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{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:
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1 10 0 2 1 1 4 2.000000 1.000000 45.000000 45 }}}}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 0 "" }{TEXT 256 71 "These are left hand approximations o
f 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 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 58 "The procedure below animates the 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 "" 0 "" {TEXT 
256 8 "Example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=x->x^
2:\na:=1:\nhmax:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sect
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