " 0 "" {MPLTEXT 1 222 42 "tplot[1]:=plot(f(x),x=xdata[1]..xdata[2]): " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 12 "with(plots):" }} {PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r edefined\n" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 18 "display(t plot[1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 133 121 121 {PLOTDATA 2 "6%-%' CURVESG6$7S7$$\"\"!F)$\"#vF)7$$\"3SLLL3x&)*3\"!#<$\"3S+y?M_Y*[(!#;7$$ \"3zmm\"H2P\"Q?F/$\"3+NLoY*HOY(F27$$\"3XLL$eRwX5$F/$\"3bq_]C_%oT(F27$$ \"3=ML$3x%3yTF/$\"3UfWF$eW;N(F27$$\"3gmm\"z%4\\Y_F/$\"3m)y)Hu&)fpsF27$ $\"34LLeR-/PiF/$\"3+MfXn*3!zrF27$$\"3;++DcmpisF/$\"3w;C)R=\\72(F27$$\" 3vLLe*)>VB$)F/$\"3Oiq2C:oXpF27$$\"3o++DJbw!Q*F/$\"3:r\"zFjqq!oF27$$\"3 %ommTIOo/\"F2$\"3Ywew&[$Q^mF27$$\"3^LL3_>jU6F2$\"3xRMa*\\sR]'F27$$\"3E ++]i^Z]7F2$\"3'HR$phIKFjF27$$\"3/++](=h(e8F2$\"3\"3(G@K=URhF27$$\"3A++ ]P[6j9F2$\"3)3[4(\\r@\\fF27$$\"3[L$e*[z(yb\"F2$\"3g^'**Q*zUpdF27$$\"3+ nm;a/cq;F2$\"3aVb;RMwZbF27$$\"3mmmm;t,mF2$\"3H1+L[U98\\F27$$\"3M+]i!f#=$ 3#F2$\"353+r!>\\\"zYF27$$\"37+](=xpe=#F2$\"3e,Rp\"HtIX%F27$$\"3-nm\"H2 8IH#F2$\"3%3(GU=kl9UF27$$\"3%om\"zpSS\"R#F2$\"3c25Ohr=%*RF27$$\"3cLL3_ ?`(\\#F2$\"3[&p35)GbbPF27$$\"3fL$e*)>pxg#F2$\"3Cr0r+&pw]$F27$$\"3D+]Pf 4t.FF2$\"3oNwpv+i#H$F27$$\"3ZLLe*Gst!GF2$\"3#*\\KdGr*=1$F27$$\"39+++DR W9HF2$\"39q1sUN/EGF27$$\"3:++DJE>>IF2$\"3;**f9p5h)f#F27$$\"35+]i!RU07$ F2$\"3;=)f\\5aCQ#F27$$\"3$)***\\(=S2LKF2$\"3m\\Rq;x&y9#F27$$\"3nmmm\"p )=MLF2$\"3LPUMZRtU>F27$$\"3U++](=]@W$F2$\"3tkx))*o<0t\"F27$$\"36L$e*[$ z*RNF2$\"3*oT$)R5A]a\"F27$$\"3e++]iC$pk$F2$\"3K)**epR]/N\"F27$$\"3Sm;H 2qcZPF2$\"3qc6APL)f<\"F27$$\"3Y+]7.\"fF&QF2$\"3sxuzn4N.5F27$$\"3amm;/O gbRF2$\"3cdcJ2n%)\\%)F/7$$\"3I+]ilAFjSF2$\"35jAI)e,3\"pF/7$$\"3)RLLL)* pp;%F2$\"3+*ph#\\Yw^bF/7$$\"3WLL3xe,tUF2$\"3%))y#*Qu%\\&H%F/7$$\"3Wn;H dO=yVF2$\"31/_0ybQ\">$F/7$$\"3a+++D>#[Z%F2$\"3B%f(G*e([3BF/7$$\"3)om;a G!e&e%F2$\"3u%GP%GXGg9F/7$$\"3wLLL$)Qk%o%F2$\"3]()=q0#4Td)!#=7$$\"3m+] iSjE!z%F2$\"3Uu60:!H#[QFiy7$$\"3u+]P40O\"*[F2$\"3U*o#HT=%o/\"Fiy7$$\"# ]F)F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fb[l-%%V IEWG6$;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 201 "" 0 "" {TEXT 220 97 "Our function looks correct. It goes throught the given peak and valley p oints with proper slope." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 204 "" 0 "" {TEXT 224 45 "IV. Calcu lation of Angle of Steepest Descent" }}{PARA 201 "" 0 "" {TEXT -1 0 " " }}{PARA 201 "" 0 "" {TEXT 220 47 "Now we must find the angle of stee pest descent." }}{PARA 201 "" 0 "" {TEXT 220 101 "So, we must find the maximum value of the absolute value of the derivative on the interval [x1,x2]. " }}{PARA 201 "" 0 "" {TEXT 220 65 "We first graph the abso lute value of f ' on the given interval." }}{PARA 201 "" 0 "" {TEXT 220 81 "From the graph below, it looks like f ' has an absolute minim um at about x = 25." }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 36 " f:=x->a[1]*x^3+b[1]*x^2+c[1]*x+d[1]:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 9 "fp:=D(f):" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 38 "plot(abs(fp(x)),x=xdata[1]..xdata[2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 130 126 126 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3S LLL3x&)*3\"!#<$\"3)46@8W$)*=>!#=7$$\"3zmm\"H2P\"Q?F-$\"3e'4UIj-\">NF07 $$\"3XLL$eRwX5$F-$\"3>Uw)pIb7C&F07$$\"3=ML$3x%3yTF-$\"3)>I=GYA@*oF07$$ \"3gmm\"z%4\\Y_F-$\"3kBMl%ofFX)F07$$\"34LLeR-/PiF-$\"3#yXVqF[i#)*F07$$ \"3;++DcmpisF-$\"3;i/L_qR<6F-7$$\"3vLLe*)>VB$)F-$\"3!4@=e[6)[7F-7$$\"3 o++DJbw!Q*F-$\"3'Gw'f_Aur8F-7$$\"3%ommTIOo/\"!#;$\"3%HK?$)\\$z*[\"F-7$ $\"3^LL3_>jU6Ffn$\"3_j\")p8(=ne\"F-7$$\"3E++]i^Z]7Ffn$\"3U:%p\\cFzo\"F -7$$\"3/++](=h(e8Ffn$\"37#pB-jE6y\"F-7$$\"3A++]P[6j9Ffn$\"3]XM_(*G&H'= F-7$$\"3[L$e*[z(yb\"Ffn$\"3pv(QZF-7$$\"3+nm;a/cq;Ffn$\"3113L93L-? F-7$$\"3mmmm;t,mFfn$\"3!fbx1D04:#F-7$$\"3M+]i!f#=$3#Ffn$\"3!*)32!pZX (=#F-7$$\"37+](=xpe=#Ffn$\"3XG,#>*fZ9AF-7$$\"3-nm\"H28IH#Ffn$\"3)f(fK3 jdMAF-7$$\"3%om\"zpSS\"R#Ffn$\"3/(=>E\\adC#F-7$$\"3cLL3_?`(\\#Ffn$\"33 4RK2y**\\AF-7$$\"3fL$e*)>pxg#Ffn$\"3q:@\"z))=eC#F-7$$\"3D+]Pf4t.FFfn$ \"3Cr\"H1td]B#F-7$$\"3ZLLe*Gst!GFfn$\"3[nY@>!))f@#F-7$$\"39+++DRW9HFfn $\"3195*Q/l\")=#F-7$$\"3:++DJE>>IFfn$\"33_g>W!eH:#F-7$$\"35+]i!RU07$Ff n$\"3fnu\"4xt86#F-7$$\"3$)***\\(=S2LKFfn$\"3cT@*Q*o`c?F-7$$\"3nmmm\"p) =MLFfn$\"3+j.h@l[**>F-7$$\"3U++](=]@W$Ffn$\"3OP(4()3Z/$>F-7$$\"36L$e*[ $z*RNFfn$\"3CskMj%R1'=F-7$$\"3e++]iC$pk$Ffn$\"35Ep_LlVw#[Z%Ffn$\"3)GXTB6#Gg%)F07$$\"3) om;aG!e&e%Ffn$\"3!>z#4VvFToF07$$\"3wLLL$)Qk%o%Ffn$\"3b58&3(>R=`F07$$\" 3m+]iSjE!z%Ffn$\"3+.1S?$[oh$F07$$\"3u+]P40O\"*[Ffn$\"3aYi7\"p@I\">F07$ $\"#]F)F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!F`[l -%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 201 "" 0 "" {TEXT 220 162 "To work exactly, we find all critical points of f ' on the given interva l and then evaluate and compare values of f ' at the critical points a nd at the endpoints." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 " " 0 "" {TEXT 225 54 "Location of angle of steepest descent/ascent and \+ slope" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 11 "fpp:=D(fp):" } }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 42 "p1:=fsolve(fpp(x)=0,x, xdata[1]..xdata[2]);" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>%#p1G$\"#D\" \"!" }{TEXT 226 1 " " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 61 "slope1:=max(abs(fp(xdata[1])),abs(fp(xdata[2])),abs(fp(p1)));" }} {PARA 205 "" 1 "" {XPPMATH 20 "6#>%'slope1G$\"++++]A!\"*" }{TEXT 226 1 " " }}}{EXCHG {PARA 201 "" 0 "" {TEXT 220 61 "The maximum value of t he absolute value of the slope is 2.25." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT -1 66 "Check: How does this slope valu e compare with your previous work?" }}}{EXCHG {PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 225 13 "Angle measure" }}{PARA 201 " " 0 "" {TEXT 220 83 "To find the associated angle in radian measure, w e use the inverse tangent function" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 24 "rangle1:=arctan(slope1);" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>%(rangle1G$\"+(*>d_6!\"*" }{TEXT 226 1 " " }}}{EXCHG {PARA 201 "" 0 "" {TEXT 220 85 "To find the associated angle in degree measure, we simply convert radians to degrees." }}}{EXCHG {PARA 201 " > " 0 "" {MPLTEXT 1 222 31 "dangle1:=evalf(rangle1*180/Pi);" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>%(dangle1G$\"+,6v.m!\")" }{TEXT 226 1 " " }}}{EXCHG {PARA 201 "" 0 "" {TEXT 220 67 "The angle of steepest descen t for this hill is 66.03751101 degrees." }}}{EXCHG {PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 221 41 "V. Safety Restrictions and Thrill Factor" }}{PARA 201 " " 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 225 15 "Safety Criteria " }}{PARA 201 "" 0 "" {TEXT 220 109 "For this hill, the angle of steep est descent is 66.03751101 degrees and so it is SAFE (less than 80 deg rees)." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 225 13 "Thrill Factor" }}{PARA 201 "" 0 "" {TEXT 220 28 "The thrill fo r this drop is " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 222 31 "rang le1*abs(ydata[2]-ydata[1]);" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#$\"+y* *GW')!\")" }{TEXT 226 1 " " }}}{EXCHG {PARA 201 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 221 35 "VI. Observation and General ization" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 97 "Build another hill using the collected peak and valley data poi nts from Steel Dragon I (Module A)" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 256 167 "Keeping in mind the coaster restric tions, build several more roller coaster drops using several different peak and valley combinations. Keep a record of your results." } {TEXT -1 0 "" }}}{PARA 206 "" 0 "" {TEXT -1 0 "" }}}{MARK "18 7 0" 38 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }