{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 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 256 "" 0 "" {TEXT 267 55 "Precalculus Exploration 6: Gen eral Sinusoidal Functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 246 "Visual exploration of the combined plot of d ifferent sinusoidal function, including function variation movies (ani mations) and applications. The students should perform these explorati ons themselves in groups of two using Maple on the computers. " } {TEXT 268 146 "Note that in order to see a little more documentation o n how to operate the movies, Precalculus Exploration 5 should be perfo rmed before this one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Complete Sine Model: asin(bx + c) + d." }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 33 "Sine Functions: a*sin(bx + c) + d" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1043 "restart:\na:=1: # Enter th e a in a*sin(bx + c) + d\nb:=3: # Enter the b in a*sin(bx + c) + d\nc :=0: # Enter the c in a*sin(bx + c) + d\nd:=-1: # Enter the d in a*si n(bx + c) + d\nk:=1: # Enter k = the number of intervals of +2*Pi on \+ the x-axis that you want.\nm:=5: # Enter m = the positive height of t he y-axis that you want.\nn:=2: # Enter n = the number of new functio ns that you want different from sin(x). \nxleft:=-2*k*Pi:\nxright:=2* k*Pi:\nybot:=-m:\nytop:=m:\nf:=(a,b,c,d,x)->a*sin(b*x+c)+d:\nList:=[]: \nfor i from 0 to n do\n#F[i]:=f(i*a,i*b,i*c,i*d,x):\nif i = 0 then\nF [i]:=f(1,1,0,0,x):\np[i]:=plot(F[i],x=xleft..xright,y=ybot..ytop,color =red):\nelse\nF[i]:=f(i*a,b,c,d,x):\np[i]:=plot(F[i],x=xleft..xright,y =ybot..ytop,color=blue):\nfi:\nList:=[op(List),F[i]]:\nod:\nSinPlot1:= plots[display](\n [seq(p[i],i=0..n)],\n labels=[`` ,``],\n tickmarks=[5,9],\n #view=[x_left..x_right, y_vertex-20..y_vertex+20],\n titlefont=[HELVETICA,DEFAULT,14],\n tit le=`Sine Functions: a*sin(bx + c) + d Plot`):\nSinPlot1;\nop(List); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 39 "Sine Functions Movie: a*sin(bx + c) + d" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1124 "restart:\na:=1: # Enter t he a in a*sin(bx + c) + d\nb:=3: # Enter the b in a*sin(bx + c) + d\n c:=0: # Enter the c in a*sin(bx + c) + d\nd:=-1: # Enter the d in a*s in(bx + c) + d\nk:=1: # Enter k = the number of intervals of +2*Pi on the x-axis that you want.\nm:=5: # Enter m = the positive height of \+ the y-axis that you want.\nn:=2: # Enter n = the number of new functi ons that you want different from sin(x). \nxleft:=-2*k*Pi:\nxright:=2 *k*Pi:\nybot:=-m:\nytop:=m:\nf:=(a,b,c,d,x)->a*sin(b*x+c)+d:\nList:=[] :\nfor i from 0 to n do\n#F[i]:=f(i*a,i*b,i*c,i*d,x):\nif i = 0 then\n F[i]:=f(1,1,0,0,x):\np[i]:=plot(F[i],x=xleft..xright,y=ybot..ytop,colo r=red):\nelse\nF[i]:=f(i*a,b,c,d,x):\np[i]:=plot(F[i],x=xleft..xright, y=ybot..ytop,color=blue):\nfi:\ncombp[i]:=plots[display]([seq(p[j],j=0 ..i)]):\nList:=[op(List),F[i]]:\nod:\nSinPlot1:=plots[display](\n \+ [seq(combp[i],i=0..n)],\n insequence = true,\n \+ labels=[``,``],\n tickmarks=[5,9],\n #view=[x_ left..x_right,y_vertex-20..y_vertex+20],\n titlefont=[HELVETICA,DEFAU LT,14],\n title=`Sine Functions: a*sin(bx + c) + d Plot`):\nSinPlot 1;\nop(List);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Complete Cosine Model: acos(bx + c) + d." }}{EXCHG {PARA 0 "" 0 "" {TEXT 258 35 "Cosine Functions: a*c os(bx + c) + d" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1045 "restart:\na:=1: # Enter the a in a*cos(bx + c) + d\nb:=3: # Enter the b in a*cos(bx + c) + d\nc:=0: # Enter the c in a*cos(bx + c) + d\nd:=-1: # Enter t he d in a*cos(bx + c) + d\nk:=1: # Enter k = the number of intervals \+ of +2*Pi on the x-axis that you want.\nm:=5: # Enter m = the positive height of the y-axis that you want.\nn:=2: # Enter n = the number of new functions that you want different from cos(x). \nxleft:=-2*k*Pi: \nxright:=2*k*Pi:\nybot:=-m:\nytop:=m:\nf:=(a,b,c,d,x)->a*cos(b*x+c)+d :\nList:=[]:\nfor i from 0 to n do\n#F[i]:=f(i*a,i*b,i*c,i*d,x):\nif i = 0 then\nF[i]:=f(1,1,0,0,x):\np[i]:=plot(F[i],x=xleft..xright,y=ybot ..ytop,color=red):\nelse\nF[i]:=f(i*a,b,c,d,x):\np[i]:=plot(F[i],x=xle ft..xright,y=ybot..ytop,color=blue):\nfi:\nList:=[op(List),F[i]]:\nod: \nCosPlot1:=plots[display](\n [seq(p[i],i=0..n)],\n \+ labels=[``,``],\n tickmarks=[5,9],\n #view=[x_le ft..x_right,y_vertex-20..y_vertex+20],\n titlefont=[HELVETICA,DEFAULT ,14],\n title=`Cosine Functions: a*cos(bx + c) + d Plot`):\nCosPlot 1;\nop(List);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 259 41 "Cosine Functions Movie: a*cos(bx \+ + c) + d" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1126 "restar t:\na:=1: # Enter the a in a*cos(bx + c) + d\nb:=3: # Enter the b in a*cos(bx + c) + d\nc:=0: # Enter the c in a*cos(bx + c) + d\nd:=-1: \+ # Enter the d in a*cos(bx + c) + d\nk:=1: # Enter k = the number of i ntervals of +2*Pi on the x-axis that you want.\nm:=5: # Enter m = the positive height of the y-axis that you want.\nn:=2: # Enter n = the \+ number of new functions that you want different from cos(x). \nxleft: =-2*k*Pi:\nxright:=2*k*Pi:\nybot:=-m:\nytop:=m:\nf:=(a,b,c,d,x)->a*cos (b*x+c)+d:\nList:=[]:\nfor i from 0 to n do\n#F[i]:=f(i*a,i*b,i*c,i*d, x):\nif i = 0 then\nF[i]:=f(1,1,0,0,x):\np[i]:=plot(F[i],x=xleft..xrig ht,y=ybot..ytop,color=red):\nelse\nF[i]:=f(i*a,b,c,d,x):\np[i]:=plot(F [i],x=xleft..xright,y=ybot..ytop,color=blue):\nfi:\ncombp[i]:=plots[di splay]([seq(p[j],j=0..i)]):\nList:=[op(List),F[i]]:\nod:\nCosPlot1:=pl ots[display](\n [seq(combp[i],i=0..n)],\n inseque nce = true,\n labels=[``,``],\n tickmarks=[5,9],\n #view=[x_left..x_right,y_vertex-20..y_vertex+20],\n titlef ont=[HELVETICA,DEFAULT,14],\n title=`Cosine Functions: a*cos(bx + c) + d Plot`):\nCosPlot1;\nop(List);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 63 "Both Sine & Cosine Models: asin(bx + c) + d, acos(bx + c) + d." }}{EXCHG {PARA 0 "" 0 "" {TEXT 260 28 "Both Sine & Cosine Functions" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1879 "restart:\na1:=1: # Enter the a in a*sin(bx + \+ c) + d\nb1:=3: # Enter the b in a*sin(bx + c) + d\nc1:=0: # Enter th e c in a*sin(bx + c) + d\nd1:=-1: # Enter the d in a*sin(bx + c) + d\n a2:=1: # Enter the a in a*cos(bx + c) + d\nb2:=3: # Enter the b in a *cos(bx + c) + d\nc2:=0: # Enter the c in a*cos(bx + c) + d\nd2:=-1: \+ # Enter the d in a*cos(bx + c) + d\nk:=1: # Enter k = the number of i ntervals of +2*Pi on the x-axis that you want.\nm:=4: # Enter m = the positive height of the y-axis that you want.\nn:=1: # Enter n = the \+ number of new functions that you want different from sin(x). \nxleft: =-2*k*Pi:\nxright:=2*k*Pi:\nybot:=-m:\nytop:=m:\nf:=(a1,b1,c1,d1,x)->a 1*sin(b1*x+c1)+d1:\ng:=(a2,b2,c2,d2,x)->a2*cos(b2*x+c2)+d2:\nList1:=[] :\nList2:=[]:\nfor i from 0 to n do\nif i = 0 then\nF[i]:=f(1,1,0,0,x) :\nG[i]:=g(1,1,0,0,x):\np[i]:=plot(F[i],x=xleft..xright,y=ybot..ytop,c olor=red):\nq[i]:=plot(G[i],x=xleft..xright,y=ybot..ytop,color=blue): \nelse\n#F[i]:=f(i*a,i*b,i*c,i*d,x):\n#G[i]:=g(i*a,i*b,i*c,i*d,x):\nF[ i]:=f(i*a1,b1,c1,d1,x):\nG[i]:=g(i*a2,b2,c2,d2,x):\np[i]:=plot(F[i],x= xleft..xright,y=ybot..ytop,color=red):\nq[i]:=plot(G[i],x=xleft..xrigh t,y=ybot..ytop,color=blue):\nfi:\nList1:=[op(List1),F[i]]:\nList2:=[op (List2),G[i]]:\nod:\ntxtp[1]:=plots[textplot]([-0.5,m-.01,`Models usin g sin(x)`],\n font=[HELVETICA,DEFAULT,12],\n \+ color=red,align=\{BELOW,LEFT\}): \ntxtp[2]:=plots [textplot]([0.5,m-.01,`Models using cos(x)`],\n \+ font=[HELVETICA,DEFAULT,12],\n color=blue,alig n=\{BELOW,RIGHT\}):\nSinCosPlot1:=plots[display](\n [seq(p[i ],i=0..n),seq(q[i],i=0..n),txtp[1],txtp[2]],\n labels=[``,`` ],\n tickmarks=[5,9],\n #view=[x_left..x_right,y_v ertex-20..y_vertex+20],\n titlefont=[HELVETICA,DEFAULT,14],\n title= `Sine & Cosine Functions Plot`):\nSinCosPlot1;\nop(List1);\nop(List2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 54 "Sine Function Voltage Model: V(t) = asin(bt + c) \+ + d." }}{EXCHG {PARA 0 "" 0 "" {TEXT 262 46 "Sine Function Voltage Mod el: a*sin(bt + c) + d" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1141 "restart: \na:=110*sqrt(2): # Enter the a in a*sin(bt + c) + d\nb:=120*Pi: \+ # Enter the b in a*sin(bt + c) + d\nc:=0: # Enter the c i n a*sin(bt + c) + d\nd:=0: # Enter the d in a*sin(bt + c) + d\nperiod:=2*Pi/b:\nfreq:=1/period:\nk:=1: # Enter k = the number of cycles that you want.\nm:=2: # Enter m = the number of positive ampl itudes of height that you want.\nn:=1: # Enter n = the number of new \+ functions that you want different from sin(x). \nxleft:=-0*period:\nx right:=k*period:\nybot:=-m*a:\nytop:=m*a:\nf:=(a,b,c,d,t)->a*sin(b*t+c )+d:\nList:=[]:\nfor i from 1 to n do\n#F[i]:=f(i*a,i*b,i*c,i*d,t):\ni f i = 0 then\nF[i]:=f(1,1,0,0,t):\np[i]:=plot(F[i],t=xleft..xright,y=y bot..ytop,color=red):\nelse\nF[i]:=f(i*a,b,c,d,t):\np[i]:=plot(F[i],t= xleft..xright,y=ybot..ytop,color=blue):\nfi:\nList:=[op(List),V(t)=F[i ]]:\nod:\nSinPlot1:=plots[display](\n [seq(p[i],i=1..n)],\n \+ labels=[`t (sec)`,`V (volts)`],\n tickmarks=[6,9], \n #view=[x_left..x_right,y_vertex-20..y_vertex+20],\n titl efont=[HELVETICA,DEFAULT,14],\n title=`Sine Function Voltage Model: V (t) = a*sin(bt + c) + d`):\nSinPlot1;\nop(List);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 52 "Sine Function Voltage Model Movie: a*sin(bt + c) + d" }{TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 1284 "restart:\na:=110*sqrt(2): # Ente r the a in a*sin(bt + c) + d\nb:=120*Pi: # Enter the b in a*sin( bt + c) + d\nc:=0: # Enter the c in a*sin(bt + c) + d\nd:=0 : # Enter the d in a*sin(bt + c) + d\nperiod:=2*Pi/b:\nfreq :=1/period:\nk:=5: # Enter k = the number of cycles that you want.\nm :=2: # Enter m = the number of positive amplitudes of height that you want.\nn:=2: # Enter n = the number of new functions that you want d ifferent from sin(x). \nxleft:=-0*period:\nxright:=k*period:\nybot:=- m*a:\nytop:=m*a:\nf:=(a,b,c,d,t)->a*sin(b*t+c)+d:\nList:=[]:\nfor i fr om 1 to n do\n#F[i]:=f(i*a,i*b,i*c,i*d,t):\nif i = 0 then\nF[i]:=f(1,1 ,0,0,t):\np[i]:=plot(F[i],t=xleft..xright,y=ybot..ytop,color=red,label s=[`t (sec)`,`V (volts)`]):\nelse\nF[i]:=f(i*a,b,c,d,t):\np[i]:=plot(F [i],t=xleft..xright,y=ybot..ytop,color=blue,labels=[`t (sec)`,`V (volt s)`]):\nfi:\ncombp[i]:=plots[display]([seq(p[j],j=1..i)]):\nList:=[op( List),V(t)=F[i]]:\nod:\nSinPlot1:=plots[display](\n [seq(com bp[i],i=1..n)],\n insequence = true,\n labels=[`t (sec)`,`V (volts)`],\n tickmarks=[5,9],\n #view=[ x_left..x_right,y_vertex-20..y_vertex+20],\n titlefont=[HELVETICA,DEF AULT,14],\n title=`Sine Function Voltage Model: V(t) = a*sin(bt + c) \+ + d`):\nSinPlot1;\nop(List);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 59 "Sine & Cosine Models: \+ Reference function shifted by [h, k]" }}{EXCHG {PARA 0 "" 0 "" {TEXT 263 79 "Sine Models: Reference function shifted by [h, k]: f(x) = a* sin(b(x - h)) + k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2621 "restart:\nma xht:=4: # Enter the max value of the graph of the data (a peak)\nmi nht:=0: # Enter the min value of the graph of the data (a valley)\np :=4: # Enter the period p = distance between two successive pea ks (or valleys)\nh:=-1: # Enter the h = the distance the refere nce sine curve is translated right(left)\nj:=2: # Enter j = the number of cycles that you want.\nn:=1: # Enter n = the number \+ of new functions that you want different from sin(x). \na:=(maxht-min ht)/2: # The amplitude a = (max-min)/2\nf:=1/p: # The freq uency f = 1/p\nb:=2*Pi/p: # The factor b = 2*Pi/p\naxis:=(max ht+minht)/2: # The central axis axis = (max+min)/2\nk:=axis: # k = the distance the reference sine curve is translated up(down)\nxleft:= -j*p:\nxright:=j*p:\nybot:=-max(2*a+1,2*a+k):\nytop:=max(2*a+1,2*a+k): \nf:=(a,b,h,k,x)->a*sin(b*(x-h))+k:\nList:=[]:\nfor i from 0 to n do\n #F[i]:=f(i*a,i*b,i*h,i*k,x):\nif i = 0 then\nF[i]:=f(a,b,0,0,x):\nFtxt [i]:=`Reference function: f`(x):\nplt[i]:=plot(F[i],x=xleft..xright,y =ybot..ytop,color=red):\nelse\nF[i]:=f(i*a,b,h,k,x):\nFtxt[i]:=` Shift ed function: f`(x-h)+k:\nplt[i]:=plot(F[i],x=xleft..xright,y=ybot..yt op,color=blue):\nfi:\nList:=[op(List),Ftxt[i]=F[i]]:\nod:\ntxtp[1]:=pl ots[textplot]([xleft/2,ytop-.01,`Reference function`],\n \+ font=[HELVETICA,DEFAULT,12],\n color= red,align=\{BELOW,CENTER\}): \ntxtp[2]:=plots[textplot]([xright/2,ytop -.01,`Shifted function`],\n font=[HELVETICA,DEF AULT,12],\n color=blue,align=\{BELOW,CENTER\}): \ntxtp[3]:=plots[textplot]([xright-.01,axis+.09,`axis `],\n \+ font=[HELVETICA,DEFAULT,12],\n col or=blue,align=\{ABOVE,LEFT\}):\nlinep[1]:=plot([[xleft,axis],[xright,a xis]],linestyle=2,color=blue):\nlinep[2]:=plot([[0,k],[h,k]],linestyle =2,color=blue):\nlinep[3]:=plot([[h,0],[h,k]],linestyle=2,color=blue): \npointp[1]:=plot([[0,0]],style=point,symbol=circle,color=blue):\npoin tp[2]:=plot([[h,k]],style=point,symbol=circle,color=blue):\nSinPlot1:= plots[display](\n [txtp[1],txtp[2],txtp[3],\n lin ep[1],linep[2],linep[3],\n pointp[1],pointp[2],\n \+ seq(plt[i],i=0..n)],\n #labels=[`t (sec)`,`V (volts)`],\n labels=[``,``],\n tickmarks=[9,9],\n \+ #view=[x_left..x_right,y_vertex-20..y_vertex+20],\n titlef ont=[HELVETICA,DEFAULT,14],\n title=`Sine Model: Reference function shifted by [h, k]`):\nSinPlot1;\n`Amplitude: a`=a,` Period : p`=p,` b`=b,` Shift right (left): h`=h,` Shift up (down): k`= k;\nop(List);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 264 82 "Cosine Models: Reference functio ns shifted by [h, k]: f(x) = a*cos(b(x - h)) + k" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 2626 "restart:\nmaxht:=4: # Enter the max value of t he graph of the data (a peak)\nminht:=0: # Enter the min value of th e graph of the data (a valley)\np:=4: # Enter the period p = di stance between two successive peaks (or valleys)\nh:=0: # Enter the h = the distance the reference cosine curve is translated right(l eft)\nj:=2: # Enter j = the number of cycles that you want.\nn: =1: # Enter n = the number of new functions that you want diffe rent from cos(x). \na:=(maxht-minht)/2: # The amplitude a = (max-min) /2\nf:=1/p: # The frequency f = 1/p\nb:=2*Pi/p: # The factor b = 2*Pi/p\naxis:=(maxht+minht)/2: # The central axis axis = (max+min)/2\nk:=axis: # k = the distance the reference cosine c urve is translated up(down)\nxleft:=-j*p:\nxright:=j*p:\nybot:=-max(2* a+1,2*a+k):\nytop:=max(2*a+1,2*a+k):\nf:=(a,b,h,k,x)->a*cos(b*(x-h))+k :\nList:=[]:\nfor i from 0 to n do\n#F[i]:=f(i*a,i*b,i*h,i*k,x):\nif i = 0 then\nF[i]:=f(a,b,0,0,x):\nFtxt[i]:=`Reference function: f`(x): \nplt[i]:=plot(F[i],x=xleft..xright,y=ybot..ytop,color=red):\nelse\nF[ i]:=f(i*a,b,h,k,x):\nFtxt[i]:=` Shifted function: f`(x-h)+k:\nplt[i]: =plot(F[i],x=xleft..xright,y=ybot..ytop,color=blue):\nfi:\nList:=[op(L ist),Ftxt[i]=F[i]]:\nod:\ntxtp[1]:=plots[textplot]([xleft/2,ytop-.01,` Reference function`],\n font=[HELVETICA,DEFAULT ,12],\n color=red,align=\{BELOW,CENTER\}): \ntx tp[2]:=plots[textplot]([xright/2,ytop-.01,`Shifted function`],\n \+ font=[HELVETICA,DEFAULT,12],\n \+ color=blue,align=\{BELOW,CENTER\}):\ntxtp[3]:=plots[textplot]([xrigh t-.01,axis+.09,`axis `],\n font=[HELVETICA,DEFA ULT,12],\n color=blue,align=\{ABOVE,LEFT\}):\nl inep[1]:=plot([[xleft,axis],[xright,axis]],linestyle=2,color=blue):\nl inep[2]:=plot([[0,k],[h,k]],linestyle=2,color=blue):\nlinep[3]:=plot([ [h,0],[h,k]],linestyle=2,color=blue):\npointp[1]:=plot([[0,0]],style=p oint,symbol=circle,color=blue):\npointp[2]:=plot([[h,k]],style=point,s ymbol=circle,color=blue):\nSinPlot1:=plots[display](\n [txtp [1],txtp[2],txtp[3],\n linep[1],linep[2],linep[3],\n \+ pointp[1],pointp[2],\n seq(plt[i],i=0..n)],\n \+ #labels=[`t (sec)`,`V (volts)`],\n labels=[``,``],\n \+ tickmarks=[7,9],\n #view=[x_left..x_right,y_vertex- 20..y_vertex+20],\n titlefont=[HELVETICA,DEFAULT,14],\n \+ title=`Cosine Model: Reference function shifted by [h, k]`): \nSinPlot1;\n`Amplitude: a`=a,` Period: p`=p,` b`=b,` Shift right (left): h`=h,` Shift up (down): k`=k;\nop(List);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 74 "Sine & Cosine Models (Tide Problem): Reference function shifte d by [h, k]" }}{EXCHG {PARA 0 "" 0 "" {TEXT 265 79 "Sine Models: Refe rence function shifted by [h, k]: f(x) = a*sin(b(x - h)) + k" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2869 "restart:\nmaxht:=58: # Enter t he max value of the graph of the data (a peak)\nminht:=2: # Enter th e min value of the graph of the data (a valley)\np:=6.2: # Ente r the period p = distance between two successive peaks (or valleys)\nh :=0: # Enter the h = the distance the reference sine curve is t ranslated right(left)\nj:=2: # Enter j = the number of cycles t hat you want.\nn:=1: # Enter n = the number of new functions th at you want different from sin(x). \na:=(maxht-minht)/2: # The amplit ude a = (max-min)/2\nf:=1/p: # The frequency f = 1/p\n#b:= evalf(2/p,4)*Pi: # The factor b = 2*Pi/p\nb:=evalf(2*Pi/p,4): # The factor b = 2*Pi/p\naxis:=(maxht+minht)/2: # The centra l axis axis = (max+min)/2\nk:=axis: # k = the distance the referen ce sine curve is translated up(down)\nxleft:=-j*p:\nxright:=j*p:\nybot :=-max(2*a+1,2*a+k):\nytop:=max(2*a+1,2*a+k):\nf:=(a,b,h,k,x)->a*sin(b *(x-h))+k:\nList:=[]:\nLList:=[]:\nfor i from 0 to n do\n#F[i]:=f(i*a, i*b,i*h,i*k,x):\nif i = 0 then\nF[i]:=f(a,b,0,0,x):\nFtxt[i]:=`Referen ce function: f`(x):\nFF[i]:=f(a,b,0,0,t):\nFFtxt[i]:=`Reference funct ion: f`(t):\nplt[i]:=plot(F[i],x=xleft..xright,y=ybot..ytop,color=red ):\nelse\nF[i]:=f(i*a,b,h,k,x):\nFtxt[i]:=` Shifted function: f`(x-h) +k:\nFF[i]:=f(i*a,b,h,k,t):\nFFtxt[i]:=` Shifted function: f`(t-h)+k: \nplt[i]:=plot(F[i],x=xleft..xright,y=ybot..ytop,color=blue):\nfi:\nLi st:=[op(List),Ftxt[i]=F[i]]:\nLList:=[op(LList),FFtxt[i]=FF[i]]:\nod: \ntxtp[1]:=plots[textplot]([xleft/2,ytop-.01,`Reference function`],\n \+ font=[HELVETICA,DEFAULT,12],\n \+ color=red,align=\{BELOW,CENTER\}): \ntxtp[2]:=plots[textplot]([ xright/2,ytop-.01,`Shifted function`],\n font=[ HELVETICA,DEFAULT,12],\n color=blue,align=\{BEL OW,CENTER\}):\ntxtp[3]:=plots[textplot]([xright-.01,axis+.09,`axis `], \n font=[HELVETICA,DEFAULT,12],\n \+ color=blue,align=\{ABOVE,LEFT\}):\nlinep[1]:=plot([[xleft,ax is],[xright,axis]],linestyle=2,color=blue):\nlinep[2]:=plot([[0,k],[h, k]],linestyle=2,color=blue):\nlinep[3]:=plot([[h,0],[h,k]],linestyle=2 ,color=blue):\npointp[1]:=plot([[0,0]],style=point,symbol=circle,color =blue):\npointp[2]:=plot([[h,k]],style=point,symbol=circle,color=blue) :\nSinPlot1:=plots[display](\n [txtp[1],txtp[2],txtp[3],\n \+ linep[1],linep[2],linep[3],\n pointp[1],pointp[2] ,\n seq(plt[i],i=0..n)],\n #labels=[`t (sec)`,`V (volts)`],\n labels=[``,``],\n tickmarks=[9,9], \n #view=[x_left..x_right,y_vertex-20..y_vertex+20],\n \+ titlefont=[HELVETICA,DEFAULT,14],\n title=`Sine Mode l: Reference function shifted by [h, k]`):\nSinPlot1;\n`Amplitude: \+ a`=a,` Period: p`=p,` b`=b,` Shift right (left): h`=h,` Shift up \+ (down): k`=k;\n#op(List);\nop(LList);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 82 "Cosine Model s: Reference functions shifted by [h, k]: f(x) = a*cos(b(x - h)) + k " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2873 "restart:\nmaxht:=58: # Ent er the max value of the graph of the data (a peak)\nminht:=2: # Ente r the min value of the graph of the data (a valley)\np:=12.4: # En ter the period p = distance between two successive peaks (or valleys) \nh:=0: # Enter the h = the distance the reference cosine curve is translated right(left)\nj:=2: # Enter j = the number of cyc les that you want.\nn:=1: # Enter n = the number of new functio ns that you want different from cos(x). \na:=(maxht-minht)/2: # The a mplitude a = (max-min)/2\nf:=1/p: # The frequency f = 1/p \n#b:=evalf(2/p,4)*Pi: # The factor b = 2*Pi/p\nb:=evalf(2*Pi /p,4): # The factor b = 2*Pi/p\naxis:=(maxht+minht)/2: # The \+ central axis axis = (max+min)/2\nk:=axis: # k = the distance the r eference cosine curve is translated up(down)\nxleft:=-j*p:\nxright:=j* p:\nybot:=-max(2*a+1,2*a+k):\nytop:=max(2*a+1,2*a+k):\nf:=(a,b,h,k,x)- >a*cos(b*(x-h))+k:\nList:=[]:\nLList:=[]:\nfor i from 0 to n do\n#F[i] :=f(i*a,i*b,i*h,i*k,x):\nif i = 0 then\nF[i]:=f(a,b,0,0,x):\nFtxt[i]:= `Reference function: f`(x):\nFF[i]:=f(a,b,0,0,t):\nFFtxt[i]:=`Referen ce function: f`(t):\nplt[i]:=plot(F[i],x=xleft..xright,y=ybot..ytop,c olor=red):\nelse\nF[i]:=f(i*a,b,h,k,x):\nFtxt[i]:=` Shifted function: \+ f`(x-h)+k:\nFF[i]:=f(i*a,b,h,k,t):\nFFtxt[i]:=` Shifted function: f` (t-h)+k:\nplt[i]:=plot(F[i],x=xleft..xright,y=ybot..ytop,color=blue): \nfi:\nList:=[op(List),Ftxt[i]=F[i]]:\nLList:=[op(LList),FFtxt[i]=FF[i ]]:\nod:\ntxtp[1]:=plots[textplot]([xleft/2,ytop-.01,`Reference functi on`],\n font=[HELVETICA,DEFAULT,12],\n \+ color=red,align=\{BELOW,CENTER\}): \ntxtp[2]:=plots[tex tplot]([xright/2,ytop-.01,`Shifted function`],\n \+ font=[HELVETICA,DEFAULT,12],\n color=blue,ali gn=\{BELOW,CENTER\}):\ntxtp[3]:=plots[textplot]([xright-.01,axis+.09,` axis `],\n font=[HELVETICA,DEFAULT,12],\n \+ color=blue,align=\{ABOVE,LEFT\}):\nlinep[1]:=plot([[ xleft,axis],[xright,axis]],linestyle=2,color=blue):\nlinep[2]:=plot([[ 0,k],[h,k]],linestyle=2,color=blue):\nlinep[3]:=plot([[h,0],[h,k]],lin estyle=2,color=blue):\npointp[1]:=plot([[0,0]],style=point,symbol=circ le,color=blue):\npointp[2]:=plot([[h,k]],style=point,symbol=circle,col or=blue):\nSinPlot1:=plots[display](\n [txtp[1],txtp[2],txtp [3],\n linep[1],linep[2],linep[3],\n pointp[1],p ointp[2],\n seq(plt[i],i=0..n)],\n #labels=[`t ( sec)`,`V (volts)`],\n labels=[``,``],\n tickmark s=[7,9],\n #view=[x_left..x_right,y_vertex-20..y_vertex+20] ,\n titlefont=[HELVETICA,DEFAULT,14],\n title=`C osine Model: Reference function shifted by [h, k]`):\nSinPlot1;\n`Amp litude: a`=a,` Period: p`=p,` b`=b,` Shift right (left): h`=h,` Shift up (down): k`=k;\n#op(List);\nop(LList);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Filename: Explo rePrecalc06.mws" }}{PARA 0 "" 0 "" {TEXT -1 36 "Copyright 2005, All Ri ghts Reserved." }}{PARA 0 "" 0 "" {TEXT -1 44 "Permission is granted t o use and modify for " }}{PARA 0 "" 0 "" {TEXT -1 37 "academic and non -commercial purposes." }}{PARA 0 "" 0 "" {TEXT -1 13 "Dr. John Pais" } }{PARA 0 "" 0 "" {TEXT -1 28 "Mathematics Department-MICDS" }}{PARA 0 "" 0 "" {TEXT -1 45 "E-mail: pais@micds.org or pais@kinetigram.com" }} {PARA 0 "" 0 "" {TEXT -1 33 "URL: http://kinetigram.com/micds" }} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} }{MARK "10 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }