{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{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 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Hea ding 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 " " 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 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 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 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 }} {SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 256 46 "Secant lin e approximations to the tangent line" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):with(plottools):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 256 72 "These are right hand approximations of the line tangent to the graph of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = \+ a;" "6#/%\"xG%\"aG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The procedure below animates the right hand approximations." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 866 "sectotanrt:=proc(f,a,hmax) \nlocal xstep,xseq,xmin,xmax,secantline,fplot,ptsline,i,j,ymin,ymax:\n xstep:=hmax/99:\nxmin:=a-hmax/2:\nxmax:=a+hmax:\nxseq:=seq(xmax-xstep* (i-1),i=1..100):\nymin:=min(minimize(f(x),x,x=xmin..a)-1,minimize(f(x) ,x,x=a..xmax)-1,-1):\nymax:=max(maximize(f(x),x,x=xmin..a)+1,maximize( f(x),x,x=a..xmax)+1,1):\nsecantline:=animate((f(a-h)-f(a))/(-h)*(x-a)+ f(a),x=xmin..xmax,h=-hmax..-0.00001,view=ymin..ymax,color=blue,thickne ss=3,frames=100):\nfplot:=plot(f(x),x=xmin..xmax,y=ymin..ymax,thicknes s=3,color=red,discont=true):\nptsline:=seq(display(point([xseq[j],f(xs eq[j])],symbol=CIRCLE,color=violet),point([a,f(a)],symbol=CIRCLE,color =violet),point([a,0],symbol=CIRCLE,color=violet),point([xseq[j],0],sym bol=CIRCLE,color=violet),line([a,0],[xseq[j],0], color=green, thicknes s=6)),j=1..100):\ndisplay(secantline,display(ptsline,insequence=true), fplot)\nend:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 8 "Ex ample:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=x->x^2:\na:=1: \nhmax:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sectotanrt(f, a,hmax);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%(A NIMATEG6`q7*-%'CURVESG6$7S7$$\"\"!F-$!\"$F-7$$\"++++]i!#6$!3+++++++]F! 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left hand approximatio ns." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 865 "sectotanlf:=proc(f, a,hmax)\nlocal xstep,xseq,xmin,xmax,secantline,fplot,ptsline,i,j,ymin, ymax:\nxstep:=hmax/99:\nxmin:=a-hmax:\nxmax:=a+hmax/2:\nxseq:=seq(xmin +xstep*(i-1),i=1..100):\nymin:=min(minimize(f(x),x,x=xmin..a)-1,minimi ze(f(x),x,x=a..xmax)-1,-1):\nymax:=max(maximize(f(x),x,x=xmin..a)+1,ma ximize(f(x),x,x=a..xmax)+1,1):\nsecantline:=animate((f(a+h)-f(a))/(h)* (x-a)+f(a),x=xmin..xmax,h=-hmax..-0.00001,view=ymin..ymax,color=blue,t hickness=3,frames=100):\nfplot:=plot(f(x),x=xmin..xmax,y=ymin..ymax,th ickness=3,color=red,discont=true):\nptsline:=seq(display(point([xseq[j ],f(xseq[j])],symbol=CIRCLE,color=violet),point([a,f(a)],symbol=CIRCLE ,color=violet),point([a,0],symbol=CIRCLE,color=violet),point([xseq[j], 0],symbol=CIRCLE,color=violet),line([a,0],[xseq[j],0], color=green, th ickness=6)),j=1..100):\ndisplay(secantline,display(ptsline,insequence= true),fplot)\nend:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 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