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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12002 "# Epsilon-Delta M
aplet\n# Copyright 2002 Waterloo Maple Inc.\n# This maplet graphically
 illustrates the delta-epsilon definition of the limit.  The definitio
n states that limit( f(x), x=a ) = L if, for all e>0, there exists d>0
 such that 0<|x-a|<d implies |f(x)-L|<e.  \n# \n# Enter a function f(x
) and values for a, L, e and d.  You can then experiment with the valu
es of a, e and d to try and find values that satisfy the definition by
 seeing the plot of f(x) around x=a annotated with these values.  Note
 that d satisfies the definition as long as the blue region fits betwe
en the two red horizontal lines on the plot.\n# \n# In the plot, x=a i
s marked by a gray vertical line, and the value L by a gray horizonal \+
line.\n# \n# The interval [a-d, a+d] is marked by a blue line on the x
-axis.\n# \n# The range of function values implied by the choice of d \+
is marked by a blue region.\n# \n# The two red horizontal lines indica
te y=L+e and y=fL-e.\n# \n# To run this maplet, click the !!! button i
n the tool bar.\n# \nrestart:\nwith(LinearAlgebra):\nwith(plots):\nwit
h(plottools):\nwith(Maplets[Elements]):\n\n###########################
#################################\n# the drawing procedure\n\ncgraph :
=  proc(f, a, L, ep, deltav, xmin, xmax, ymin, ymax)\n\n  local fc, g,
 aline, Lline, e, ept1, ept2, elinem, elineu, elinel, asd, \n        a
ad, dptr1, dptr2, dliner, dptb1, dptb2, dlineb, j, k, p1, p2, delta;\n
\n  #plotting the graph\n  fc := plot(f, x=xmin..xmax, y=ymin..ymax, c
olor=navy);\n\n  #drawing a vertical and horizonal line though (a,f(a)
)\n  g := unapply(f, x):\n  aline := line([a, 0], [a, L], thickness=2,
 color=gray):\n  Lline := line([xmin, L], [xmax, L], color=gray):\n\n \+
 #scaling the value from the epsilon-slider to between 0 and 1\n  e :=
 ep/100:\n\n  #drawing the epsilon neighborhood and horizonal lines at
 the epsilon end-points\n  ept1 := [0, L-e]:\n  ept2 := [0, L+e]:\n  e
linem := line(ept1, ept2, thickness=4, color=red):\n  elineu := line([
xmin, L+e], [xmax, L+e], color=red):\n  elinel := line([xmin, L-e], [x
max, L-e], color=red):\n\n  #scaling the value from the delta-slider t
o between 0 and 1\n  delta := deltav/100:\n\n  asd := a-delta:\n  aad \+
:= a+delta: \n\n  #the shaded in parts\n  p1 := polygon([[asd, 0], [as
d, g(aad)], [aad, g(aad)], [aad, 0]], \n                 color=turquoi
se, style=patchnogrid):\n  p2 := polygon([[0, g(aad)], [0, g(asd)], [a
ad, g(asd)], [aad, g(aad)]], \n                 color=turquoise, style
=patchnogrid):\n\n  #drawing the vertical lines from a+-delta to the f
unction\n\n  dptr1 := [asd, 0]:\n  dptr2 := [aad, 0]:\n  dliner := lin
e(dptr1, dptr2, thickness=4, color=blue):\n  dptb1 := [0, g(asd)]:\n  \+
dptb2 := [0, g(aad)]:\n\n\n  plots[display](aline, Lline, fc, elinem, \+
elineu, elinel, dliner, p1, p2): \n\nend proc:\n\n####################
########################################\n# converts the value chosen \+
on the slider into the real value \n# chosen for epsilon and delta\n\n
realvalue := proc(rvalue)\n  evalf[3](rvalue/100.0):\nend proc:\n\n###
#########################################################\n# zoom in o
n the graph; make the axes shorter\n\nzoomin := proc(normaxes)\n  loca
l getsmaller:\n  getsmaller := normaxes/2:\nend proc:\n\n#############
###############################################\n# zoom out on the gra
ph; make the axes longer\n\nzoomout := proc(normaxes)\n  local getbigg
er:\n  getbigger :=  normaxes * 2:\nend proc:\n\n#####################
#######################################\n# move a-line back .1\n\nales
stenth :=  proc(a)\n  local newa:\n  newa := a-.1;\nend proc:\n\n#####
#######################################################\n# move a-line
 back .5\n\nalesshalf :=  proc(a)\n  local newa:\n  newa := a-.5;\nend
 proc:\n\n############################################################
\n# move a-line back 1\n\naless1 :=  proc(a)\n  local newa:\n  newa :=
 a-1;\nend proc:\n\n##################################################
##########\n# move a-line forward .1\n\naaddtenth :=  proc(a)\n  local
 newa:\n  newa := a+.1;\nend proc:\n\n################################
############################\n# move a-line forward .5\n\naaddhalf := \+
 proc(a)\n  local newa:\n  newa := a+.5;\nend proc:  \n\n#############
###############################################\n\n#move a-line forwar
d 1\naadd1 :=  proc(a)\n  local newa:\n  newa := a+1;\nend proc:  \n\n
############################################################\n\nhelpSt
r := \n\"This maplet graphically illustrates the delta-epsilon definit
ion of limits.  The definition states that limit( f(x), x=a ) = L if, \+
for all e>0, there exists d>0 such that 0<|x-a|<d implies |f(x)-L|<e.
\n\nEnter a function f(x) and values for a, L, e and d.  You can then \+
experiment with the values of a, L, e and d to try and find values tha
t satisify the definition by seeing the plot of f(x) around x=a annota
ted with these values.  Note that d satisfies the definition as long a
s the blue region fits between the two red horizontal lines on the plo
t.  \n \nIn the plot, x=a is marked by a gray vertical line, and the v
alue L by a gray horizontal line. \nThe interval [a-d, a+d] is marked \+
by a blue line on the x-axis.\n\nThe range of function values implied \+
by the choice of d is marked by a blue region. \n\nThe two red horizon
tal lines indicate y=L+e and y=L-e.\":\n\n############################
################################\n\nContinuityMaplet := Maplet( 'onsta
rtup'=RunWindow('mainWin'),\n\nFont['F2']('family'=\"Default\", 'bold'
='true', 'size'=14),\n\nMenuBar['MB'](Menu(\"File\", \n               \+
 MenuItem(\"New\", 'onclick'='Aclear'),\n                MenuSeparator
(),  \n                MenuItem(\"Close\", Shutdown())\n              \+
    ),#end menu\n              Menu(\"Help\", \n                MenuIt
em(\"About this maplet\", 'onclick'=RunWindow('helpWin') ))\n         \+
    ),\n \nWindow['mainWin']('layout'='BL1', 'menubar'='MB', \n       \+
            'title'=\"Calculus 1 - EpsilonDelta\"),\n \n  BoxLayout['B
L1'](\n    BoxRow('background'=\"#DDFFFF\", 'inset'=0, 'spacing'=0,\n \+
  \n  BoxColumn('inset'=0, 'spacing'=0, 'background'=\"#DDFFFF\", 'cap
tion'=\"Plot Window\", 'border'=true,\n          Plotter['pl1']()\n   \+
    ), # end BoxColumn\n \n   BoxColumn('background'=\"#DDFFFF\",\n   \+
\n    #user enters ranges\n        BoxRow('background'=\"#DDFFFF\",\n \+
              \"Enter a function = \", TextField['TF1'](\"1/x\", 15),
\n               \"a = \", TextField['TFa'](\"1/2\", 5),\n            \+
   \"L = \", TextField['TFL'](\"2\", 5)), \n    BoxRow('background'=\"
#DDFFFF\",\n      BoxRow('background'=\"#DDFFFF\", 'border'=true, 'cap
tion'=\"Change a\", 'halign'='left',\n        BoxColumn('background'=
\"#DDFFFF\",\n               Button(\"left .1\", 'background'=\"#CCFFF
F\",'onclick'='Aalesstenth'), \n               Button(\"right .1\", 'b
ackground'=\"#CCFFFF\",'onclick'='Aaaddtenth')), \n        BoxColumn('
background'=\"#DDFFFF\",\n               Button(\"left .5\", 'backgrou
nd'=\"#CCFFFF\",'onclick'='Aalesshalf'), \n               Button(\"rig
ht .5\", 'background'=\"#CCFFFF\",'onclick'='Aaaddhalf')), \n        B
oxColumn('background'=\"#DDFFFF\",\n               Button(\"left 1\", \+
'background'=\"#CCFFFF\",'onclick'='Aaless1'), \n               Button
(\"right 1\", 'background'=\"#CCFFFF\",'onclick'='Aaadd1')))),\n      \+
\n        BoxRow('background'=\"#DDFFFF\", 'border'=true, 'caption'=\"
Plot ranges\",'halign'='left',\n               \"xmin = \", TextField[
'TFxmin'](\"1/10\", 3), \n               \"xmax = \", TextField['TFxma
x'](\"1\", 3), \n               \"ymin = \", TextField['TFymin'](\"0\"
, 3), \n               \"ymax = \", TextField['TFymax'](\"10\", 3)), \+
\n\n    #user types in the 'a' value and selects epsilon and delta fro
m the slider\n\n     BoxRow('background'=\"#DDFFFF\", \"epsilon=\", \n
            Slider['SLe'](0..100, 75, 'filled'='true','snapticks'='fal
se', \n                          'minorticks'=5, 'background'=\"#CCFFF
F\", 'onchange'='Asle'), \n            TextField['TFsle'](\".750\", 5,
 'editable'=false)), \n     BoxRow('background'=\"#DDFFFF\", \"delta= \+
\", \n             Slider['SLd'](0..100, 10, 'filled'='true','snaptick
s'='false',  \n                          'minorticks'=5, 'background'=
\"#CCFFFF\", 'onchange'='Asld'), \n             TextField['TFsld'](\".
100\", 5, 'editable'=false)), \n  \n  BoxRow('background'=\"#DDFFFF\",
 'inset'=0, 'spacing'=5,\n           Button(\"Plot\", 'background'=\"#
CCFFFF\",\n                  Evaluate('pl1'='cgraph(TF1, TFa, TFL, SLe
, SLd, TFxmin, TFxmax, \n                                         TFym
in, TFymax)')), \n           Button(\"Zoom in\", 'background'=\"#CCFFF
F\",'onclick'='Azoomin'), \n           Button(\"Zoom out\", 'backgroun
d'=\"#CCFFFF\",'onclick'='Azoomout'), \n           Button(\"New\", 'ba
ckground'=\"#CCFFFF\",' onclick'='Aclear'), \n           Button(\"Clos
e\", 'background'=\"#CCFFFF\", Shutdown()))))), \n\n##################
##########################################\n\n  Window['helpWin']( 're
sizable'='false',\n    'title'=\"About the Delta-Epsilon Continuity Ma
plet\",\n    BoxColumn('inset'=0, 'spacing'=10,\n      'background'=\"
#CCFFFF\",\n'border'='true',\n      BoxCell(\n        TextBox('height'
=20, 'width'=48,\n          'background'=\"#EEFFFF\", 'font'='F2', 'fo
reground'=\"#333399\", \n          'editable'='false', \n          'va
lue'=helpStr\n        ) # end TextBox\n      ), # end BoxCell\n\n     \+
 BoxRow('inset'=0, 'spacing'=0, 'background'=\"#CCFFFF\",\n        But
ton(\"Close\", \n          CloseWindow('helpWin'), \n          'backgr
ound'=\"#DDFFFF\")\n      ) # end BoxRow\n    ) # end BoxColumn\n  ), \+
# end helpWin\n\n#####################################################
#######\n\n  Action[Aalesstenth](\n  Evaluate('function'='alesstenth',
 'target'='TFa', Argument('TFa')), \n  Evaluate('pl1'='cgraph(TF1, TFa
, TFL, SLe, SLd, TFxmin, TFxmax, TFymin, TFymax)')\n  ),\n \n  Action[
Aalesshalf](\n  Evaluate('function'='alesshalf', 'target'='TFa', Argum
ent('TFa')), \n  Evaluate('pl1'='cgraph(TF1, TFa, TFL, SLe, SLd, TFxmi
n, TFxmax, TFymin, TFymax)')\n  ),\n \n  Action[Aaless1](\n  Evaluate(
'function'='aless1', 'target'='TFa', Argument('TFa')), \n  Evaluate('p
l1'='cgraph(TF1, TFa, TFL, SLe, SLd, TFxmin, TFxmax, TFymin, TFymax)')
\n  ), \n\n  Action[Aaaddtenth](\n  Evaluate('function'='aaddtenth', '
target'='TFa', Argument('TFa')), \n  Evaluate('pl1'='cgraph(TF1, TFa, \+
TFL, SLe, SLd, TFxmin, TFxmax, TFymin, TFymax)')\n  ), \n\n  Action[Aa
addhalf](\n  Evaluate('function'='aaddhalf', 'target'='TFa', Argument(
'TFa')), \n  Evaluate('pl1'='cgraph(TF1, TFa, TFL, SLe, SLd, TFxmin, T
Fxmax, TFymin, TFymax)')\n  ),\n \n  Action[Aaadd1](\n  Evaluate('func
tion'='aadd1', 'target'='TFa', Argument('TFa')), \n  Evaluate('pl1'='c
graph(TF1, TFa, TFL, SLe, SLd, TFxmin, TFxmax, TFymin, TFymax)')\n  ),
 \n\n  Action[Azoomin](\n  Evaluate('function'='zoomin', 'target'='TFx
min', Argument('TFxmin')), \n  Evaluate('function'='zoomin', 'target'=
'TFxmax', Argument('TFxmax')), \n  Evaluate('function'='zoomin', 'targ
et'='TFymin', Argument('TFymin')), \n  Evaluate('function'='zoomin', '
target'='TFymax', Argument('TFymax')), \n  Evaluate('pl1'='cgraph(TF1,
 TFa, TFL, SLe, SLd, TFxmin, TFxmax, TFymin, TFymax)')\n  ), \n\n  Act
ion[Azoomout](\n  Evaluate('function'='zoomout', 'target'='TFxmin', Ar
gument('TFxmin')), \n  Evaluate('function'='zoomout', 'target'='TFxmax
', Argument('TFxmax')), \n  Evaluate('function'='zoomout', 'target'='T
Fymin', Argument('TFymin')), \n  Evaluate('function'='zoomout', 'targe
t'='TFymax', Argument('TFymax')), \n  Evaluate('pl1'='cgraph(TF1, TFa,
 TFL, SLe, SLd, TFxmin, TFxmax, TFymin, TFymax)')\n  ), \n\n  Action[A
sle](\n  Evaluate('function'='realvalue', 'target'='TFsle', Argument('
SLe')), \n  Evaluate('pl1'='cgraph(TF1, TFa, TFL, SLe, SLd, TFxmin, TF
xmax, TFymin, TFymax)')\n  ), \n\n  Action[Asld](\n  Evaluate('functio
n'='realvalue', 'target'='TFsld', Argument('SLd')), \n  Evaluate('pl1'
='cgraph(TF1, TFa, TFL, SLe, SLd, TFxmin, TFxmax, TFymin, TFymax)')\n \+
 ), \n\n  Action[Aclear](\n    SetOption('TF1'=\"\"), \n    SetOption(
'TFxmin'=\"\"), \n    SetOption('TFxmax'=\"\"), \n    SetOption('TFymi
n'=\"\"), \n    SetOption('TFymax'=\"\"), \n    SetOption('TFa'=\"0\")
,\n    SetOption('TFL'=\"\"), \n    SetOption('TFsle'=\"\"), \n    Set
Option('TFsld'=\"\"),\n    Evaluate('pl1'='plot(0)')\n  )\n\n):\n\n###
#########################################################\n\nMaplets[D
isplay]( ContinuityMaplet );\n\n######################################
######################" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the na
me changecoords has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 43 "W
arning, the name arrow has been redefined\n" }}{PARA 6 "" 1 "" {TEXT 
-1 38 "Initializing Java runtime environment." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }