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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "Graphs of derivatives" }}{PARA 0 
"" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 
"" 0 "" {TEXT -1 18 "Version: 14.7.2005" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 24 "load calculus procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Ma
ple m-file " }{TEXT 266 10 "calculus.m" }{TEXT -1 32 " is required by \+
this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into \+
a Maple session by a command similar to the one that follows, where th
e file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "read \"K:\\\\Maple/procdrs/calculus.m\";" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Graphs of derivatives" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 119 "It is sometimes illuminating to draw the graph of a function t
ogether with the graph of its derivative.\nThe graphs of  " }{XPPEDIT 
18 0 "f(x) = x-x^3/6;" "6#/-%\"fG6#%\"xG,&F'\"\"\"*&F'\"\"$\"\"'!\"\"F
-" }{TEXT -1 22 "  and its derivative  " }{XPPEDIT 18 0 "`f '`(x) = 1-
x^2/2;" "6#/-%$f~'G6#%\"xG,&\"\"\"F)*&F'\"\"#F+!\"\"F," }{TEXT -1 26 "
  can be drawn as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "f := x->x-x^3/6;\ndf := D(f):\ndf(x
);\nplot([f(x),df(x)],x=-3..3,color=[red,blue],linestyle=[1,3]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow
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\"-%%VIEWG6$;F(Fb_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 199 "We can plot some of the tangen
t lines along with the graph to give an indication of how the gradient
 changes along the curve in line with the values of the derivative as \+
seen from the previous graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "f := x->x-x^3/6: 'f(x)'=f(x
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red,cyan,brown,green,blue,magenta,maroon,tan]:\ntangents := seq(df(b[i
])*(x-b[i])+f(b[i]),i=1..7):\nplot([f(x),tangents],x=-3..3,y=-2..2,col
or=colors,\n                               linestyle=[1,4$7]);" }}
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#F.$\"\"#F." 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "
Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Another way to obtain suc
h an animation is to use the " }{TEXT 0 7 "display" }{TEXT -1 18 " pro
cedure in the " }{TEXT 0 5 "plots" }{TEXT -1 26 " package with the opt
ion \"" }{TEXT 266 15 "insequence=true" }{TEXT -1 2 "\"." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 528 "f :=
 x->x-x^3/6: 'f(x)'=f(x);\ndf := D(f): `f '`(x)=df(x);\na := -3:\nb :=
 3:\nd := .8:\np1 := plot(f(x),x=a..b,color=red):\nfrms := NULL:\nnumf
rms := 50:\nh := abs(b-a)/(numfrms-1):\nfor i to numfrms do\n   u := a
+i*h;\n   fu := f(u);\n   dfu := df(u);\n   dd := d/sqrt(1+dfu^2);\n  \+
 p2 := plot(dfu*(x-u)+fu,x=u-dd..u+dd,color=blue,thickness=2);\n   p3 \+
:= plot([[[u,fu]]$3],style=point,symbol=[circle,diamond,cross],color=b
lack); \n   frms := frms,plots[display]([p1,p2,p3]);\nend do:\nplots[d
isplay]([frms],insequence=true,view=[a..b,-2..2]);\n\n" }}{PARA 11 "" 
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Y\"F17$$\"3d=dW<#[i+$F1$!3,5!=zC36]\"F17$$\"3Q><M>%3R,$F1$!3;qp&QL!zI:
F17$$\"3&=tiB*>`AIF1$!3)eITCL0Uc\"F17$$\"30/Jgu4>JIF1$!3%\\F?i`dxf\"F1
7$$\"3(\\4i%3b`RIF1$!3P'[L;I\"4I;F17$$\"3!))oEY98r/$F1$!3+n(e#eMXf;F17
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$\"3YgkA@fjsIF1$!3O')*48B]$e<F17$$\"3Ox9g\\a\\!3$F1$!3%HMBin/))y\"F17$
$\"3S2r*yf=\"*3$F1$!3mcJ1H!=A#=F17$$\"3'Q[rU()Ht4$F1$!3/(HODWNS&=F17$$
\"3-)o[(zu*e5$F1$!3?j'HcdLs)=F17$$\"3![t$4;_w8JF1$!3,B\\9D)>x\">F17$$
\"3rGhjK;DAJF1$!3+'Q'zjLg]>F17$$\"3ud(fVjm58$F1$!3_%30(=,w%)>F17$$\"3v
kMF>,uQJF1$!3#)[IEuO\\9?F17$$\"3'pD-KpFq9$F1$!33AI%)GngY?F17$$\"3g\"R5
!o&*ebJF1$!3o'y)[cEyz?F17$$\"3;@?9#plR;$F1$!34\\XFH)QA6#F17$$\"3'4lJH-
q?<$F1$!3h&35`%=kV@F17$$\"33hH_+&o5=$F1$!3A#))\\0a4&y@F17$$\"3w7A'G.a
\"*=$F1$!3#zBEbrR)4AF17$$\"3C&ocW1(y(>$F1$!3wVagk8HVAF17$$\"36j??u)4c?
$F1$!3oYQcINgtAF17$$\"3D'eG:FiT@$F1$!3n1'[WtUnI#F17$$\"3kN*34T4AA$F1$!
3V:cqUT#zL#F17$$\"3y<#R-*4iIKF1$!3?*e[))p<0P#F17$$\"3z*GJl%[%)QKF1$!3@
2!Qv&QQ-CF17$$\"3!ev@>YauC$F1$!3sYR[sZuNCF17$$\"3;O)>[aYdD$F1$!37zch)H
vyY#F17$$\"3/$>NsREUE$F1$!3X!4+BSL2]#F17$$\"31m6>KgjsKF1$!3>)H'HC%>L`#
F17$$\"3_0[K8OO!G$F1$!3g([dUfiKc#F17$$\"3sR5O$H?#*G$F1$!3a1!=h%3e(f#F1
7$$\"3+(pmN\"=9(H$F1$!331Hp*\\v#GEF17$$\"3&)>[J1ye0LF1$!3Ib^X-C+hEF17$
$\"3fk\"f!*prOJ$F1$!39DtDCiK#p#F17$$\"3-++#e%*eBK$F1$!3&pfQ'[z)fs#F1Fj
_mF\\`m-F(6&7#7$$\"3(o$=fz*[C7$F1$!3%[8FC0o8&>F1Fh`mF\\amF^am-F(6&Fjb_
nFdamF\\amF^am-F(6&Fjb_nFiamF\\amF^amF\\bm-Fgbm6$;F,$\"+]*eBK$F\\cmFeb
m-Fgbm6$F^do;$!\"#F.$F_`mF." 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "A
 more elaborate animation is considered in the next section." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "An animation procedure for gr
aphs of derivatives: " }{TEXT 0 9 "derivplot" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "derivplot: usage" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Ca
lling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 261 4 "    " }{TEXT -1 18 "
derivplot( f, p ) " }{TEXT 263 1 "\n" }{TEXT -1 4 "    " }}{PARA 0 "" 
0 "" {TEXT -1 21 "    deriv( f, p, q ) " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 178 "     f   -     an expr
ession involving a single variable, say x, or a function x -> f(x)\n\n
    p   -     horizontal plotting range in the form x=a..b, when x is \+
an expression in x" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "                " }{TEXT 259 2 "OR" }{TEXT -1 49 "   in t
he form a..b when f is afunction x -> f(x)" }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }{TEXT 23 4 "q - " }
{TEXT -1 92 "     vertical range (optional), which can be given in the
 form s..t, or in the form y=s..t.\n" }}{PARA 0 "" 0 "" {TEXT -1 5 "  \+
   " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 
0 9 "derivplot" }{TEXT -1 181 " provides an animation which shows the \+
graph of the derivative being drawn as if by a pencil from left to rig
ht on the same diagram as a pre-existing graph of the original functio
n." }}{PARA 15 "" 0 "" {TEXT -1 259 "A mini-tangent segment traces alo
ng the graph of the original function to indicate the gradient of the \+
tangent at a particular point at the same time that the corresponding \+
point recording the value of this gradient is plotted on the graph of \+
the derivative.\n" }}{PARA 0 "" 0 "" {TEXT 262 8 "Options:" }{TEXT -1 
1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 76 "frames=n\nThe number of frames f
or the animation. The default is \"frames=40\"." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "color=[clr1,clr2]\nclr1 \+
is the colour of the original curve and clr2 is the colour of the grap
h of the derivative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 13 "tgt_color=clr" }}{PARA 0 "" 0 "" {TEXT -1 98 "clr is th
e colour of the tangent line which follows the graph of the function d
uring the animation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 12 "tgt_length=n" }}{PARA 0 "" 0 "" {TEXT -1 115 "The tange
nt line can be specified with this option. The length is given as a fr
action 1/n of the width of the plot." }}{PARA 0 "" 0 "" {TEXT -1 31 "T
he default is \"tgt_length=10\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 15 "tgt_thickness=n" }}{PARA 0 "" 0 "" {TEXT 
-1 102 "The thickness of the tangent line can be specified with this o
ption. The default is \"tgt_thickness=2\"." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "tgt_scaling=s" }}{PARA 0 "" 0 "
" {TEXT -1 153 "If the plot is re-sized, the length of the tangent may
 appear to change as the animation progresses. This can be compensated
 for by means of this option." }}{PARA 0 "" 0 "" {TEXT -1 67 "The defa
ult is given by the plot width divided by the plot height. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Other options f
or plot can be used just as with the plot command." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To ma
ke the procedure active open the subsection, place the cursor anywhere
 after the prompt [ >  and press [Enter].\nYou can then close up the s
ubsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "derivplot: implemen
tation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10915 "deriv_plot := proc() derivplot(args[1..nargs]) end
:\n\nderivplot := proc(ff,rng)\n   local h,xi,yi,frms,x,y,seqn,xmax,xm
in,vd,hd,x1,x2,y1,y2,dx,dy,mi,\n      frame,clr,curve,deriv,tgt,lbls,o
pts,startoptions,xrange,scl,\n      yrange,rs,crvdat,derivdat,ymin,yma
x,fderivdat,j,pt,Options,i,\n      proctype,vars,Df,crvdatj,derivdatj,
k,n,dat,temp,tgtclr,tgtthk,\n      tgtlen,tgtscl,opt,tgc,dc,bignum;\n
\n   if nargs<2 then\n      error \"at least 2 arguments are required;
 the basic syntax is: 'derivplot(f(x),x=a..b)'.\"\n   end if;\n\n   # \+
Collect all the input data.\n   if type(ff,procedure) or patmatch(ff,a
a::procedure@@bb::posint) then\n      proctype := true;\n      if type
(rng,range) then\n         rs := rng;\n      else\n         error \"2n
d argument must have the form 'a..b' to provide a horizontal range ove
r which to plot the graph of %1\",ff;\n      end if;\n   elif type(ff,
algebraic) then \n      vars := indets(ff,name) minus indets(ff,realco
ns);\n      if nops(vars)<>1 and not type(ff,realcons) then \n        \+
 if not has(indets(ff),\{Int,Sum\}) then\n            error \"1st argu
ment must be an expression which depends only on a single variable\"\n
         end if;\n      end if;\n      if type(rng,name=range) then\n \+
        proctype := false;\n         x := op(1,rng);\n         if not \+
member(x,vars)  and not type(ff,realcons) then\n            error \"1s
t argument, %1, must be an expression which depends only on the variab
le %2\",ff,x;\n         end if;\n         rs := op(2,rng);\n      else
\n         error \"2nd argument must have the form '%1=a..b' to provid
e a horizontal range over which to plot the graph of %2\",x,ff;\n     \+
 end if;\n   else\n      error \"the 1st argument, %1, must be an alge
braic expression in a single variable, or a numerical valued procedure
 with a single argument\",ff;      \n   end if;\n   \n   xmin := evalf
(op(1,rs));\n   xmax := evalf(op(2,rs));\n   if not type(xmin,numeric)
 or not type(xmax,numeric) then\n      error \"each end point of the h
orizontal range %1 must evaluate to a numeric\",rs;\n   end if;\n   if
 xmin>=xmax then\n      error \"2nd argument horizontal range is inval
id\";\n   end if;\n   if proctype then\n      xrange := xmin..xmax;\n \+
  else\n      xrange := x=xmin..xmax;\n   end if;\n\n   startoptions :
= 3;  \n   yrange := NULL;\n   ymax := infinity;\n   ymin := -infinity
;\n   if nargs>2 then\n      if type(args[3],range) or type(args[3],na
me=range) then\n         startoptions := 4;\n         if type(args[3],
range) then\n            rs := args[3]; \n         else\n            r
s := op(2,args[3]);\n            y := op(1,args[3]);\n         end if;
\n         ymin := evalf(op(1,rs));\n         ymax := evalf(op(2,rs));
\n         if not type(ymin,numeric) or not type(ymax,numeric) then\n \+
           error \"each end point of the vertical range %1 must evalua
te to a numeric\",rs;\n         end if;\n         if ymin>=ymax then\n
            error \"3rd argument vertical range is invalid\";\n       \+
  end if;\n         if type(args[3],range) then\n            yrange :=
 ymin..ymax; \n         else\n            yrange := y=ymin..ymax;\n   \+
      end if;\n      end if;\n   end if;\n\n   # Get the options we ne
ed to process.\n   frms := 40;\n   Options := [];\n   opts := [];\n   \+
clr := [red,blue];\n   tgtclr := black;\n   tgtthk := 2;\n   tgtlen :=
 10;\n   if nargs>=startoptions then\n      Options:=[args[startoption
s..nargs]];\n      if not type(Options,list(equation)) then\n         \+
error \"each optional argument must be an equation\"\n      end if;\n \+
     if hasoption(Options,frames,'frms','Options') then\n         if n
ot type(frms,posint) or frms<2 then\n            error \"\\\"frames\\
\" must be an integer greater than 1\"\n         end if;\n      end if
;\n      if hasoption(Options,color,'temp','Options') or \n         ha
soption(Options,colour,'temp','Options') then\n         if type(temp,l
ist) and nops(temp)=2 then\n            clr := temp;\n         elif ty
pe(temp,name) or (type(temp,function) and \n           member(op(0,tem
p),\{COLOR,COLOUR\})) then\n            clr[1] := temp;\n            c
lr[2] := temp;\n         else\n            error \"\\\"color\\\" must \+
be a single color, or a list of two colours\"\n         end if;\n     \+
 end if;\n      hasoption(Options,tgt_color,'tgtclr','Options');\n    \+
  hasoption(Options,tgt_colour,'tgtclr','Options');\n      if hasoptio
n(Options,tgt_thickness,'tgtthk','Options') then\n         if not type
(tgtthk,posint) then\n            error \"\\\"tgt_thickness\\\" must b
e a positive integer\"\n         end if;\n      end if;\n      if haso
ption(Options,tgt_length,'tgtlen','Options') then\n         if not typ
e(tgtlen,posint) or tgtlen<2 then\n            error \"\\\"tgt_length
\\\" must be an integer greater than 1\"\n         end if;\n      end \+
if;\n      if hasoption(Options,tgt_scaling,'tgtscl','Options') then\n
         if not type(tgtscl,numeric) or tgtscl<0 then\n            err
or \"\\\"tgt_scaling\\\" must be positive number\"\n         end if;\n
      end if;\n      if hasoption(Options,discont,'temp','Options') th
en\n         opts := [op(opts),'discont'=temp];\n      end if;\n      \+
if hasoption(Options,resolution,'temp','Options') then\n         opts \+
:= [op(opts),'resolution'=temp];\n      end if;\n      if hasoption(Op
tions,adaptive,'temp','Options') then\n         opts := [op(opts),'ada
ptive'=temp];\n      end if;\n      if hasoption(Options,numpoints,'te
mp','Options') then\n         opts := [op(opts),'numpoints'=temp];\n  \+
    end if;\n      for i to nops(Options) do\n         opt := op(i,Opt
ions); \n         if not member(op(1,opt),\n            \{'color','col
our','linestyle','line_style','thickness',\n            'scaling','xti
ckmarks','ytickmarks','tickmarks','labels',\n            'style','symb
ol','symbolsize','title','axes','font',\n            'labelfont','labe
l_font','titlefont','title_font',\n            'axesfont','axes_font',
'view','labeldirections'\}) then\n            error \"unknown or inval
id option: %1\",opt;\n         end if;\n      end do;\n   end if;\n\n \+
  # Construct the curves\n   if proctype then\n      Df := D(ff);\n   \+
   curve := plot(ff,xrange,yrange,color=clr[1],op(opts));\n      crvda
t := [op(op(1,curve))];\n      derivdat := [op(op(1,plot(Df,xrange,yra
nge,op(opts))))];\n   else\n      Df := diff(ff,x);\n      curve := pl
ot(ff,xrange,yrange,color=clr[1],op(opts));\n      crvdat := [op(op(1,
curve))];\n      derivdat := [op(op(1,plot(Df,xrange,yrange,op(opts)))
)];\n   end if;\n\n   dat := [];\n   for j to nops(crvdat) do\n      i
f type(crvdat[j],list) then\n         crvdatj := [];\n         for i t
o nops(crvdat[j]) do\n            if type(crvdat[j,i],list(numeric))\n
                              and nops(crvdat[j,i])=2 then\n          \+
     crvdatj := [op(crvdatj),crvdat[j,i]];\n            end if;\n     \+
    end do;\n         dat := [op(dat),crvdatj];\n      end if;\n   end
 do;\n   crvdat := dat;\n   dat := [];\n   for j to nops(derivdat) do
\n      if type(derivdat[j],list) then\n         derivdatj := [];\n   \+
      for i to nops(derivdat[j]) do\n            if type(derivdat[j,i]
,list(numeric))\n                             and nops(derivdat[j,i])=
2 then\n               derivdatj := [op(derivdatj),derivdat[j,i]];\n  \+
          end if;\n         end do;\n         dat := [op(dat),derivdat
j];\n      end if;\n   end do;\n   derivdat := dat;\n\n   # If no vert
ical range is given calculate the default range.\n   if yrange=NULL th
en\n      ymin := crvdat[1,1,2];\n      ymax := ymin;\n      for j to \+
nops(crvdat) do\n         for i to nops(crvdat[j]) do\n            yi \+
:= crvdat[j,i,2];\n            if yi>ymax then ymax := yi end if;\n   \+
         if yi<ymin then ymin := yi end if;\n         end do;\n      e
nd do:\n      for j to nops(derivdat) do\n         for i to nops(deriv
dat[j]) do\n            yi := derivdat[j,i,2];\n            if yi>ymax
 then ymax := yi end if;\n            if yi<ymin then ymin := yi end i
f;\n         end do;\n      end do;\n      vd := (ymax-ymin)/tgtlen;\n
      yrange := ymin-vd..ymax+vd;\n   end if;\n\n   # Calculate x spac
ing and length of mini-tangent\n   h := (xmax-xmin)/(frms-1);\n   hd :
= (xmax-xmin)/tgtlen;\n   if not assigned(tgtscl) then\n      scl := (
xmax-xmin)/(ymax-ymin)\n   else\n      scl := tgtscl\n   end if;\n\n  \+
 if type(xrange,equation) and type(yrange,equation) then\n      if not
 hasoption(Options,labels,'lbls','Options') then\n         lbls := [op
(1,xrange),op(1,yrange)];\n      end if;\n      xrange := op(2,xrange)
;\n      yrange := op(2,yrange);\n   elif type(xrange,equation) then\n
      if not hasoption(Options,labels,'lbls','Options') then\n        \+
 lbls := [op(1,xrange),``];\n      end if;\n      xrange := op(2,xrang
e);\n   elif type(yrange,equation) then\n      if not hasoption(Option
s,labels,'lbls','Options') then\n         lbls := [``,op(1,yrange)];\n
      end if;\n      yrange := op(1,yrange);\n   else      \n     if n
ot hasoption(Options,labels,'lbls','Options') then\n         lbls := [
``,``];\n      end if;         \n   end if;\n\n   # Construct the inte
rmediate frames.\n   seqn := NULL;\n   bignum := Float(1,9);\n   for i
 from 1 to frms do\n      xi := evalf(xmin+(i-1)*h);\n      if proctyp
e then\n         yi := ff(xi);\n         mi := Df(xi); # slope of tang
ent \n      else\n         yi := evalf(eval(subs(x=xi,ff)));\n        \+
 mi := evalf(eval(subs(x=xi,Df))); # slope of tangent\n      end if;\n
      if mi=Float(infinity) then mi := bignum\n      elif mi=Float(inf
inity) then mi := -bignum end if;\n      if yi=Float(infinity) then yi
 := bignum\n      elif yi=Float(infinity) then yi := -bignum end if;\n
 \n      # Collect points on derivative graph for frame\n      fderivd
at := [];\n      n := nops(derivdat);\n      for j from 1 to n do\n   \+
      derivdatj := [];\n         for k to nops(derivdat[j]) do\n      \+
      pt := derivdat[j,k];\n            if pt[1]>xi then break end if;
\n            derivdatj := [op(derivdatj),pt];\n         end do;\n    \+
     if derivdatj<>[] then\n            if j=n then\n               de
rivdatj := [op(derivdatj),[xi,mi]];\n            end if;\n            \+
fderivdat := [op(fderivdat),derivdatj];\n         end if;\n      end d
o;\n \n      dx := hd/sqrt(1.0+mi^2*scl^2);  # half x width of mini-ta
ngent\n      dy := dx*mi;\n      x1 := xi-dx;\n      x2 := xi+dx;\n   \+
   y1 := yi-dy;\n      y2 := yi+dy;\n      tgc := `plot/color`(tgtclr)
;\n      dc := `plot/color`(clr[2]);\n      if fderivdat<>[] then\n   \+
      deriv := traperror(plot(fderivdat,color=dc));\n         if deriv
=lasterror then deriv := NULL end if;      \n      else\n         deri
v := NULL;\n      end if;\n      if type(yi,numeric) and type(mi,numer
ic) then \n         tgt := PLOT(POINTS([xi,yi],SYMBOL(CIRCLE),tgc),\n \+
                 POINTS([xi,yi],SYMBOL(DIAMOND),tgc),\n               \+
   POINTS([xi,yi],SYMBOL(CROSS),tgc),\n                  POINTS([xi,mi
],SYMBOL(CIRCLE),dc),\n                  CURVES([[x1,y1],[x2,y2]],\n  \+
                THICKNESS(tgtthk),`plot/color`(tgtclr)));\n      else
\n         tgt := NULL;\n      end if;\n      frame:= plots[display]([
curve,deriv,tgt],labels=lbls,\n              view=[xrange,yrange]);\n \+
     seqn := seqn,frame;\n   end do;\n   plots[display]([seqn],inseque
nce=true,op(Options));\nend proc:  # derivplot" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are gi
ven in the next section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "derivplot" }{TEXT -1 11 ": ex
amples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "f(x) = x^2/2-x;" "6#/-%\"fG6#%\"xG,&*&F'\"\"#F*!\"
\"\"\"\"F'F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := x -> x^2/2-x:\nDiff(f(x
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)*****z*******HF4Fh[l-F[\\l6%7$$\"+)*******RF`\\l$\"+#*******RF`\\lFa
\\lFe\\l-F[\\l6%F]crFi\\lFe\\l-F[\\l6%F]crF^]lFe\\l-F[\\l6%7$F^cr$\"+)
*******HF`\\lFa\\lFh[l-F(6%7$7$$\"+TA`#y$F`\\l$\"+AnfZLF`\\l7$$\"+bxY<
UF`\\l$\"+iKS_YF`\\lF`^lFe\\lFd^lF]_l" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "f(x) = x-x^3/6" "6#/-%\"fG6#%\"xG,&F'\"\"\"*&F'\"\"$\"\"'!\"\"F-
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 108 "f := x -> x-x^3/6:\nDiff(f(x),x)=diff(f(x),x)
;\nderivplot(f(x),x=-3.5..3.5,color=[magenta,COLOR(RGB,0,.7,0)]);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
y = 1/(x^2-1)" "6#/%\"yG*&\"\"\"F&,&*$%\"xG\"\"#F&F&!\"\"F+" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 216 "f := x -> 1/(x^2-1):\nDiff(f(x),x)=diff(f(x),x);\np1
 := plot([[[-1,-4],[-1,4]],[[1,-4],[1,4]]],\n linestyle=3,color=black)
:\np2 := derivplot(f(x),x=-3..3,y=-4..4,discont=true):\nplots[display]
([p1,p2],view=[-3..3,-4..4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "f(x) = sqrt(1-x^2);" "6#/-%\"fG
6#%\"xG-%%sqrtG6#,&\"\"\"F,*$F'\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "f :=
 x -> sqrt(1-x^2):\nDiff(f(x),x)=diff(f(x),x);\nderivplot(f(x),x=-1..1
,y=-2..2,scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x) = sin(x);" "6#/-%\"fG6#%\"xG
-%$sinG6#F'" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f := sin:\nDiff(f(x),x)=diff
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0 "" {TEXT -1 10 "Example 6 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = 1/(1+x^2);" "6#/-%\"fG
6#%\"xG*&\"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "f := \+
x -> 1/(1+x^2):\nDiff(f(x),x)=diff(f(x),x);\nderivplot(f(x),x=-3.5..3.
5,color=[coral,COLOR(RGB,.5,0,1)],\n        tgt_length=15,tgt_scaling=
2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&\"\"\"F(,&F(F(*$)%
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